Gamma(qp,KM), where

are matrix elements of the one-body property
operator W in a basis set of molecular spin-orbitals used
in the calculations. The calculation of reduced density
matrices provides the most convenient way of calculating CC
and EOMCC properties of ground and excited states. In
addition, by having reduced density matrices, one can
calculate CC and EOMCC electron densities,
rhoK(x) = Sum_pq Gamma(qp,K) (phi_q(x))* phi_p(x),
where phi_p(x) and phi_q(x) are molecular spin-orbitals and
x represents the electronic (spatial and spin) coordinates.
By diagonalizing Gamma(qp,K), one can determine the natural
occupation numbers and natural orbitals for the CC or EOMCC
state |PsiK>.
The above strategy of handling molecular properties
analytically by determining one-body reduced density
matrices was implemented in the CC/EOMCC programs
incorporated in GAMESS. At this time, the calculations of
reduced density matrices and selected properties are
possible at the CCSD (ground states) and EOMCCSD (ground
and excited states) levels of theory (T=T1+T2, R=R1+R2,
L=L0+L1+L2). Currently, in the main output the program
prints the CCSD and EOMCCSD electric multipole (dipole,
quadrupole, etc.) moments and several other one-electron
properties that one can extract from the CCSD/EOMCCSD
density matrices, the EOMCCSD transition dipole moments and
the corresponding dipole and oscillator strengths, and the
natural occupation numbers characterizing the CCSD/EOMCCSD
wave functions. In addition, the complete CCSD/EOMCCSD one-
body reduced density matrices and transition density
matrices in the RHF molecular orbital basis and the CCSD
and EOMCCSD natural orbital occupation numbers are printed
in the PUNCH output file. The eigenvalues of the density
matrix (natural occupation numbers) are ordered such that
the corresponding eigenvectors (CCSD or EOMCCSD natural
orbitals) have the largest overlaps with the consecutive
ground-state RHF MOs. Thus, the first eigenvalue of the
density matrix corresponds to the CCSD or EOMCCSD natural
orbital that has the largest overlap with the RHF MO 1, the
second with RHF MO 2, etc. This ordering is particularly
useful for analyzing excited states, since in this way one
can easily recognize orbital excitations that define a
given excited state.
One has to keep in mind that the reduced density
matrices originating from CC and EOMCC calculations are not
symmetric. Thus, if we, for example, want to calculate the
dipole strength between states K and M for the x component
of the dipole mu_x, |~~ values for
unrestricted DFT are smaller than for unrestricted HF.
2. GAMESS computes the ~~~~ quantity in an approximate
way, namely it pretend that the Kohn-Shan orbitals can be
used to form a determinant (WRONG, WRONG, WRONG, there is
no wavefunction in DFT!!!) and then uses the same formula
that a UHF job uses to evaluate that determinant's spin
expectation value.
G.J.Laming, N.C.Handy, R.D.Amos
Mol.Phys. 80, 1121-1134(1993)
J.Baker, A.Scheiner, J.Andzelm
Chem.Phys.Lett. 216, 380-388(1993)
C.Adamo, V.Barone, A.Fortunelli
J.Chem.Phys. 98, 8648-8652(1994)
J.A.Pople, P.M.W.Gill, N.C.Handy
Int.J.Quantum Chem. 56, 303-305(1995)
J.Wang, A.D.Becke, V.H.Smith
J.Chem.Phys. 102, 3477-3480(1995)
J.M.Wittbrodt, H.B.Schlegel
J.Chem.Phys. 105, 6574-6577(1996)
J.Grafenstein, D.Cremer
Mol.Phys. 99, 981-989(2001)
and commentary in Koch & Holthausen, pp 52-54.
Orbital energies:
The discussion on page 49-50 of Koch and Holthausen shows
that although the highest occupied orbital's eigenvalue
should be the ionization potential for exact Kohn-Sham
calculations, the functionals which we actually can use
normally greatly underestimate the IP values. The first
two papers below connect the HOMO eigenvalue to the IP, and
the third shows that while the band gap is underestimated
by existing functionals, the gap's center is correctly
predicted. The final paper is recent work using SCF
densities to generate exchange-correlation potentials that
actually give fairly good IP values:
J.F.Janak Phys.Rev.B 18, 7165-7168(1978)
M.Levy, J.P.Perdew, V.Sahni
Phys.Rev.A 30, 2745-2748(1984)
J.P.Perdew, M.Levy Phys.Rev.Lett. 51, 1884-1887(1983)
A.Nagy, M.Levy Chem.Phys.Lett. 296, 313-315(1998)
Geometry Searches and Internal Coordinates
Stationary points are places on the potential energy
surface with a zero gradient vector (first derivative of
the energy with respect to nuclear coordinates). These
include minima (whether relative or global), better known
to chemists as reactants, products, and intermediates; as
well as transition states (also known as saddle points).
The two types of stationary points have a precise
mathematical definition, depending on the curvature of the
potential energy surface at these points. If all of the
eigenvalues of the hessian matrix (second derivative
of the energy with respect to nuclear coordinates) are
positive, the stationary point is a minimum. If there is
one, and only one, negative curvature, the stationary
point is a transition state. Points with more than one
negative curvature do exist, but are not important in
chemistry. Because vibrational frequencies are basically
the square roots of the curvatures, a minimum has all
real frequencies, and a saddle point has one imaginary
vibrational "frequency".
GAMESS locates minima by geometry optimization, as
RUNTYP=OPTIMIZE, and transition states by saddle point
searches, as RUNTYP=SADPOINT. In many ways these are
similar, and in fact nearly identical FORTRAN code is used
for both. The term "geometry search" is used here to
describe features which are common to both procedures.
The input to control both RUNTYPs is found in the $STATPT
group.
As will be noted in the symmetry section below, an
OPTIMIZE run does not always find a minimum, and a
SADPOINT run may not find a transtion state, even though
the gradient is brought to zero. You can prove you have
located a minimum or saddle point only by examining the
local curvatures of the potential energy surface. This
can be done by following the geometry search with a
RUNTYP=HESSIAN job, which should be a matter of routine.
quasi-Newton Searches
Geometry searches are most effectively done by what is
called a quasi-Newton-Raphson procedure. These methods
assume a quadratic potential surface, and require the
exact gradient vector and an approximation to the hessian.
It is the approximate nature of the hessian that makes the
method "quasi". The rate of convergence of the geometry
search depends on how quadratic the real surface is, and
the quality of the hessian. The latter is something you
have control over, and is discussed in the next section.
GAMESS contains different implementations of quasi-
Newton procedures for finding stationary points, namely
METHOD=NR, RFO, QA, and the seldom used SCHLEGEL. They
differ primarily in how the step size and direction are
controlled, and how the Hessian is updated. The CONOPT
method is a way of forcing a geometry away from a minimum
towards a TS. It is not a quasi-Newton method, and is
described at the very end of this section.
The NR method employs a straight Newton-Raphson step.
There is no step size control, the algorithm will simply
try to locate the nearest stationary point, which may be a
minimum, a TS, or any higher order saddle point. NR is
not intended for general use, but is used by GAMESS in
connection with some of the other methods after they have
homed in on a stationary point, and by Gradient Extremal
runs where location of higher order saddle points is
common. NR requires a very good estimate of the geometry
in order to converge on the desired stationary point.
The RFO and QA methods are two different versions of
the so-called augmented Hessian techniques. They both
employ Hessian shift parameter(s) in order to control the
step length and direction.
In the RFO method, the shift parameter is determined by
approximating the PES with a Rational Function, instead of
a second order Taylor expansion. For a RUNTYP=SADPOINT,
the TS direction is treated separately, giving two shift
parameters. This is known as a Partitioned Rational
Function Optimization (P-RFO). The shift parameter(s)
ensure that the augmented Hessian has the correct eigen-
value structure, all positive for a minimum search, and
one negative eigenvalue for a TS search. The (P)-RFO step
can have any length, but if it exceeds DXMAX, the step is
simply scaled down.
In the QA (Quadratic Approximation) method, the shift
parameter is determined by the requirement that the step
size should equal DXMAX. There is only one shift
parameter for both minima and TS searches. Again the
augmented Hessian will have the correct structure. There
is another way of describing the same algorithm, namely as
a minimization on the "image" potential. The latter is
known as TRIM (Trust Radius Image Minimization). The
working equation is identical in these two methods.
When the RFO steplength is close to DXMAX, there is
little difference between the RFO and QA methods. However,
the RFO step may in some cases exceed DXMAX significantly,
and a simple scaling of the step will usually not produce
the best direction. The QA step is the best step on the
hypersphere with radius DXMAX. For this reason QA is the
default algorithm.
Near a stationary point the straight NR algorithm is
the most efficient. The RFO and QA may be viewed as
methods for guiding the search in the "correct" direction
when starting far from the stationary point. Once the
stationary point is approached, the RFO and QA methods
switch to NR, automatically, when the NR steplength drops
below 0.10 or DXMAX, whichever is the smallest.
The QA method works so well that we use it exclusively,
and so the SCHLEGEL method will probably be omitted from
some future version of GAMESS.
You should read the papers mentioned below in order to
understand how these methods are designed to work. The
first 3 papers describe the RFO and TRIM/QA algorithms. A
good but slightly dated summary of search procedures is
given by Bell and Crighton, and see also the review by
Schlegel. Most of the FORTRAN code for geometry searches,
and some of the discussion in this section was written by
Frank Jensen of the University of Southern Denmark, whose
paper compares many of the algorithms implemented in
GAMESS:
1. J.Baker J.Comput.Chem. 7, 385-395(1986)
2. T.Helgaker Chem.Phys.Lett. 182, 305-310(1991)
3. P.Culot, G.Dive, V.H.Nguyen, J.M.Ghuysen
Theoret.Chim.Acta 82, 189-205(1992)
4. H.B.Schlegel J.Comput.Chem. 3, 214-218(1982)
5. S.Bell, J.S.Crighton
J.Chem.Phys. 80, 2464-2475(1984).
6. H.B.Schlegel Advances in Chemical Physics (Ab Initio
Methods in Quantum Chemistry, Part I), volume 67,
K.P.Lawley, Ed. Wiley, New York, 1987, pp 249-286.
7. F.Jensen J.Chem.Phys. 102, 6706-6718(1995).
the nuclear Hessian
Although quasi-Newton methods require only an
approximation to the true hessian, the quality of this
matrix has a great affect on convergence of the geometry
search.
There is a procedure contained within GAMESS for
guessing a positive definite hessian matrix, HESS=GUESS.
If you are using Cartesian coordinates, the guess hessian
is based on pairwise atom stretches. The guess is more
sophisticated when internal coordinates are defined, as
empirical rules will be used to estimate stretching and
bending force constants. Other angular force constants are
set to 1/4. The guess often works well for minima, but
cannot possibly find transition states (because it is
positive definite). Therefore, GUESS may not be selected
for SADPOINT runs.
Two options for providing a more accurate hessian are
HESS=READ and CALC. For the latter, the true hessian is
obtained by direct calculation at the initial geometry, and
then the geometry search begins, all in one run. The READ
option allows you to feed in the hessian in a $HESS group,
as obtained by a RUNTYP=HESSIAN job. The second procedure
is actually preferable, as you get a chance to see the
frequencies. Then, if the local curvatures look good, you
can commit to the geometry search. Be sure to include a
$GRAD group (if the exact gradient is available) in the
HESS=READ job so that GAMESS can take its first step
immediately.
Note also that you can compute the hessian at a lower
basis set and/or wavefunction level, and read it into a
higher level geometry search. In fact, the $HESS group
could be obtained at the semiempirical level. This trick
works because the hessian is 3Nx3N for N atoms, no matter
what atomic basis is used. The gradient from the lower
level is of course worthless, as the geometry search must
work with the exact gradient of the wavefunction and basis
set in current use. Discard the $GRAD group from the lower
level calculation!
You often get what you pay for. HESS=GUESS is free, but
may lead to significantly more steps in the geometry
search. The other two options are more expensive at the
beginning, but may pay back by rapid convergence to the
stationary point.
The hessian update frequently improves the hessian for a
few steps (especially for HESS=GUESS), but then breaks
down. The symptoms are a nice lowering of the energy or
the RMS gradient for maybe 10 steps, followed by crazy
steps. You can help by putting the best coordinates into
$DATA, and resubmitting, to make a fresh determination of
the hessian.
The default hessian update for OPTIMIZE runs is BFGS,
which is likely to remain positive definite. The POWELL
update is the default for SADPOINT runs, since the hessian
can develop a negative curvature as the search progresses.
The POWELL update is also used by the METHOD=NR and CONOPT
since the Hessian may have any number of negative
eigenvalues in these cases. The MSP update is a mixture of
Murtagh-Sargent and Powell, suggested by Josep Bofill,
(J.Comput.Chem., 15, 1-11, 1994). It sometimes works
slightly better than Powell, so you may want to try it.
coordinate choices
Optimization in cartesian coordinates has a reputation
of converging slowly. This is largely due to the fact
that translations and rotations are usually left in the
problem. Numerical problems caused by the small eigen-
values associated with these degrees of freedom are the
source of this poor convergence. The methods in GAMESS
project the hessian matrix to eliminate these degrees of
freedom, which should not cause a problem. Nonetheless,
Cartesian coordinates are in general the most slowly
convergent coordinate system.
The use of internal coordinates (see NZVAR in $CONTRL
as well as $ZMAT) also eliminates the six rotational and
translational degrees of freedom. Also, when internal
coordinates are used, the GUESS hessian is able to use
empirical information about bond stretches and bends.
On the other hand, there are many possible choices for the
internal coordinates, and some of these may lead to much
poorer convergence of the geometry search than others.
Particularly poorly chosen coordinates may not even
correspond to a quadratic surface, thereby ending all hope
that a quasi-Newton method will converge.
Internal coordinates are frequently strongly coupled.
Because of this, Jerry Boatz has called them "infernal
coordinates"! A very common example to illustrate this
might be a bond length in a ring, and the angle on the
opposite side of the ring. Clearly, changing one changes
the other simultaneously. A more mathematical definition
of "coupled" is to say that there is a large off-diagonal
element in the hessian. In this case convergence may be
unsatisfactory, especially with a diagonal GUESS hessian,
where a "good" set of internals is one with a diagonally
dominant hessian. Of course, if you provide an accurately
computed hessian, it will have large off-diagonal values
where those are truly present. Even so, convergence may
be poor if the coordinates are coupled through large 3rd
or higher derivatives. The best coordinates are therefore
those which are the most "quadratic".
One very popular set of internal coordinates is the
usual "model builder" Z-matrix input, where for N atoms,
one uses N-1 bond lengths, N-2 bond angles, and N-3 bond
torsions. The popularity of this choice is based on its
ease of use in specifying the initial molecular geometry.
Typically, however, it is the worst possible choice of
internal coordinates, and in the case of rings, is not
even as good as Cartesian coordinates.
However, GAMESS does not require this particular mix
of the common types. GAMESS' only requirement is that you
use a total of 3N-6 coordinates, chosen from these 3 basic
types, or several more exotic possibilities. (Of course,
we mean 3N-5 throughout for linear molecules). These
additional types of internal coordinates include linear
bends for 3 collinear atoms, out of plane bends, and so on.
There is no reason at all why you should place yourself in
a straightjacket of N-1 bonds, N-2 angles, and N-3
torsions.
If the molecule has symmetry, be sure to use internals
which are symmetrically related.
For example, the most effective choice of coordinates
for the atoms in a four membered ring is to define all
four sides, any one of the internal angles, and a dihedral
defining the ring pucker. For a six membered ring, the
best coordinates seem to be 6 sides, 3 angles, and 3
torsions. The angles should be every other internal
angle, so that the molecule can "breathe" freely. The
torsions should be arranged so that the central bond of
each is placed on alternating bonds of the ring, as if
they were pi bonds in Kekule benzene. For a five membered
ring, we suggest all 5 sides, 2 internal angles, again
alternating every other one, and 2 dihedrals to fill in.
The internal angles of necessity skip two atoms where the
ring closes. Larger rings should generalize on the idea
of using all sides but only alternating angles. If there
are fused rings, start with angles on the fused bond, and
alternate angles as you go around from this position.
Rings and more especially fused rings can be tricky.
For these systems, especially, we suggest the Cadillac of
internal coordinates, the "natural internal coordinates"
of Peter Pulay. For a description of these, see
P.Pulay, G.Fogarosi, F.Pang, J.E.Boggs,
J.Am.Chem.Soc. 101, 2550-2560 (1979).
G.Fogarasi, X.Zhou, P.W.Taylor, P.Pulay
J.Am.Chem.Soc. 114, 8191-8201 (1992).
These are linear combinations of local coordinates, except
in the case of rings. The examples given in these two
papers are very thorough.
An illustration of these types of coordinates is given
in the example job EXAM25.INP, distributed with GAMESS.
This is a nonsense molecule, designed to show many kinds
of functional groups. It is defined using standard bond
distances with a classical Z-matrix input, and the angles
in the ring are adjusted so that the starting value of
the unclosed OO bond is also a standard value.
Using Cartesian coordinates is easiest, but takes a very
large number of steps to converge. This however, is better
than using the classical Z-matrix internals given in $DATA,
which is accomplished by setting NZVAR to the correct 3N-6
value. The geometry search changes the OO bond length to
a very short value on the 1st step, and the SCF fails to
converge. (Note that if you have used dummy atoms in the
$DATA input, you cannot simply enter NZVAR to optimize in
internal coordinates, instead you must give a $ZMAT which
involves only real atoms).
The third choice of internal coordinates is the best set
which can be made from the simple coordinates. It follows
the advice given above for five membered rings, and because
it includes the OO bond, has no trouble with crashing this
bond. It takes 20 steps to converge, so the trouble of
generating this $ZMAT is certainly worth it compared to the
use of Cartesians.
Natural internal coordinates are defined in the final
group of input. The coordinates are set up first for the
ring, including two linear combinations of all angles and
all torsions withing the ring. After this the methyl is
hooked to the ring as if it were a NH group, using the
usual terminal methyl hydrogen definitions. The H is
hooked to this same ring carbon as if it were a methine.
The NH and the CH2 within the ring follow Pulay's rules
exactly. The amount of input is much greater than a normal
Z-matrix. For example, 46 internal coordinates are given,
which are then placed in 3N-6=33 linear combinations. Note
that natural internals tend to be rich in bends, and short
on torsions.
The energy results for the three coordinate systems
which converge are as follows:
NSERCH Cart good Z-mat nat. int.
0 -48.6594935049 -48.6594935049 -48.6594935049
1 -48.6800538676 -48.6806631261 -48.6838361406
2 -48.6822702585 -48.6510215698 -48.6874045449
3 -48.6858299354 -48.6882945647 -48.6932811528
4 -48.6881499412 -48.6849667085 -48.6946836332
5 -48.6890226067 -48.6911899936 -48.6959800274
6 -48.6898261650 -48.6878047907 -48.6973821465
7 -48.6901936624 -48.6930608185 -48.6987652146
8 -48.6905304889 -48.6940607117 -48.6996366016
9 -48.6908626791 -48.6949137185 -48.7006656309
10 -48.6914279465 -48.6963767038 -48.7017273728
11 -48.6921521142 -48.6986608776 -48.7021504975
12 -48.6931136707 -48.7007305310 -48.7022405019
13 -48.6940437619 -48.7016095285 -48.7022548935
14 -48.6949546487 -48.7021531692 -48.7022569328
15 -48.6961698826 -48.7022080183 -48.7022570260
16 -48.6973813002 -48.7022454522 -48.7022570662
17 -48.6984850655 -48.7022492840
18 -48.6991553826 -48.7022503853
19 -48.6996239136 -48.7022507037
20 -48.7002269303 -48.7022508393
21 -48.7005379631
22 -48.7008387759
...
50 -48.7022519950
from which you can see that the natural internals are
actually the best set. The $ZMAT exhibits upward burps
in the energy at step 2, 4, and 6, so that for the
same number of steps, these coordinates are always at a
higher energy than the natural internals.
The initial hessian generated for these three columns
contains 0, 33, and 46 force constants. This assists
the natural internals, but is not the major reason for
its superior performance. The computed hessian at the
final geometry of this molecule, when transformed into the
natural internal coordinates is almost diagonal. This
almost complete uncoupling of coordinates is what makes
the natural internals perform so well. The conclusion
is of course that not all coordinate systems are equal,
and natural internals are the best. As another example,
we have run the ATCHCP molecule, which is a popular
geometry optimization test, due to its two fused rings:
H.B.Schlegel, Int.J.Quantum Chem., Symp. 26, 253-264(1992)
T.H.Fischer and J.Almlof, J.Phys.Chem. 96, 9768-9774(1992)
J.Baker, J.Comput.Chem. 14, 1085-1100(1993)
Here we have compared the same coordinate types, using a
guess hessian, or a computed hessian. The latter set of
runs is a test of the coordinates only, as the initial
hessian information is identical. The results show clearly
the superiority of the natural internals, which like the
previous example, give an energy decrease on every step:
HESS=GUESS HESS=READ
Cartesians 65 41 steps
good Z-matrix 32 23
natural internals 24 13
A final example is phosphinoazasilatrane, with three rings
fused on a common SiN bond, in which 112 steps in Cartesian
space became 32 steps in natural internals. The moral is:
"A little brain time can save a lot of CPU time."
In late 1998, a new kind of internal coordinate method
was included into GAMESS. This is the delocalized
internal
coordinate (DLC) of
J.Baker, A. Kessi, B.Delley
J.Chem.Phys. 105, 192-212(1996)
although as is the usual case, the implementation is not
exactly the same. Bonds are kept as independent
coordinates,
while angles are placed in linear combination by the DLC
process. There are some interesting options for applying
constraints, and other options to assist the automatic DLC
generation code by either adding or deleting coordinates.
It is simple to use DLCs in their most basic form:
$contrl nzvar=xx $end
$zmat dlc=.true. auto=.true. $end
Our initial experience is that the quality of DLCs is
not as good as explicitly constructed natural internals,
which benefit from human chemical knowledge, but are almost
always better than carefully crafted $ZMATs using only the
primitive internal coordinates (although we have seen a few
exceptions). Once we have more numerical experience with
the use of DLC's, we will come back and revise the above
discussion of coordinate choices. In the meantime, they
are quite simple to choose, so give them a go.
the role of symmetry
At the end of a succesful geometry search, you will
have a set of coordinates where the gradient of the energy
is zero. However your newly discovered stationary point
is not necessarily a minimum or saddle point!
This apparent mystery is due to the fact that the
gradient vector transforms under the totally symmetric
representation of the molecular point group. As a direct
consequence, a geometry search is point group conserving.
(For a proof of these statements, see J.W.McIver and
A.Komornicki, Chem.Phys.Lett., 10,303-306(1971)). In
simpler terms, the molecule will remain in whatever point
group you select in $DATA, even if the true minimum is in
some lower point group. Since a geometry search only
explores totally symmetric degrees of freedom, the only
way to learn about the curvatures for all degrees of
freedom is RUNTYP=HESSIAN.
As an example, consider disilene, the silicon analog
of ethene. It is natural to assume that this molecule is
planar like ethene, and an OPTIMIZE run in D2h symmetry
will readily locate a stationary point. However, as a
calculation of the hessian will readily show, this
structure is a transition state (one imaginary frequency),
and the molecule is really trans-bent (C2h). A careful
worker will always characterize a stationary point as
either a minimum, a transition state, or some higher order
stationary point (which is not of great interest!) by
performing a RUNTYP=HESSIAN.
The point group conserving properties of a geometry
search can be annoying, as in the preceeding example, or
advantageous. For example, assume you wish to locate the
transition state for rotation about the double bond in
ethene. A little thought will soon reveal that ethene is
D2h, the 90 degrees twisted structure is D2d, and
structures in between are D2. Since the saddle point is
actually higher symmetry than the rest of the rotational
surface, you can locate it by RUNTYP=OPTIMIZE within D2d
symmetry. You can readily find this stationary point with
the diagonal guess hessian! In fact, if you attempt to do
a RUNTYP=SADPOINT within D2d symmetry, there will be no
totally symmetric modes with negative curvatures, and it
is unlikely that the geometry search will be very well
behaved.
Although a geometry search cannot lower the symmetry,
the gain of symmetry is quite possible. For example, if
you initiate a water molecule optimization with a trial
structure which has unequal bond lengths, the geometry
search will come to a structure that is indeed C2v (to
within OPTTOL, anyway). However, GAMESS leaves it up to
you to realize that a gain of symmetry has occurred.
In general, Mother Nature usually chooses more
symmetrical structures over less symmetrical structures.
Therefore you are probably better served to assume the
higher symmetry, perform the geometry search, and then
check the stationary point's curvatures. The alternative
is to start with artificially lower symmetry and see if
your system regains higher symmetry. The problem with
this approach is that you don't necessarily know which
subgroup is appropriate, and you lose the great speedups
GAMESS can obtain from proper use of symmetry. It is good
to note here that "lower symmetry" does not mean simply
changing the name of the point group and entering more
atoms in $DATA, instead the nuclear coordinates themselves
must actually be of lower symmetry.
practical matters
Geometry searches do not bring the gradient exactly to
zero. Instead they stop when the largest component of the
gradient is below the value of OPTTOL, which defaults to
a reasonable 0.0001. Analytic hessians usually have
residual frequencies below 10 cm**-1 with this degree of
optimization. The sloppiest value you probably ever want
to try is 0.0005.
If a geometry search runs out of time, or exceeds
NSTEP, it can be restarted. For RUNTYP=OPTIMIZE, restart
with the coordinates having the lowest total energy
(do a string search on "FINAL"). For RUNTYP=SADPOINT,
restart with the coordinates having the smallest gradient
(do a string search on "RMS", which means root mean
square).
These are not necessarily at the last geometry!
The "restart" should actually be a normal run, that is
you should not try to use the restart options in $CONTRL
(which may not work anyway). A geometry search can be
restarted by extracting the desired coordinates for $DATA
from the printout, and by extracting the corresponding
$GRAD group from the PUNCH file. If the $GRAD group is
supplied, the program is able to save the time it would
ordinarily take to compute the wavefunction and gradient
at the initial point, which can be a substantial savings.
There is no input to trigger reading of a $GRAD group: if
found, it is read and used. Be careful that your $GRAD
group actually corresponds to the coordinates in $DATA, as
GAMESS has no check for this.
Sometimes when you are fairly close to the minimum, an
OPTIMIZE run will take a first step which raises the
energy, with subsequent steps improving the energy and
perhaps finding the minimum. The erratic first step is
caused by the GUESS hessian. It may help to limit the size
of this wrong first step, by reducing its radius, DXMAX.
Conversely, if you are far from the minimum, sometimes you
can decrease the number of steps by increasing DXMAX.
When using internals, the program uses an iterative
process to convert the internal coordinate change into
Cartesian space. In some cases, a small change in the
internals will produce a large change in Cartesians, and
thus produce a warning message on the output. If these
warnings appear only in the beginning, there is probably
no problem, but if they persist you can probably devise
a better set of coordinates. You may in fact have one of
the two problems described in the next paragraph. In
some cases (hopefully very few) the iterations to find
the Cartesian displacement may not converge, producing a
second kind of warning message. The fix for this may
very well be a new set of internal coordinates as well,
or adjustment of ITBMAT in $STATPT.
There are two examples of poorly behaved internal
coordinates which can give serious problems. The first
of these is three angles around a central atom, when
this atom becomes planar (sum of the angles nears 360).
The other is a dihedral where three of the atoms are
nearly linear, causing the dihedral to flip between 0 and
180. Avoid these two situations if you want your geometry
search to be convergent.
Sometimes it is handy to constrain the geometry search
by freezing one or more coordinates, via the IFREEZ array.
For example, constrained optimizations may be useful while
trying to determine what area of a potential energy surface
contains a saddle point. If you try to freeze coordinates
with an automatically generated $ZMAT, you need to know
that the order of the coordinates defined in $DATA is
y
y x r1
y x r2 x a3
y x r4 x a5 x w6
y x r7 x a8 x w9
and so on, where y and x are whatever atoms and molecular
connectivity you happen to be using.
saddle points
Finding minima is relatively easy. There are large
tables of bond lengths and angles, so guessing starting
geometries is pretty straightforward. Very nasty cases
may require computation of an exact hessian, but the
location of most minima is straightforward.
In contrast, finding saddle points is a black art.
The diagonal guess hessian will never work, so you must
provide a computed one. The hessian should be computed at
your best guess as to what the transition state (T.S.)
should be. It is safer to do this in two steps as outlined
above, rather than HESS=CALC. This lets you verify you
have guessed a structure with one and only one negative
curvature. Guessing a good trial structure is the hardest
part of a RUNTYP=SADPOINT!
This point is worth iterating. Even with sophisticated
step size control such as is offered by the QA/TRIM or RFO
methods, it is in general very difficult to move correctly
from a region with incorrect curvatures towards a saddle
point. Even procedures such as CONOPT or RUNTYP=GRADEXTR
will not replace your own chemical intuition about where
saddle points may be located.
The RUNTYP=HESSIAN's normal coordinate analysis is
rigorously valid only at stationary points on the surface.
This means the frequencies from the hessian at your trial
geometry are untrustworthy, in particular the six "zero"
frequencies corresponding to translational and rotational
(T&R) degrees of freedom will usually be 300-500 cm**-1,
and possibly imaginary. The Sayvetz conditions on the
printout will help you distinguish the T&R "contaminants"
from the real vibrational modes. If you have defined a
$ZMAT, the PURIFY option within $STATPT will help zap out
these T&R contaminants).
If the hessian at your assumed geometry does not have
one and only one imaginary frequency (taking into account
that the "zero" frequencies can sometimes be 300i!), then
it will probably be difficult to find the saddle point.
Instead you need to compute a hessian at a better guess
for the initial geometry, or read about mode following
below.
If you need to restart your run, do so with the
coordinates which have the smallest RMS gradient. Note
that the energy does not necessarily have to decrease in a
SADPOINT run, in contrast to an OPTIMIZE run. It is often
necessary to do several restarts, involving recomputation
of the hessian, before actually locating the saddle point.
Assuming you do find the T.S., it is always a good
idea to recompute the hessian at this structure. As
described in the discussion of symmetry, only totally
symmetric vibrational modes are probed in a geometry
search. Thus it is fairly common to find that at your
"T.S." there is a second imaginary frequency, which
corresponds to a non-totally symmetric vibration. This
means you haven't found the correct T.S., and are back to
the drawing board. The proper procedure is to lower the
point group symmetry by distorting along the symmetry
breaking "extra" imaginary mode, by a reasonable amount.
Don't be overly timid in the amount of distortion, or the
next run will come back to the invalid structure.
The real trick here is to find a good guess for the
transition structure. The closer you are, the better. It
is often difficult to guess these structures. One way
around this is to compute a linear least motion (LLM)
path. This connects the reactant structure to the product
structure by linearly varying each coordinate. If you
generate about ten structures intermediate to reactants
and products, and compute the energy at each point, you
will in general find that the energy first goes up, and
then down. The maximum energy structure is a "good" guess
for the true T.S. structure. Actually, the success of
this method depends on how curved the reaction path is.
A particularly good paper on the symmetry which a
saddle point (and reaction path) can possess is by
P.Pechukas, J.Chem.Phys. 64, 1516-1521(1976)
mode following
In certain circumstances, METHOD=RFO and QA can walk
from a region of all positive curvatures (i.e. near a
minimum) to a transition state. The criteria for whether
this will work is that the mode being followed should be
only weakly coupled to other close-lying Hessian modes.
Especially, the coupling to lower modes should be almost
zero. In practise this means that the mode being followed
should be the lowest of a given symmetry, or spatially far
away from lower modes (for example, rotation of methyl
groups at different ends of the molecule). It is certainly
possible to follow also modes which do not obey these
criteria, but the resulting walk (and possibly TS location)
will be extremely sensitive to small details such as the
stepsize.
This sensitivity also explain why TS searches often
fail, even when starting in a region where the Hessian has
the required one negative eigenvalue. If the TS mode is
strongly coupled to other modes, the direction of the mode
is incorrect, and the maximization of the energy along
that direction is not really what you want (but what you
get).
Mode following is really not a substitute for the
ability to intuit regions of the PES with a single local
negative curvature. When you start near a minimum, it
matters a great deal which side of the minima you start
from, as the direction of the search depends on the sign
of the gradient. We strongly urge that you read before
trying to use IFOLOW, namely the papers by Frank Jensen
and Jon Baker mentioned above, and see also Figure 3 of
C.J.Tsai, K.D.Jordan, J.Phys.Chem. 97, 11227-11237 (1993)
which is quite illuminating on the sensitivity of mode
following to the initial geometry point.
Note that GAMESS retains all degrees of freedom in its
hessian, and thus there is no reason to suppose the lowest
mode is totally symmetric. Remember to lower the symmetry
in the input deck if you want to follow non-symmetric
modes. You can get a printout of the modes in internal
coordinate space by a EXETYP=CHECK run, which will help
you decide on the value of IFOLOW.
* * *
CONOPT is a different sort of saddle point search
procedure. Here a certain "CONstrained OPTimization" may
be considered as another mode following method. The idea
is to start from a minimum, and then perform a series of
optimizations on hyperspheres of increasingly larger
radii. The initial step is taken along one of the Hessian
modes, chosen by IFOLOW, and the geometry is optimized
subject to the constraint that the distance to the minimum
is constant. The convergence criteria for the gradient
norm perpendicular to the constraint is taken as 10*OPTTOL,
and the corresponding steplength as 100*OPTTOL.
After such a hypersphere optimization has converged, a
step is taken along the line connecting the two previous
optimized points to get an estimate of the next hyper-
sphere geometry. The stepsize is DXMAX, and the radius of
hyperspheres is thus increased by an amount close (but not
equal) to DXMAX. Once the pure NR step size falls below
DXMAX/2 or 0.10 (whichever is the largest) the algorithm
switches to a straight NR iterate to (hopefully) converge
on the stationary point.
The current implementation always conducts the search
in cartesian coordinates, but internal coordinates may be
printed by the usual specification of NZVAR and ZMAT. At
present there is no restart option programmed.
CONOPT is based on the following papers, but the actual
implementation is the modified equations presented in
Frank Jensen's paper mentioned above.
Y. Abashkin, N. Russo,
J.Chem.Phys. 100, 4477-4483(1994).
Y. Abashkin, N. Russo, M. Toscano,
Int.J.Quant.Chem. 52, 695-704(1994).
There is little experience on how this method works in
practice, experiment with it at your own risk!
Intrinisic Reaction Coordinate Methods
The Intrinsic Reaction Coordinate (IRC) is defined as
the minimum energy path connecting the reactants to
products
via the transition state. In practice, the IRC is found by
first locating the transition state for the reaction. The
IRC is then found in halves, going forward and backwards
from the saddle point, down the steepest descent path in
mass weighted Cartesian coordinates. This is accomplished
by numerical integration of the IRC equations, by a variety
of methods to be described below.
The IRC is becoming an important part of polyatomic
dynamics research, as it is hoped that only knowledge of
the
PES in the vicinity of the IRC is needed for prediction of
reaction rates, at least at threshhold energies. The IRC
has a number of uses for electronic structure purposes as
well. These include the proof that a certain transition
structure does indeed connect a particular set of reactants
and products, as the structure and imaginary frequency
normal mode at the saddle point do not always unambiguously
identify the reactants and products. The study of the
electronic and geometric structure along the IRC is also of
interest. For example, one can obtain localized orbitals
along the path to determine when bonds break or form.
The accuracy to which the IRC is determined is dictated
by the use one intends for it. Dynamical calculations
require a very accurate determination of the path, as
derivative information (second derivatives of the PES at
various IRC points, and path curvature) is required later.
Thus, a sophisticated integration method (such as AMPC4 or
RK4), and small step sizes (STRIDE=0.05, 0.01, or even
smaller) may be needed. In addition to this, care should
be taken to locate the transition state carefully (perhaps
decreasing OPTTOL by a factor of 10), and in the initiation
of the IRC run. The latter might require a hessian matrix
obtained by double differencing, certainly the hessian
should be PURIFY'd. Note also that EVIB must be chosen
carefully, as decribed below.
On the other hand, identification of reactants and
products allows for much larger step sizes, and cruder
integration methods. In this type of IRC one might want to
be careful in leaving the saddle point (perhaps STRIDE
should be reduced to 0.10 or 0.05 for the first few steps
away from the transition state), but once a few points have
been taken, larger step sizes can be employed. In general,
the defaults in the $IRC group are set up for this latter,
cruder quality IRC. The STRIDE value for the GS2 method
can usually be safely larger than for other methods, no
matter what your interest in accuracy is.
The simplest method of determining an IRC is linear
gradient following, PACE=LINEAR. This method is also known
as Euler's method. If you are employing PACE=LINEAR, you
can select "stabilization" of the reaction path by the
Ishida, Morokuma, Komornicki method. This type of
corrector
has no apparent mathematical basis, but works rather well
since the bisector usually intersects the reaction path at
right angles (for small step sizes). The ELBOW variable
allows for a method intermediate to LINEAR and stabilized
LINEAR, in that the stabilization will be skipped if the
gradients at the original IRC point, and at the result of a
linear prediction step form an angle greater than ELBOW.
Set ELBOW=180 to always perform the stabilization.
A closely related method is PACE=QUAD, which fits a
quadratic polynomial to the gradient at the current and
immediately previous IRC point to predict the next point.
This pace has the same computational requirement as LINEAR,
and is slightly more accurate due to the reuse of the old
gradient. However, stabilization is not possible for this
pace, thus a stabilized LINEAR path is usually more
accurate
than QUAD.
Two rather more sophisticated methods for integrating
the IRC equations are the fourth order Adams-Moulton
predictor-corrector (PACE=AMPC4) and fourth order Runge-
Kutta (PACE=RK4). AMPC4 takes a step towards the next IRC
point (prediction), and based on the gradient found at this
point (in the near vincinity of the next IRC point) obtains
a modified step to the desired IRC point (correction).
AMPC4 uses variable step sizes, based on the input STRIDE.
RK4 takes several steps part way toward the next IRC point,
and uses the gradient at these points to predict the next
IRC point. RK4 is the most accurate integration method
implemented in GAMESS, and is also the most time consuming.
The Gonzalez-Schlegel 2nd order method finds the next
IRC point by a constrained optimization on the surface of
a hypersphere, centered at 1/2 STRIDE along the gradient
vector leading from the previous IRC point. By
construction,
the reaction path between two successive IRC points is
thus a circle tangent to the two gradient vectors. The
algorithm is much more robust for large steps than the
other
methods, so it has been chosen as the default method.
Thus,
the default for STRIDE is too large for the other methods.
The number of energy and gradients need to find the next
point varies with the difficulty of the constrained
optimization, but is normally not very many points. Be
sure
to provide the updated hessian from the previous run when
restarting PACE=GS2.
The number of wavefunction evaluations, and energy
gradients needed to jump from one point on the IRC to the
next
point are summarized in the following table:
PACE # energies # gradients
---- ---------- -----------
LINEAR 1 1
stabilized
LINEAR 3 2
QUAD 1 1 (+ reuse of historical
gradient)
AMPC4 2 2 (see note)
RK4 4 4
GS2 2-4 2-4 (equal numbers)
Note that the AMPC4 method sometimes does more than one
correction step, with each such corection adding one more
energy and gradient to the calculation. You get what you
pay for in IRC calculations: the more energies and
gradients which are used, the more accurate the path found.
A description of these methods, as well as some others
that were found to be not as good is geven by Kim Baldridge
and Lisa Pederson, Pi Mu Epsilon Journal, 9, 513-521
(1993).
* * *
All methods are initiated by jumping from the saddle
point, parallel to the normal mode (CMODE) which has an
imaginary frequency. The jump taken is designed to lower
the energy by an amount EVIB. The actual distance taken is
thus a function of the imaginary frequency, as a smaller
FREQ will produce a larger initial jump. You can simply
provide a $HESS group instead of CMODE and FREQ, which
involves less typing. To find out the actual step taken
for
a given EVIB, use EXETYP=CHECK. The direction of the jump
(towards reactants or products) is governed by FORWRD.
Note
that if you have decided to use small step sizes, you must
employ a smaller EVIB to ensure a small first step. The
GS2 method begins by following the normal mode by one half
of STRIDE, and then performing a hypersphere minimization
about that point, so EVIB is irrelevant to this PACE.
The only method which proves that a properly converged
IRC has been obtained is to regenerate the IRC with a
smaller step size, and check that the IRC is unchanged.
Again, note that the care with which an IRC must be
obtained
is highly dependent on what use it is intended for.
Some key IRC references are:
K.Ishida, K.Morokuma, A.Komornicki
J.Chem.Phys. 66, 2153-2156 (1977)
K.Muller
Angew.Chem., Int.Ed.Engl.19, 1-13 (1980)
M.W.Schmidt, M.S.Gordon, M.Dupuis
J.Am.Chem.Soc. 107, 2585-2589 (1985)
B.C.Garrett, M.J.Redmon, R.Steckler, D.G.Truhlar,
K.K.Baldridge, D.Bartol, M.W.Schmidt, M.S.Gordon
J.Phys.Chem. 92, 1476-1488(1988)
K.K.Baldridge, M.S.Gordon, R.Steckler, D.G.Truhlar
J.Phys.Chem. 93, 5107-5119(1989)
C.Gonzales, H.B.Schlegel
J.Chem.Phys. 90, 2154-2161(1989)
The IRC discussion closes with some practical tips:
The $IRC group has a confusing array of variables, but
fortunately very little thought need be given to most of
them. An IRC run is restarted by moving the coordinates of
the next predicted IRC point into $DATA, and inserting the
new $IRC group into your input file. You must select the
desired value for NPOINT. Thus, only the first job which
initiates the IRC requires much thought about $IRC.
The symmetry specified in the $DATA deck should be the
symmetry of the reaction path. If a saddle point happens
to have higher symmetry, use only the lower symmetry in
the $DATA deck when initiating the IRC. The reaction path
will have a lower symmetry than the saddle point whenever
the normal mode with imaginary frequency is not totally
symmetric. Be careful that the order and orientation of
the
atoms corresponds to that used in the run which generated
the hessian matrix.
If you wish to follow an IRC for a different isotope,
use the $MASS group. If you wish to follow the IRC in
regular Cartesian coordinates, just enter unit masses for
each atom. Note that CMODE and FREQ are a function of the
atomic masses, so either regenerate FREQ and CMODE, or
more simply, provide the correct $HESS group.
Gradient Extremals
This section of the manual, as well as the source code
to trace gradient extremals was written by Frank Jensen of
the University of Southern Denmark.
A Gradient Extremal (GE) curve consists of points where
the gradient norm on a constant energy surface is
stationary. This is equivalent to the condition that
the gradient is an eigenvector of the Hessian. Such GE
curves radiate along all normal modes from a stationary
point, and the GE leaving along the lowest normal mode
from a minimum is the gentlest ascent curve. This is not
the same as the IRC curve connecting a minimum and a TS,
but may in some cases be close.
GEs may be divided into three groups: those leading
to dissociation, those leading to atoms colliding, and
those which connect stationary points. The latter class
allows a determination of many (all?) stationary points on
a PES by tracing out all the GEs. Following GEs is thus a
semi-systematic way of mapping out stationary points. The
disadvantages are:
i) There are many (but finitely many!) GEs for a
large molecule.
ii) Following GEs is computationally expensive.
iii) There is no control over what type of
stationary point (if any) a GE will lead to.
Normally one is only interested in minima and TSs, but
many higher order saddle points will also be found.
Furthermore, it appears that it is necessary to follow GEs
radiating also from TSs and second (and possibly also
higher) order saddle point to find all the TSs.
A rather complete map of the extremals for the H2CO
potential surface is available in a paper which explains
the points just raised in greater detail:
K.Bondensgaard, F.Jensen,
J.Chem.Phys. 104, 8025-8031(1996).
An earlier paper gives some of the properties of GEs:
D.K.Hoffman, R.S.Nord, K.Ruedenberg,
Theor. Chim. Acta 69, 265-279(1986).
There are two GE algorithms in GAMESS, one due to Sun
and Ruedenberg (METHOD=SR), which has been extended to
include the capability of locating bifurcation points and
turning points, and another due to Jorgensen, Jensen, and
Helgaker (METHOD=JJH):
J. Sun, K. Ruedenberg, J.Chem.Phys. 98, 9707-9714(1993)
P. Jorgensen, H. J. Aa. Jensen, T. Helgaker
Theor. Chim. Acta 73, 55 (1988).
The Sun and Ruedenberg method consist of a predictor
step taken along the tangent to the GE curve, followed by
one or more corrector steps to bring the geometry back to
the GE. Construction of the GE tangent and the corrector
step requires elements of the third derivative of the
energy, which is obtained by a numerical differentiation
of two Hessians. This puts some limitations on which
systems the GE algorithm can be used for. First, the
numerical differentiation of the Hessian to produce third
derivatives means that the Hessian should be calculated by
analytical methods, thus only those types of wavefunctions
where this is possible can be used. Second, each
predictor/corrector step requires at least two Hessians,
but often more. Maybe 20-50 such steps are necessary for
tracing a GE from one stationary point to the next. A
systematic study of all the GE radiating from a stationary
point increases the work by a factor of ~2*(3N-6). One
should thus be prepared to invest at least hundreds, and
more likely thousands, of Hessian calculations. In other
words, small systems, small basis sets, and simple wave-
functions.
The Jorgensen, Jensen, and Helgaker method consists of
taking a step in the direction of the chosen Hessian
eigenvector, and then a pure NR step in the perpendicular
modes. This requires (only) one Hessian calculation for
each step. It is not suitable for following GEs where the
GE tangent forms a large angle with the gradient, and it
is incapable of locating GE bifurcations.
Although experience is limited at present, the JJH
method does not appear to be suitable for following GEs in
general (at least not in the current implementation).
Experiment with it at your own risk!
The flow of the SR algorithm is as follows: A
predictor geometry is produced, either by jumping away
from a stationary point, or from a step in the tangent
direction from the previous point on the GE. At the
predictor geometry, we need the gradient, the Hessian, and
the third derivative in the gradient direction. Depending
on HSDFDB, this can be done in two ways. If .TRUE. the
gradient is calculated, and two Hessians are calculated at
SNUMH distance to each side in the gradient direction.
The Hessian at the geometry is formed as the average of
the two displaced Hessians. This corresponds to a double-
sided differentiation, and is the numerical most stable
method for getting the partial third derivative matrix.
If HSDFDB = .FALSE., the gradient and Hessian are
calculated at the current geometry, and one additional
Hessian is calculated at SNUMH distance in the gradient
direction. This corresponds to a single-sided differen-
tiation. In both cases, two full Hessian calculations are
necessary, but HSDFDB = .TRUE. require one additional
wavefunction and gradient calculation. This is usually
a fairly small price compared to two Hessians, and the
numerically better double-sided differentiation has
therefore been made the default.
Once the gradient, Hessian, and third derivative is
available, the corrector step and the new GE tangent are
constructed. If the corrector step is below a threshold,
a new predictor step is taken along the tangent vector.
If the corrector step is larger than the threshold, the
correction step is taken, and a new micro iteration is
performed. DELCOR thus determines how closely the GE will
be followed, and DPRED determine how closely the GE path
will be sampled.
The construction of the GE tangent and corrector step
involve solution of a set of linear equations, which in
matrix notation can be written as Ax=B. The A-matrix is
also the second derivative of the gradient norm on the
constant energy surface.
After each corrector step, various things are printed
to monitor the behavior: The projection of the gradient
along the Hessian eigenvalues (the gradient is parallel
to an eigenvector on the GE), the projection of the GE
tangent along the Hessian eigenvectors, and the overlap
of the Hessian eigenvectors with the mode being followed
from the previous (optimzed) geometry. The sign of these
overlaps are not significant, they just refer to an
arbitrary phase of the Hessian eigenvectors.
After the micro iterations has converged, the Hessian
eigenvector curvatures are also displayed, this is an
indication of the coupling between the normal modes. The
number of negative eigenvalues in the A-matrix is denoted
the GE index. If it changes, one of the eigenvalues must
have passed through zero. Such points may either be GE
bifurcations (where two GEs cross) or may just be "turning
points", normally when the GE switches from going uphill
in energy to downhill, or vice versa. The distinction is
made based on the B-element corresponding to the A-matrix
eigenvalue = 0. If the B-element = 0, it is a bifurcation,
otherwise it is a turning point.
If the GE index changes, a linear interpolation is
performed between the last two points to locate the point
where the A-matrix is singular, and the corresponding
B-element is determined. The linear interpolation points
will in general be off the GE, and thus the evaluation of
whether the B-element is 0 is not always easy. The
program additionally evaluates the two limiting vectors
which are solutions to the linear sets of equations, these
are also used for testing whether the singular point is a
bifurcation point or turning point.
Very close to a GE bifurcation, the corrector step
become numerically unstable, but this is rarely a problem
in practice. It is a priori expected that GE bifurcation
will occur only in symmetric systems, and the crossing GE
will break the symmetry. Equivalently, a crossing GE may
be encountered when a symmetry element is formed, however
such crossings are much harder to detect since the GE
index does not change, as one of the A-matrix eigenvalues
merely touches zero. The program prints an message if
the absolute value of an A-matrix eigenvalue reaches a
minimum near zero, as such points may indicate the
passage of a bifurcation where a higher symmetry GE
crosses. Run a movie of the geometries to see if a more
symmetric structure is passed during the run.
An estimate of the possible crossing GE direction is
made at all points where the A-matrix is singular, and two
perturbed geometries in the + and - direction are written
out. These may be used as predictor geometries for
following a crossing GE. If the singular geometry is a
turning point, the + and - geometries are just predictor
geometries on the GE being followed.
In any case, a new predictor step can be taken to trace
a different GE from the newly discovered singular point,
using the direction determined by interpolation from the
two end point tangents (the GE tangent cannot be uniquely
determined at a bifurcation point). It is not possible to
determine what the sign of IFOLOW should be when starting
off along a crossing GE at a bifurcation, one will have to
try a step to see if it returns to the bifurcation point
or not.
In order to determine whether the GE index change it
is necessary to keep track of the order of the A-matrix
eigenvalues. The overlap between successive eigenvectors
are shown as "Alpha mode overlaps".
Things to watch out for:
1) The numerical differentiation to get third derivatives
requires more accuracy than usual. The SCF convergence
should be at least 100 times smaller than SNUMH, and
preferably better. With the default SNUMH of 10**(-4)
the SCF convergence should be at least 10**(-6). Since
the last few SCF cycles are inexpensive, it is a good idea
to tighten the SCF convergence as much as possible, to
maybe 10**(-8) or better. You may also want to increase
the integral accuracy by reducing the cutoffs (ITOL and
ICUT) and possibly also try more accurate integrals
(INTTYP=HONDO). The CUTOFF in $TRNSFM may also be reduced
to produce more accurate Hessians. Don't attempt to use a
value for SNUMH below 10**(-6), as you simply can't get
enough accuracy. Since experience is limited at present,
it is recommended that some tests runs are made to learn
the sensitivity of these factors for your system.
2) GEs can be followed in both directions, uphill or
downhill. When stating from a stationary point, the
direction is implicitly given as away from the stationary
point. When starting from a non-stationary point, the "+"
and "-" directions (as chosen by the sign of IFOLOW)
refers to the gradient direction. The "+" direction is
along the gradient (energy increases) and "-" is opposite
to the gradient (energy decreases).
3) A switch from one GE to another may be seen when two
GE come close together. This is especially troublesome
near bifurcation points where two GEs actually cross. In
such cases a switch to a GE with -higher- symmetry may
occur without any indication that this has happened,
except possibly that a very large GE curvature suddenly
shows up. Avoid running the calculation with less
symmetry than the system actually has, as this increases
the likelihood that such switches occuring. Fix: alter
DPRED to avoid having the predictor step close to the
crossing GE.
4) "Off track" error message: The Hessian eigenvector
which is parallel to the gradient is not the same as
the one with the largest overlap to the previous
Hessian mode. This usually indicate that a GE switch
has occured (note that a switch may occur without this
error message), or a wrong value for IFOLOW when starting
from a non-stationary point. Fix: check IFOLOW, if it is
correct then reduce DPRED, and possibly also DELCOR.
5) Low overlaps of A-matrix eigenvectors. Small overlaps
may give wrong assignment, and wrong conclusions about GE
index change. Fix: reduce DPRED.
6) The interpolation for locating a point where one of the
A-matrix eigenvalues is zero fail to converge. Fix:
reduce DPRED (and possibly also DELCOR) to get a shorther
(and better) interpolation line.
7) The GE index changes by more than 1. A GE switch may
have occured, or more than one GE index change is located
between the last and current point. Fix: reduce DPRED to
sample the GE path more closely.
8) If SNRMAX is too large the algorithm may try to locate
stationary points which are not actually on the GE being
followed. Since GEs often pass quite near a stationary
point, SNRMAX should only be increased above the default
0.10 after some consideration.
Continuum Solvation Methods
In a very thorough 1994 review of continuum solvation
models, Tomasi and Persico divide the possible approaches
to the treatment of solvent effects into four categories:
a) virial equations of state, correlation functions
b) Monte Carlo or molecular dynamics simulations
c) continuum treatments
d) molecular treatments
The Effective Fragment Potential method, documented in the
following section of this chapter, falls into the latter
category, as each EFP solvent molecule is modeled as a
distinct object (discrete solvation). This section
describes the four continuum models which are implemented
in the standard version of GAMESS, and a fifth model which
can be interfaced.
Continuum models typically form a cavity of some sort
containing the solute molecule, while the solvent outside
the cavity is thought of as a continuous medium and is
categorized by a limited amount of physical data, such as
the dielectric constant. The electric field of the charged
particles comprising the solute interact with this
background medium, producing a polarization in it, which in
turn feeds back upon the solute's wavefunction.
Self Consistent Reaction Field (SCRF)
A simple continuum model is the Onsager cavity model,
often called the Self-Consistent Reaction Field, or SCRF
model. This represents the charge distribution of the
solute in terms of a multipole expansion. SCRF usually
uses an idealized cavity (spherical or ellipsoidal) to
allow an analytic solution to the interaction energy
between the solute multipole and the multipole which this
induces in the continuum. This method is implemented in
GAMESS in the simplest possible fashion:
i) a spherical cavity is used
ii) the molecular electrostatic potential of the
solute is represented as a dipole only, except
a monopole is also included for an ionic solute.
The input for this implementation of the Kirkwood-Onsager
model is provided in $SCRF.
Some references on the SCRF method are
1. J.G.Kirkwood J.Chem.Phys. 2, 351 (1934)
2. L.Onsager J.Am.Chem.Soc. 58, 1486 (1936)
3. O.Tapia, O.Goscinski Mol.Phys. 29, 1653 (1975)
4. M.M.Karelson, A.R.Katritzky, M.C.Zerner
Int.J.Quantum Chem., Symp. 20, 521-527 (1986)
5. K.V.Mikkelsen, H.Agren, H.J.Aa.Jensen, T.Helgaker
J.Chem.Phys. 89, 3086-3095 (1988)
6. M.W.Wong, M.J.Frisch, K.B.Wiberg
J.Am.Chem.Soc. 113, 4776-4782 (1991)
7. M.Szafran, M.M.Karelson, A.R.Katritzky, J.Koput,
M.C.Zerner J.Comput.Chem. 14, 371-377 (1993)
8. M.Karelson, T.Tamm, M.C.Zerner
J.Phys.Chem. 97, 11901-11907 (1993)
The method is very sensitive to the choice of the solute
RADIUS, but not very sensitive to the particular DIELEC of
polar solvents. The plots in reference 7 illustrate these
points very nicely. The SCRF implementation in GAMESS is
Zerner's Method A, described in the same reference. The
total solute energy includes the Born term, if the solute
is an ion. Another limitation is that a solute's electro-
static potential is not likely to be fit well as a dipole
moment only, for example see Table VI of reference 5 which
illustrates the importance of higher multipoles. Finally,
the restriction to a spherical cavity may not be very
representative of the solute's true shape. However, in the
special case of a roundish molecule, and a large dipole
which is geometry sensitive, the SCRF model may include
sufficient physics to be meaningful:
M.W.Schmidt, T.L.Windus, M.S.Gordon
J.Am.Chem.Soc. 117, 7480-7486(1995).
Polarizable Continuum Model (PCM)
A much more sophisticated continuum method, named the
Polarizable Continuum Model, is also available. The PCM
method places a solute in a cavity formed by a union of
spheres centered on each atom. PCM also includes a more
exact treatment of the electrostatic interaction with the
surrounding medium, as the electrostatic potential of the
solute generates an 'apparent surface charge' on the
cavity's surface. The computational procedure divides this
surface into small tesserae, on which the surface charge
(and contributions to the gradient) are evaluated.
Typically the spheres defining the cavity are taken to be
1.2 times the van der Waals radii. A technical difficulty
caused by the penetration of the solute's charge density
outside this cavity is dealt with by a renormalization.
The solvent is characterized by its dielectric constant,
surface tension, size, density, and so on. Procedures are
provided not only for the computation of the electrostatic
interaction of the solute with the apparent surface
charges, but also for the cavitation energy, and dispersion
and repulsion contributions to the solvation free energy.
The main input group is $PCM, with $PCMCAV providing
auxiliary cavity information. If any of the optional
energy computations are requested in $PCM, the additional
input groups $IEFPCM, $NEWCAV, $DISBS, or $DISREP may be
required.
Solvation of course affects the non-linear optical
properties of molecules. The PCM implementation extends
RUNTYP=TDHF to include solvent effects. Both static and
frequency dependent hyperpolarizabilities can be found.
Besides the standard PCM electrostatic contribution, the
IREP and IDP keywords can be used to determine the effects
of repulsion and dispersion on the polarizabilities.
Due to its sophistication, users of the PCM model are
strongly encouraged to read the primary literature. The
first references use the field method to solve the apparent
surface charge problem. Recently the integral equation
formalism and conductor-like PCM have been developed as
more numerically suitable methods.
The implementation of the PCM model in GAMESS has
received considerable attention from Hui Li and Jan Jensen
at the University of Iowa. This includes new techniques
for solving the surface charge problem, new tessellations
that provide for numerically stable nuclear gradients,
extension to all SCFTYP values, and modification of the
interface with the EFP model (quo vadis). See
1) H.Li, C.S.Pomelli, J.H.Jensen
Theoret.Chim.Acta 109, 71-84(2003)
2) H.Li, J.H.Jensen, J.Comput.Chem. 25, 1449-1462 (2004)
General papers on PCM (review paper 4 is a landmark):
3) S.Miertus, E.Scrocco, J.Tomasi
Chem.Phys. 55, 117-129(1981)
4) J.Tomasi, M.Persico Chem.Rev. 94, 2027-2094(1994)
5) R.Cammi, J.Tomasi J.Comput.Chem. 16, 1449-1458(1995)
The GEPOL-GB method for cavity construction:
6) J.L.Pascual-Ahuir, E.Silla, J.Tomasi, R.Bonaccorsi
J.Comput.Chem. 8, 778-787(1987)
Charge renormalization (see also ref. 5):
7) B.Mennucci, J.Tomasi J.Chem.Phys. 106, 5151-5158(1997)
Derivatives with respect to nuclear coordinates:
(energy gradient and hessian) See also paper 2.
8) R.Cammi, J.Tomasi J.Chem.Phys. 100, 7495-7502(1994)
9) R.Cammi, J.Tomasi J.Chem.Phys. 101, 3888-3897(1995)
10) M.Cossi, B.Mennucci, R.Cammi
J.Comput.Chem. 17, 57-73(1996)
Derivatives with respect to applied electric fields:
(polarizabilities and hyperpolarizabilities)
11) R.Cammi, J.Tomasi
Int.J.Quantum Chem. Symp. 29, 465-474(1995)
12) R.Cammi, M.Cossi, J.Tomasi
J.Chem.Phys. 104, 4611-4620(1996)
13) R.Cammi, M.Cossi, B.Mennucci, J.Tomasi
J.Chem.Phys. 105, 10556-10564(1996)
14) B. Mennucci, C. Amovilli, J. Tomasi
Chem.Phys.Lett. 286, 221-225(1998)
Cavitation energy:
15) R.A.Pierotti Chem.Rev. 76, 717-726(1976)
16) J.Langlet, P.Claverie, J.Caillet, A.Pullman
J.Phys.Chem. 92, 1617-1631(1988)
Dispersion and repulsion energies:
17) F.Floris, J.Tomasi J.Comput.Chem. 10, 616-627(1989)
18) C.Amovilli, B.Mennucci
J.Phys.Chem.B 101, 1051-1057(1997)
Integral Equation Formalism papers. The first of these
deals with anisotropies, the last 2 with nuclear gradients.
19) E.Cances, B.Mennucci, J.Tomasi
J.Chem.Phys. 107, 3032-3041(1997)
20) B.Mennucci, E.Cances, J.Tomasi
J.Phys.Chem.B 101, 10506-17(1997)
21) B.Mennucci, R.Cammi, J.Tomasi
J.Chem.Phys. 109, 2798-2807(1998)
22) J.Tomasi, B.Mennucci, E.Cances
J.Mol.Struct.(THEOCHEM) 464, 211-226(1999)
23) E.Cances, B.Mennucci J.Chem.Phys. 109, 249-259(1998)
24) E.Cances, B.Mennucci, J.Tomasi
J.Chem.Phys. 109, 260-266(1998)
Conductor PCM (C-PCM):
25. V.Barone, M.Cossi J.Phys.Chem.A 102, 1995-2001(1998)
26. M.Cossi, N.Rega, G.Scalmani, V.Barone
J.Comput.Chem. 24, 669-681(2003)
At the present time, the PCM model in GAMESS has the
following limitations:
a) Although any SCFTYP may be used, along with their
matching DFT methods, none of the following may be
used: MP2, CI, or Coupled Cluster.
b) semi-empirical methods may not be used
c) the only other solvent method that may be used at
used with PCM is the EFP model.
d) point group symmetry is switched off internally
during PCM.
e) The PCM model runs in parallel for IEF=3, -3, 10,
or -10 and for all 5 wavefunctions (energy or
gradient), but not for TDHF jobs.
f) electric field integrals at normals to the surface
elements are stored on disk, even during DIRSCF
runs. The file size may be considerable.
g) To minimize common block storage, the maximum
number of spheres forming the cavity is 500, with
an upper limit on the number of surface tesserae
set to 20000. These may be raised by the 'mung'
script listed in the Programming chapter.
h) nuclear derivatives are limited to gradients,
although theory for hessians is given in paper 10.
The calculation shown next illustrates the use of most
PCM options. Since methane is non-polar, its internal
energy change and the direct PCM electrostatic interaction
is smaller than the cavitation, repulsion, and dispersion
corrections. Note that the use of ICAV, IREP, and IDP are
currently incompatible with gradients, so a reasonable
calculation sequence might be to perform the geometry
optimization with PCM electrostatics turned on, then
perform an additional calculation to include the other
solvent effects, adding extra functions to improve the
dispersion correction.
! calculation of CH4 (metano) in PCM water.
! This input reproduces the data in Table 2, line 6, of
! C.Amovilli, B.Mennucci J.Phys.Chem. B101, 1051-7(1997)
!
! The gas phase FINAL energy is -40.2075980280
! The FINAL energy in PCM water= -40.2143590161
! (lit.)
! FREE ENERGY IN SOLVENT = -25234.89 KCAL/MOL
! INTERNAL ENERGY IN SOLVENT = -25230.64 KCAL/MOL
! DELTA INTERNAL ENERGY = .01 KCAL/MOL ( 0.0)
! ELECTROSTATIC INTERACTION = -.22 KCAL/MOL (-0.2)
! PIEROTTI CAVITATION ENERGY = 5.98 KCAL/MOL ( 6.0)
! DISPERSION FREE ENERGY = -6.00 KCAL/MOL (-6.0)
! REPULSION FREE ENERGY = 1.98 KCAL/MOL ( 2.0)
! TOTAL INTERACTION = 1.73 KCAL/MOL ( 1.8)
! TOTAL FREE ENERGY IN SOLVENT= -25228.91 KCAL/MOL
!
$contrl scftyp=rhf runtyp=energy $end
$guess guess=huckel $end
$system memory=300000 $end
! the W1 basis input here exactly matches HONDO's DZP
$DATA
CH4...gas phase geometry...in PCM water
Td
Carbon 6.
DZV
D 1 ; 1 0.75 1.0
Hydrogen 1. 0.6258579976 0.6258579976 0.6258579976
DZV 0 1.20 1.15 ! inner and outer scale factors
P 1 ; 1 1.00 1.0
$END
! reference cited used value for H2O's solvent radius
! which differs from the built in constants.
! D-PCM with the GEPOL-GB tessellation is chosen to
! reproduce the literature calculation.
$PCM IEF=0 ICOMP=2 IREP=1 IDP=1 ICAV=1
SOLVNT=WATER RSOLV=1.35 $END
$NEWCAV IPTYPE=2 ITSNUM=540 $END
$TESCAV MTHALL=1 $END
! dispersion W2 basis uses exponents which are
! 1/3 of smallest exponent in W1 basis of $DATA.
$DISBS NADD=11 NKTYP(1)=0,1,2, 0,1, 0,1, 0,1, 0,1
XYZE(1)=0.0,0.0,0.0, 0.0511
0.0,0.0,0.0, 0.0382
0.0,0.0,0.0, 0.25
1.1817023, 1.1817023, 1.1817023, 0.05435467
1.1817023, 1.1817023, 1.1817023, 0.33333333
-1.1817023, 1.1817023,-1.1817023, 0.05435467
-1.1817023, 1.1817023,-1.1817023, 0.33333333
1.1817023,-1.1817023,-1.1817023, 0.05435467
1.1817023,-1.1817023,-1.1817023, 0.33333333
-1.1817023,-1.1817023, 1.1817023, 0.05435467
-1.1817023,-1.1817023, 1.1817023, 0.33333333 $end
SVPE and SS(V)PE.
The Surface Volume Polarization for Electrostatics
(SVPE), and an approximation to SVPE called the Surface and
Simulation of Volume Polarization for Electrostatics
(SS(V)PE) are continuum solvation models. Compared to
other continuum models, SVPE and SS(V)PE pay careful
attention to the problems of escaped charge, the shape of
the surface cavity, and to integration of the Poisson
equation for surface charges.
The original references for what is now called the
SVPE (surface and volume polarization for electrostatics)
method are the theory paper:
"Charge penetration in Dielectric Models of Solvation"
D.M.Chipman, J.Chem.Phys. 106, 10194-10206 (1997)
and the two implementation papers:
"Volume Polarization in Reaction Field Theory"
C.-G.Zhan, J.Bentley, D.M.Chipman
J.Chem.Phys. 108, 177-192 (1998)
and
"New Formulation and Implementation for Volume
Polarization in Dielectric Continuum Theory"
D.M.Chipman, J.Chem.Phys. 124, 224111-1/10 (2006)
which should be cited in any publications that utilize the
SVPE code.
The original reference for the SS(V)PE (surface and
simulation of volume polarization for electrostatics)
method is:
"Reaction Field Treatment of Charge Penetration"
D.M.Chipman, J.Chem.Phys. 112, 5558-5565 (2000)
which should be cited in any publications that utilize the
SS(V)PE code.
Further information on the performance of SVPE and of
SS(V)PE can be found in:
"Comparison of Solvent Reaction Field Representations"
D.M.Chipman, Theor.Chem.Acc. 107, 80-89 (2002).
Details of the SS(V)PE convergence behavior and programming
strategy are in:
"Implementation of Solvent Reaction Fields for
Electronic Structure" D.M.Chipman, M.Dupuis,
Theor.Chem.Acc. 107, 90-102 (2002).
The SVPE and SS(V)PE models are like PCM and COSMO in
that they treat solvent as a continuum dielectric residing
outside a molecular-shaped cavity, determining the apparent
charges that represent the polarized dielectric by solving
Poisson's equation. The main difference between SVPE and
SS(V)PE is in treatment of volume polarization effects that
arise because of the tail of the electronic wave function
that penetrates outside the cavity, sometimes referred to
as the "escaped charge." SVPE treats volume polarization
effects explicitly by including apparent charges in the
volume outside the cavity as well as on the cavity surface.
With a sufficient number of grid points, SVPE can then
provide an exact treatment of charge penetration effects.
SS(V)PE, like PCM and COSMO, is an approximate treatment
that only uses apparent charges located on the cavity
surface. The SS(V)PE equation is particularly designed to
simulate as well as possible the influence of the missing
volume charges. For more information on the similarities
and differences of the SVPE and SS(V)PE models with other
continuum methods, see the paper "Comparison of Solvent
Reaction Field Representations" cited just above.
In addition, the cavity construction and Poisson
solver used in this implementation of SVPE and SS(V)PE also
receive careful numerical treatment. For example, the
cavity may be chosen to be an isodensity contour surface,
and the Lebedev grids for the Poisson solver can be chosen
very densely. The Lebedev grids used for surface
integration are taken from the Fortran translation by C.
van Wuellen of the original C programs developed by D.
Laikov. They were obtained from the CCL web site
www.ccl.net/cca/software/SOURCES/FORTRAN/Lebedev-Laikov-
Grids. A recent leading reference is V. I. Lebedev and D.
N. Laikov, Dokl. Math. 59, 477-481 (1999). All these grids
have octahedral symmetry and so are naturally adapted for
any solute having an Abelian point group. The larger and/or
the less spherical the solute may be, the more surface
points are needed to get satisfactory precision in the
results. Further experience will be required to develop
detailed recommendations for this parameter. Values as
small as 110 are usually sufficient for simple diatomics
and triatomics. The default value of 1202 has been found
adequate to obtain the energy to within 0.1 kcal/mol for
solutes the size of monosubstituted benzenes. The SVPE
method uses additional layers of points outside the cavity.
Typically just two layers are sufficient to converge the
direct volume polarization contribution to better than 0.1
kcal/mol.
The SVPE and SS(V)PE codes both report the amount of
solute charge penetrating outside the cavity as calculated
by Gauss' Law. The SVPE code additionally reports the same
quantity as alternatively calculated from the explicit
volume charges, and any substantial discrepancy between
these two determinations indicates that more volume
polarization layers should have been included for better
precision. The energy contribution from the outermost
volume polarization layer is also reported. If it is
significant then again more layers should have been
included. However, these tests are only diagnostic.
Passing them does not guarantee that enough layers are
included.
The SVPE and SS(V)PE models treat the electrostatic
interaction between a quantum solute and a classical
dielectric continuum solvent. No treatment is yet
implemented for cavitation, dispersion, or any of a variety
of other specific solvation effects. Note that corrections
for these latter effects that might be reported by other
programs are generally not transferable. The reason is
that they are usually parameterized to improve the ultimate
agreement with experiment. In addition to providing
corrections for the physical effects advertised, they
therefore also inherently include contributions that help
to make up for any deficiencies in the electrostatic
description. Consequently, they are appropriate only for
use with the particular electrostatic model in which they
were originally developed.
Analytic nuclear gradients are not yet available for
the SVPE or SS(V)PE energy, but numerical differentiation
will permit optimization of small solute molecules.
Wavefunctions may be any of the SCF type: RHF, UHF, ROHF,
GVB, and MCSCF, or the DFT analogs of some of these. In the
MCSCF implementation, no initial wavefunction is available
so the solvation code does not kick in until the second
iteration.
We close with a SVPE example. The gas phase energy,
obtained with no $SVP group, is -207.988975, and the run
just below gives the SVPE energy -208.006282. The free
energy of solvation, -10.860 kcal/mole, is the difference
of these, and is quoted at the right side of the 3rd line
from the bottom of Table 2 in the paper cited. The
"REACTION FIELD FREE ENERGY" for SVPE is -12.905 kcal/mole,
which is only part of the solvation free energy. There is
also a contribution due to the SCRF procedure polarizing
the wave function from its gas phase value, causing the
solute internal energy in dielectric to differ from that in
gas. Evaluating this latter contribution is what requires
the separate gas phase calculation. Changing the number of
layers (NVLPL) to zero produces the SS(V)PE approximation
to SVPE, E= -208.006208.
! SVPE solvation test...acetamide
! reproduce data in Table 2 of the paper on SVPE,
! D.M.Chipman J.Chem.Phys. 124, 224111/1-10(2006)
!
$contrl scftyp=rhf runtyp=energy $end
$system mwords=4 $end
$basis gbasis=n31 ngauss=6 ndfunc=1 npfunc=1 $end
$guess guess=moread norb=16 $end
$scf nconv=8 $end
$svp nvlpl=3 rhoiso=0.001 dielst=78.304 nptleb=1202 $end
$data
CH3CONH2 cgz geometry RHF/6-31G(d,p)
C1
C 6.0 1.361261 -0.309588 -0.000262
C 6.0 -0.079357 0.152773 -0.005665
H 1.0 1.602076 -0.751515 0.962042
H 1.0 1.537200 -1.056768 -0.767127
H 1.0 2.002415 0.542830 -0.168045
O 8.0 -0.387955 1.310027 0.002284
N 7.0 -1.002151 -0.840834 -0.011928
H 1.0 -1.961646 -0.589397 0.038911
H 1.0 -0.752774 -1.798630 0.035006
$end
gas phase vectors, E(RHF)= -207.9889751769
$VEC
1 1 1.18951670E-06 1.74015997E-05
...snipped...
$END
Conductor-like screening model (COSMO)
COSMO (conductor-like screening model) represents a
different approach for carrying out polarized continuum
calculations. The model was originally developed by
Andreas Klamt, with extensions to ab initio computation
in GAMESS by Kim Baldridge.
In the COSMO method, the surrounding medium is modeled
as a conductor rather than as a dielectric in order to
establish the initial boundary conditions. The assumption
that the surrounding medium is well modelled as a conductor
simplifies the electrostatic computations and corrections
may be made a posteriori for dielectric behavior.
The current implementation of COSMO involves the
computation of distributed multipoles up to hexadecapoles
to represent the charge distribution of the molecule
within the cavity. The multipole moments induce the
formation of charges on the surface of the cavity that
contains the molecule. These charges are then fed back
into the SCF, and both the molecular wavefunction and
the surface charges are iterated to self-consistency.
The original model of Klamt was introduced using a
molecular shaped cavity which had open parts along the
crevices of intersecting atomic spheres. While having
considerable technical advantages, this approximation
causes artifacts in the context of the more generalized
theory, so the current method for cavity construction
includes a closure of the cavity to eliminate crevices or
pockets.
At present, the COSMO model accounts only for the
electrostatic interactions between solvent and solute.
Klamt has proposed a novel statistical scheme to compute
the full solvation free energy for neutral solutes, which
will be formulated for GAMESS by Baldridge et al.
The simplicity of the COSMO model allows computation of
gradients, allowing optimization within the context of the
solvent. The method is programmed for closed shell RHF
energy and gradient, and the MP2 energy correction may be
obtained.
Some references on the COSMO model are:
A.Klamt, G.Schuurman
J.Chem.Soc.Perkin Trans 2, 799-805(1993)
A.Klamt J.Phys.Chem. 99, 2224-2235(1995)
K.Baldridge, A.Klamt
J.Chem.Phys. 106, 6622-6633 (1997)
SMx Solvation Models
A final possible continuum treatment is the "solvation
model" approach. Ab initio SMx is described in
SM5.42:
J.Li, G.D.Hawkins, C.J.Cramer, D.G.Truhlar
Chem.Phys.Lett. 288, 293-298(1998)
SM5.43:
J.D.Thompson, C.J.Cramer, D.G.Truhlar
J.Phys.Chem.A 108, 6532-6542 (2004)
SM6:
C.P.Kelly, C.J.Cramer, D.G.Truhlar
J.Theor.Comput.Chem. 1, 1133-1152(2005)
SM6T:
A.C.Chamberlin, C.J.Cramer, D.G.Truhlar
J.Phys.Chem.B 110, 5665-5675(2006)
SMx represents the molecule's electrostatics as a set of
atomic point charges. These are chosen by a procedure
based on correcting Lowdin atomic charges according to a
quadratic function of the computed Mayer bond orders,
which is designed to better reproduce experimental dipole
moments. These charges are called "charge model 2", and
CM2 is described in
J.Li, T.Zhu, C.J.Cramer, D.G.Truhlar
J.Phys.Chem.A 102, 1820-1831(1998)
In addition to a self-consistent reaction field treatment
of the CM2 electrostatics, SMx includes a term accounting
for the following first solvation shell effects: cavity
creation, dispersion, and changes in solvent structure,
which are modeled by atomic surface tension parameters.
It is possible to use this code simply to extract gas phase
CM2 charges. The implementation is termed GAMESSPLUS, and
is available at
http://comp.chem.umn.edu/gamessplus
After signing a license not much more stringent than the
license for GAMESS itself, you can obtain the new source
code from U. Minnesota. The interface is not clean, as
considerable code is inserted directly into RHFUHF and
other GAMESS modules, so you must be very careful to obtain
code that matches the dates on the top of your original
GAMESS source files.
The Effective Fragment Potential Method
The basic idea behind the effective fragment potential
(EFP) method is to replace the chemically inert part of a
system by EFPs, while performing a regular ab initio
calculation on the chemically active part. Here "inert"
means that no covalent bond breaking process occurs. This
"spectator region" consists of one or more "fragments",
which interact with the ab initio "active region" through
non-bonded interactions, and so of course these EFP
interactions affect the ab initio wavefunction. A simple
example of an active region might be a solute molecule,
with a surrounding spectator region of solvent molecules
represented by fragments. Each discrete solvent molecule
is represented by a single fragment potential, in marked
contrast to continuum models for solvation.
The quantum mechanical part of the system is entered in
the $DATA group, along with an appropriate basis. The EFPs
defining the fragments are input by means of a $EFRAG
group, and one or more $FRAGNAME groups describing each
fragment's EFP. These groups define non-bonded
interactions between the ab initio system and the
fragments, and also between the fragments. The former
interactions enter via one-electron operators in the ab
initio Hamiltonian, while the latter interactions are
treated by analytic functions. The only electrons
explicitly treated (with basis functions used to expand
occupied orbitals) are those in the active region, so there
are no new two electron terms. Thus the use of EFPs leads
to significant time savings, compared to full ab initio
calculations on the same system.
There are two types of EFP available in GAMESS, EFP1 and
EFP2. EFP1, the original method, employs a fitted
repulsive potential. EFP1 is primarily used to model water
molecules to study aqueous solvation effects, at the
RHF/DZP or DFT/DZP (specifically, B3LYP) levels, see
references 1-3 and 26, respectively. EFP2 is a more
general method that is applicable to any species, including
water, and its repulsive potential is obtained from first
principles. EFP2 has been extended to include other
effects as well, such as charge transfer and dispersion.
EFP2 forms the basis of the covalent EFP method described
below for modeling enzymes, see reference 14.
Parallelization of the EFP1 and EFP2 models is described
in reference 32.
MD simulations with EFP are described in reference 31.
The ab initio/EFP1, or pure EFP system can be wrapped in
a Polarizable Continuum Model, see references 23 and 43.
terms in an EFP
The non-bonded interactions currently implemented are:
1) Coulomb interaction. The charge distribution of the
fragments is represented by an arbitrary number of charges,
dipoles, quadrupoles, and octopoles, which interact with
the ab initio hamiltonian as well as with multipoles on
other fragments (see reference 2 and 18). It is possible
to use a screening term that accounts for the charge
penetration (reference 17 and 42). This screening term is
automatically included for EFP1. Typically the multipole
expansion points are located on atomic nuclei and at bond
midpoints.
2) Dipole polarizability. An arbitrary number of dipole
polarizability tensors can be used to calculate the induced
dipole on a fragment due to the electric field of the ab
initio system as well as all the other fragments. These
induced dipoles interact with the ab initio system as well
as the other EFPs, in turn changing their electric fields.
All induced dipoles are therefore iterated to self-
consistency. Typically the polarizability tensors are
located at the centroid of charge of each localized orbital
of a fragment. See reference 41.
3) Repulsive potential. Two different forms are used in
EFP1: one for ab initio-EFP repulsion and one for EFP-EFP
repulsion. The form of the potentials is empirical, and
consists of distributed Gaussian or exponential functions,
respectively. The primary contribution to the repulsion is
the quantum mechanical exchange repulsion, but the fitting
technique used to develop this term also includes the
effects of charge transfer. Typically these fitted
potentials are located on each atomic nucleus within the
fragment (see reference 3). In EFP2, polarization energies
can also be augmented by screening terms, analogous to the
electrostatic screening, to prevent "polarization collapse"
(MS in preparation)
For EFP2, the third term is divided into separate analytic
formulae for different physical interactions:
a) exchange repulsion
b) dispersion
c) charge transfer
A summary of EFP2, and its contrast to EFP1 can be found in
reference 18 and 44. The repulsive potential for EFP2 is
based on an overlap expansion using localized molecular
orbitals, as described in references 5, 6, and 9.
Dispersion energy is described in reference 34, and charge
transfer in reference 39 (which supercedes reference 22's
formulae).
EFP2 potentials have no fitted parameters, and can be
automatically generated during a RUNTYP=MAKEFP job, as
described below.
constructing an EFP1
RUNTYP=MOROKUMA assists in the decomposition of inter-
molecular interaction energies into electrostatic,
polarization, charge transfer, and exchange repulsion
contributions. This is very useful in developing EFPs
since potential problems can be attributed to a particular
term by comparison to these energy components for a
particular system.
A molecular multipole expansion can be obtained using
$ELMOM. A distributed multipole expansion can be obtained
by either a Mulliken-like partitioning of the density
(using $STONE) or by using localized molecular orbitals
($LOCAL: DIPDCM and QADDCM). The dipole polarizability
tensor can be obtained during a Hessian run ($CPHF), and a
distributed LMO polarizability expression is also available
($LOCAL: POLDCM).
In EFP1, the repulsive potential is derived by fitting
the difference between ab initio computed intermolecular
interaction energies, and the form used for Coulomb and
polarizability interactions. This difference is obtained
at a large number of different interaction geometries, and
is then fitted. Thus, the repulsive term is implicitly a
function of the choices made in representing the Coulomb
and polarizability terms. Note that GAMESS currently does
not provide a way to obtain these EFP1 repulsive potential.
Since a user cannot generate all of the EFP1 terms
necessary to define a new $FRAGNAME group using GAMESS, in
practice the usage of EFP1 is limited to the internally
stored H2ORHF or H2ODFT potentials mentioned below.
constructing an EFP2
As noted above, the repulsive potential for EFP2 is
derived from a localized orbital overlap expansion. It is
generally recommended that one use at least a double zeta
plus diffuse plus polarization basis set, e.g. 6-31++G(d,p)
to generate the EFP2 repulsive potential. However, it has
been observed that 6-31G(d) works reasonably well due to a
fortuitous cancellation of errors. The EFP2 potential for
any molecule can be generated as follows:
(a) Choose a basis set and geometry for the molecule of
interest. The geometry is ordinarily optimized at the
Hartree-Fock level of theory with the chosen basis set, but
this is not a requirement. It is good to recall, however,
that EFP internal geometries are fixed, so it is important
to give some thought to the chosen geometry.
(b) Perform a RUNTYP=MAKEFP run for the chosen molecule
using the chosen geometry in $DATA and the chosen basis set
in $BASIS. This will generate the entire EFP2 potential in
the run's .dat file. The only user-defined variable that
must be filled in is changing the FRAGNAME's group name, to
$C2H5OH or $DMSO, etc.
(c) Transfer the entire fragment potential for the molecule
to any input file in which this fragment is to be used.
Since the internal geometry of an EFP is fixed, one need
only specify the first three atoms of any fragment in order
to position them in $EFRAG. Coordinates of any other atoms
in the rigid fragment will be automatically determined by
the program.
If the EFP contains less than three atoms, you can still
generate a fragment potential. After a normal MAKEFP run,
add dummy atoms (e.g. in the X and/or Y directions) with
zero nuclear charges, and add corresponding dummy bond
midpoints too. Carefully insert zero entries in the
multipole sections for each such dummy point, but don't add
data to any other kind of EFP term such as polarizability.
This trick gives the necessary 3 points for use in $EFRAG
groups to specify "rotational" positions of fragments.
current limitations
1. For EFP1, the energy and energy gradient are programmed,
which permits RUNTYP=ENERGY, GRADIENT, and numerical
HESSIAN. The necessary programing to use the EFP gradients
to move on the potential surface are programmed for
RUNTYP=OPTIMIZE, SADPOINT, and IRC, but the other gradient
based potential surface explorations such as DRC or VSCF
are not yet available. Finally, RUNTYP=PROP is also
permissible.
For EFP2, the gradient terms for ab initio-EFP interactions
have not yet been coded, so geometry optimizations are only
sensible for a COORD=FRAGONLY run; that is, a run in which
only EFP2 fragments are present.
2. The ab initio part of the system must be treated with
RHF, ROHF, UHF, the open shell SCF wavefunctions permitted
by the GVB code, or MCSCF. DFT analogs of RHF, ROHF, and
UHF may also be used. Correlated methods such as MP2 and
CI should not be used.
3. EFPs can move relative to the ab initio system and
relative to each other, but the internal structure of an
EFP is frozen.
4. The boundary between the ab initio system and EFP1's
must not be placed across a chemical bond. However, see
the discussion below regarding covalent bonds.
5. Calculations must be done in C1 symmetry at present.
6. Reorientation of the fragments and ab initio system is
not well coordinated. If you are giving Cartesian
coordinates for the fragments (COORD=CART in $EFRAG), be
sure to use $CONTRL's COORD=UNIQUE option so that the ab
initio molecule is not reoriented.
7. If you need IR intensities, you have to use NVIB=2. The
potential surface is usually very soft for EFP motions, and
double differenced Hessians should usually be obtained.
practical hints for using EFPs
At the present time, we have only two internally stored
EFP potentials suitable for general use. These model
water, using the fragment name H2ORHF or H2ODFT. The
H2ORHF numerical parameters are improved values over the
values which were presented and used in reference 2, and
they also include the improved EFP-EFP repulsive term
defined in reference 3. The H2ORHF water EFP was derived
from RHF/DH(d,p) computations on the water dimer system.
When you use it, therefore, the ab initio part of your
system should be treated at the SCF level, using a basis
set of the same quality (ideally DH(d,p), but probably
other DZP sets such as 6-31G(d,p) will give good results as
well). Use of better basis sets than DZP with this water
EFP has not been tested. Similarly, H2ODFT was developed
using B3LYP/DZP water wavefunctions, so should be used with
B3LYP treatments of solutes.
As noted, effective fragments have frozen internal
geometries, and therefore only translate and rotate with
respect to the ab initio region. An EFP's frozen
coordinates are positioned to the desired location(s) in
$EFRAG as follows:
a) the corresponding points are found in $FRAGNAME.
b) Point -1- in $EFRAG and its FRAGNAME equivalent are
made to coincide.
c) The vector connecting -1- and -2- is aligned with
the corresponding vector connecting FRAGNAME points.
d) The plane defined by -1-, -2-, and -3- is made to
coincide with the corresponding FRAGNAME plane.
Therefore the 3 points in $EFRAG define only the relative
position of the EFP, and not its internal structure. So, if
the "internal structure" given by points in $EFRAG differs
from the true values in $FRAGNAME, then the order in which
the points are given in $EFRAG can affect the positioning
of the fragment. It may be easier to input water EFPs if
you use the Z-matrix style to define them, because then you
can ensure you use the actual frozen geometry in your
$EFRAG. Note that the H2ORHF EFP uses the frozen geometry
r(OH)=0.9438636, a(HOH)=106.70327, and the names of its 3
fragment points are ZO1, ZH2, ZH3.
The translations and rotations of EFPs with respect to
the ab initio system and one another are automatically
quite soft degrees of freedom. After all, the EFP model is
meant to handle weak interactions! Therefore the
satisfactory location of structures on these flat surfaces
will require use of a tight convergence on the gradient:
OPTTOL=0.00001 in the $STATPT group.
The effect of a bulk continuum surrounding the solute
plus EFP waters can be obtained by using the PCM model, see
reference 23 and 43. To do this, simply add a $PCM group
to your input, in addition to the $EFRAG. The simultaneous
use of EFP and PCM is presently limited to energy
calculations, so any geometry optimization must be done
with only $EFRAG input.
global optimization
If there are a large number of effective fragments, it
is difficult to locate the lowest energy structures by
hand. Typically these are numerous, and one would like to
have a number of them, not just the very lowest energy.
The RUNTYP of GLOBOP contains a Monte Carlo procedure to
generate a random set of starting structures to look for
those with the lowest energy at a single temperature. If
desired, a simulated annealing protocol to cool the
temperature may be used. These two procedures may be
combined with a local minimum search, at some or all of the
randomly generated structures. The local minimum search is
controlled by the usual geometry optimizer, namely $STATPT
input, and thus permits the optimization of any ab initio
atoms.
The Monte Carlo procedure by default uses a Metropolis
algorithm to move just one of the effective fragments. If
desired, the method of Parks to move all fragments at once
may be tried, by changing ALPHA from zero and setting
BOLTWT=AVESTEP instead of STANDARD.
The present program was used to optimize the structure
of water clusters. Let us consider the case of the twelve
water cluster, for which the following ten structures were
published by Day, Pachter, Gordon, and Merrill:
1. (D2d)2 -0.170209 6. (D2d)(C2) -0.167796
2. (D2d)(S4) -0.169933 7. S6 -0.167761
3. (S4)2 -0.169724 8. cage b -0.167307
4. D3 -0.168289 9. cage a -0.167284
5. (C1c)(Cs) -0.167930 10. (C1c)(C1c) -0.167261
A test input using Metropolis style Monte Carlo to examine
300 geometries at each temperature value, using simulated
annealing cooling from 200 to 50 degrees, and with local
minimization every 10 structures was run ten times. Each
run sampled about 7000 geometries. One simulation found
structure 2, while two of the runs found structure 3. The
other seven runs located structures with energy values in
the range -0.163 to -0.164. In all cases the runs began
with the same initial geometry, but produced different
results due to the random number generation used in the
Monte Carlo. Clearly one must try a lot of simulations to
be confident about having found most of the low energy
structures. In particular, it is good to try more than one
initial structure, unlike what was done in this test.
If there is an ab initio molecule in your system, it is
probably impractical to carry out a simulated annealing
protocol. However, a single temperature Monte Carlo
calculation may be feasible. In particular, you may wish
to avoid the local minimization steps, and instead manually
examine the structures from the Monte Carlo steps in order
to choose a few for full geometry optimization. Note that
SMODIF input can allow the ab initio part of the system to
participate in the Monte Carlo jumps. However, this should
be done with caution.
Monte Carlo references:
N.Metropolis, A.Rosenbluth, A.Teller
J.Chem.Phys. 21, 1087(1953).
G.T.Parks Nucl.Technol. 89, 233(1990).
Monte Carlo with local minimization:
Z.Li, H.A.Scheraga
Proc.Nat.Acad.Sci. USA 84, 6611(1987).
Simulated annealing reference:
S.Kirkpatrick, C.D.Gelatt, M.P.Vecci
Science 220, 671(1983).
The present program is described in reference 15. It is
pattened on the work of
D.J.Wales, M.P.Hodges Chem.Phys.Lett. 286, 65-72 (1998).
QM/MM across covalent bonds
Recent work by Visvaldas Kairys and Jan Jensen has made
it possible to extend the EFP methodology beyond the simple
solute/solvent case described above. When there is a
covalent bond between the portion of the system to be
modeled by quantum mechanics, and the portion which is to
be treated by EFP multipole and polarizability terms, an
additional layer is needed in the model. The covalent
linkage is not so simple as the interactions between closed
shell solute and solvent molecules. The "buffer zone"
between the quantum mechanics and the EFP consists of
frozen nuclei, and frozen localized orbitals, so that the
quantum mechanical region sees a orbital representation of
the closest particles, and multipoles etc. beyond that.
Since the orbitals in the buffer zone are frozen, it need
extend only over a few atoms in order to keep the orbitals
in the fully optimized quantum region within that region.
The general outline of this kind of computation is as
follows:
a) a full quantum mechanics computation on a system
containing the quantum region, the buffer region,
and a few atoms into the EFP region, to obtain the
frozen localized orbitals in the buffer zone.
This is called the "truncation run".
b) a full quantum mechanics computation on a system
with all quantum region atoms removed, and with
the frozen localized orbitals in the buffer zone.
The necessary multipole and polarizability data
to construct the EFP that will describes the EFP
region will be extracted from the wavefunction.
This is called the "MAKEFP run". It is possible
to use several such runs if the total EFP region
is quite large.
c) The intended QM/MM run(s), after combining the
information from these first two types of runs.
As an example, consider a protonated lysine residue
which one might want to consider quantum mechanically in a
protein whose larger parts are to be treated with an EFP.
The protonated lysine is
NH2
+ /
H3N(CH2)(CH2)(CH2)--(CH2)(CH)
\
COOH
The bonds which you see drawn show how the molecule is
partitioned between the quantum mechanical side chain, a
CH2CH group in the buffer zone, and eventually two
different EFPs may be substituted in the area of the NH2
and COOH groups to form the protein backbone.
The "truncation run" will be on the entire system as you
see it, with the 13 atoms in the side chain first in $DATA,
the 5 atoms in the buffer zone next in $DATA, and the
simplified EFP region at the end. This run will compute
the full quantum wavefunction by RUNTYP=ENERGY, followed by
the calculation of localized orbitals, and then truncation
of the localized orbitals that are found in the buffer zone
so that they contain no contribution from AOs outside the
buffer zone. The key input groups for this run are
$contrl
$truncn doproj=.true. plain=.true. natab=13 natbf=5 $end
This will generate a total of 6 localized molecular
orbitals in the buffer zone (one CC, three CH, two 1s inner
shells), expanded in terms of atomic orbitals located only
on those atoms.
The truncation run prepares template input files for
the next run, including adjustments of nuclear charges at
boundaries, etc.
The "MAKEFP" run drops all 13 atoms in the quantum
region, and uses the frozen orbitals just prepared to
obtain a wavefunction for the EFP region. The carbon atom
in the buffer zone that is connected to the now absent QM
region will have its nuclear charge changed from 6 to 5 to
account for a missing electron. The key input for this
RUNTYP=MAKEFP job is the six orbitals in $VEC, plus the
groups
$guess guess=huckel insorb=6 $end
$mofrz frz=.true. ifrz(1)=1,2,3,4,5,6 $end
$stone
QMMMbuf
$end
which will cause the wavefunction optimization for the
remaining atoms to optimize orbitals only in the NH2 and
COOH pieces. After this wavefunction is found, the run
extracts the EFP information needed for the QM/MM third
run(s). This means running the Stone analysis for
distributed multipoles, and obtaining a polarizability
tensor for each localized orbital in the EFP region.
The QM/MM run might be RUNTYP=OPTIMIZE, etc. depending
on what you want to do with the quantum atoms, and its
$DATA group will contain both the 13 fully optimized atoms,
and the 5 buffer atoms, and a basis set will exist on both
sets of atoms. The carbon atom in the buffer zone that
borders the EFP region will have its nuclear charge set to
4 since now two bonding electrons to the EFP region are
lost. $VEC input will provide the six frozen orbitals in
the buffer zone. The EFP atoms are defined in a fragment
potential group.
The QM/MM run could use RHF or ROHF wavefunctions, to
geometry optimize the locations of the quantum atoms (but
not of course the frozen buffer zone or the EFP piece). It
could remove the proton to compute the proton affinity at
that terminal nitrogen, hunt for transition states, and so
on. Presently the gradient for GVB and MCSCF is not quite
right, so their use is discouraged.
Input to control the QM/MM preparation is $TRUNCN and
$MOFRZ groups. There are a number of other parameters in
various groups, namely QMMMBUF in $STONE, MOIDON and POLNUM
in $LOCAL, NBUFFMO in $EFRAG, and INSORB in $GUESS that are
relevant to this kind of computation. For RUNTYP=MAKEFP,
the biggest choices are LOCAL=RUEDENBRG vs. BOYS, and
POLNUM in $LOCAL, otherwise this is pretty much a standard
RUNTYP=ENERGY input file.
Source code distributions of GAMESS contain a directory
named ~/gamess/tools/efp, which has various tools for EFP
manipulation in it, described in file readme.1st. A full
input file for the protonated lysine molecule is included,
with instructions about how to proceed to the next steps.
Tips on more specialized input possibilities are appended
to the file readme.1st.
references
The first paper is more descriptive, while the second
presents a very detailed derivation of the EFP1 method.
Reference 18 is an overview article on EFP2.
The model development papers are: 1, 2, 3, 5, 6, 9, 14,
17, 18, 22, 23, 26, 31, 32, 34, 39, 41, 42, 43.
1. "Effective fragment method for modeling intermolecular
hydrogen bonding effects on quantum mechanical
calculations"
J.H.Jensen, P.N.Day, M.S.Gordon, H.Basch, D.Cohen,
D.R.Garmer, M.Krauss, W.J.Stevens in "Modeling the
Hydrogen Bond" (D.A. Smith, ed.) ACS Symposium Series
569, 1994, pp 139-151.
2. "An effective fragment method for modeling solvent
effects in quantum mechanical calculations".
P.N.Day, J.H.Jensen, M.S.Gordon, S.P.Webb,
W.J.Stevens, M.Krauss, D.Garmer, H.Basch, D.Cohen
J.Chem.Phys. 105, 1968-1986(1996).
3. "The effective fragment model for solvation: internal
rotation in formamide"
W.Chen, M.S.Gordon, J.Chem.Phys., 105, 11081-90(1996)
4. "Transphosphorylation catalyzed by ribonuclease A:
Computational study using ab initio EFPs"
B.D.Wladkowski, M. Krauss, W.J.Stevens
J.Am.Chem.Soc. 117, 10537-10545(1995)
5. "Modeling intermolecular exchange integrals between
nonorthogonal orbitals"
J.H.Jensen J.Chem.Phys. 104, 7795-7796(1996)
6. "An approximate formula for the intermolecular Pauli
repulsion between closed shell molecules"
J.H.Jensen, M.S.Gordon Mol.Phys. 89, 1313-1325(1996)
7. "A study of aqueous glutamic acid using the effective
fragment potential model"
P.N.Day, R.Pachter J.Chem.Phys. 107, 2990-9(1997)
8. "Solvation and the excited states of formamide"
M.Krauss, S.P.Webb J.Chem.Phys. 107, 5771-5(1997)
9. "An approximate formula for the intermolecular Pauli
repulsion between closed shell molecules. Application
to the effective fragment potential method"
J.H.Jensen, M.S.Gordon
J.Chem.Phys. 108, 4772-4782(1998)
10. "Study of small water clusters using the effective
fragment potential method"
G.N.Merrill, M.S.Gordon J.Phys.Chem.A 102, 2650-7(1998)
11. "Solvation of the Menshutkin Reaction: A Rigourous
test of the Effective Fragement Model"
S.P.Webb, M.S.Gordon J.Phys.Chem.A 103, 1265-73(1999)
12. "Evaluation of the charge penetration energy between
nonorthogonal molecular orbitals using the Spherical
Gaussian Overlap approximation"
V.Kairys, J.H.Jensen
Chem.Phys.Lett. 315, 140-144(1999)
13. "Solvation of Sodium Chloride: EFP study of NaCl(H2O)n"
C.P.Petersen, M.S.Gordon
J.Phys.Chem.A 103, 4162-6(1999)
14. "QM/MM boundaries across covalent bonds: frozen LMO
based approach for the Effective Fragment Potential
method"
V.Kairys, J.H.Jensen J.Phys.Chem.A 104, 6656-65(2000)
15. "A study of water clusters using the effective fragment
potential and Monte Carlo simulated annealing"
P.N.Day, R.Pachter, M.S.Gordon, G.N.Merrill
J.Chem.Phys. 112, 2063-73(2000)
16. "A combined discrete/continuum solvation model:
Application to glycine" P.Bandyopadhyay, M.S.Gordon
J.Chem.Phys. 113, 1104-9(2000)
17. "Evaluation of charge penetration between distributed
multipolar expansions"
M.A.Freitag, M.S.Gordon, J.H.Jensen, W.J.Stevens
J.Chem.Phys. 112, 7300-7306(2000)
18. "The Effective Fragment Potential Method: a QM-based MM
approach to modeling environmental effects in
chemistry"
M.S.Gordon, M.A.Freitag, P.Bandyopadhyay, J.H.Jensen,
V.Kairys, W.J.Stevens J.Phys.Chem.A 105, 293-307(2001)
19. "Accurate Intraprotein Electrostatics derived from
first principles: EFP study of proton affinities of
lysine 55 and tyrosine 20 in Turkey Ovomucoid"
R.M.Minikis, V.Kairys, J.H.Jensen
J.Phys.Chem.A 105, 3829-3837(2001)
20. "Active site structure & mechanism of Human Glyoxalase"
U.Richter, M.Krauss J.Am.Chem.Soc. 123, 6973-6982(2001)
21. "Solvent effect on the global and atomic DFT-based
reactivity descriptors using the EFP model. Solvation
of ammonia." R.Balawender, B.Safi, P.Geerlings
J.Phys.Chem.A 105, 6703-6710(2001)
22. "Intermolecular exchange-induction and charge transfer:
Derivation of approximate formulas using nonorthogonal
localized molecular orbitals."
J.H.Jensen J.Chem.Phys. 114, 8775-8783(2001)
23. "An integrated effective fragment-polarizable continuum
approach to solvation: Theory & application to glycine"
P.Banyopadhyay, M.S.Gordon, B.Mennucci, J.Tomasi
J.Chem.Phys. 116, 5023-5032(2002)
24. "The prediction of protein pKa's using QM/MM: the pKa
of Lysine 55 in turkey ovomucoid third domain"
H.Li, A.W.Hains, J.E.Everts, A.D.Robertson, J.H.Jensen
J.Phys.Chem.B 106, 3486-3494(2002)
25. "Computational studies of aliphatic amine basicity"
D.C.Caskey, R.Damrauer, D.McGoff
J.Org.Chem. 67, 5098-5105(2002)
26. "Density Functional Theory based Effective Fragment
Potential" I.Adamovic, M.A.Freitag, M.S.Gordon
J.Chem.Phys. 118, 6725-6732(2003)
27. "Intraprotein electrostatics derived from first
principles: Divid-and-conquer approaches for QM/MM
calculations" P.A.Molina, H.Li, J.H.Jensen
J.Comput.Chem. 24, 1791-1799(2003)
28. "Formation of alkali metal/alkaline earth cation water
clusters, M(H2O)1-6, M=Li+, K+, Mg+2, Ca+2: an
effective fragment potential caase study"
G.N.Merrill, S.P.Webb, D.B.Bivin
J.Phys.Chem.A 107, 386-396(2003)
29. "Anion-water clusters A-(H2O)1-6, A=OH, F, SH, Cl, and
Br. An effective fragment potential test case"
G.N.Merrill, S.P.Webb
J.Phys.Chem.A 107,7852-7860(2003)
30. "The application of the Effective Fragment Potential to
molecular anion solvation: a study of ten oxyanion-
water clusters, A-(H2O)1-4"
G.N.Merrill, S.P.Webb J.Phys.Chem.A 108, 833-839(2004)
31. "The effective fragment potential: small clusters and
radial distribution functions"
H.M.Netzloff, M.S.Gordon J.Chem.Phys. 121, 2711-4(2004)
32. "Fast fragments: the development of a parallel
effective fragment potential method"
H.M.Netzloff, M.S.Gordon
J.Comput.Chem. 25, 1926-36(2004)
33. "Theoretical investigations of acetylcholine (Ach) and
acetylthiocholine (ATCh) using ab initio and effective
fragment potential methods"
J.Song, M.S.Gordon, C.A.Deakyne, W.Zheng
J.Phys.Chem.A 108, 11419-11432(2004)
34. "Dynamic polarizability, dispersion coefficient C6, and
dispersion energy in the effective fragment potential
method"
I.Adamovic, M.S.Gordon Mol.Phys. 103, 379-387(2005)
35. "Solvent effects on the SN2 reaction: Application of
the density functional theory-based effective fragment
potential method"
I.Adamovic, M.S.Gordon J.Phys.Chem.A 109, 1629-36(2005)
36. "Theoretical study of the solvation of fluorine and
chlorine anions by water"
D.D.Kemp, M.S.Gordon J.Phys.Chem.A 109, 7688-99(2005)
37. "Modeling styrene-styrene interactions"
I.Adamovic, H.Li, M.H.Lamm, M.S.Gordon
J.Phys.Chem.A 110, 519-525(2006)
38. "Methanol-water mixtures: a microsolvation study using
the Effective Fragment Potential method"
I.Adamovic, M.S.Gordon
J.Phys.Chem.A 110, 10267-10273(2006)
39. "Charge transfer interaction in the effective fragment
potential method" H.Li, M.S.Gordon, J.H.Jensen
J.Chem.Phys. 124, 214108/1-16(2006)
40. "Incremental solvation of nonionized and zwitterionic
glycine"
C.M.Aikens, M.S.Gordon
J.Am.Chem.Soc. 128, 12835-12850(2006)
41. "Gradients of the polarization energy in the Effective
Fragment Potential method"
H.Li, H.M.Netzloff, M.S.Gordon
J.Chem.Phys. 125, 194103/1-9(2006)
42. "Electrostatic energy in the Effective Fragment
Potential method: Theory and application to benzene
dimer"
L.V.Slipchenko, M.S.Gordon
J.Comput.Chem. 28, 276-291(2007)
43. "Polarization energy gradients in combined Quantum
Mechanics, Effective Fragment Potential, and
Polarizable Continuum Model Calculations"
H.Li, M.S.Gordon J.Chem.Phys. submitted
44. "The Effective Fragment Potential: a general method
for predicting intermolecular interactions"
M.S.Gordon, L.V.Slipchenko, H.Li, J.H.Jensen
Annual Reports in Computational Chemistry, in press.
The Fragment Molecular Orbital method
coded by D. G. Fedorov and K. Kitaura at
Research Institute for Computational Sciences (RICS)
National Institute of Advanced Industrial Science
and Technology (AIST)
AIST Tsukuba Central 2, Umezono 1-1-1,
Tsukuba, 305-8568, Japan.
The method was proposed by Professor Kitaura and coworkers
in 1999, based on the Energy Decomposition Analysis (EDA,
sometimes called the Morokuma-Kitaura energy
decomposition). The FMO method is completely independent of
and bears no relation to:
1. Frontier molecular orbitals (FMO),
2. Fragment molecular orbitals (FMO).
The latter name is often used for the process of
construction of full molecular orbitals by combining MO
diagrams for parts of a molecule, ala Roald Hoffmann.
The FMO program was interfaced with GAMESS and follows
general GAMESS guidelines for code distribution and usage.
The users of the FMO program are requested to cite the
FMO3-RHF paper as the basic FMO reference,
D.G. Fedorov, K. Kitaura,
J. Chem. Phys. 120, 6832-6840(2004)
and other papers as appropriate (see below).
The basis idea of the method is to acknowledge the fact the
exchange and self-consistency are local in most molecules
(and clusters and molecular crystals), which permits
treating remote parts with Coulomb operators only, ignoring
the exchange. This idea further evolves into doing
molecular calculations, piecewise, with Coulomb fields due
to the remaining parts. In practice one divides the
molecule into fragments and performs n-mer calculations of
these in the Coloumb field of other fragments (n=1,2,3).
There are no empirical parameters, and the only departure
from ab initio rigor is the subjective fragmentation. It
has been observed that if performed physically reasonably,
the fragmentation scheme alters the results very little.
What changes the accuracy the most is the fragment size,
which also determines the computational efficiency of the
method.
The first question is how to get started. If your molecule
is a protein found in the PDB (http://www.rcsb.org/pdb),
you are lucky because you can simply use the fragmentation
program "fmoutil" that is provided with GAMESS, or
http://staff.aist.go.jp/d.g.fedorov/fmo/main.html
for the latest version. If you have a cluster of identical
molecules, you are lucky as well, since you can perform
fragmentation with just one keyword ($FMO nacut=). In other
cases you need to fragment manually. At the moment not
much further advice can be given here other than studying
the provided sample files, and reading the keyword
descriptions. Some modelling software can aid in the
fragmentation, e.g. you may be able to select some atoms on
the screen and dump their ordinal numbers to a file. You
may be able to use graphical tools developed for ABINIT-MP
(another FMO program):
http://moldb.nihs.go.jp/abinitmp
or Ghemical-GMS developed at the University of Iowa,
http://www.uiowa.edu/~ghemical.
Computationally, it is always better to partition in a
geometrical way (close parts together), so that the
distance-based approximations are more effficient. The
accuracy depends mainly upon the locality of the density
distribution, and the appropriateness of partitioning it
into fragments. There is no simple connexion between the
geometrical proximity of fragmentation and accuracy.
Supposing you know how to fragment, you should choose a
basis set and fragment size. We recommend 2 amino acid
residues or 2-4 water molecules per fragment for final
energetics (or, even better, three-body with 1 molecule or
residue per fragment). For geometry optimizations one may
be able to use 1 res/mol per fragment, especially if
gradient convergence to about 0.001 is desired. Note that
although it was claimed that FMO gradient is analytic
(Chem. Phys. Lett., 336 (2001), 163.) it is not so. Neither
theory nor program for fully analytic gradient has been
developed, to the best of our knowledge up to this day
(December 21, 2006). The gradient implementation is nearly
analytic, meaning three small terms are missing. The
magnitude of these small terms depends upon the fragment
size (larger fragments have smaller errors). It has been
our experience that in proteins with 1 residue per fragment
one gets 1e-3...1e-4 error in the gradient, and with 2
residues per fragment it is about 1e-4...1e-5. If you
experience energy rising during geometry optimizations, you
can consider two countermeasures:
1. increase approximation thresholds, e.g. RESPPC from
2.0->2.5, RESDIM from 2.0 -> 2.5.
2. increase fragment size (e.g. by merging very small
fragments with their neighbors).
Finally a word of caution: optimizing systems with charged
fragments in the absence of solvent is frequently not a
good idea: oppositely charged fragments will most likely
seek each other, unless there is some conformational
barrier.
One thing you should clearly understand about gradients is
that if you compare FMO gradients with full molecule ab
initio gradients, there are two sources of errors:
a) error of the analytic FMO gradient compared to ab
initio.
b) error of a "nearly analytic" FMO gradient compared
to the analytic FMO gradient.
Since the analytic FMO gradient is not available, these two
are not separable at the moment. If FMO gradients were
fully analytic, geometry optimization and dynamics would
have run perfectly, irrespective of error a).
For basis sets you should use general guidelines and your
experience developed for ab initio methods. There is a file
provided (HOs.txt) that contains hybrid molecular orbitals
(HO) used to divide the MO space along fragmentation points
at covalent bonds. If your basis set is not there you need
to construct your own set of HOs. See the example file
makeLMO.inp for this purpose.
Next you choose a wavefunction type. At present one can use
RHF, DFT, MP2, CC, and MCSCF (all except MCSCF support the
3-body expansion). Geometry optimization can be performed
with all of these methods, except CC.
Note that presence of $FMO turns FMO on.
Guidelines for approximations with FMO3
Three sets are suggested, for various accuracies:
low: resppc=2.5 resdim=2.5 ritrim(1)=0.001,-1,1.25
medium: resppc=2.5 resdim=3.25 ritrim(1)=1.25,-1,2.0
high: resppc=2.5 resdim=4.0 ritrim(1)=2,2,2
Note that gradient runs do not support nonzero RESDIM and
thus use RESDIM=0 if gradient is to be computed. The "low"
level of accuracy for FMO3 has an error versus full ab
initio similar to FMO2, except for extended basis sets (6-
311G** etc) where it is substantially better than FMO2.
Thus the low level is only recommended for those large
basis sets, and if a better level cannot be afforded. The
medium level is recommended for production FMO3 runs; the
high level is mostly for accuracy evaluation in FMO
development. The cost is roughly: 3(low), 6(medium),
12(high). This means that FMO3 with the medium level takes
roughly six times longer than FMO2.
How to perform FMO-MCSCF calculations
Assuming that you are reasonably acquainted with ab initio
MCSCF, only FMO-specific points are highlighted. The active
space (the number of orbitals/electrons) is specified for
the MCSCF fragment. The number of core/virtual orbitals for
MCSCF dimers will be automatically computed. The most
important issue is the initial orbitals for the MCSCF
monomer. Just as for ab initio MCSCF, you should execise
chemical knowledge and provide appropriate orbitals. There
are two basic ways to input MCSCF initial orbitals:
A) through the FMO monomer density binary file
B) by providing a text $VEC group.
The former way is briefly described in INPUT.DOC (see
orbital conversion). The latter way is really identical to
ab initio MCSCF, except the orbitals should be prepared for
the fragment (so in many cases you would have to get them
from an FMO calculation). Once you have the orbitals, put
them into $VEC1, and use the IJVEC option in $FMOPRP (e.g.,
if your MCSCF fragment is number 5, you would use $VEC1 and
ijvec(1)=5,0). For two-layer MCSCF the following
conditions apply. Usually one cannot simply use F40
restart, because its contents will be overwritten with RHF
orbitals and this will mess up your carefully chosen MCSCF
orbitals. Therefore, two ways exist. One is to modify A)
above by reordering the orbitals with something like
$guess guess=skip norder=1 iorder(28)=29,30,31,32,28 $end
Then the lower RHF layer will converge RHF orbitals that
you reorder with iorder in the same run (add 512 to nguess
in $FMO). This requires you know how to reorder before
running the job so it is not always convenient. Probably
the best way to run two-layer MCSCF is verbatim B) above,
so just provide MCSCF monomer orbitals in $VEC1. Finally,
it may happen that some MCSCF dimer will not converge.
Beside the usual MCSCF tricks to gain convergence as the
last resort you may be able to prepare good initial dimer
orbitals, put them into $VEC2 ($VEC3 etc) and read them
with ijvec. SOSCF is the preferred converger in FMO, and
the other one (FULLNR) has not been modified to eradicate
the artefacts of convergence (due to fractioned bonds). In
the bad cases you can try running one or two monomer SCF
iterations with FULLNR, stop the job and use its orbitals
in F40 to do a restart with SOSCF. We also found useful to
set CASDII=0.005 and nofo=10 in some cases running FOCAS
longer to get better orbitals for SOSCF.
How to perform multilayer runs
For some fragments you may like to specify a different
level of electron correlation and/or basis set. In a
typical case, you would use high level for the reaction
center and a lower level for the remaining part of the
system. The set up for multilayer runs is very similar to
the unilayer case. You only have to specify to what layer
each fragment belongs and for each layer define DFTTYP,
MPLEVL, SCFTYP as well as a basis set. If fractioned bonds
are present, appropriate HOs should be defined. See the
paragraph above for multilayer MCSCF. Currently geometry
optimizations of multilayer runs require adding 128 to
NGUESS, if basis sets in layers differ from each other.
How to mix basis sets in FMO
You can mix basis sets in both uni and multilayer cases.
The difference between a 2-layer run with one basis set per
layer and a 1-layer run with 2-basis sets is significant:
in the former case the lower level densities are converged
with all fragments computed at the lower level. In the
latter case, the fragments are converged simultaneously,
each with its own basis set. In addition, dimer corrections
between layers will be computed differently: with the lower
basis set in the former case and with mixed basis set in
the latter. The latter approach may result in unphysical
polarization, so mixing basis sets is mainly intended to
add diffuse functions to anionic (e.g., carboxyl) groups,
not as a substitute for two-layer runs.
How to perform FMO/PCM calculations
Solvent effects can be taken into account with PCM. PCM in
FMO is very similar to regular PCM. There is one basic
difference: in FMO/PCM the total electron density that
determines the electrostatic interaction is computed using
the FMO density expansion up to n-body terms. The cavity
is constructed surrounding the whole molecule, and the
whole cavity is used in each individual m-mer calculation.
There are several levels of accuracy (determined by the "n"
above), and the recommended level is FMO/PCM[1(2)],
specified by:
$pcm ief=-10 icomp=2 icav=1 idisp=1 ifmo=2 $end
$fmoprp npcmit=2 $end
$tescav ntsall=240 $end
$pcmcav radii=suahf $end
Many PCM options can be used as in the regular PCM. The
following restrictions apply:
IEF may be only -3 or -10, IDP must be 0.
No FMO/PCM gradients are available. Multilayer FMO runs
are supported. Restarts are only limited to IREST=2, and
in this case PCM charges (the ASCs) are not recycled.
However, the initial guess for the charges is fairly
reasonable, so IREST=2 may be useful although reading the
ASCs may be implemented in future.
Note for advanced users. IFMO < NBODY runs are permitted.
They are denoted by FMOm/PCM[n], where m=NBODY and n=IFMO.
In FMOm/PCM[n], the ASCs are computed with n-body level.
The difference between FMO2/PCM[1] and FMO2/PCM[1(2)] is
that in the former the ASCs are computed at the 1-body
level, whereas for the former at the 2-body level, but
without self-consistency (which would be FMO2/PCM[2]).
Probably, FMO3/PCM[2] should be regarded as the most
accurate and still affordable (with a few thousand nodes)
method. However, FMO3/PCM[1(2)] (specified with NBODY=3,
IFMO=2 and NPCMIT=2) is much cheaper and slightly less
accurate than FMO3/PCM[2]. FMO3/PCM[3] is the most
accurate and expensive level of all.
How to perform FMO/EFP calculations
Solvent effects can also be taken into account with the
Effective Fragment Potential model. The presence of both
$FMO and $EFRAG groups selects FMO/EFP calculations. See
the $EFRAG group and the $FMO group for details.
In the FMO/EFP method, the Quantum Mechanical part of the
calculation in the usual EFP method is replaced by the FMO
method, which may save time for large molecules such as
proteins. FMO/EFP energies should have almost the same
accuracy as the usual EFP runs.
In the present version, both FMOn/EFP1 (water solvent only)
and FMOn/EFP2 (n=2,3) are available for RHF, DFT and MP2
energy calculations, while only FMOn/EFP1 is allowed for
RHF, DFT and MP2 gradients. For FMO/EFP geometry
optimizations, you can only use the solver provided by
GAMESS, which means that the maximum number of atoms is
2000. Also, you can use the MC global optimization
technique for FMO/EFP by setting RUNTYP=GLOBOP. Of course,
the group DDI (GDDI) parallelization technique for the FMO
method can be used. For any other information of FMO and
EFP, follow the instructions in this file.
Geometry optimizations for FMO
Due to the latest improvements, the standard optimizers in
GAMESS are now well parallelized, and thus recommended to
be used with FMO up to the limit hardwired in GAMESS (2000
atoms). That applies to Cartesian coordinates only. Other
types have not been tested and should be assumed to be not
usable with FMO. If your system has more than 2000 atoms
you can consider RUNTYP=OPTFMO, which can now use Hessian
updates and provides reasonable way to optimize although it
is not as good as the standard means in RUNTYP=OPTIMIZE.
Pair interaction energy decomposition analysis (PIEDA)
PIEDA can be performed for the PL0 and PL states. The PL0
state is the electronic state in which fragments are
polarised by the environment in its free (noninteracting)
state. The simplest example is that in a water cluster,
each molecule is computed in the electrostatic field
exerted by the electron densities of free water molecules.
The PL state is the FMO converged monomer state, that is,
the state in which fragments are polarised by the self-
consistently converged environment. Namely, following the
FMO prescription, fragments are recomputed in the external
field, until the latter converges. Using the PL0 state
requires a series of separate runs; and it also relies on a
"free state" which can be defined in many ways for
molecules with fractioned covalent bonds.
What should be done to do the PL0 state analysis?
1. run FMO0.
This computes the free state for each fragment, and those
electron densities are stored on file 30 (to be renamed
file 40 and reused in step 3).
2. compute BDA energies (if fractioned bonds are present),
using sample files in tools/fmo/pieda. This corrects for
artifacts of bond fractioning, and involves running a model
system like H3C-CH3, to amend for C-C bond fractioning.
3. Using results of (1) and (2), one can do the PL0
analysis. In addition to pasting the data from the two
punch files in steps 1,2 and the density file in step 1
should be provided.
What should be done to do the PL state analysis? The PL
state itself does not need either the free state or PL0
results. However, if the PL0 results are available,
coupling terms can be computed, and in this case IPIEDA is
set to 2; otherwise to 1.
So the easiest and frequently sufficient way to run PIEDA
is to set IPIEDA=1 and do not provide any data from
preceding calculations. The result of a PIEDA calculation
is a set of pair interaction energies (interfragment
interaction energies), decomposed into electrostatic,
exchange-repulsion, charge transfer and dispersion
contributions.
Finally, PIEDA (especially for the PL state) can be thought
of as FMO-EDA, EDA being the Kitaura-Morokuma decomposition
(RUNTYP=MOROKUMA). In fact, PIEDA (for the PL state) in
the case of just two fragments of standalone molecules is
entirely equivalent to EDA, which can be easily verified,
by running the full PIEDA analysis (ipieda=2). Note that
PIEDA can be run as direct SCF, whereas EDA cannot be, and
for large fragments PIEDA code can be used to perform EDA.
Also, EDA in GAMESS has no explicit dispersion.
Graphical representation of results
To plot orbitals for an n-mer, set NPUNCH=2 in $SCF and
PLTORB=.T.. There are several ways to produce cube files
with electron densities. They are described in detail in
tools/fmo/fmocube/README. To plot pair interaction maps,
use tools/fmo/fmograbres and Gnuplot or Excel.
Parallelization of FMO runs with GDDI
The FMO method has been developed within a 2-level
hierarchical parallelization scheme, group DDI (GDDI),
allowing massively parallel calculations. Different groups
of processors handle the various monomer, dimer, and maybe
trimer computations. The processor groups should be sized
so that GAMESS' innate scaling is fairly good, and the
fragments should be mapped onto the processor groups in an
intelligent fashion.
This is a very important and seemingly difficult issue. It
is very common to be able to speed up parallel runs at
least several times just by using GDDI better. First of
all, do not use plain DDI and always employ GDDI when
running FMO calculations. Next, learn that you can and
should divide nodes into groups to achieve better
performance. The very basic rule of thumb is to try to have
several times fewer groups than jobs. Since the number of
monomers and dimers is different, group division should
reflect that fact. Ergo, find a small parallel computer
with 8-32 nodes and experiment changing just two numbers:
ngrfmo(1)=N1,N2 and see how performance changes for your
particular system.
Limitations of the FMO method in GAMESS
1. dimensions: in general none, except that geometry
optimizations use the standard GAMESS engine which means
you are limited to 2000 atoms. This can be changed by
changing the source and recompiling GAMESS (see elsewhere).
2. Almost none of the "SCF extensions" in GAMESS can be
used with FMO. This means, in particular, no ECPs, no
MCPs, no CHARMM, no SIMOMM, etc. Anything that is not a
plain basis set, with atoms given only by those legally
entered in $FMOXYZ, will not work. The only "SCF
extension" supported are the PCM and EFP solvent models.
Not every illegal combination is trapped, caveat emptor!
3. RUNTYP is limited to ENERGY, GRADIENT, OPTIMIZE, OPTFMO,
FMO0, and GLOBOP only! Do not even try other ones.
4. For the three-body FMO expansion ($FMO NBODY=3). RESDIM
may be used with RUNTYP=ENERGY. Three-body FMO-MCSCF is
not implemented.
5. No semiempirical methods may be used.
What will work the same way as ab initio:
The various SCF convergers, all DFT functionals, in-core
integrals, direct SCF.
Restarts with the FMO method
RUNTYP=ENERGY can be restarted from anywhere before
trimers. To restart monomer SCF, copy file F40 with monomer
densities to the grandmaster node. To restart dimers,
provide file F40 and monomer energies ($FMOENM).
Optionally, some dimer energies can be supplied ($FMOEND)
to skip computation of corresponding dimers.
RUNTYP=GRADIENT can be easily restarted from monomer SCF
(which really means it is a restart of RUNTYP=ENERGY, since
gradient is computed at the end of this step). Provide
file F40. There is another restart option (1024 in $FMOPRP
irest=), supporting full gradient restart, requiring,
however, that you set this option in the original run
(whose results you use to restart). To use this option, you
would also need to keep (or save and restore) F38 files on
each node (they are different).
RUNTYP=OPTIMIZE can be restarted from anywhere within the
first RUNTYP=GRADIENT run (q.v.). In addition, by
replacing FMOXYZ group, one can restart at a different
geometry.
RUNTYP=OPTFMO can be restarted by providing a new set of
coordinates in $FMOXYZ and, optionally, by transferring
$OPTRST from the punch into the input file.
Note on accuracy
The FMO method is aimed at computation of large molecules.
This means that the total energy is large, for example, a
6646 atom molecule has a total energy of -165,676 Hartrees.
If one uses the standard accuracy of roughly 1e-9 (that
should be taken relatively), one gets an error as much as
0.001 hartree, in a single calculation. FMO involves many
ab initio single point calculations of fragments and their
n-mers, thus it can be expected that numeric accuracy is 1-
2 orders lower than that given by 1e-9. Therefore, it is
compulsory that accuracy should be raised, which is done by
default.
The following default parameters are reset by FMO:
ICUT/$CONTRL (9->12), ITOL/$CONTRL(20->24),
CONV/$SCF(1e-5 -> 1e-7),
CUTOFF/$MP2 (1e-9->1e-12), CUTTRF/$TRANS(1e-9->1e-10).
CVGTOL/$DET,$GUGDIA (1e-5 -> 1e-6)
This to some extent slows down the calculation (perhaps on
the order of 10-15%). It is suggested that you maintain
this accuracy for all final energetics. However, you may
be able to drop the accuracy a bit for the initial part of
geometry optimization if you are willing to do manual work
of adjusting accuracy in the input. It is recommended to
keep high accuracy at the flat surfaces (the final part of
optimizations) though. For DFT the numeric grid's accuracy
may be increased in accordance with the molecule size, e.g.
extending the default grid of 96*12*24 to 96*20*40.
However, some tests indicate that energy differences are
quite insensitive to this increase.
FMO References
I. Basic FMO papers
The book chapter is the best source for an introduction,
and concise summary of the FMO development:
Theoretical development of the fragment molecular orbital
(FMO) method, D. G. Fedorov, K. Kitaura, in "Modern methods
for theoretical physical chemistry of biopolymers", E. B.
Starikov, J. P. Lewis, S. Tanaka, Eds., pp 3-38, Elsevier,
Amsterdam, 2006.
1. Fragment molecular orbital method: an approximate
computational method for large molecules"
K. Kitaura, E. Ikeo, T. Asada, T. Nakano, M. Uebayasi
Chem. Phys. Lett., 313 (1999), 701.
2. Fragment molecular orbital method: application to
polypeptides
T. Nakano, T. Kaminuma, T. Sato, Y. Akiyama,
M. Uebayasi, K. Kitaura Chem.Phys.Lett. 318 (2000), 614.
3. Fragment molecular orbital method: analytical energy
gradients
K. Kitaura, S.-I. Sugiki, T. Nakano, Y. Komeiji,
M. Uebayasi, Chem. Phys. Lett., 336 (2001), 163.
4. Fragment molecular orbital method: use of approximate
electrostatic potential
T. Nakano, T. Kaminuma, T. Sato, K. Fukuzawa,
Y. Akiyama, M. Uebayasi, K. Kitaura
Chem. Phys. Lett., 351 (2002), 475.
5. The extension of the fragment molecular orbital method
with the many-particle Green's function,
K. Yasuda, D. Yamaki, J. Chem. Phys. 125 (2006) 154101.
II. FMO in GAMESS
1. A new hierarchical parallelization scheme: generalized
distributed data interface (GDDI), and an application to
the fragment molecular orbital method (FMO).
D. G. Fedorov, R. M. Olson, K. Kitaura, M. S. Gordon,
S. Koseki J. Comput. Chem. 25, 872-880(2004).
2. The importance of three-body terms in the fragment
molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 120, 6832-6840(2004).
3. On the accuracy of the 3-body fragment molecular orbital
method (FMO) applied to density functional theory
D. G. Fedorov and K. Kitaura
Chem. Phys. Lett. 389, 129-134(2004).
4. Second order Moller-Plesset perturbation theory based
upon the fragment molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 121, 2483-2490(2004).
5. Multiconfiguration self-consistent-field theory based
upon the fragment molecular orbital method.
D. G. Fedorov and K. Kitaura
J. Chem. Phys. 122, 054108/1-10(2005).
6. Multilayer Formulation of the Fragment Molecular Orbital
Method (FMO).
D. G. Fedorov, T. Ishida, K. Kitaura
J. Phys. Chem. A. 109, 2638-2646(2005).
7. Coupled-cluster theory based upon the Fragment Molecular
Orbital method.
D. G. Fedorov, K. Kitaura
J. Chem. Phys. 123, 134103/1-11 (2005)
8. The polarizable continuum model (PCM) interfaced with
the fragment molecular orbital method (FMO).
D. G. Fedorov, K. Kitaura, H. Li, J. H. Jensen,
M. S. Gordon, J. Comput. Chem., 27, 976-985(2006)
9. The three-body fragment molecular orbital method for
accurate calculations of large systems,
D. G. Fedorov, K. Kitaura
Chem. Phys. Lett. 433, 182-187(2006).
10. Pair interaction energy decomposition analysis,
D. G. Fedorov, K. Kitaura
J. Comp. Chem. 28, 222-237(2007).
11. On the accuracy of the three-body fragment molecular
orbital method (FMO) applied to Moeller-Plesset
perturbation theory,
D. G. Fedorov, K. Ishimura, T. Ishida, K. Kitaura,
P. Pulay, S. Nagase
J. Comput. Chem., in press.
12. The Fragment Molecular Orbital method for geometry
optimizations of polypeptides and proteins,
D.G.Fedorov, T. Ishida, M. Uebayasi, K. Kitaura
J.Phys.Chem.A, in press.
Other FMO references including applications can be found
at:
http://staff.aist.go.jp/d.g.fedorov/fmo/main.html
MOPAC Calculations within GAMESS
Parts of MOPAC 6.0 have been included in GAMESS so
that the GAMESS user has access to three semiempirical
wavefunctions: MNDO, AM1 and PM3. These wavefunctions
are quantum mechanical in nature but neglect most two
electron integrals, a deficiency that is (hopefully)
compensated for by introduction of empirical parameters.
The quantum mechanical nature of semiempirical theory
makes it quite compatible with the ab initio methodology
in GAMESS. As a result, very little of MOPAC 6.0 actually
is incorporated into GAMESS. The part that did make it in
is the code that evaluates
1) the one- and two-electron integrals,
2) the two-electron part of the Fock matrix,
3) the cartesian energy derivatives, and
4) the ZDO atomic charges and molecular dipole.
Everything else is actually GAMESS: coordinate input
(including point group symmetry), the SCF convergence
procedures, the matrix diagonalizer, the geometry
searcher, the numerical hessian driver, and so on. Most
of the output will look like an ab initio output.
It is extremely simple to perform one of these
calculations. All you need to do is specify GBASIS=MNDO,
AM1, or PM3 in the $BASIS group. Note that this not only
picks a particular Slater orbital basis, it also selects a
particular "hamiltonian", namely a particular parameter
set.
MNDO, AM1, and PM3 will not work with every option in
GAMESS. Currently the semiempirical wavefunctions support
SCFTYP=RHF, UHF, and ROHF in any combination with
RUNTYP=ENERGY, GRADIENT, OPTIMIZE, SADPOINT, HESSIAN, and
IRC. Note that all hessian runs are numerical finite
differencing. The MOPAC CI and half electron methods are
not supported.
Because the majority of the implementation is GAMESS
rather than MOPAC you will notice a few improvments.
Dynamic memory allocation is used, so that GAMESS uses far
less memory for a given size of molecule. The starting
orbitals for SCF calculations are generated by a Huckel
initial guess routine. Spin restricted (high spin) ROHF
can be performed. Converged SCF orbitals will be labeled
by their symmetry type. Numerical hessians will make use
of point group symmetry, so that only the symmetry unique
atoms need to be displaced. Infrared intensities will be
calculated at the end of hessian runs. We have not at
present used the block diagonalizer during intermediate
SCF iterations, so that the run time for a single geometry
point in GAMESS is usually longer than in MOPAC. However,
the geometry optimizer in GAMESS can frequently optimize
the structure in fewer steps than the procedure in MOPAC.
Orbitals and hessians are punched out for convenient reuse
in subsequent calculations. Your molecular orbitals can
be drawn with the PLTORB graphics program, which has been
taught about s and p STO basis sets.
However, because of the STO basis set used in semi-
empirical runs, the various property calculations coded for
Gaussian basis sets are unavailable. This means $ELMOM,
$ELPOT, etc. properties are unavailable. Likewise the
solvation models do not work with semi-empirical runs.
Note that MOPAC6 did not include d STO functions, and it
is therefore quite impossible to run transition metals.
To reduce CPU time, only the EXTRAP convergence
accelerator is used by the SCF procdures. For difficult
cases, the DIIS, RSTRCT, and/or SHIFT options will work,
but may add significantly to the run time. With the
Huckel guess, most calculations will converge acceptably
without these special options.
MOPAC parameters exist for the following elements. The
printout when you run will give you specific references for
each kind of atom. The quote on alkali's below means that
these elements are treated as "sparkles", rather than as
atoms with genuine basis functions.
For MNDO:
H
Li * B C N O F
Na' * Al Si P S Cl
K' * ... Zn * Ge * * Br
Rb' * ... * * Sn * * I
* * ... Hg * Pb *
For AM1: For PM3:
H H
* * B C N O F Li Be * C N O
F
Na Mg Al Si P S Cl Na Mg Al Si P S
Cl
K Ca ... Zn * Ge * * Br K Ca ... Zn Ga Ge As Se
Br
Rb' * ... * * Sn * * I Rb' * ... Cd In Sn Sb Te
I
* * ... Hg * * * * * ... Hg Tl Pb Bi
Semiempirical calculations are very fast. One of the
motives for the MOPAC implementation within GAMESS is to
take advantage of this speed. Semiempirical models can
rapidly provide reasonable starting geometries for ab
initio optimizations. Semiempirical hessian matrices are
obtained at virtually no computational cost, and may help
dramatically with an ab initio geometry optimization.
Simply use HESS=READ in $STATPT to use a MOPAC $HESS group
in an ab initio run.
It is important to exercise caution as semiempirical
methods can be dead wrong! The reasons for this are bad
parameters (in certain chemical situations), and the
underlying minimal basis set. A good question to ask
before using MNDO, AM1 or PM3 is "how well is my system
modeled with an ab initio minimal basis sets, such as
STO-3G?" If the answer is "not very well" there is a good
chance that a semiempirical description is equally poor.
Molecular Properties and Conversion Factors
These two papers are of general interest:
A.D.Buckingham, J.Chem.Phys. 30, 1580-1585(1959).
D.Neumann, J.W.Moskowitz J.Chem.Phys. 49, 2056-
2070(1968).
The first deals with multipoles, and the second with other
properties such as electrostatic potentials.
All units are derived from the atomic units for distance
and the monopole electric charge, as given below.
distance - 1 au = 5.291771E-09 cm
monopole - 1 au = 4.803242E-10 esu
1 esu = sqrt(g-cm**3)/sec
dipole - 1 au = 2.541766E-18 esu-cm
1 Debye = 1.0E-18 esu-cm
quadrupole - 1 au = 1.345044E-26 esu-cm**2
1 Buckingham = 1.0E-26 esu-cm**2
octopole - 1 au = 7.117668E-35 esu-cm**3
electric potential - 1 au = 9.076814E-02 esu/cm
electric field - 1 au = 1.715270E+07 esu/cm**2
1 esu/cm**2 = 1 dyne/esu
electric field gradient- 1 au = 3.241390E+15 esu/cm**3
The atomic unit for electron density is electron/bohr**3
for the total density, and 1/bohr**3 for an orbital
density.
The atomic unit for spin density is excess alpha spins per
unit volume, h/4*pi*bohr**3. Only the expectation value
is computed, with no constants premultiplying it.
IR intensities are printed in Debye**2/amu-Angstrom**2.
These can be converted into intensities as defined by
Wilson, Decius, and Cross's equation 7.9.25, in km/mole,
by multiplying by 42.255. If you prefer 1/atm-cm**2, use
a conversion factor of 171.65 instead. A good reference
for deciphering these units is A.Komornicki, R.L.Jaffe
J.Chem.Phys. 1979, 71, 2150-2155. A reference showing
how IR intensities change with basis improvements at the
HF level is Y.Yamaguchi, M.Frisch, J.Gaw, H.F.Schaefer,
J.S.Binkley, J.Chem.Phys. 1986, 84, 2262-2278.
Raman intensities in A**4/amu multiply by 6.0220E-09 for
units of cm**4/g.
Localized Molecular Orbitals
Three different orbital localization methods are
implemented in GAMESS. The energy and dipole based
methods normally produce similar results, but see
M.W.Schmidt, S.Yabushita, M.S.Gordon in J.Chem.Phys.,
1984, 88, 382-389 for an interesting exception. You can
find references to the three methods at the beginning of
this chapter.
The method due to Edmiston and Ruedenberg works by
maximizing the sum of the orbitals' two electron self
repulsion integrals. Most people who think about the
different localization criteria end up concluding that
this one seems superior. The method requires the two
electron integrals, transformed into the molecular orbital
basis. Because only the integrals involving the orbitals
to be localized are needed, the integral transformation is
actually not very time consuming.
The Boys method maximizes the sum of the distances
between the orbital centroids, that is the difference in
the orbital dipole moments.
The population method due to Pipek and Mezey maximizes
a certain sum of gross atomic Mulliken populations. This
procedure will not mix sigma and pi bonds, so you will not
get localized banana bonds. Hence it is rather easy to
find cases where this method give different results than
the Ruedenberg or Boys approach.
GAMESS will localize orbitals for any kind of RHF, UHF,
ROHF, or MCSCF wavefunctions. The localizations will
automatically restrict any rotation that would cause the
energy of the wavefunction to be changed (the total
wavefunction is left invariant). As discussed below,
localizations for GVB or CI functions are not permitted.
The default is to freeze core orbitals. The localized
valence orbitals are scarcely changed if the core orbitals
are included, and it is usually convenient to leave them
out. Therefore, the default localizations are: RHF
functions localize all doubly occupied valence orbitals.
UHF functions localize all valence alpha, and then all
valence beta orbitals. ROHF functions localize all valence
doubly occupied orbitals, and all singly occupied orbitals,
but do not mix these two orbital spaces. MCSCF functions
localize all valence MCC type orbitals, and localize all
active orbitals, but do not mix these two orbital spaces.
To recover the invariant MCSCF function, you must be using
a FORS=.TRUE. wavefunction, and you must set GROUP=C1 in
$DRT, since the localized orbitals possess no symmetry.
In general, GVB functions are invariant only to
localizations of the NCO doubly occupied orbitals. Any
pairs must be written in natural form, so pair orbitals
cannot be localized. The open shells may be degenerate, so
in general these should not be mixed. If for some reason
you feel you must localize the doubly occupied space, do a
RUNTYP=PROP job. Feed in the GVB orbitals, but tell the
program it is SCFTYP=RHF, and enter a negative ICHARG so
that GAMESS thinks all orbitals occupied in the GVB are
occupied in this fictitous RHF. Use NINA or NOUTA to
localize the desired doubly occupied orbitals. Orbital
localization is not permitted for CI, because we cannot
imagine why you would want to do that anyway.
Boys localization of the core orbitals in molecules
having elements from the third or higher row almost never
succeeds. Boys localization including the core for second
row atoms will often work, since there is only one inner
shell on these. The Ruedenberg method should work for any
element, although including core orbitals in the integral
transformation is more expensive.
The easiest way to do localization is in the run which
generates the wavefunction, by selecting LOCAL=xxx in the
$CONTRL group. However, localization may be conveniently
done at any time after determination of the wavefunction,
by executing a RUNTYP=PROP job. This will require only
$CONTRL, $BASIS/$DATA, $GUESS (pick MOREAD), the converged
$VEC, possibly $SCF or $DRT to define your wavefunction,
and optionally some $LOCAL input.
There is an option to restrict all rotations that would
mix orbitals of different symmetries. SYMLOC=.TRUE. yields
only partially localized orbitals, but these still possess
symmetry. They are therefore very useful as starting
orbitals for MCSCF or GVB-PP calculations. Because they
still have symmetry, these partially localized orbitals run
as efficiently as the canonical orbitals. Because it is
much easier for a user to pick out the bonds which are to
be correlated, a significant number of iterations can be
saved, and convergence to false solutions is less likely.
* * *
The most important reason for localizing orbitals is to
analyze the wavefunction. A simple example is to look at
shapes of the orbitals with the MacMolPlt program. Or, you
might read the localized orbitals in during a RUNTYP=PROP
job to examine their Mulliken populations.
Localized orbitals are a particularly interesting way
to analyze MCSCF computations. The localized orbitals may
be oriented on each atom (see option ORIENT in $LOCAL) to
direct the orbitals on each atom towards their neighbors
for maximal bonding, and then print a bond order analysis.
The orientation procedure is newly programmed by J.Ivanic
and K.Ruedenberg, to deal with the situation of more than
one localized orbital occuring on any given atom. Some
examples of this type of analysis are
D.F.Feller, M.W.Schmidt, K.Ruedenberg
J.Am.Chem.Soc. 104, 960-967 (1982)
T.R.Cundari, M.S.Gordon
J.Am.Chem.Soc. 113, 5231-5243 (1991)
N.Matsunaga, T.R.Cundari, M.W.Schmidt, M.S.Gordon
Theoret.Chim.Acta 83, 57-68 (1992).
In addition, the energy of your molecule can be
partitioned over the localized orbitals so that you may
be able to understand the origin of barriers, etc. This
analysis can be made for the SCF energy, and also the MP2
correction to the SCF energy, which requires two separate
runs. An explanation of the method, and application to
hydrogen bonding may be found in J.H.Jensen, M.S.Gordon,
J.Phys.Chem. 99, 8091-8107(1995).
Analysis of the SCF energy is based on the localized
charge distribution (LCD) model: W.England and M.S.Gordon,
J.Am.Chem.Soc. 93, 4649-4657 (1971). This is implemented
for RHF and ROHF wavefunctions, and it requires use of
the Ruedenberg localization method, since it needs the
two electron integrals to correctly compute energy sums.
All orbitals must be included in the localization, even
the cores, so that the total energy is reproduced.
The LCD requires both electronic and nuclear charges
to be partitioned. The orbital localization automatically
accomplishes the former, but division of the nuclear
charge may require some assistance from you. The program
attempts to correctly partition the nuclear charge, if you
select the MOIDON option, according to the following: a
Mulliken type analysis of the localized orbitals is made.
This determines if an orbital is a core, lone pair, or
bonding MO. Two protons are assigned to the nucleus to
which any core or lone pair belongs. One proton is
assigned to each of the two nuclei in a bond. When all
localized orbitals have been assigned in this manner, the
total number of protons which have been assigned to each
nucleus should equal the true nuclear charge.
Many interesting systems (three center bonds, back-
bonding, aromatic delocalization, and all charged species)
may require you to assist the automatic assignment of
nuclear charge. First, note that MOIDON reorders the
localized orbitals into a consistent order: first comes
any core and lone pair orbitals on the 1st atom, then
any bonds from atom 1 to atoms 2, 3, ..., then any core
and lone pairs on atom 2, then any bonds from atom 2 to
3, 4, ..., and so on. Let us consider a simple case
where MOIDON fails, the ion NH4+. Assuming the nitrogen
is the 1st atom, MOIDON generates
NNUCMO=1,2,2,2,2
MOIJ=1,1,1,1,1
2,3,4,5
ZIJ=2.0,1.0,1.0,1.0,1.0,
1.0,1.0,1.0,1.0
The columns (which are LMOs) are allowed to span up to 5
rows (the nuclei), in situations with multicenter bonds.
MOIJ shows the Mulliken analysis thinks there are four
NH bonds following the nitrogen core. ZIJ shows that
since each such bond assigns one proton to nitrogen, the
total charge of N is +6. This is incorrect of course,
as indeed will always happen to some nucleus in a charged
molecule. In order for the energy analysis to correctly
reproduce the total energy, we must ensure that the
charge of nitrogen is +7. The least arbitrary way to
do this is to increase the nitrogen charge assigned to
each NH bond by 1/4. Since in our case NNUCMO and MOIJ
and much of ZIJ are correct, we need only override a
small part of them with $LOCAL input:
IJMO(1)=1,2, 1,3, 1,4, 1,5
ZIJ(1)=1.25, 1.25, 1.25, 1.25
which changes the charge of the first atom of orbitals
2 through 5 to 5/4, changing ZIJ to
ZIJ=2.0,1.25,1.25,1.25,1.25,
1.0, 1.0, 1.0, 1.0
The purpose of the IJMO sparse matrix pointer is to let
you give only the changed parts of ZIJ and/or MOIJ.
Another way to resolve the problem with NH4+ is to
change one of the 4 equivalent bond pairs into a "proton".
A "proton" orbital AH treats the LMO as if it were a
lone pair on A, and so assigns +2 to nucleus A. Use of
a "proton" also generates an imaginary orbital, with
zero electron occupancy. For example, if we make atom
2 in NH4+ a "proton", by
IPROT(1)=2
NNUCMO(2)=1
IJMO(1)=1,2,2,2 MOIJ(1)=1,0 ZIJ(1)=2.0,0.0
the automatic decomposition of the nuclear charges will be
NNUCMO=1,1,2,2,2,1
MOIJ=1,1,1,1,1,2
3,4,5
ZIJ=2.0,2.0,1.0,1.0,1.0,1.0
1.0,1.0,1.0
The 6th column is just a proton, and the decomposition
will not give any electronic energy associated with
this "orbital", since it is vacant. Note that the two ways
we have disected the nuclear charges for NH4+ will both
yield the correct total energy, but will give very
different individual orbital components. Most people
will feel that the first assignment is the least arbitrary,
since it treats all four NH bonds equivalently.
However you assign the nuclear charges, you must
ensure that the sum of all nuclear charges is correct.
This is most easily verified by checking that the energy
sum equals the total SCF energy of your system.
As another example, H3PO is studied in EXAM26.INP.
Here the MOIDON analysis decides the three equivalent
orbitals on oxygen are O lone pairs, assigning +2 to
the oxygen nucleus for each orbital. This gives Z(O)=9,
and Z(P)=14. The least arbitrary way to reduce Z(O)
and increase Z(P) is to recognize that there is some
backbonding in these "lone pairs" to P, and instead
assign the nuclear charge of these three orbitals by
1/3 to P, 5/3 to O.
Because you may have to make several runs, looking
carefully at the localized orbital output before the
correct nuclear assignments are made, there is an
option to skip directly to the decomposition when the
orbital localization has already been done. Use
$CONTRL RUNTYP=PROP
$GUESS GUESS=MOREAD NORB=
$VEC containing the localized orbitals!
$TWOEI
The latter group contains the necessary Coulomb and
exchange integrals, which are punched by the first
localization, and permits the decomposition to begin
immediately.
SCF level dipoles can also be analyzed using the
DIPDCM flag in $LOCAL. The theory of the dipole
analysis is given in the third paper of the LCD
sequence. The following list includes application of
the LCD analysis to many problems of chemical interest:
W.England, M.S.Gordon J.Am.Chem.Soc. 93, 4649-4657 (1971)
W.England, M.S.Gordon J.Am.Chem.Soc. 94, 4818-4823 (1972)
M.S.Gordon, W.England J.Am.Chem.Soc. 94, 5168-5178 (1972)
M.S.Gordon, W.England Chem.Phys.Lett. 15, 59-64 (1972)
M.S.Gordon, W.England J.Am.Chem.Soc. 95, 1753-1760 (1973)
M.S.Gordon J.Mol.Struct. 23, 399 (1974)
W.England, M.S.Gordon, K.Ruedenberg,
Theoret.Chim.Acta 37, 177-216 (1975)
J.H.Jensen, M.S.Gordon, J.Phys.Chem. 99, 8091-8107(1995)
J.H.Jensen, M.S.Gordon, J.Am.Chem.Soc. 117, 8159-8170(1995)
M.S.Gordon, J.H.Jensen, Acc.Chem.Res. 29, 536-543(1996)
* * *
It is also possible to analyze the MP2 correlation
correction in terms of localized orbitals, for the RHF
case. The method is that of G.Peterssen and M.L.Al-Laham,
J.Chem.Phys., 94, 6081-6090 (1991). Any type of localized
orbital may be used, and because the MP2 calculation
typically omits cores, the $LOCAL group will normally
include only valence orbitals in the localization. As
mentioned already, the analysis of the MP2 correction must
be done in a separate run from the SCF analysis, which must
include cores in order to sum up to the total SCF energy.
* * *
Typically, the results are most easily interpreted
by looking at "the bigger picture" than at "the details".
Plots of kinetic and potential energy, normally as a
function of some coordinate such as distance along an
IRC, are the most revealing. Once you determine, for
example, that the most significant contribution to the
total energy is the kinetic energy, you may wish to look
further into the minutia, such as the kinetic energies
of individual localized orbitals, or groups of LMOs
corresponding to an entire functional group.
Transition Moments and Spin-Orbit Coupling
A review of various ways of computing spin-orbit coupling:
D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon,
Int.Rev.Phys.Chem. 22, 551-592(2003)
GAMESS can compute transition moments and oscillator
strengths for the radiative transitions between states
written in terms of CI wavefunctions (GUGA only). The
moments are computed using both the "length form" and the
"velocity form". In a.u., where h-bar=m=1, we start from
[A,q] = -i dA/dp
For A=H, dH/dp=p, and p= -i d/dq,
[H,q] = -i p = -d/dq.
For non-degenerate states,
=
(Ea-Eb) = -
This relates the dipole to the velocity form,
= -1/(Ea-Eb)
but the CI states will give different numbers for each
side, since the states aren't exact eigenfunctions.
Transition moment computation is OPERAT=DM in $TRANST. For
transition moments, the CI is necessarily performed on
states of the same multiplicity.
All other operators are various spin-orbit coupling
options. There are two kinds of calculations possible,
which we will call SO-CI and SO-MCQDPT. Note that there
is a hyphen in "spin-orbit CI" to avoid confusion with
"second order CI" in the sense of the SOCI keyword in $DRT
input. For SO-CI, the initial states may be any CI wave-
function that the GUGA package can generate. For SO-MCQDPT
the initial states for spin-orbit coupling are of CAS type,
and the operator mixing them corresponds to MCQDPT
generalised for spin-dependent operators (with certain
approximations).
GAMESS can compute the "microscopic Breit-Pauli
spin-orbit operator", which includes both a one and two
electron operator. The full Breit-Pauli operator can be
computed exactly (OPERAT=HSO2), or approximated in two
ways: complete elimination of the 2e- term, whose absence
can be approximately accounted for by means of effective
nuclear charges (HSO1), or by inclusion of only the core-
active 2e- terms which typically account for 90% or more
of the two electron term, while saving most of the 2e-
terms' CPU cost (HSO2P).
Spin-orbit runs can be done for general spins, for
more than two different spin multiplicities at once, for
general active spaces. At times, when the spatial wave-
function is degenerate, a spin-orbit run may involve only
one spin multiplicity, e.g. a triplet-pi state in a linear
molecule. The most common case is two different spins.
It is also possible to obtain the dipole transition moments
between the final spin-mixed wavefunctions, which of course
do not any longer have a rigourous S quantum no. When the
run is SO-MCQDPT, the transition moment are first computed
only between CAS states, and then combined with the spin-
mixed SO-MCQDPT coefficients. Compared to older versions,
the basis set has been fully generalized to allow any s, p,
d, f, g, or L functions.
states
For transition moments, the states are generated by
CI calculations using the GUGA package. These states are
the final states, and the results are just the transition
moments between these states. The states are defined by
$DRTx input groups.
For SO-CI, the energy of the CI states forms the
diagonal of a spin-orbit Hamiltonian, as in the state basis
the spin-free Hamiltonian is of course diagonal. Addition
of the Pauli-Breit operator does not change the diagonal,
but does add H-so elements off diagonal. For SO-MCQDPT,
the spin-free MCQDPT matrix elements are expanded into
matrices corresponding to all Ms values for a pair of
multiplicities. These matrices are block-diagonal before
the addition of spin-orbit coupling terms, coupling Ms
values. The diagonalization of this spin-orbit Hamiltonian
gives new energy levels, and spin-mixed final states.
Optionally, the transition dipoles between the final states
can be computed. The input requirements are $DRTx or
$MCQDx groups which define the original pure spin states.
We will call the initial states CAS-CI, since most of
the time they will be MCSCF states. There may be cases
such as the Na example below where SCF orbitals are used,
or other cases where a FOCI or SOCI wavefunction might be
used for the initial states. Please keep in mind that the
term does not imply the states must be MCSCF states, just
that they commonly are.
In the above, x may vary from 1 to 64. The reason for
allowing such a large range is to permit the use of Abelian
point group symmetry during the generation of the initial
states. The best explanation will be an example, but the
number of these input groups depends on both the number of
orbital sets input, and how much symmetry is present. The
next two subsections discuss these points.
orbitals
The orbitals for transition moments or for SO-CI can be
one common set of orbitals used by all CI states. If one
set of orbitals is used, the transition moment or spin-
orbit coupling can be found for any type of GUGA CI wave-
function. Alternatively, two sets of orbitals (obtained by
separate MCSCF orbital optimizations) can be used. Two or
more separate CIs will be carried out. The two MO sets
must share a common set of frozen core orbitals, and the
CI must be of the complete active space type. These
restrictions are needed to leave the CI wavefunctions
invariant under the necessary rotation to corresponding
orbitals. The non-orthogonal procedure implemented is a
GUGA driven equivalent to the method of Lengsfield, et al.
Note that the FOCI and SOCI methods described by these
workers are not available in GAMESS.
If you would like to use separate orbitals during the
CI, they may be generated with the FCORE option in $MCSCF.
Typically you would optimize the ground state completely,
then use these MCSCF orbitals in an optimization of the
excited state, under the constraint of FCORE=.TRUE.
For SO-MCQDPT calculations, only one set of orbitals
may be input to describe all CAS-CI states. Typically that
orbital set will be obtained by state-averaged MCSCF, see
WSTATE in $DET/$DRT, and also in the $MCQDx input. Note
that although the RUNTYP=TRANSITN driver is tied to the
GUGA CI package, there is no reason the orbitals cannot be
obtained using the determinant CI package. In fact, for
the case of spin-orbit coupling, you might want to utilize
the ability to state average over several spins, see PURES
in $DET.
If there is no molecular symmetry present, transition
moment calculations will provide $DRT1 if there is one set
of orbitals, otherwise $DRT1 defines the CI based on $VEC1
and $DRT2 the CI based on $VEC2. Also for the case of no
symmetry, a spin-orbit job should enter one $DRTx or $MCQDx
for every spin multiplicity, and all states of the same
multiplicity have to be generated from $VEC1 or $VEC2,
according to IVEX input.
symmetry
The CAS-CI states are most efficiently generated using
symmetry, since states of different symmetry have zero
Hamiltonian matrix elements. It is probably more efficient
to do four CI calculations in the group C2v on A1, A2, B1,
and B2 symmetry, than one CI with a combined Hamiltonian
in C1 symmetry (unless the active space is very small), and
similar remarks apply to the SO-MCQDPT case. In order to
avoid repeatedly saying $DRTx or $MCQDx, the following few
paragraphs say $DRTx only.
Again supposing the group is C2v, and you are
interested in singlet-triplet coupling. After some
preliminary CI calculations, you might know that the lowest
8 states are two 1-a1, 1-b1, two 1-b2, one 3-a1, and two 3-
b2 states. In this case your input would consist of five
$DRTx, of which you can give the three singlets in any
order but these must preceed the two triplet input groups
to follow the rule of increasing multiplicity. Clearly it
is not possible to write a formula for how many $DRTx there
will be, this depends not only on the point group, but also
the chemistry of the situation.
If you are using two sets of orbitals, the generation
of the corresponding orbitals for the two sets will permute
the active orbitals in an unpredictable way. Use ISTSYM to
define the desired state symmetry, rather than relying on
the orbital order. It is easy and safer to be explicit
about the spatial orbital symmetry.
The users are encouraged to specify full symmetry in
their $DATA input even though they may choose to set the
symmetry in $DRTx to C1. The CI states will be labelled in
the group given in $DATA. The use of non-Abelian symmetry
is limited by the absence of non-Abelian CI or MCQDPT. In
this case the users can choose between setting full non-
Abelian symmetry in $DATA and C1 in $DRT or else an Abelian
subgroup in both $DATA and $DRT. The latter choice appears
to be most efficient at present.
An example of SO-MCQDPT illustrating how the carbon
atom of Kh symmetry (full rotation-reflection group) can be
entered in D2h, Kh's highest Abelian group. The run time is
considerably longer in C1 symmetry.
As another example, consider an organic molecule with a
singly excited state, where that state might be coupled to
low or high spin, and where these two states might be close
enough to have a strong spin-orbit coupling. If it happens
that the S1 and S0 states possess different symmetry, a
very reaasonable calculation would be to treat the S1 and
T1 state with the same $VEC2 orbitals, leaving the ground
state described by $VEC1. After doing an MCSCF on the S0
ground state for $VEC1, you could do a state-averaged MCSCF
for $VEC2 optimized for T1 and S1 simultaneously, using
PURES. The spin orbit job would obtain its initial states
from three GUGA CI computations, S0 from $VEC1 and $DRT1,
S1 from $VEC2 and $DRT2, and T1 from $VEC2 and $DRT3. Your
$TRANST would be NUMCI=3, IROOTS(1)=1,1,1, IVEX(1)=1,2,2.
Note that the second IROOTS value is 1 because S1 was
presumed to have a different symmetry than S0, so ISTSYM in
$DRT1 and $DRT2 will differ. The calculation just outlined
cannot be done if S0 and S1 have the same spatial symmetry,
as IROOTS(1)=1,2,1 to obtain S1 during the second CI will
bring in an additional S0 state (one expressed in terms of
the $VEC2, at slightly higher energy). This problem is the
origin of the statement several paragraphs above that a
system with no symmetry will have one $DRTx for every spin
multiplicity included.
For transition moments, which do not diagonalize a
matrix containing these duplicated states, it is OK to
proceed, provided you ignore all transition moments between
the same states obtained in the two different CIs.
spin orbit details
Spin-orbit coupling is always performed in a quasi-
degenerate perturbative manner. Typically the states close
in energy are included into the spin-orbit coupling matrix.
"Close" has a easily understandable meaning, since in the
limit of small coupling the quasi-degenerate treatment is
reduced to a second order perturbative treatment, that is,
the affect of a state upon the state of primary interest is
given by the square of the spin-orbit coupling matrix
element divided by the difference of the adibatic energies.
This is useful to keep in mind when deciding how many CI
states to include in the matrix. The states that are
included are treated in a fashion that is equivalent to
infinite order perturbation theory (exact) whereas the
states that are not included make no contribution.
The choice between HSO2 and HSO2FF is very often in
favour of the former. HSO2 computes the matrix elements in
CSF basis and then contracts them with CI coefficients,
whereas HSO2FF uses a generalized density in AO basis
computed for each pair of states, thus HSO2 is much more
efficient in case of multiple states given in IROOTS.
HSO2FF takes less memory for integral storage, thus it can
be superior in case of small active spaces and large basis
sets, in part because it does not store 2e SOC integrals on
disk and secondly, it does not redundantly treat the same
pair of determinants if they appear in different CSFs. The
numerical results with HSO2 and HSO2FF should be identical
within machine and algorithmic accuracy.
The spin-orbit operator contains a one electron term
arising from Pauli's reduction of the hydrogenic Dirac
equation to one-component form, and a two electron term
added by Breit. The only practical limitation on the
computation of the Breit term is that HSO2FF is limited to
10 active orbitals on 32 bit machines, and to about 26
active orbitals on 64 bit machines. The spin-orbit matrix
elements vanish for |delta-S| > 1, but it is possible to
include three or more spins in the computation. Since
singlets interact with triplets, and triplets interact
with pentuplets, inclusion of S=0,1,2 simultaneously lets
you pick up the indirect interaction between singlets and
pentuplets that the intermediate triplets afford.
As an approximation, the nuclear charge appearing in
the one electron term can be regarded as an empirical scale
factor, compensating for the omission of the two electron
operator. In addition, these effective charges are often
used to compensate for missing nodes in valence orbitals
of ECP runs, in which case the ZEFF are typically very far
from the two nuclear charges. ZEFTYP selects some built
in values obtained by S.Koseki et al, but if you have some
favorite parameters, they can be read in as the ZEFF input
array. Effective charges may be used for any OPERAT, but
are most often used with HSO1.
Various symmetries are used to avoid computing zero
spin-orbit matrix elements. NOSYM in $TRANST allows some
control over this: NOSYM=1 gives up point group symmetry
completely, while 2 turns off additional symmetries such
as spin selection rules. HSO1,2,2P compute all matrix
elements in a group (i.e. between two $DRTx groups with
fixed Ms(ket)-Ms(bra)) if at least one of them does not
vanish by symmetry, and HSO2PP actually avoids computation
for each pair of states if forbidden by symmetry. Setting
NOSYM=2 will cause HSO2FF to consider the elements between
two singlets, which are always calculated for HSO1,2,2P
when transition dipoles are requested as well.
SYMTOL has a dramatic effect on the run speed. This
cutoff is applied to CSF coefficcients, their products,
and these products times CSF orbital overlaps. The value
permits a tradeoff of accuracy for run time, and since the
error in the spin-orbit properties approaches SYMTOL mainly
for SOCI functions, it may be useful to increase SYMTOL to
save time for CAS or FOCI functions. Some experimenting
will tell you what you can get away with. SYMTOL is also
used during CI state symmetry assignment, for NOIRR=-1
in $DRT.
In case if you do not provide enough storage for the
form factors sorting then some extra disk space will be
used; the extra disk space can be eliminated if you set
SAVDSK=.TRUE. (the amount of savings depends on the active
space and memory provided, it some cases it can decrease
the disk space up to one order of magnitude). The form
factors are in binary format, and so can be transfered
between computers only if they have compatible binary
files. There is a built-in check for consistency of a
restart file DAFL30 with the current run parameters.
input nitty-gritty
The transition moment and spin-orbit coupling driver
is a rather restricted path through the GUGA CI part of
GAMESS. Note that $GUESS is not read, instead the MOs will
be MOREAD in a $VEC1 and perhaps a $VEC2 group. It is not
possible to reorder MOs. For SO-CI,
1) Give SCFTYP=NONE CITYP=GUGA MPLEVL=0.
2) $CIINP is not read. The CI is hardwired to consist
of CI DRT generation, integral transformation/sorting,
Hamiltonian generation, and diagonalization. This
means $DRT1 (and maybe $DRT2,...), $TRANS, $CISORT,
$GUGEM, and $GUGDIA input is read, and acted upon.
3) The density matrices are not generated, and so no
properties (other than the transition moment or the
spin-orbit coupling) are computed.
4) There is no restart capability provided, except for
saving some form-factor information.
5) $DRT1, $DRT2, $DRT3, ... must go from lowest to highest
multiplicity.
6) IROOTS will determine the number of CI states in each
CI for which the properties are calculated. Use
NSTATE to specify the number of CI states for the
CI Hamiltonian diagonalization. Sometimes the CI
convergence is assisted by requesting more roots
to be found in the diagonalization than you want to
include in the property calculation.
For SO-MCQDPT, the steps are
1) Give SCFTYP=NONE CITYP=NONE MPLEVL=2.
2) the number of roots in each MCQDPT is controlled by
$TRANST's IROOTS, and each such calculation is
defined by $MCQD1, $MCQD2, ... input. These must go
from lowest multiplicity to highest.
references
The review already mentioned:
"Spin-orbit coupling in molecules: chemistry beyond the
adiabatic approximation".
D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon,
Int.Rev.Phys.Chem. 22, 551-592(2003)
Reference for separate active orbital optimization:
1. B.H.Lengsfield, III, J.A.Jafri, D.H.Phillips,
C.W.Bauschlicher, Jr. J.Chem.Phys. 74,6849-6856(1981)
References for transition moments:
2. F.Weinhold, J.Chem.Phys. 54,1874-1881(1970)
3. C.W.Bauschlicher, S.R.Langhoff
Theoret.Chim.Acta 79:93-103(1991)
4. "Intramediate Quantum Mechanics, 3rd Ed." Hans A.
Bethe, Roman Jackiw Benjamin/Cummings Publishing,
Menlo Park, CA (1986), chapters 10 and 11.
5. S.Koseki, M.S.Gordon J.Mol.Spectrosc. 123, 392-
404(1987)
References for Zeff spin-orbit coupling, and ZEFTYP values:
6. S.Koseki, M.W.Schmidt, M.S.Gordon
J.Phys.Chem. 96, 10768-10772 (1992)
7. S.Koseki, M.S.Gordon, M.W.Schmidt, N.Matsunaga
J.Phys.Chem. 99, 12764-12772 (1995)
8. N.Matsunaga, S.Koseki, M.S.Gordon
J.Chem.Phys. 104, 7988-7996 (1996)
9. S.Koseki, M.W.Schmidt, M.S.Gordon
J.Phys.Chem.A 102, 10430-10435 (1998)
10. S.Koseki, D.G.Fedorov, M.W.Schmidt, M.S.Gordon
J.Phys.Chem.A 105, 8262-8268 (2001)
References for full Breit-Pauli spin-orbit coupling:
11. T.R.Furlani, H.F.King
J.Chem.Phys. 82, 5577-5583 (1985)
12. H.F.King, T.R.Furlani
J.Comput.Chem. 9, 771-778 (1988)
13. D.G.Fedorov, M.S.Gordon
J.Chem.Phys. 112, 5611-5623 (2000)
with the latter including information on the partial
two electron operator method.
Reference for SO-MCQDPT:
14. D.G.Fedorov, J.P.Finley Phys.Rev.A 64, 042502 (2001)
Reference for Spin-Orbit with Model Core Potentials:
D.G.Fedorov, M.Klobukowski
Chem.Phys.Lett. 360, 223-228(2002)
Recent applications:
16. D.G.Fedorov, M.Evans, Y.Song, M.S.Gordon, C.Y.Ng
J.Chem.Phys. 111, 6413-6421 (1999)
17. D.G.Fedorov, M.S.Gordon, Y.Song, C.Y.Ng
J.Chem.Phys. 115, 7393-7400 (2001)
18. B.J.Duke J.Comput.Chem. 22, 1552-1556 (2001)
19. D.G.Fedorov, S.Koseki, M.W.Schmidt, M.S.Gordon
K.Hirao and Y.Ishikawa (eds.) Recent Advances in
Relativistic Molecular Theory, Vol. 5,
(World Scientific, Singapore), 2004, pp 107-136.
* * *
Special thanks to Bob Cave and Dave Feller for their
assistance in performing check spin-orbit coupling runs
with the MELDF programs. Special thanks to Tom Furlani
for contributing his 2e- spin-orbit code and answering
many questions about its interface. Special thanks to
Haruyuki Nakano for explaining the spin functions used
in the MCQDPT package.
examples
We end with 2 examples. Note that you must know what
you are doing with term symbols, J quantum numbers, point
group symmetry, and so on in order to make skillful use of
this part of the program. Seeing your final degeneracies
turn out like a text book says it should is beautiful!
! Compute the splitting of the famous sodium D line.
! Joseph von Fraunhofer (Denkschriften der Koeniglichen
! Akademie der Wissenschf. zu Muenchen, 5, 193(1814-1815))
! observed the sun through good prisms, finding 700 lines,
! and named the brightest ones A, B, C... just in order.
! He was able to resolve the D line into two lines, which
! occur at 5895.940 and 5889.973 Angstroms. It would take
! a century to understand the D line is Na's 3s <-> 3p
! transition, and that spin-orbit coupling is what splits
! the D line into two. Charlotte Moore's Atomic Energy
! Levels, volume 1, gives the experimental 2-P interval
! as 17.1963, since the three relevent levels are at
! 2-S-1/2= 0.0, 2-P-1/2= 16,956.183, 2-P-3/2= 16,973.379.
1. generate ground state 2-S orbitals by conventional ROHF.
the energy of the ground state is -161.8413919816
--- $contrl scftyp=rohf mult=2 $end
--- $system kdiag=3 memory=300000 $end
--- $guess guess=huckel $end
2. generate excited state 2-P orbitals, using a state-
averaged SCF wavefunction to ensure radial degeneracy of
the 3p shell is preserved. The open shell SCF energy is
-161.7682895801. The computation is both spin and space
restricted open shell SCF on the 2-P Russell-Saunders term.
Starting orbitals are reordered orbitals from step 1.
--- $contrl scftyp=gvb mult=2 $end
--- $system kdiag=3 memory=300000 $end
--- $guess guess=moread norb=13
--- norder=1 iorder(6)=7,8,9,6 $end
--- $scf nco=5 nseto=1 no(1)=3 rstrct=.t. couple=.true.
--- f(1)= 1.0 0.16666666666667
--- alpha(1)= 2.0 0.33333333333333 0.0
--- beta(1)= -1.0 -0.16666666666667 0.0 $end
3. compute spin-orbit coupling in the 2-P term. The use of
C1 symmetry in $DRT1 ensures that all three spatial CSFs
are kept in the CI function. In the preliminary CI, the
spin function is just the alpha spin doublet, and all three
roots should be degenerate, and furthermore equal to the
GVB energy at step 2. The spin-orbit coupling code uses
both doublet spin functions with each of the three spatial
wavefunctions, so the spin-orbit Hamiltonian is a 6x6
matrix. The two lowest roots of the full 6x6 spin-orbit
Hamiltonian are the doubly degenerate 2-P-1/2 level, while
the other four roots are the degenerate 2-P-3/2 level.
$contrl scftyp=none cityp=guga runtyp=transitn mult=2 $end
$system memory=2000000 $end
$basis gbasis=n31 ngauss=6 $end
$gugdia nstate=3 $end
$transt operat=hso1 numvec=1 numci=1 nfzc=5 nocc=8
iroots=3 zeff=10.04 $end
$drt1 group=c1 fors=.true. nfzc=5 nalp=1 nval=2 $end
$data
Na atom...2-P excited state...6-31G basis
Dnh 2
Na 11.0
$end
--- GVB ORBITALS --- GENERATED AT 7:46:08 CST 30-MAY-1996
Na atom...2-P excited state
E(GVB)= -161.7682895801, E(NUC)= .0000000000, 5
ITERS
$VEC1
1 1 9.97912679E-01 8.83038094E-03 0.00000000E+00...
... orbitals from step 2 go here ...
13 3-1.10674398E+00 0.00000000E+00 0.00000000E+00
$END
As an example of both SO-MCQDPT, and the use of as much
symmetry as possible, consider carbon. The CAS-CI uses
an active space of 2s,2p,3s,3p orbitals, and the spin-orbit
job includes all terms from the lowest configuration,
2s2,2p2. These terms are 3-P, 1-D, and 1-S. If you look
at table 58 in Herzberg's book on electronic spectra, you
will be able to see how the Kh spatial irreps P, D, S are
partitioned into the D2h irreps input below.
! C SO-MRMP on all levels in the s**2,p**2 configuration.
!
! levels CAS and MCQDPT
! 1 .0000 .0000 cm-1 3-P-0
! 2-4 12.6879-12.8469 13.2721-13.2722 3-P-1
! 5-9 37.8469-37.8470 39.5638-39.5639 3-P-2
! 10-14 12169.1275 10251.7910 1-D-2
! 15 19264.4221 21111.5130 1-S-0
!
! The active space consists of (2s,2p,3s,3p) with 4 e-.
! D2h symmetry speeds up the calculation considerably,
! on the same computer D2h = 78 and C1 = 424 seconds.
$contrl scftyp=none cityp=none mplevl=2 runtyp=transitn
$end
$system memory=5000000 $end
!
! below is input to run in C1 subgroup
!
--- $transt operat=hso2 numvec=-2 numci=2 nfzc=1 nocc=9
--- iroots(1)=6,3 parmp=3
--- ivex(1)=1,1 $end
--- $MCQD1 nosym=1 nstate=6 mult=1 INORB=1 iforb=3
--- nmofzc=1 nmodoc=0 nmoact=8
--- wstate(1)=1,1,1,1,1,1 thrcon=1e-8 thrgen=1e-10
$END
--- $MCQD2 nosym=1 nstate=3 mult=3 INORB=1 iforb=3
--- nmofzc=1 nmodoc=0 nmoact=8
--- wstate(1)=1,1,1 thrcon=1e-8 thrgen=1e-10 $END
!
! below is input to run in D2h subgroup
!
$transt operat=hso2 numvec=-7 numci=7 nfzc=1 nocc=9
iroots(1)=3,1,1,1, 1,1,1 parmp=3
ivex(1)=1,1,1,1,1,1,1 $end
$MCQD1 nosym=-1 mult=1 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=1 wstate(1)=1,1,1 thrcon=1e-8 thrgen=1e-10
$END
$MCQD2 nosym=-1 mult=1 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=2 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD3 nosym=-1 mult=1 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=3 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD4 nosym=-1 mult=1 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=4 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD5 nosym=-1 mult=3 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=2 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD6 nosym=-1 mult=3 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=3 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
$MCQD7 nosym=-1 mult=3 INORB=1 iforb=3
nmofzc=1 nmodoc=0 nmoact=8
istsym=4 wstate(1)=1 thrcon=1e-8 thrgen=1e-10 $END
!
! input to prepare the 3-P ground state orbitals
! great care is taken to create symmetry equivalent p's
!
--- $contrl scftyp=mcscf cityp=none mplevl=0
--- runtyp=energy mult=3 $end
--- $guess guess=moread norb=55 purify=.t. $end
--- $mcscf cistep=guga fullnr=.t. $end
--- $drt group=c1 fors=.true. nmcc=1 ndoc=1 nalp=2
nval=5 $end
--- $gugdia nstate=9 maxdia=1000 $end
--- $gugdm2 wstate(1)=1,1,1 $end
!
$data
C...aug-cc-pvtz (10s,5p,2d,1f) -> [4s,3p,2d,1f]
(1s,1p,1d,1f)
Dnh 2
C 6.0
S 8
1 8236.000000 0.5310000000E-03
2 1235.000000 0.4108000000E-02
3 280.8000000 0.2108700000E-01
4 79.27000000 0.8185300000E-01
5 25.59000000 0.2348170000
6 8.997000000 0.4344010000
7 3.319000000 0.3461290000
8 0.3643000000 -0.8983000000E-02
S 8
1 8236.000000 -0.1130000000E-03
2 1235.000000 -0.8780000000E-03
3 280.8000000 -0.4540000000E-02
4 79.27000000 -0.1813300000E-01
5 25.59000000 -0.5576000000E-01
6 8.997000000 -0.1268950000
7 3.319000000 -0.1703520000
8 0.3643000000 0.5986840000
S 1
1 0.9059000000 1.000000000
S 1
1 0.1285000000 1.000000000
P 3
1 18.71000000 0.1403100000E-01
2 4.133000000 0.8686600000E-01
3 1.200000000 0.2902160000
P 1
1 0.3827000000 1.000000000
P 1
1 0.1209000000 1.000000000
D 1
1 1.097000000 1.000000000
D 1
1 0.3180000000 1.000000000
F 1
1 0.7610000000 1.000000000
S 1
1 0.440200000E-01 1.00000000
P 1
1 0.356900000E-01 1.00000000
D 1
1 0.100000000 1.00000000
F 1
1 0.268000000 1.00000000
$end
--- OPTIMIZED MCSCF MO-S --- GENERATED 22-AUG-2000
E(MCSCF)= -37.7282408589, 11 ITERS
$VEC1
1 1 9.75511467E-01 ...
$END
~~