( 2 Jun 94)
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* Section 4 - Further Information *
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This section of the manual contains both references, and
hints on how to do things. The following is a list of
the topics covered:
o Computational References.
o Basis Set References, and descriptions.
o How to do RHF, ROHF, UHF, and GVB calculations.
General considerations
Direct SCF
SCF convergence accelerators
High spin open shell SCF (ROHF)
Other open shell SCF cases (GVB)
True GVB perfect pairing runs
The special case of TCSCF
A caution about symmetry with GVB
o How to do CI and MCSCF calculations.
Specification of the wavefunction
Use of symmetry
Selection of orbitals
The iterative process
Miscellaneous hints
MCSCF implementation
References
o Geometry Searches and Internal Coordinates.
Quasi-Newton searches
The hessian
Coordinates choices
The role of symmetry
Practical matters
Saddle points
Mode following
o MOPAC calculations within GAMESS
o Molecular Properties, and conversion factors.
o Localization tips.
o Intrinsic Reaction Coordinate (IRC) methods
o Transition moments and spin-orbit coupling.
Computational References
------------- ----------
GAMESS -
M.W.Schmidt, K.K.Baldridge, J.A.Boatz, S.T.Elbert,
M.S.Gordon, J.J.Jensen, S.Koseki, N.Matsunaga,
K.A.Nguyen, S.Su, T.L.Windus, M.Dupuis, J.A.Montgomery
J.Comput.Chem. 14, 1347-1363 (1993)
HONDO -
"The General Atomic and Molecular Electronic Structure
System: HONDO 7.0" M.Dupuis, J.D.Watts, H.O.Villar,
G.J.B.Hurst Comput.Phys.Comm. 52, 415-425(1989)
"HONDO: A General Atomic and Molecular Electronic
Structure System" M.Dupuis, P.Mougenot, J.D.Watts,
G.J.B.Hurst, H.O.Villar in "MOTECC: Modern Techniques
in Computational Chemistry" E.Clementi, Ed.
ESCOM, Leiden, the Netherlands, 1989, pp 307-361.
"HONDO: A General Atomic and Molecular Electronic
Structure System" M.Dupuis, A.Farazdel, S.P.Karna,
S.A.Maluendes in "MOTECC: Modern Techniques in
Computational Chemistry" E.Clementi, Ed.
ESCOM, Leiden, the Netherlands, 1990, pp 277-342.
M.Dupuis, S.Chin, A.Marquez in Relativistic and Electron
Correlation Effects in Molecules, G.Malli, Ed. Plenum
Press, NY 1994.
The final three papers describes many of the algorithms
in detail, and much of this paper applies also to GAMESS.
sp integrals -
J.A.Pople, W.J.Hehre J.Comput.Phys. 27, 161-168(1978)
sp gradient integrals -
A.Komornicki, K.Ishida, K.Morokuma, R.Ditchfield,
M.Conrad Chem.Phys.Lett. 45, 595-602(1977)
spdfg integrals -
"Numerical Integration Using Rys Polynomials"
H.F.King and M.Dupuis J.Comput.Phys. 21,144(1976)
"Evaluation of Molecular Integrals over Gaussian
Basis Functions"
M.Dupuis,J.Rys,H.F.King J.Chem.Phys. 65,111-116(1976)
"Molecular Symmetry and Closed Shell HF Calculations"
M.Dupuis and H.F.King Int.J.Quantum Chem. 11,613(1977)
"Computation of Electron Repulsion Integrals using
the Rys Quadrature Method"
J.Rys,M.Dupuis,H.F.King J.Comput.Chem. 4,154-157(1983)
spd gradient integrals -
"Molecular Symmetry. II. Gradient of Electronic Energy
with respect to Nuclear Coordinates"
M.Dupuis and H.F.King J.Chem.Phys. 68,3998(1978)
spd hessian integrals -
"Molecular Symmetry. III. Second derivatives of Electronic
Energy with respect to Nuclear Coordinates"
T.Takada, M.Dupuis, H.F.King
J.Chem.Phys. 75, 332-336 (1981)
RHF -
C.C.J. Roothaan Rev.Mod.Phys. 23, 69(1951)
UHF -
J.A. Pople, R.K. Nesbet J.Chem.Phys 22, 571 (1954)
GVB and OCBSE-ROHF -
F.W.Bobrowicz and W.A.Goddard, in Modern Theoretical
Chemistry, Vol 3, H.F.Schaefer III, Ed., Chapter 4.
ROHF -
R.McWeeny, G.Diercksen J.Chem.Phys. 49,4852-4856(1968)
M.F.Guest, V.R.Saunders, Mol.Phys. 28, 819-828(1974)
J.S.Binkley, J.A.Pople, P.A.Dobosh
Mol.Phys. 28, 1423-1429 (1974)
E.R.Davidson Chem.Phys.Lett. 21,565(1973)
K.Faegri, R.Manne Mol.Phys. 31,1037-1049(1976)
H.Hsu, E.R.Davidson, and R.M.Pitzer
J.Chem.Phys. 65,609(1976)
Direct SCF -
J.Almlof, K.Faegri, K.Korsell
J.Comput.Chem. 3, 385-399 (1982)
M.Haser, R.Ahlrichs
J.Comput.Chem. 10, 104-111 (1989)
GUGA CI -
B.Brooks and H.F.Schaefer J.Chem. Phys. 70,5092(1979)
B.Brooks, W.Laidig, P.Saxe, N.Handy, and H.F.Schaefer,
Physica Scripta 21,312(1980).
2nd order Moller-Plesset -
M.Dupuis, S.Chin, A.Marquez, see above
MOPAC 6 -
J.J.P.Stewart J.Computer-Aided Molecular Design
4, 1-105 (1990)
References for parameters for individual atoms may be
found on the printout from your runs.
DIIS (Direct Inversion in the Iterative Subspace) -
P.Pulay J.Comput.Chem. 3, 556-560(1982)
RHF/ROHF/TCSCF coupled perturbed Hartree Fock -
"Single Configuration SCF Second Derivatives on a Cray"
H.F.King, A.Komornicki in "Geometrical Derivatives of
Energy Surfaces and Molecular Properties" P.Jorgensen
J.Simons, Ed. D.Reidel, Dordrecht, 1986, pp 207-214.
Y.Osamura, Y.Yamaguchi, D.J.Fox, M.A.Vincent, H.F.Schaefer
J.Mol.Struct. 103, 183-186 (1983)
M.Duran, Y.Yamaguchi, H.F.Schaefer
J.Phys.Chem. 92, 3070-3075 (1988)
Davidson eigenvector method -
E.R.Davidson J.Comput.Phys. 17,87(1975) and "Matrix
Eigenvector Methods" p. 95 in "Methods in Computational
Molecular Physics" ed. by G.H.F.Diercksen and S.Wilson
EVVRSP in memory diagonalization -
S.T.Elbert Theoret.Chim.Acta 71,169-186(1987)
Energy localization-
C.Edmiston, K.Ruedenberg Rev.Mod.Phys. 35, 457-465(1963).
Boys localization-
S.F.Boys, "Quantum Science of Atoms, Molecules, and Solids"
P.O.Lowdin, Ed, Academic Press, NY, 1966, pp 253-262.
Population localization -
J.Pipek, P.Z.Mezey J.Chem.Phys. 90, 4916(1989).
Geometry optimization and saddle point location -
J.Baker J.Comput.Chem. 7, 385-395(1986).
T.Helgaker Chem.Phys.Lett. 182, 503-510(1991).
P.Culot, G.Dive, V.H.Nguyen, J.M.Ghuysen
Theoret.Chim.Acta 82, 189-205(1992).
Vibrational Analysis in Cartesian coordinates -
W.D.Gwinn J.Chem.Phys. 55,477-481(1971)
Normal coordinate decomposition analysis -
J.A.Boatz and M.S.Gordon,
J.Phys.Chem. 93, 1819-1826(1989).
Modified Virtual Orbitals (MVOs) -
C.W.Bauschlicher, Jr. J.Chem.Phys. 72,880-885(1980)
Effective core potentials (ECP) -
L.R.Kahn, P.Baybutt, D.G.Truhlar
J.Chem.Phys. 65, 3826-3853 (1976)
See also the papers listed for SBK and HW basis sets.
TCGMSG
R.J.Harrison, Int.J.Quantum Chem. 40, 847-863(1991)
Bond orders
I.Mayer, Chem.Phys.Lett., 97,270-274(1983), 117,396(1985).
I.Mayer, Theoret.Chim.Acta, 67,315-322(1985).
Basis Set References
----- --- ----------
An excellent review of the relationship between
the atomic basis used, and the accuracy with which
various molecular properties will be computed is:
E.R.Davidson, D.Feller Chem.Rev. 86, 681-696(1986).
STO-NG H-Ne Ref. 1 and 2
Na-Ar, Ref. 2 and 3 **
K,Ca,Ga-Kr Ref. 4
Rb,Sr,In-Xe Ref. 5
Sc-Zn,Y-Cd Ref. 6
1) W.J.Hehre, R.F.Stewart, J.A.Pople
J.Chem.Phys. 51, 2657-2664(1969).
2) W.J.Hehre, R.Ditchfield, R.F.Stewart, J.A.Pople
J.Chem.Phys. 52, 2769-2773(1970).
3) M.S.Gordon, M.D.Bjorke, F.J.Marsh, M.S.Korth
J.Am.Chem.Soc. 100, 2670-2678(1978).
** the valence scale factors for Na-Cl are taken
from this paper, rather than the "official"
Pople values in Ref. 2.
4) W.J.Pietro, B.A.Levi, W.J.Hehre, R.F.Stewart,
Inorg.Chem. 19, 2225-2229(1980).
5) W.J.Pietro, E.S.Blurock, R.F.Hout,Jr., W.J.Hehre, D.J.
DeFrees, R.F.Stewart Inorg.Chem. 20, 3650-3654(1980).
6) W.J.Pietro, W.J.Hehre J.Comput.Chem. 4, 241-251(1983).
MINI/MIDI H-Xe Ref. 9
9) "Gaussian Basis Sets for Molecular Calculations"
S.Huzinaga, J.Andzelm, M.Klobukowski, E.Radzio-Andzelm,
Y.Sakai, H.Tatewaki Elsevier, Amsterdam, 1984.
The MINI bases are three gaussian expansions of each
atomic orbital. The exponents and contraction
coefficients are optimized for each element, and s and p
exponents are not constrained to be equal. As a result
these bases give much lower energies than does STO-3G.
The valence MINI orbitals of main group elements are
scaled by factors optimized by John Deisz at North Dakota
State University. Transition metal MINI bases are not
scaled. The MIDI bases are derived from the MINI sets by
floating the outermost primitive in each valence orbitals,
and renormalizing the remaining 2 gaussians. MIDI bases
are not scaled by GAMESS. The transition metal bases are
taken from the lowest SCF terms in the s**1,d**n
configurations.
3-21G H-Ne Ref. 10 (also 6-21G)
Na-Ar Ref. 11 (also 6-21G)
K,Ca,Ga-Kr,Rb,Sr,In-Xe Ref. 12
Sc-Zn Ref. 13
Y-Cd Ref. 14
10) J.S.Binkley, J.A.Pople, W.J.Hehre
J.Am.Chem.Soc. 102, 939-947(1980).
11) M.S.Gordon, J.S.Binkley, J.A.Pople, W.J.Pietro,
W.J.Hehre J.Am.Chem.Soc. 104, 2797-2803(1982).
12) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 7,359-378(1986)
13) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 8,861-879(1987)
14) K.D.Dobbs, W.J.Hehre J.Comput.Chem. 8,880-893(1987)
N-31G references for 4-31G 5-31G 6-31G
H 15 15 15
He 23 23 23
Li 19,24 19
Be 20,24 20
B 17 19
C-F 15 16 16
Ne 23 23
Na-Ga 22
Si 21 **
P-Cl 18 22
Ar 22
15) R.Ditchfield, W.J.Hehre, J.A.Pople
J.Chem.Phys. 54, 724-728(1971).
16) W.J.Hehre, R.Ditchfield, J.A.Pople
J.Chem.Phys. 56, 2257-2261(1972).
17) W.J.Hehre, J.A.Pople J.Chem.Phys. 56, 4233-4234(1972).
18) W.J.Hehre, W.A.Lathan J.Chem.Phys. 56,5255-5257(1972).
19) J.D.Dill, J.A.Pople J.Chem.Phys. 62, 2921-2923(1975).
20) J.S.Binkley, J.A.Pople J.Chem.Phys. 66, 879-880(1977).
21) M.S.Gordon Chem.Phys.Lett. 76, 163-168(1980)
** - Note that the built in 6-31G basis for Si is
not that given by Pople in reference 22.
The Gordon basis gives a better wavefunction,
for a ROHF calculation in full atomic (Kh)
symmetry,
6-31G Energy virial
Gordon -288.828573 1.999978
Pople -288.828405 2.000280
See the input examples for how to run in Kh.
22) M.M.Francl, W.J.Pietro, W.J.Hehre, J.S.Binkley,
M.S.Gordon, D.J.DeFrees, J.A.Pople
J.Chem.Phys. 77, 3654-3665(1982).
23) Unpublished, copied out of GAUSSIAN82.
24) For Li and Be, 4-31G is actually a 5-21G expansion.
Extended basis sets
6-311G
28) R.Krishnan, J.S.Binkley, R.Seeger, J.A.Pople
J.Chem.Phys. 72, 650-654(1980).
valence double zeta "DZV" sets:
"DH" basis - DZV for H, Li-Ne, Al-Ar
30) T.H.Dunning, Jr., P.J.Hay Chapter 1 in "Methods of
Electronic Structure Theory", H.F.Shaefer III, Ed.
Plenum Press, N.Y. 1977, pp 1-27.
Note that GAMESS uses inner/outer scale factors of
1.2 and 1.15 for DH's hydrogen (since at least 1983).
To get Thom's usual basis, scaled 1.2 throughout:
HYDROGEN 1.0 x, y, z
DH 0 1.2 1.2
"BC" basis - DZV for Ga-Kr
31) R.C.Binning, Jr., L.A.Curtiss
J.Comput.Chem. 11, 1206-1216(1990)
valence triple zeta "TZV" sets:
TZV for Li-Ne
40) T.H. Dunning, J.Chem.Phys. 55 (1971) 716-723.
TZV for Na-Ar - also known as the "MC" basis
41) A.D.McLean, G.S.Chandler
J.Chem.Phys. 72,5639-5648(1980).
TZV for K,Ca
42) A.J.H. Wachters, J.Chem.Phys. 52 (1970) 1033-1036.
(see Table VI, Contraction 3).
TZV for Sc-Zn (taken from HONDO 7)
This is Wachters' (14s9p5d) basis (ref 42) contracted
to (10s8p3d) with the following modifications
1. the most diffuse s removed;
2. additional s spanning 3s-4s region;
3. two additional p functions to describe the 4p;
4. (6d) contracted to (411) from ref 43,
except for Zn where Wachter's (5d)/[41]
and Hay's diffuse d are used.
43) A.K. Rappe, T.A. Smedley, and W.A. Goddard III,
J.Phys.Chem. 85 (1981) 2607-2611
Valence only basis sets (for use with corresponding ECPs)
SBK -31G splits, bigger for trans. metals (available Li-Rn)
50) W.J.Stevens, H.Basch, M.Krauss
J.Chem.Phys. 81, 6026-6033 (1984)
51) W.J.Stevens, H.Basch, M.Krauss, P.Jasien
Can.J.Chem, 70, 612-630 (1992)
52) T.R.Cundari, W.J.Stevens
J.Chem.Phys. 98, 5555-5565(1993)
HW -21 splits (sp exponents not shared)
transition metals (not built in at present, although
they will work if you type them in).
53) P.J.Hay, W.R.Wadt J.Chem.Phys. 82, 270-283 (1985)
main group (available Na-Xe)
54) W.R.Wadt, P.J.Hay J.Chem.Phys. 82, 284-298 (1985)
see also
55) P.J.Hay, W.R.Wadt J.Chem.Phys. 82, 299-310 (1985)
Polarization exponents
STO-NG*
60) J.B.Collins, P. von R. Schleyer, J.S.Binkley,
J.A.Pople J.Chem.Phys. 64, 5142-5151(1976).
3-21G*. See also reference 12.
61) W.J.Pietro, M.M.Francl, W.J.Hehre, D.J.DeFrees, J.A.
Pople, J.S.Binkley J.Am.Chem.Soc. 104,5039-5048(1982)
6-31G* and 6-31G**. See also reference 22 above.
62) P.C.Hariharan, J.A.Pople
Theoret.Chim.Acta 28, 213-222(1973)
STO-NG* means d orbitals are used on third row atoms only.
The original paper (ref 60) suggested z=0.09 for
Na and Mg, and z=0.39 for Al-Cl.
At NDSU we prefer to use the same exponents as
in 3-21G* and 6-31G*, so we know we're looking
at changes in the sp basis, not the d exponent.
3-21G* means d orbitals on main group elements in the
third and higher periods. Not defined for the
transition metals, where there are p's already
in the basis. Except for alkalis and alkali
earths, the 4th and 5th row zetas are from
Huzinaga, et al (ref 9). The exponents are
normally the same as for 6-31G*.
6-31G* means d orbitals on second and third row atoms.
We use Mark Gordon's z=0.395 for silicon, as well
as his fully optimized sp basis (ref 21).
This is often written 6-31G(d) today.
6-31G** means the same as 6-31G*, except that p functions
are added on hydrogens.
This is often written 6-31G(d,p) today.
6-311G** means p orbitals on H, and d orbitals elsewhere.
The exponents were derived from correlated atomic
states, and so are considerably tighter than the
polarizing functions used in 6-31G**, etc.
This is often written 6-311G(d,p) today.
The definitions for 6-31G* for C-F are disturbing in
that they treat these atoms the same. Dunning and Hay
(ref 30) have recommended a better set of exponents for
second row atoms and a slightly different value for H.
2p, 3p, 2d, 3p polarization sets are usually thought
of as arising from applying splitting factors to the
1p and 1d values. For example, SPLIT2=2.0, 0.5 means
to double and halve the single value. The default
values for SPLIT2 and SPLIT3 are taken from reference
72, and were derived with correlation in mind. The
SPLIT2 values often produce a higher (!) HF energy than
the singly polarized run, because the exponents are
split too widely. SPLIT2=0.4,1.4 will always lower the
SCF energy (the values are the unpublished personal
preference of MWS), and for SPLIT3 we might suggest
3.0,1.0,1/3.
With all this as background, we are ready to present
the table of polarization exponents built into GAMESS.
Built in polarization exponents, chosen by POLAR=
in the $BASIS group. The values are for d functions
unless otherwise indicated.
Please note that the names associated with each
column are only generally descriptive. For example, the
column marked "Pople" contains a value for Si with which
John Pople would not agree, and that the K-Xe values in
this column were actually originally from the Huzinaga
group. The exponents for Ga-Kr are from the Binning and
Curtiss paper, not Thom Dunning. And so on.
POPLE POPN311 DUNNING HUZINAGA HONDO7
------ ------- ------- -------- ------
H 1.1(p) 0.75(p) 1.0(p) 1.0(p) 1.0(p)
He 1.1(p) 0.75(p) 1.0(p) 1.0(p) 1.0(p)
Li 0.2 0.200 0.076(p)
Be 0.4 0.255 0.164(p) 0.32
B 0.6 0.401 0.70 0.388 0.50
C 0.8 0.626 0.75 0.600 0.72
N 0.8 0.913 0.80 0.864 0.98
O 0.8 1.292 0.85 1.154 1.28
F 0.8 1.750 0.90 1.496 1.62
Ne 0.8 2.304 1.00 1.888 2.00
Na 0.175 0.061(p) 0.157
Mg 0.175 0.101(p) 0.234
Al 0.325 0.198 0.311
Si 0.395 0.262 0.388
P 0.55 0.340 0.465
S 0.65 0.421 0.542
Cl 0.75 0.514 0.619
Ar 0.85 0.617 0.696
K 0.1 0.039(p)
Ca 0.1 0.059(p)
Ga 0.207 0.141
Ge 0.246 0.202
As 0.293 0.273
Se 0.338 0.315
Br 0.389 0.338
Kr 0.443 0.318
Rb 0.11 0.034(p)
Sr 0.11 0.048(p)
A blank means the value equals the "Pople" column.
Common d polarization for all sets:
In Sn Sb Te I Xe
0.160 0.183 0.211 0.237 0.266 0.297
Tl Pb Bi Po At Rn
0.146 0.164 0.185 0.204 0.225 0.247
Anion diffuse functions
3-21+G, 3-21++G, etc.
70) T.Clark, J.Chandrasekhar, G.W.Spitznagel, P. von R.
Schleyer J.Comput.Chem. 4, 294-301(1983)
71) G.W.Spitznagel, Diplomarbeit, Erlangen, 1982.
72) M.J.Frisch, J.A.Pople, J.S.Binkley J.Chem.Phys.
80, 3265-3269 (1984).
Anions usually require diffuse basis functions to
properly represent their spatial diffuseness. The use of
diffuse sp shells on atoms in the second and third rows is
denoted by a + sign, also adding diffuse s functions on
hydrogen is symbolized by ++. These designations can be
applied to any of the Pople bases, e.g. 3-21+G, 3-21+G*,
6-31++G**. The following exponents are for L shells,
except for H. For H-F, they are taken from ref 70. For
Na-Cl, they are taken directly from reference 71. These
values may be found in footnote 13 of reference 72.
For Ga-Br, In-I, and Tl-At these were optimized for the
atomic ground state anion, using ROHF with a flexible ECP
basis set, by Ted Packwood at NDSU.
H
0.0360
Li Be B C N O F
0.0074 0.0207 0.0315 0.0438 0.0639 0.0845 0.1076
Na Mg Al Si P S Cl
0.0076 0.0146 0.0318 0.0331 0.0348 0.0405 0.0483
Ga Ge As Se Br
0.0205 0.0222 0.0287 0.0318 0.0376
In Sn Sb Te I
0.0223 0.0231 0.0259 0.0306 0.0368
Tl Pb Bi Po At
0.0170 0.0171 0.0215 0.0230 0.0294
Additional information about diffuse functions and also
Rydberg type exponents can be found in reference 30.
The following atomic energies are from UHF
calculations (RHF on 1-S states), with p orbitals not
symmetry equivalenced, and using the default molecular
scale factors. They should be useful in picking a basis
of the desired energy accuracy, and estimating the correct
molecular total energies.
Atom state STO-2G STO-3G 3-21G 6-31G
H 2-S -.454397 -.466582 -.496199 -.498233
He 1-S -2.702157 -2.807784 -2.835680 -2.855160
Li 2-S -7.070809 -7.315526 -7.381513 -7.431236
Be 1-S -13.890237 -14.351880 -14.486820 -14.566764
B 2-P -23.395284 -24.148989 -24.389762 -24.519492
C 3-P -36.060274 -37.198393 -37.481070 -37.677837
N 4-S -53.093007 -53.719010 -54.105390 -54.385008
O 3-P -71.572305 -73.804150 -74.393657 -74.780310
F 2-P -95.015084 -97.986505 -98.845009 -99.360860
Ne 1-S -122.360485 -126.132546 -127.803825 -128.473877
Na 2-S -155.170019 -159.797148 -160.854065 -161.841425
Mg 1-S -191.507082 -197.185978 -198.468103 -199.595219
Al 2-P -233.199965 -239.026471 -240.551046 -241.854186
Si 3-P -277.506857 -285.563052 -287.344431 -288.828598
P 4-S -327.564244 -336.944863 -339.000079 -340.689008
S 3-P -382.375012 -393.178951 -395.551336 -397.471414
Cl 2-P -442.206260 -454.546015 -457.276552 -459.442939
Ar 1-S -507.249273 -521.222881 -524.342962 -526.772151
SCF *
Atom state DH 6-311G MC limit
H 2-S -.498189 -.499810 -- -0.5
He 1-S -- -2.859895 -- -2.861680
Li 2-S -7.431736 -7.432026 -- -7.432727
Be 1-S -14.570907 -14.571874 -- -14.573023
B 2-P -24.526601 -24.527020 -- -24.529061
C 3-P -37.685571 -37.686024 -- -37.688619
N 4-S -54.397260 -54.397980 -- -54.400935
O 3-P -74.802707 -74.802496 -- -74.809400
F 2-P -99.395013 -99.394158 -- -99.409353
Ne 1-S -128.522354 -128.522553 -- -128.547104
Na 2-S -- -- -161.845587 -161.858917
Mg 1-S -- -- -199.606558 -199.614636
Al 2-P -241.855079 -- -241.870014 -241.876699
Si 3-P -288.829617 -- -288.847782 -288.854380
P 4-S -340.689043 -- -340.711346 -340.718798
S 3-P -397.468667 -- -397.498023 -397.504910
Cl 2-P -459.435938 -- -459.473412 -459.482088
Ar 1-S -- -- -526.806626 -526.817528
* M.W.Schmidt and K.Ruedenberg, J.Chem.Phys. 71,
3951-3962(1979). These are ROHF in Kh energies.
How to do RHF, ROHF, UHF, and GVB calculations
--- -- -- --- ---- --- --- --- ------------
* * * General considerations * * *
These four SCF wavefunctions are all based on Fock
operator techniques, even though some GVB runs use more
than one determinant. Thus all of these have an intrinsic
N**4 time dependence, because they are all driven by
integrals in the AO basis. This similarity makes it
convenient to discuss them all together. In this section
we will use the term HF to refer generically to any of
these four wavefunctions, including the multi-determinate
GVB-PP functions. $SCF is the main input group for all
these HF wavefunctions.
As will be discussed below, in GAMESS the term ROHF
refers to high spin open shell SCF only, but other open
shell coupling cases are possible using the GVB code.
Analytic gradients are implemented for every possible
HF type calculation possible in GAMESS, and therefore
numerical hessians are available for each.
Analytic hessian calculation is implemented for RHF,
ROHF, and any GVB case with NPAIR=0 or NPAIR=1. Analytic
hessians are more accurate, and much more quickly computed
than numerical hessians, but require additional disk
storage to perform an integral transformation, and also
more physical memory.
The second order Moller-Plesset energy correction
(MP2) is implemented for RHF, UHF, and ROHF, but not any
GVB case. Note that since calculation of the first order
wavefunction's density matrix is not implemented, the
properties during an MP2 run are for the underlying HF
(zeroth order) wavefunction.
Direct SCF is implemented for every possible HF type
calculation involving only the energy or gradient (and
thus numerical hessians). The direct method may not be
used with DEM convergence, or when forming MVOs. Direct
SCF may be used when analytic hessians or the MP2 energy
correction is being computed.
* * * direct SCF * * *
Normally, HF calculations proceed by evaluating a
large number of two electron repulsion integrals, and
storing these on a disk. This integral file is read back
in during each HF iteration to form the appropriate Fock
operators. In a direct HF, the integrals are not stored
on disk, but are instead reevaluated during each HF
iteration. Since the direct approach *always* requires
more CPU time, the default for DIRSCF in $SCF is .FALSE.
Even though direct SCF is slower, there are at least
three reasons why you may want to consider using it. The
first is that it may not be possible to store all of the
integrals on the disk drives attached to your computer.
Secondly, the index label packing scheme used by GAMESS
restricts the basis set size to no more than 361 if the
integrals are stored on disk, whereas for direct HF you
can (in principle) use up to 2047 basis functions.
Finally, what you are really interested in is reducing the
wall clock time to obtain your answer, not the CPU time.
Workstations have modest hardware (and sometimes software)
I/O capabilities. Other environments such as an IBM
mainframe shared by many users may also have very poor
CPU/wall clock performance for I/O bound jobs such as
conventional HF.
You can estimate the disk storage requirements for
conventional HF using a P or PK file by the following
formulae:
nint = 1/sigma * 1/8 * N**4
Mbytes = nint * x / 1024**2
Here N is the total number of basis functions in your
run, which you can learn from an EXETYP=CHECK run. The
1/8 accounts for permutational symmetry within the
integrals. Sigma accounts for the point group symmetry,
and is difficult to estimate accurately. Sigma cannot be
smaller than 1, in no symmetry (C1) calculations. For
benzene, sigma would be almost six, since you generate 6
C's and 6 H's by entering only 1 of each in $DATA. For
water sigma is not much larger than one, since most of the
basis set is on the unique oxygen, and the C2v symmetry
applies only to the H atoms. The factor x is 12 bytes per
integral for RHF, and 20 bytes per integral for ROHF, UHF,
and GVB. Finally, since integrals very close to zero need
not be stored on disk, the actual power dependence is not
as bad as N**4, and in fact in the limit of very large
molecules can be as low as N**2. Thus plugging in sigma=1
should give you an upper bound to the actual disk space
needed. If the estimate exceeds your available disk
storage, your only recourse is direct HF.
What are the economics of direct HF? Naively, if we
assume the run takes 10 iterations to converge, we must
spend 10 times more CPU time doing the integrals on each
iteration. However, we do not have to waste any CPU time
reading blocks of integrals from disk, or in unpacking
their indices. We also do not have to waste any wall
clock time waiting for a relatively slow mechanical device
such as a disk to give us our data.
There are some less obvious savings too, as first
noted by Almlof. First, since the density matrix is known
while we are computing integrals, we can use the Schwarz
inequality to avoid doing some of the integrals. In a
conventional SCF this inequality is used to avoid doing
small integrals. In a direct SCF it can be used to avoid
doing integrals whose contribution to the Fock matrix is
small (density times integral=small). Secondly, we can
form the Fock matrix by calculating only its change since
the previous iteration. The contributions to the change
in the Fock matrix are equal to the change in the density
times the integrals. Since the change in the density goes
to zero as the run converges, we can use the Schwarz
screening to avoid more and more integrals as the
calculation progresses. The input option FDIFF in $SCF
selects formation of the Fock operator by computing only
its change from iteration to iteration. The FDIFF option
is not implemented for GVB since there are too many density
matrices from the previous iteration to store, but is the
default for direct RHF, ROHF, and UHF.
So, in our hypothetical 10 iteration case, we do not
spend as much as 10 times more time in integral
evaluation. Additionally, the run as a whole will not
slow down by whatever factor the integral time is
increased. A direct run spends no additional time summing
integrals into the Fock operators, and no additional time
in the Fock diagonalizations. So, generally speaking, a
RHF run with 10-15 iterations will slow down by a factor
of 2-4 times when run in direct mode. The energy gradient
time is unchanged by direct HF, and this is a large time
compared to HF energy, so geometry optimizations will be
slowed down even less. This is really the converse of
Amdahl's law: if you slow down only one portion of a
program by a large amount, the entire program slows down
by a much smaller factor.
To make this concrete, here are some times for GAMESS
benchmark BENCH12, which is a RHF energy for a moderately
large molecule. The timings shown were obtained on a
DECstation 3100 under Ultrix 3.1, which was running only
these tests, so that the wall clock times are meaningful.
This system is typical of Unix workstations in that it
uses SCSI disks, and the operating system is not terribly
good at disk I/O. By default GAMESS stores the integrals
on disk in the form of a P supermatrix, because this will
save time later in the SCF cycles. By choosing NOPK=1 in
$INTGRL, an ordinary integral file can be used, which
typically contains many fewer integrals, but takes more
CPU time in the SCF. Because the DECstation is not
terribly good at I/O, the wall clock time for the ordinary
integral file is actually less than when the supermatrix
is used, even though the CPU time is longer. The run
takes 13 iterations to converge, the times are in seconds.
P supermatrix ordinary file
# nonzero integrals 8,244,129 6,125,653
# blocks skipped 55,841 55,841
CPU time (ints) 709 636
CPU time (SCF) 1289 1472
CPU time (job total) 2123 2233
wall time (job total) 3468 3200
When the same calculation is run in direct mode
(integrals are processed like an ordinary integral disk
file when running direct),
iteration 1: FDIFF=.TRUE. FDIFF=.FALSE.
# nonzero integrals 6,117,416 6,117,416
# blocks skipped 60,208 60,208
iteration 13:
# nonzero integrals 3,709,733 6,122,912
# blocks skipped 105,278 59,415
CPU time (job total) 6719 7851
wall time (job total) 6764 7886
Note that elimination of the disk I/O dramatically
increases the CPU/wall efficiency. Here's the bottom line
on direct HF:
best direct CPU / best disk CPU = 6719/2123 = 3.2
best direct wall/ best disk wall= 6764/3200 = 2.1
Direct SCF is slower than conventional SCF, but not
outrageously so! From the data in the tables, we can see
that the best direct method spends about 6719-1472 = 5247
seconds doing integrals. This is an increase of about
5247/636 = 8.2 in the time spent doing integrals, in a run
which does 13 iterations (13 times evaluating integrals).
8.2 is less than 13 because the run avoids all CPU charges
related to I/O, and makes efficient use of the Schwarz
inequality to avoid doing many of the integrals in its
final iterations.
* * * SCF convergence accelerators * * *
Generally speaking, the simpler the HF function, the
better its convergence. In our experience, the majority
of RHF, ROHF, and UHF runs will converge readily from
GUESS=HUCKEL. GVB runs typically require GUESS=MOREAD,
although the Huckel guess occasionally works. RHF
convergence is the best, closely followed by ROHF. In the
current implementation in GAMESS, ROHF is somewhat better
convergent than the closely related unrestricted high spin
UHF. For example, the radical cation of diphosphine (test
job BENCH10) converges in 12 iterations for ROHF and 15
iterations for UHF, starting from the neutral's closed
shell orbitals. GVB calculations require much more care,
and cases with NPAIR greater than one are particularly
difficult.
Unfortunately, not all HF runs converge readily. The
best way to improve your convergence is to provide better
starting orbitals! In many cases, this means to MOREAD
orbitals from some simpler HF case. For example, if you
want to do a doublet ROHF, and the HUCKEL guess does not
seem to converge, do this: Do an RHF on the +1 cation.
RHF is typically more stable than ROHF, UHF, or GVB, and
cations are usually readily convergent. Then MOREAD the
cation's orbitals into the neutral calculation which you
wanted to do at first.
GUESS=HUCKEL does not always guess the correct
electronic configuration. It may be useful to use PRTMO
in $GUESS during a CHECK run to examine the starting
orbitals, and then reorder them with NORDER if that seems
appropriate.
Of course, by default GAMESS uses the convergence
procedures which are usually most effective. Still, there
are cases which are difficult, so the $SCF group permits
you to select several alternative methods for improving
convergence. Briefly, these are
EXTRAP. This extrapolates the three previous Fock
matrices, in an attempt to jump ahead a bit faster. This
is the most powerful of the old-fashioned accelerators,
and normally should be used at the beginning of any SCF
run. When an extrapolation occurs, the counter at the
left of the SCF printout is set to zero.
DAMP. This damps the oscillations between several
successive Fock matrices. It may help when the energy is
seen to oscillate wildly. Thinking about which orbitals
should be occupied initially may be an even better way to
avoid oscillatory behaviour.
SHIFT. This shifts the diagonal elements of the virtual
part of the Fock matrix up, in an attempt to uncouple the
unoccupied orbitals from the occupied ones. At convergence,
this has no effect on the orbitals, just their orbital
energies, but will produce different (and hopefully better)
orbitals during the iterations.
RSTRCT. This limits mixing of the occupied orbitals
with the empty ones, especially the flipping of the HOMO
and LUMO to produce undesired electronic configurations or
states. This should be used with caution, as it makes it
very easy to converge on incorrect electronic configurations,
especially if DIIS is also used. If you use this, be sure
to check your final orbital energies to see if they are
sensible. A lower energy for an unoccupied orbital than
for one of the occupied ones is a sure sign of problems.
DIIS. Direct Inversion in the Iterative Subspace is
a modern method, due to Pulay, using stored error and Fock
matrices from a large number of previous iterations to
interpolate an improved Fock matrix. This method was
developed to improve the convergence at the final stages
of the SCF process, but turns out to be quite powerful at
forcing convergence in the initial stages of SCF as well.
By giving ETHRSH as 10.0 in $SCF, you can practically
guarantee that DIIS will be in effect from the first
iteration. The default is set up to do a few iterations
with conventional methods (extrapolation) before engaging
DIIS. This is because DIIS can sometimes converge to
solutions of the SCF equations that do not have the lowest
possible energy. For example, the 3-A-2 small angle state
of SiLi2 (see M.S.Gordon and M.W.Schmidt, Chem.Phys.Lett.,
132, 294-8(1986)) will readily converge with DIIS to a
solution with a reasonable S**2, and an energy about 25
milliHartree above the correct answer. A SURE SIGN OF
TROUBLE WITH DIIS IS WHEN THE ENERGY RISES TO ITS FINAL
VALUE. However, if you obtain orbitals at one point on a
PES without DIIS, the subsequent use of DIIS with MOREAD
will probably not introduce any problems. Because DIIS
is quite powerful, EXTRAP, DAMP, and SHIFT are all turned
off once DIIS begins to work. DEM and RSTRCT will still
be in use, however.
DEM. Direct energy minimization should be your last
recourse. It explores the "line" between the current
orbitals and those generated by a conventional change in
the orbitals, looking for the minimum energy on that line.
DEM should always lower the energy on every iteration,
but is very time consuming, since each of the points
considered on the line search requires evaluation of a
Fock operator. DEM will be skipped once the density
change falls below DEMCUT, as the other methods should
then be able to affect final convergence. While DEM is
working, RSTRCT is held to be true, regardless of the
input choice for RSTRCT. Because of this, it behooves
you to be sure that the initial guess is occupying the
desired orbitals. DEM is available only for RHF. The
implementation in GAMESS resembles that of R.Seeger and
J.A.Pople, J.Chem.Phys. 65, 265-271(1976). Simultaneous
use of DEM and DIIS resembles the ADEM-DIOS method of
H.Sellers, Chem.Phys.Lett. 180, 461-465(1991).
* * * High spin open shell SCF (ROHF) * * *
Open shell SCF calculations are performed in GAMESS by
both the ROHF code and the GVB code. Note that when the
GVB code is executed with no pairs, the run is NOT a true
GVB run, and should be referred to in publications and
discussion as a ROHF calculation.
The ROHF module in GAMESS can handle any number of
open shell electrons, provided these have a high spin
coupling. Some commonly occurring cases are:
one open shell, doublet:
$CONTRL SCFTYP=ROHF MULT=2 $END
two open shells, triplet:
$CONTRL SCFTYP=ROHF MULT=3 $END
m open shells, high spin:
$CONTRL SCFTYP=ROHF MULT=m+1 $END
John Montgomery of United Technologies is responsible
for the current ROHF implementation in GAMESS. The
following discussion is due to him:
The Fock matrix in the MO basis has the form
closed open virtual
closed F2 | Fb | (Fa+Fb)/2
-----------------------------------
open Fb | F1 | Fa
-----------------------------------
virtual (Fa+Fb)/2 | Fa | F0
where Fa and Fb are the usual alpha and beta Fock
matrices any UHF program produces. The Fock operators
for the doubly, singly, and zero occupied blocks can be
written as
F2 = Acc*Fa + Bcc*Fb
F1 = Aoo*Fa + Boo*Fb
F0 = Avv*Fa + Bvv*Fb
Some choices found in the literature for these
canonicalization coefficients are
Acc Bcc Aoo Boo Avv Bvv
Guest and Saunders 1/2 1/2 1/2 1/2 1/2 1/2
Roothaan single matrix -1/2 3/2 1/2 1/2 3/2 -1/2
Davidson 1/2 1/2 1 0 1 0
Binkley, Pople, Dobosh 1/2 1/2 1 0 0 1
McWeeny and Diercksen 1/3 2/3 1/3 1/3 2/3 1/3
Faegri and Manne 1/2 1/2 1 0 1/2 1/2
The choice of the diagonal blocks is arbitrary, as
ROHF is converged when the off diagonal blocks go to zero.
The exact choice for these blocks can however have an
effect on the convergence rate. This choice also affects
the MO coefficients, and orbital energies, as the
different choices produce different canonical orbitals
within the three subspaces. All methods, however, will
give identical total wavefunctions, and hence identical
properties such as gradients and hessians.
The default coupling case in GAMESS is the Roothaan
single matrix set. Note that pre-1988 versions of GAMESS
produced "Davidson" orbitals. If you would like to fool
around with any of these other canonicalizations, the Acc,
Aoo, Avv and Bcc, Boo, Bvv parameters can be input as the
first three elements of ALPHA and BETA in $SCF.
* * * Other open shell SCF cases (GVB) * * *
Genuine GVB-PP runs will be discussed later in this
section. First, we will consider how to do open shell SCF
with the GVB part of the program.
It is possible to do other open shell cases with the
GVB code, which can handle the following cases:
one open shell, doublet:
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO(1)=1 $END
two open shells, triplet:
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=2 NO(1)=1,1 $END
two open shells, singlet:
$CONTRL SCFTYP=GVB MULT=1 $END
$SCF NCO=xx NSETO=2 NO(1)=1,1 $END
Note that the first two cases duplicate runs which the
ROHF module can do better. Note that all of these cases
are really ROHF, since the default for NPAIR in $SCF is 0.
Many open shell states with degenerate open shells
(for example, in diatomic molecules) can be treated as
well. There is a sample of this in the 'Input Examples'
section of this manual.
If you would like to do any cases other than those
shown above, you must derive the coupling coefficients
ALPHA and BETA, and input them with the occupancies F in
the $SCF group.
Mariusz Klobukowski of the University of Alberta has
shown how to obtain coupling coefficients for the GVB open
shell program for many such open shell states. These can
be derived from the values in Appendix A of the book "A
General SCF Theory" by Ramon Carbo and Josep M. Riera,
Springer-Verlag (1978). The basic rule is
(1) F(i) = 1/2 * omega(i)
(2) ALPHA(i) = alpha(i)
(3) BETA(i) = - beta(i),
where omega, alpha, and beta are the names used by Ramon
in his Tables.
The variable NSETO should give the number of open
shells, and NO should give the degeneracy of each open
shell. Thus the 5-S state of carbon would have NSETO=2,
and NO(1)=1,3.
Some specific examples, for the lowest term in each
of the atomic P**N configurations are
! p**1 2-P state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.16666666666667
ALPHA(1)= 2.0 0.33333333333333 0.00000000000000
BETA(1)= -1.0 -0.16666666666667 -0.00000000000000 $END
! p**2 3-P state
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.333333333333333
ALPHA(1)= 2.0 0.66666666666667 0.16666666666667
BETA(1)= -1.0 -0.33333333333333 -0.16666666666667 $END
! p**3 4-S state
$CONTRL SCFTYP=ROHF MULT=4 $END
! p**4 3-P state
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.66666666666667
ALPHA(1)= 2.0 1.33333333333333 0.83333333333333
BETA(1)= -1.0 -0.66666666666667 -0.50000000000000 $END
! p**5 2-P state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=3 COUPLE=.TRUE.
F(1)= 1.0 0.83333333333333
ALPHA(1)= 2.0 1.66666666666667 1.33333333333333
BETA(1)= -1.0 -0.83333333333333 -0.66666666666667 $END
Coupling constants for d**N configurations are
! d**1 2-D state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.1
ALPHA(1)= 2.0, 0.20, 0.00
BETA(1)=-1.0,-0.10, 0.00 $END
! d**2 average of 3-F and 3-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.2
ALPHA(1)= 2.0, 0.40, 0.05
BETA(1)=-1.0,-0.20,-0.05 $END
! d**3 average of 4-F and 4-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.3
ALPHA(1)= 2.0, 0.60, 0.15
BETA(1)=-1.0,-0.30,-0.15 $END
! d**4 5-D state
$CONTRL SCFTYP=GVB MULT=5 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.4
ALPHA(1)= 2.0, 0.80, 0.30
BETA(1)=-1.0,-0.40,-0.30 $END
! d**5 6-S state
$CONTRL SCFTYP=ROHF MULT=6 $END
! d**6 5-D state
$CONTRL SCFTYP=GVB MULT=5 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.6
ALPHA(1)= 2.0, 1.20, 0.70
BETA(1)=-1.0,-0.60,-0.50 $END
! d**7 average of 4-F and 4-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.7
ALPHA(1)= 2.0, 1.40, 0.95
BETA(1)=-1.0,-0.70,-0.55 $END
! d**8 average of 3-F and 3-P states
$CONTRL SCFTYP=GVB MULT=3 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.8
ALPHA(1)= 2.0, 1.60, 1.25
beta(1)=-1.0,-0.80,-0.65 $end
! d**9 2-D state
$CONTRL SCFTYP=GVB MULT=2 $END
$SCF NCO=xx NSETO=1 NO=5 COUPLE=.TRUE. F(1)=1.0,0.9
ALPHA(1)= 2.0, 1.80, 1.60
BETA(1)=-1.0,-0.90,-0.80 $END
The source for these values is R.Poirier, R.Kari, and
I.G.Csizmadia's book "Handbook of Gaussian Basis Sets",
Elsevier, Amsterdam, 1985.
Note that GAMESS can do a proper calculation on the ground
terms for the d**2, d**3, d**7, and d**8 configurations
only by means of state averaged MCSCF. For d**8, use
$CONTRL SCFTYP=MCSCF MULT=3 $END
$DRT GROUP=C1 FORS=.TRUE. NMCC=xx NDOC=3 NALP=2 $END
$GUGDIA NSTATE=10 $END
$GUGDM2 WSTATE(1)=1,1,1,1,1,1,1,0,0,0 $END
* * * True GVB perfect pairing runs * * *
True GVB runs are obtained by choosing NPAIR nonzero.
If you wish to have some open shell electrons in addition
to the geminal pairs, you may add the pairs to the end of
any of the GVB coupling cases shown above. The GVB module
assumes that you have reordered your MOs into the order:
NCO double occupied orbitals, NSETO sets of open shell
orbitals, and NPAIR sets of geminals (with NORDER=1 in the
$GUESS group).
Each geminal consists of two orbitals and contains two
singlet coupled electrons (perfect pairing). The first MO
of a geminal is probably heavily occupied (such as a
bonding MO u), and the second is probably weakly occupied
(such as an antibonding, correlating orbital v). If you
have more than one pair, you must be careful that the
initial MOs are ordered u1, v1, u2, v2..., which is -NOT-
the same order that RHF starting orbitals will be found
in. Use NORDER=1 to get the correct order.
These pair wavefunctions are actually a limited form
of MCSCF. GVB runs are much faster than MCSCF runs,
because the natural orbital u,v form of the wavefunction
permits a Fock operator based optimization. However,
convergence of the GVB run is by no means assured. The
same care in selecting the correlating orbitals that you
would apply to an MCSCF run must also be used for GVB
runs. In particular, look at the orbital expansions when
choosing the starting orbitals, and check them again after
the run converges.
GVB runs will be carried out entirely in orthonormal
natural u,v form, with strong orthogonality enforced on
the geminals. Orthogonal orbitals will pervade your
thinking in both initial orbital selection, and the entire
orbital optimization phase (the CICOEF values give the
weights of the u,v orbitals in each geminal). However,
once the calculation is converged, the program will
generate and print the nonorthogonal, generalized valence
bond orbitals. These GVB orbitals are an entirely
equivalent way of presenting the wavefunction, but are
generated only after the fact.
Convergence of GVB runs is by no means as certain as
convergence of RHF, UHF, or ROHF. This is especially true
when extended bases are used. You can assist convergence
by doing a preliminary RHF or UHF calculation (take either
the alpha orbitals, or NOs of the latter), and use these
orbitals for GUESS=MOREAD. Generation of MVOs during the
prelimnary SCF can also be advantageous.
The total number of electrons in the GVB wavefunction
is given by the following formula:
NE = 2*NCO + sum 2*F(i)*NO(i) + 2*NPAIR
i
The charge is obtained by subtracting the total number of
protons given in $DATA. The multiplicity is implicit in
the choice of alpha and beta constants. Note that ICHARG
and MULT must be given correctly in $CONTRL anyway.
* * * the special case of TCSCF * * *
The wavefunction with NSETO=0 and NPAIR=1 is called
GVB-PP(1) by Goddard, two configuration SCF (TCSCF) by
Schaefer or Davidson, and CAS-SCF with two electrons in
two orbitals by others. Note that this is just semantics,
as all of these are the identical wavefunction. It is
also a very important wavefunction, as TCSCF is the
minimum treatment required to describe singlet biradicals.
This function can be obtained with SCFTYP=MCSCF, but it is
usually much faster to use the Fock based SCFTYP=GVB. Due
to its importance, the TCSCF function has been singled
out for special attention. The DIIS method is implemented
for the case NSETO=0 and NPAIR=1. Analytic hessians can
be obtained when NPAIR=1, and in this case NSETO may be
nonzero.
* * * A caution about symmetry in GVB * * *
Caution! Some exotic calculations with the GVB
program do not permit the use of symmetry. The symmetry
algorithm in GAMESS was "derived assuming that the
electronic charge density transforms according to the
completely symmetric representation of the point group",
Dupuis/King, JCP, 68, 3998(1978). This may not be true
for certain open shell cases. Consider the following
correct input for the singlet-delta state of NH:
$CONTRL SCFTYP=GVB NOSYM=1 $END
$SCF NCO=3 NSETO=2 NO(1)=1,1 $END
for the x**1y**1 state, or for the x**2-y**2 state,
$CONTRL SCFTYP=GVB NOSYM=1 $END
$SCF NCO=3 NPAIR=1 $END
Neither example will work, unless you enter NOSYM=1.
Note that if you are using coupling constants which
have been averaged over all degenerate states which
comprise a single term, as is done in EXAM15 and EXAM16,
the charge density is symmetric, and these runs can
exploit symmetry.
Since GAMESS cannot at present detect that the GVB
electronic state is not totally symmetric (or averaged
to at least have a totally symmetric density), it is
---your--- responsibility to recognize that NOSYM=1
must be chosen in the input.
How to do CI and MCSCF calculations
--- -- -- -- --- ----- ------------
The majority of the input to a CI calculation is in
the $DRT group, with $INTGRL, $CIINP, $TRANS, $CISORT,
$GUGEM, $GUGDIA, $GUGDM, $GUGDM2, $LAGRAN groups relevant,
but hardly ever given. The defaults for all these latter
groups are usually correct. Perhaps the most interesting
variable outside the $DRT group is NSTATE in $GUGDIA to
include excited states in the CI computation.
The MCSCF part of GAMESS is based on the CI package,
so that most of each iteration is just a CI using the
current set of orbitals. The final step in each MCSCF
iteration is the generation of improved MOs by means of a
Newton-Raphson optimization. Thus, all of the input just
mentioned for a CI run except $GUGDM is still relevant,
with the $MCSCF group being of primary concern to users.
The $CIINP group is largely irrelevant, while the $TRFDM2
group is possibly relevant to MCSCF gradient calculations.
Once again, the defaults in any of the groups other than
$DRT and $MCSCF are probably appropriate for your run.
The most interesting variable outside $DRT and $MCSCF is
probably WSTATE in $GUGDM2.
It is very helpful to execute a EXETYP=CHECK run
before doing any MCSCF or CI run. The CHECK run will tell
you the total number of CSFs, the electronic state, and
the charge and multiplicity your $DRT generated. The
CHECK run also lets the program feel out the memory that
will be required to actually do the run. Thus the CHECK
run can potentially prevent costly mistakes, or tell you
when a calculation is prohibitively large.
--- Specification of the wavefunction ---
The configuration state functions (CSFs) to be used
in a CI or MCSCF calculation are specified in the $DRT by
giving a single reference CSF, and the maximum degree of
electron excitation from that CSF.
The MOs are filled in the order FZC or MCC first,
followed by DOC, AOS, BOS, ALP, VAL, EXT, FZV (Aufbau
principle). AOS, BOS, and ALP are singly occupied MOs.
The open shell singlet couplings NAOS and NBOS must give
the same value, and their MOs alternate. An example is
the input NFZC=1 NDOC=2 NAOS=2 NBOS=2 NALP=1 NVAL=3 which
gives the reference CSF
FZC,DOC,DOC,AOS,BOS,AOS,BOS,ALP,VAL,VAL,VAL
This is a doublet state with five unpaired electrons. VAL
orbitals are unoccupied only in the reference CSF, they
will become occupied as the other CSFs are generated.
To perform a singles and doubles CI calculation with
any such reference, select the electron excitation level
IEXCIT=2. For a normal CI, you would give all of the
virtual MOs as VALs.
Another example, for a MCSCF run, might be NMCC=3
NDOC=3 NVAL=2, which gives the reference
MCC,MCC,MCC,DOC,DOC,DOC,VAL,VAL
MCSCF calculations are usually of the Full Optimized
Reaction Space (FORS) type. Choosing FORS=.TRUE. leads
to an excitation level of 4, as the 6 valence electrons
have only 4 holes available for excitation. MCSCF runs
typically have only a small number of VAL orbitals. If
you choose an excitation level IEXCIT that is smaller than
that needed to generate the FORS space, be sure to set
FORS=.FALSE. in $MCSCF or else VERY POOR or even NO
CONVERGENCE will result.
The next example is a first or second order CI
wavefunction, NFZC=3 NDOC=3 NVAL=2 NEXT=-1 leads to
FZC,FZC,FZC,DOC,DOC,DOC,VAL,VAL,EXT,EXT,...
FOCI or SOCI is chosen by selecting the appropriate flag,
the correct excitation level is automatically generated.
Note that the negative number for NEXT causes all
remaining MOs to be included in the external orbital
space. One way of viewing FOCI and SOCI wavefunctions is
as all singles, or all singles and doubles from the entire
MCSCF wavefunction as a reference. An equivalent way of
saying this is that all CSFs with N electrons (in this
case N=6) distributed in the valence orbitals in all ways
(that is the FORS MCSCF wavefunction) to make the base
wavefunction. To this, FOCI adds all CSFs with N-1
electrons in active and 1 electron in external orbitals.
SOCI adds all possible CSFs with N-2 electrons in active
orbitals and 2 in external orbitals. SOCI is often
prohibitively large, but is also a very accurate
wavefunction.
--- use of symmetry ---
The CI part of the program can work with D2h, and any
of its Abelian subgroups. However, $DRT allows the input
of some (but not all) higher point groups. For these
non-Abelian groups, the program automatically assigns the
orbitals to an irrep in the highest possible Abelian
subgroup. For the other non-Abelian groups, you must at
present select GROUP=C1. Note that when you are computing
a Hessian matrix, many of the displaced geometries are
asymmetric, hence you must choose C1 in $DRT (however, be
sure to use the highest symmetry possible in $DATA!).
As an example, consider a molecule with Cs symmetry,
and these two reference CSFs
...MCC...DOC DOC VAL VAL
...MCC...DOC AOS BOS VAL
Suppose that the 2nd and 3rd active MOs have symmetries a'
and a". Both of these generate singlet wavefunctions,
with 4 electrons in 4 active orbitals, but the former
constructs 1-A' CSFs, while the latter generates 1-A"
CSFs. On the other hand, if the 2nd and 3rd orbitals had
the same symmetry type, the references are equivalent.
--- selection of orbitals ---
The first step is to partition the orbital space into
core, active, and external sets, in a manner which is
sensible for your chemical problem. This is a bit of an
art, and the user is referred to the references quoted at
the end of this section. Having decided what MCSCF to
perform, you now must consider the more pedantic problem
of what orbitals to begin the MCSCF calculation with.
You should always start an MCSCF run with orbitals
from some other run, by means of GUESS=MOREAD. Do not
expect to be able to use HCORE, MINGUESS, or EXTGUESS!
Example 6 is a poor example, converging only because
methylene has so much symmetry, and the basis is so small.
If you are just beginning your MCSCF problem, use orbitals
from some appropriate converged SCF run. Thus, a
realistic example of beginning a MCSCF calculation is
examples 8 and 9. Once you get an MCSCF to converge, you
can and should use these MCSCF MOs (which will be Schmidt
orthogonalized) at other nearby geometries.
Starting from SCF orbitals can take a little bit of
care. Most of the time (but not always) the orbitals you
want to correlate will be the highest occupied orbitals in
the SCF. Fairly often, however, the correlating orbitals
you wish to use will not be the lowest unoccupied virtuals
of the SCF. You will soon become familiar with NORDER=1
in $GUESS, as well as the nearly mandatory GUESS=MOREAD.
You should also explore SYMLOC in $LOCAL, as the canonical
SCF MOs are often spread out over regions of the molecule
other than "where the action is".
Convergence of MCSCF is by no means guaranteed. Poor
convergence can invariably be traced back to a poor
initial selection of orbitals. My advice is, before you
even start: "Look at the orbitals. Then look at the
orbitals again". Later, if you have any trouble: "Look
at the orbitals some more".
Even if you don't have any trouble, look at the
orbitals to see if they converged to what you expected,
and have reasonable occupation numbers. MCSCF is by no
means the sort of "black box" that RHF is these days, so
LOOK VERY CAREFULLY AT YOUR RESULTS. Since orbitals are
very hard to look at on the screen, this means print them
out on paper, and inspect the individual MO expansion
coefficients! Better yet, plot them and look at their
nodal characteristics.
--- the iterative process ---
Each MCSCF iteration consists of the following steps:
transformation of the integrals to the current MO basis,
sorting of these integrals, generation of the Hamiltonian
matrix, optimization of the CI coefficients by a Davidson
method diagonalization, generation of the second order
density matrix, and finally orbital improvment by the
Newton-Raphson method. On the first iteration at the
first geometry, you will receive the normal amount of
printout from each of these stages, while subsequent
iterations will produce only a single line summarizing the
iteration.
Each iteration contains MICIT Newton-Raphson orbital
improvements. After the first such NR step, a micro-
iteration consists solely of an integral transformation
over the new MOs, followed immediately by a NR orbital
improvment, reusing the old second order density matrix.
This avoids the CI diagonalization step, so may be of some
use in MCSCF calculations with large CSF expansions. If
the number of CSFs is small, using additional micro-
iterations is a stupid idea.
--- miscellaneous hints ---
Old versions of the program required that you give
correct input data in groups like $TRANS. Nowadays this
is not required, usually only $DRT and $MCSCF are given.
Old versions of the program forced you to assign each
orbital a symmetry code number, this is now done
automatically.
A very common MCSCF wavefunction, the simplest
possible function describing a singlet diradical, has 2
electrons in 2 active MOs. While this wavefunction can be
obtained in a MCSCF run (using NDOC=1 NVAL=1), it can be
obtained much faster by use of the GVB code, with one GVB
pair. This GVB-PP(1) wavefunction is also known in the
literature as two configuration SCF (TCSCF). The two
configuration form of this GVB is equivalent to the three
configurations obtained in this MCSCF, as optimization in
natural form (20 and 02) causes the coefficient of the 11
CSF to vanish.
One way to improve upon the SCF orbitals as starting
MOs is to generate modified virtual orbitals (MVOs).
MVOs are obtained by diagonalizing the Fock operator of a
very positive ion, within the virtual orbital space only.
As implemented in GAMESS, MVOs can be obtained at the end
of any RHF, ROHF, or GVB run by setting MVOQ in $SCF
nonzero, at the cost of a single SCF cycle. (Generating
MVOs in a run feeding in converged orbitals is more
expensive, as the 2 electron integrals must be
recomputed). Generating MVOs will never change any of
the occupied orbitals, but will generate more valence like
LUMOs.
It is easy to generate large numbers of CSFs with the
$DRT input (especially SOCI), which produces a bottleneck
in the CI diagonalization step. If the number of CSFs is
small (say under 10,000), then the most time consuming
step in a CI run is the integral transformation, and in a
MCSCF run is both the integral transformation and the
Newton-Raphson orbital improvement. However, larger
numbers of CSFs will require a great deal of time in the
diagonalization step, as well as external disk storage for
the pieces of the Hamiltonian matrix. The largest CI
calculation performed at NDSU had about 85,000 CSFs,
although we have heard of a CISD with over 200,000 CSFs
that required 8 IBM 3380 disks to hold the Hamiltonian
matrix. The largest MCSCF performed at NDSU had about
20,000 CSFs, although we have heard of one with about
35,000 CSFs. MCSCF typically is smaller, simply because
this kind of run must perform a CI diagonalization once
every iteration.
--- MCSCF implementation ---
The MCSCF code in GAMESS is adapted from the HONDO7
program, and was written by Michel Dupuis of IBM. The
implementation is of the type termed "unfolded two-step
Newton-Raphson" by Roos. This means that the orbital and
CI coefficient optimizations are separated. The latter
are obtained in a conventional CI step, and the former by
means of a NR orbital improvment (using the augmented
hessian matrix). This means convergence is not quite
second order, but that only the orbital-orbital hessian
terms must be computed.
There are three different methods for obtaining the
two electron contributions to the Lagrangian matrix and
the orbital hessian in GAMESS (collectively called the NR
matrices). These methods compute exactly the same NR
matrices, so the overall number of iterations is the same
for all methods.
The computational problem in forming the NR matrices
is the combination of the second order density matrix, and
the transformed two electron integrals. Both of these are
4 index matrices, but the DM2 is much smaller since its
indices run only over occupied orbitals. The TEIs are
much more numerous, as two of the indices run over all
orbitals, including unoccupied ones.
METHOD=TEI is driven by the two electron integrals.
After reading all of the DM2 into memory, this method
begins to read the TEI file, and combines these integrals
with the available DM2 elements. This method uses the
least memory, reads the DM2 and TEI files only once each,
and is usually very much slower than the other two
methods.
METHOD=DM2 is the reverse strategy, with all the
integrals in core, and is driven by reading DM2 elements
from disk. Because the number of TEIs is large, the
implementation actually divides the TEIs into four sets,
(ij/kl), (aj/kl), (ab/kl), and (aj/bl), where i,j,k,l run
over the internal orbitals, and a and b are external
orbitals. The program must have enough memory to hold all
the TEI's belonging to a particular set in memory. Thus,
construction of the NR matrices requires 4 reads of the
TEI file, and 4 reads of the DM2. METHOD=DM2 requires
more memory than METHOD=TEI, and has 4 times the I/O, but
is much faster.
METHOD=FORMULA tries to combine the merits of both
methods. This method constructs a Newton formula tape on
the first MCSCF iteration at the first geometry computed.
This formula tape contains integer pointers telling how
TEIs should be combined with DM2 elements to form the NR
matrices. If all of the TEIs will not fit into memory,
the formula tape must be sorted after it is generated.
The formula tape can be saved for use in later runs on the
same molecule, see FMLFIL in $MCSCF. This can be a very
large disk file, which may diminish your enthusiasm for
saving it.
Once the formula tape is constructed, the "passes"
(say M in number) begin. A pass consists of reading as
large a block of the TEI file as will fit into the
available memory, then reading the related portion of the
formula tape and the entire DM2 file for information about
how to efficiently combine the TEI's and the DM2. When
all of this TEI block is processed, the next block of
TEI's is read to begin the next "pass". An iteration thus
performs one complete read through the TEI and formula
files, and M reads of the DM2. METHOD=FORMULA uses as
much memory as the program has available (above a certain
minimum), by adjusting the chunk size used for the TEI
file), has an I/O requirement somewhat larger than
METHOD=TEI, and is perhaps a bit faster than METHOD=DM2.
It does require additional disk space for the Newton
formula tape.
Unlike METHOD=DM2 and METHOD=TEI, METHOD=FORMULA
requires a canonical order file of transformed integrals.
Thus, this method does not work with the current integral
transformation, which produces only a reverse canonical
order file.
As all three methods have advantages and
disadvantages, all three are available. The amount of
memory required for each method will be included in your
output (including EXETYP=CHECK jobs) to assist you in
selecting a method.
--- references ---
There are several review articles about second order
MCSCF methodologies. Of these, Roos' is a nice overview
of the subject, while the other 3 are more detailed.
1. "The Multiconfiguration (MC) SCF Method"
B.O.Roos, in "Methods in Computational Molecular
Physics", edited by G.H.F.Diercksen and S.Wilson
D.Reidel Publishing, Dordrecht, Netherlands, 1983,
pp 161-187.
2. "Optimization and Characterization of a MCSCF State"
Olsen, Yeager, Jorgensen in "Advances in Chemical
Physics", vol.54, edited by I.Prigogine and S.A.Rice,
Wiley Interscience, New York, 1983, pp 1-176.
3. "Matrix Formulated Direct MCSCF and Multiconfiguration
Reference CI Methods"
H.-J.Werner, in "Advances in Chemical Physics", vol.69,
edited by K.P.Lawley, Wiley Interscience, New York,
1987, pp 1-62.
4. "The MCSCF Method" R.Shepard, ibid., pp 63-200.
Some papers particularly germane to the implementation
in GAMESS are
5. "General second order MCSCF theory: A Density Matrix
Directed Algorithm"
B.H.Lengsfield, III, J.Chem.Phys. 73,382-390(1980).
6. "The use of the Augmented Matrix in MCSCF Theory"
D.R.Yarkony, Chem.Phys.Lett. 77,634-635(1981).
The next set of references discuss the choice of
orbitals for the active space. This is a matter of some
importance, and needs to be understood well by the GAMESS
user.
7. "The CASSCF Method and its Application in Electronic
Structure Calculations"
B.O.Roos, in "Advances in Chemical Physics", vol.69,
edited by K.P.Lawley, Wiley Interscience, New York,
1987, pp 339-445.
8. "Are Atoms Intrinsic to Molecular Electronic
Wavefunctions?"
K.Ruedenberg, M.W.Schmidt, M.M.Dombek, S.T.Elbert
Chem.Phys. 71, 41-49, 51-64, 65-78 (1982).
The final references are simply some examples of FORS
MCSCF applications, the latter done with GAMESS.
9. D.F.Feller, M.W.Schmidt, K.Ruedenberg,
J.Am.Chem.Soc. 104, 960-967(1982).
10. M.W.Schmidt, P.N.Truong, M.S.Gordon,
J.Am.Chem.Soc. 109, 5217-5227(1987).
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 two different implementations of
quasi-Newton procedures for finding stationary points,
namely METHOD=STANDARD and METHOD=SCHLEGEL. For minima
searches, the two procedures vary in the algorithm used
to update the hessian, and in how large steps are scaled.
For saddle point runs, the standard method treats the
unique direction of smallest (hopefully negative) curvature
in a more sophisticated way than the Schlegel algorithm.
For both kinds of searches, the standard method seems to
be more robust, boasting the ability (in some cases) to
locate transition states even when started in a region of
all positive local curvatures.
The "standard method" in GAMESS has largely been
developed by Frank Jensen of Odense University, by
combining several variants of the quasi-Newton method.
First, if the local curvatures of the approximate hessian
are correct for the given RUNTYP, a conventional Newton-
Raphson (NR) step is computed. If the NR step is longer
than the trust radius, or if the hessian curvatures are
incorrect, a Rational Function Optimization (RFO or P-RFO)
step will be computed. If the RFO step is also too long,
a restricted step is computed to ensure that the step
will not exceed the trust radius. There are two rather
different ways of thinking about this, as a min-max type
problem (the Quadratic Approximation (QA) step) or as a
minimization problem on the image surface (the Trust
Radius Image Minimization (TRIM) step). The working
equation is actually identical in these two methods, and
the two rather different derivations give us more than
one way to see why it works. The trust radius increases
if the ratio of the actual energy change between points
to the predicted change is near unity, and vice versa.
The STANDARD 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
which form the "standard method" for GAMESS' geometry
searches. A good summary of various search procedures is
given by Bell and Crighton, and in the review by Schlegel.
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(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.
* * * the Hessian * * *
Although quasi-Newton methods require only an
approximation to the true hessian, the choice of this
matrix has a great affect on convergence of the geometry
search.
There is a procedure contained within GAMESS for
guessing a diagonal positive definite hessian matrix,
HESS=GUESS. If you are using Cartesian coordinates, the
guess hessian is 1/3 times the unit matrix. The guess is
more sophisticated when internal coordinates are defined,
as empirical rules will be used to estimate stretching
and bending force constants. Other force constants are set
to 1/4. The diagonal 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.
* * * 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 "standard method"
in GAMESS projects 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,
but possibly even with an accurately calculated one. Thus,
a "good" set of internals is one with a diagonally dominant
hessian.
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 -48.6844234867
2 -48.6822702585 -48.6510215698 -48.6881175652
3 -48.6858299354 -48.6882945647 -48.6933816338
4 -48.6881499412 -48.6849667085 -48.6947421032
5 -48.6890226067 -48.6911899936 -48.6960508492
6 -48.6898261650 -48.6878047907 -48.6974654893
7 -48.6901936624 -48.6930608185 -48.6988318245
8 -48.6905304889 -48.6940607117 -48.6996847663
9 -48.6908626791 -48.6949137185 -48.7007217630
10 -48.6914279465 -48.6963767038 -48.7017826506
11 -48.6921521142 -48.6986608776 -48.7021265394
12 -48.6931136707 -48.7007305310 -48.7022386031
13 -48.6940437619 -48.7016095285 -48.7022533741
14 -48.6949546487 -48.7021531692 -48.7022564052
15 -48.6961698826 -48.7022080183 -48.7022564793
16 -48.6973813002 -48.7022454522
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."
* * * 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!
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.
* * * Mode Following * * *
In certain circumstances, METHOD=STANDARD can walk from
a region of all positive curvatures to a transition state.
This is a tricky business, and should be attempted only
after reading Baker's paper carefully. 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. Of course, you should attempt to follow only
totally symmetric modes with IFOLOW. 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.
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. Be sure to use your brain when you use
IFOLOW!
Figure 3 of the paper C.J.Tsai, K.D.Jordan, J.Phys.Chem.
97, 11227-11237 (1993) is illuminating on the sensitivity
of mode following to the initial geometry point.
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.
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 quote 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 * Be * C N O F
Na' * Al Si P S Cl Na' Mg Al Si P S Cl
K' * ... Zn * Ge * * Br K' * ... 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
--------- ----------
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).
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
octupole - 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.
Localization tips
------------ ----
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 to yield
localized banana bonds. Hence it is rather easy to find
cases where this method give rather 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 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 invariant an MCSCF function, you must be using a
FORS=.TRUE. wavefunction, and you must set GROUP=C1 in
$DRT as 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
determines 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.
IRC 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.
A small program which converts the IRC results punched
to file IRCDATA into a format close to that required by the
POLYRATE VTST dynamics program written in Don Truhlar's
group is available. For a copy of this IRCED program,
contact Mike Schmidt. The POLYRATE program must be obtained
from the Truhlar group.
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.Phys.Chem. 94, 5523-5527(1990)
J.Chem.Phys. 95, 5853-5860(1991)
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.
Transition Moments and Spin-Orbit Coupling
---------- ------- --- ---------- --------
GAMESS can compute transition moments and oscillator
strengths for the radiative transition between two CI
wavefunctions. The moments are computed using both the
"length (dipole) form" and "velocity form". GAMESS can
also compute the one electron portion of the "microscopic
Breit-Pauli spin orbit operator". The two electron terms
can be approximately accounted for by means of effective
nuclear charges.
The orbitals for the CI can be one common set of
orbitals used by both CI states. If one set of orbitals
is used, the transition moment or spin-orbit coupling can
be found for any type of CI wavefunction GAMESS can
compute.
Alternatively, two sets of orbitals (obtained from
separate MCSCF orbital optimizations) can be used. Two
separate CIs will be carried out (in 1 run). 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 transformation to
corresponding orbitals. The non-orthogonal procedure
implemented in GAMESS 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 for the
states, use the FCORE option in $MCSCF in a SCFTYP=MCSCF
optimization of the orbitals. Typically you would
optimize the ground state completely, and use these MCSCF
orbitals in an optimization of the excited state, under
the constraint of FCORE=.TRUE. The core orbitals of such
a MCSCF optimization should be declared MCC, rather than
FZC, even though they are frozen.
In the case of transition moments either one or two CI
calculations are performed, necessarily on states of the
same multiplicity. Thus, only a $DRT1 is read. A
spin-orbit coupling run almost always does two CI
calculations, as the states are usually of different
multiplicities. So, spin-orbit runs might read only
$DRT1, but usually read both $DRT1 and $DRT2. The first
CI calculation, defined by $DRT1, must be for the state of
lower multiplicity, with $DRT2 defining the state of
higher multiplicity.
You will probably have to lower the symmetry in $DRT1
and $DRT2 to C1. You may use full spatial symmetry in the
DRT groups only if the two states happen to have the same
total spatial symmetry.
The transition moment and spin orbit coupling driver
is a rather restricted path through GAMESS, in that
1) GUESS=SKIP is forced, regardless of what you put in a
$GUESS group. The rest of this group is ignored. The
NOCC orbitals are read (as GUESS=MOREAD) in a $VEC1
group, and possibly a $VEC2 group. It is not possible
to reorder the orbitals.
2) $CIINP is not read. The CI is hardwired to consist
of DRT generation, integral transformation and 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.
5) DRT input is given in $DRT1 and maybe $DRT2.
6) IROOTS will determine the number of eigenvectors found
in each CI calculation, except NSTATE in $GUGDIA will
override IROOTS if it is larger.
References:
F.Weinhold, J.Chem.Phys. 54,1874-1881(1970)
B.H.Lengsfield, III, J.A.Jafri, D.H.Phillips,
C.W.Bauschlicher, Jr. J.Chem.Phys. 74,6849-6856(1981)
"Intramediate Quantum Mechanics, 3rd Ed." Hans A. Bethe,
Roman Jackiw Benjamin/Cummings Publishing, Menlo Park,
CA (1986), chapters 10 and 11.
For an application of the transition moment code:
S.Koseki, M.S.Gordon J.Mol.Spectrosc. 123, 392-404(1987)
For more information on effective nuclear charges:
S.Koseki, M.W.Schmidt, M.S.Gordon
J.Phys.Chem. 96, 10768-10772 (1992)
* * *
Special thanks to Bob Cave and Dave Feller for their
assistance in performing check spin-orbit coupling runs
with the MELDF programs.
