Appendix: Quick overview of the OBS-GMCSC method
In the GMCSC method, the wavefunction is a linear combination of configurations. Each
configuration is the anti-symmetrized product of an 'orbital string' and a spin part. Here, the term
'orbital string' is used to denote a product of as many orbitals as there are electrons, while, in
general, the 'spin part' is a linear combination of all possible modes of spin coupling compatible
with given values of the total spin and its projection along a 'quantization axis'. The latter are of
course both zero for a closed-shell molecule. Spin-coupling modes can be expressed by different
equivalent schemes, essentially different basis sets for the same finite-dimensional spin space; the
GMCSC program adopts, as spin basis set, the Yamanouchi-Kotani (YK) functions1. They enter an
arbitrary spin function with adjustable coefficients that, by analogy with single-configuration SC
theory, are referred to as 'spin-coupling coefficients'. A single set of spin-coupling coefficients,
common to all configurations, was assumed in the original formulation of the method, under the
'MCSC' acronym2. This was later generalized to the GMCSC method3, which allows for a separate
set of spin-coupling coefficients for each configuration. In the following, the two closely related
approaches will be referred to as (G)MCSC for brevity.
As is usually done in most ab initio approaches, orbitals are in turn expressed as linear
combinations, with adjustable coefficients (orbital coefficients), of basis functions, usually atomic
orbitals of Gaussian or Slater type. Linear combinations of these can also be adopted, as is usually
the case in frozen-core calculations, when SCF MOs often are the most convenient choice of basis
set. In general, G(MCSC) orbitals are not restricted by double-occupancy or orthogonality
requirements. However, selective such constraints may be imposed on one or more configurations4.
To the point that, in single-configuration mode, the method, and the program that embodies it, can
be used to compute GVB or SCF wavefunctions, though this is surely not the most efficient way to
do so.
The 'weight' of each configuration in (G)MCSC wavefunctions can be expressed in terms of the
configuration's Chirgwin-Coulson5 'occupation number' wi, defined as
wi = ci Σj cj <i|j>
(A1)
where ci and cj are the normalized coefficients of configurations i and j, and <i|j> is their overlap.
It is easy to see that Chirgwin-Coulson occupation numbers sum up to exactly 1.0. They are perhaps
the most straightforward generalization of the configuration weights in fully-orthogonal
wavefunctions, for which of course one simply uses the squares of the configurations' coefficients
(Eq. A1 obviously reduces to wi = ci2 when <i|j> = δij). In general, Chirgwin-Coulson occupation
numbers are not guaranteed to be positive, but usually only small ones may happen to be negative,
unless the set of configurations is very nearly linearly dependent.
Alternatively, one may use Gallup-Norbeck6 occupation numbers, which are defined as:
ni = N |ci|2 / (S-1)ii
(A2)
where S-1 is the inverse of the matrix of configuration overlaps Sij ≡ <i|j>, and N is a normalization
constant given by
N-1 = Σi |ci|2 / (S-1)ii
(A3)
Though this is not obvious by their definition, it can be shown6 that Gallup-Norbeck occupation
numbers, which are always positive, measure a configuration's unique contribution to the overall
wavefunction, i.e. the contribution from that part of the configuration which is orthogonal to all
other configurations included in the wavefunction. Use of a normalization constant uniformly
1
rescales these contributions and ensures the Gallup-Norbeck occupation numbers always sum up to
exactly 1.0. The normalization constant can be large, especially in the presence of large
configuration overlaps.
The (G)MCSC approach can be used for the simultaneous optimization of basis-set exponential
parameters, which are then treated as variational parameters, on the same footing as spin-coupling,
configuration and orbital coefficients. Such calculations are referred to as OBS-(G)MCSC, where
OBS stands for Optimized Basis Set. They are admittedly much more demanding, from a
computational viewpoint, than those carried out with a fixed basis set.
All variational parameters are of course determined by energy minimization, as appropriate for
electronic ground states. This is carried out by use of an iterative second-order method7,8, which
requires the availability of first and second derivatives of the energy with respect to all variational
parameters, cross terms included. In (G)MCSC calculations, and their OBS counterparts, these are
computed analytically at each iteration. Convergence on an energy minimum is assumed when the
gradient is essentially zero, and the second-derivative matrix is positive definite. The zero-gradient
condition is usually deemed to be met, in (G)MCSC calculations, when the largest gradient
component does not exceed 10-8 atomic units (e.g. 10-8 Eh a0 for STO exponential parameters, where
Eh denotes the atomic unit of energy, the Hartree, 1 Eh = 4.35975 x 10-18 J, and a0 that of distance,
the Bohr, 1 a0 = 0.5291772 x 10-10 m) 9.
1
M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, Tables of Molecular Integrals ( Maruzen,
Tokyo, 1955), Chapter 1, pp. 2-28.
2
F. E. Penotti, Intl. J. Quant. Chem. 46, 535-576 (1993).
3
F. E. Penotti, Intl. J. Quantum Chem. 59, 349-378 (1996).
4
F.E. Penotti, J. Comput. Chem. 27, 762-772 (2006).
5
B. A. Chirgwin and C. A. Coulson, Proc. R. Soc. Lond. A 201, 196-209 (1950).
6
G. A. Gallup, and J. M. Norbeck, Chem. Phys. Letters 21, 495-500 (1973).
7
S. M. Goldfeld, R. E. Quandt, and H. F. Trotter, Econometrica 34, 541-551 (1966).
8
N. C. Pyper and J. Gerratt, Proc. R. Soc. Lond. A 355, 407-439 (1977).
9
E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 17, 1795-1803 (1988).
2