Page 298 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

COUPLED HARTREE-FOCK APPROACH 281
where the different blocks of are classified according to the irreducible representa-
tion with frequency of G, and its appearence. Accordingly, in the new
basis, the symmetry adapted tensor components are [4]

for every operation of the group and for each block This implies either that
or that, being invariant under the operations of G, it carries the one-
dimensional totally symmetric representation, Thus, if
the totally symmetric representation occurs m times in the direct product represen-
tation, then the tensor is fully determined by just m numbers. Therefore theoretical
procedures for evaluating the higher-rank polarizability tensors appearing in (1) and
(2) should efficiently exploit the symmetry properties of a given molecule to save
computer effort. The number of independent parameters can be conveniently eval-
uated a priori via simple techniques based on symmetrized Kronecker products [4].
Tables reporting data for a number of groups are available [1].
Besides the elementary properties of index permutational symmetry considered in
eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs.
(8)-(14), much more powerful group-theoretical tools [6] can be developed to speed
up coupled Hartree-Fock (CHF) calculations [7–11] of hyperpolarizabilities, which are
nowadays almost routinely performed in a number of studies dealing with non linear
response of molecular systems [12–35], in particular at the self-consistent-field (SCF)
level of accuracy.
The present paper is aimed at developing an efficient CHF procedure [6–11] for the
entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up
to the 5-th rank. Owing to the 2n + 1 theorem of perturbation theory [36], only 2-nd
order perturbed wavefunctions and density matrices need to be calculated. Explicit
expressions for the perturbed energy up to the 4-th order are given in Sec. IV.
A computer program for the theoretical determination of electric polarizabilities and
hyperpolarizabilitieshas been implemented at the ab initio level using a computa-
tional scheme based on CHF perturbation theory [7–11]. Zero-order SCF, and first-
and second-order CHF equations are solved to obtain the corresponding perturbed
wavefunctions and density matrices, exploiting the entire molecular symmetry to re-
duce the number of matrix element which are to be stored in, and processed by,
computer. Then and tensors are evaluated. This method has been
applied to evaluate the second hyperpolarizability of benzene using extended basis
sets of Gaussian functions, see Sec. VI.

2. Solution of first-order CHF equation

The Hartree-Fock equations for the i-th element of a set containing occ occupied
molecular orbitals in a closed shell system with n = 2occ electrons are [8]

where the orthonormality conditions are written