Wave function symmetry reduction

The full Cl expansion within the active space severely restricts the number of orbitals and electrons that can be treated by CASSCF methods. Table 4.3 shows how many singlet CSFs are generated for an [n, n]-CASSCF wave function (eq. (4.13)), without any reductions arising from symmetry. 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

The crystal field theory. The basics of the CFT were introduced in the classical work by Bethe [150] devoted to the description of splitting atomic terms in crystal environments of various symmetry. The splitting pattern itself is established by considering the reduction in the symmetry of atomic wave functions while the spatial symmetry of the system goes down from the spherical (in the case of a free atom) to that of a point group of the crystal environment. It is widely described in inorganic chemistry textbooks (seee.g. [152]). 

This form of the Hamiltonian shows explicitly the couplings between wave functions with different Mk values and makes possible to factorize occupation vectors in alpha- and beta-strings like done in non-relativistic Cl theory. The difference with non-relativistic theory is that calculations are not restricted to one value of Mk. Applied without further approximation the formalism gives therefore no dramatic reduction in operation count over the symmetry-adapted unrestricted scheme described in the previous section. An advantage of the formalism is, however, that it facilitates incorporation of the relativistic scheme in non-relativistic Cl or MCSCF implementations [35] and that the scheme gives a natural subdivision of the full Cl matrix. 

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