Hamiltonians and Operators

The first step in any quantum mechanical evaluation of properties is the construaion of appropriate operators for the properties. Of course, even if not done explicitly, this means developing the Hamiltonian with the various embedded parameters. Electrical properties illustrate what is to be done in general. Using V, the rank-one polytensor ( q. [1], a molecular Hamiltonian for an applied electrical potential is 

Mo is the total charge, M, is the x-component of the dipole moment, M an element of the second moment, 

This says that the dipole moment operator will be needed for the derivative Schrodinger equations involving derivatives with respect to V (the -com-ponent of a uniform field), or that the second moment operator will be needed for derivative Schrodinger equations involving differentiation with respect to field gradient components such as V. In general, there will be operators combined with parameters in the Hamiltonians, and then the derivative Hamiltonians will be operators of some sort. These must be constructed. 

The primed variables may have a separate origin. A set of three integers such as kx, ky, kg) is colleaively represented on the left-hand side of Eq. [10] by one character, in this case k. Table 2 shows how choices of the integers lead to the various kinds of operators needed for properties. 

To generate the matrix representations of these operators, expectation values of the operator must be computed for all combinations of the basis functions. Considering Gaussian basis functions, the task is to compute integrals that are elements of the desired matrix, such as 

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The downside to the (spin)-unrestricted Hartree-Fock (UHF) method is that the unrestricted wavefunction usually will not be an eigenfunction of the operator. Since the Hamiltonian and operators commnte, the true wavefunction must be an eigenfunction of both of these operators. The UHF wavefunction is typically contaminated with higher spin states for singlet states, the most important contaminant is the triplet state. A procedure called spin projection can be used to remove much of this contamination. However, geometry optimization is difficult to perform with spin projection. Therefore, great care is needed when an unrestricted wavefunction is utilized, as it must be when the molecule of interest is inherently open shell, like in radicals. 

THE ALGEBRA OF EFFECTIVE HAMILTONIANS AND OPERATORS EXACT OPERATORS... 

Effective Hamiltonians and effective operators are used to provide a theoretical justification and, when necessary, corrections to the semi-empirical Hamiltonians and operators of many fields. In such applications, Hq may, but does not necessarily, correspond to a well defined model. For example. Freed and co-workers utilize ab initio DPT and QDPT calculations to study some semi-empirical theories of chemical bonding [27-29] and the Slater-Condon parameters of atomic physics [30]. Lindgren and his school employ a special case of DPT to analyze atomic hyperfine interaction model operators [31]. Ellis and Osnes [32] review the extensive body of work on the derivation of the nuclear shell model. Applications to other problems of nuclear physics, to solid state, and to statistical physics are given in reviews by Brandow [33, 34], while... 

Semi-empirical Hamiltonians and operators are taken to be state independent [56] and have the same Hermiticity as their true counterparts. Consequently, the valence shell effective Hamiltonians and operators they mimic must also have these two properties. Table I shows that the effective Hani iltonian and operator definitions H and A, as well as H and either A or a fulfill these criteria. Thus, these definition pairs may be used to derive the valence shell effective Hamiltonians and operators mimicked by the semi-empirical methods. Table III indicates that the commutation relation (4.12) is preserved by all three definition pairs. Hence, the validity of the relations derived from the semi-empirical version of (4.12) depends on the extent to which the semi-empirical Hamiltonians and operators actually mimic, respectively, exact valence shell effective Hamiltonians and operators. In particular, the latter Hamiltonians and operators contain higher-body terms which are neglected, or ignored, in semi-empirical theories. These nonclassical higher body interactions have been shown to be nonnegligible for the valence shell Hamiltonians of many atoms and molecules [27, 145-149] and for the dipole moment operators of some small molecules [56-58]. There is no a... 

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