The AO basis

In cases where these guidelines cannot be met, one must use the largest abelian subgroup from the true Gs of the molecule. We will show some examples later. 

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The oharge- and exohange-density matrix elements in the AO basis are ... 

Essentially all of the teclmiques discussed above require the evaluation of one- and two-electron integrals over the AO basis fiinctions (x l./lx ) and mentioned earlier, there are of the order of A /8... 

We see therefore that, however desirable it is to abandon symmetry constraints from the point of view of the variation method, we shall be involved in radical departures from the conventional AO expansion method - particularly in its minimal basis form. Indeed most of the changes required to the usual AO basis method are already implicit in any decision to lower the symmetry of the AO basis from its local spherically symmetric form to that of the molecular point group. We shall see later that these considerations are too pessimistic. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

The upshot of these points is that, in throwing the emphasis on extending the AO basis and thereby generating a mass of Unear coefficients by the variation process, much of the interpretation of the phenomena occurring on bond formation is rendered very difficult. It is not easy, for example, to unambiguously separate the three effects discussed above from each other. 

The AO-basis Bloch functions are, as sum over reciprocal lattice vectors,... 

For the valence bond orbitals themselves, it is generally natural to specify a starting guess in the AO basis. Such a guess might, of course, not lie entirely inside the space spanned by the active space, and it must therefore be projected onto the space of the active MOs. This is achieved trivially in CASVB, by multiplication by the inverse of the matrix of MO coefficients. 

G The AO basis in this case is the same as that for the 6-31G set with a set of d orbitals added. In these calculations the d and the d oibitals are included in the virtual orbital set in which single excitations are included in generating stmctmes. The ds orbitals were not used. The inclusion of these d orbitals provides polarization when the molecule is formed. 

We show the results of calculations at the ST03G and 6-3 IG levels of the AO basis. Table 16.8 shows the orbitals used and the number of functions produced for each case. These statistics apply to each of the calculations we give. 

The TMCs electronic wave function formalizing the CFT ionic model is one with a fixed number of electrons in the d-shell. In the EHCF method it is used as a zero approximation. The interactions responsible for electron transfers between the d-shell and the ligands are treated as perturbations. Following the standards semiempirical setting we restrict the AO basis for all atoms of the TMC by the valence orbitals. All the AOs of the TMC are... 

All of these AO integrals can be calculated and stored, to be called up when needed to evaluate the electronic energy. The closed shell energy in the AO basis can be written as... 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

Let us now consider integrals over a totally symmetric Hermitian one-electron operator O. Over the AO basis we must have... 

In 1989, however, a practical resolution of this problem was derived independently by Haser and by Almlof [16]. They obtain the necessary matrix representation information by utilizing the reducible representation matrices obtained from symmetry transformations on the AO basis as in Eq. 5.1. Full details of their procedure is beyond the scope of this course, but, as would be expected, it has many similarities to the non-totally symmetric operators discussed in the previous section. 

The general matrix representation for the hamiltonian in the AO basis is then35... 

Here the subscript i refers to the state that is chosen as the origin of energies, (e.g., the HL state). The final features of some interest are the three states that result from diagonalization of the VB Hamiltonian in the structure set and the AO basis set. It is seen that there are three roots the lowest is the ground state representing the H H bond (see Eq. 2.2) while the other two represent excited... 

Slater functions The general arguments concerning the physically sound form of the states to be included in the AOs basis sets given above have been implemented in the Slater type AOs ... 

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