Slater-type orbitals orthonormality

Let us consider now, the optimization - by a combined intra- and inter-orbit variation - of the energy density functional given by Eq. (Ill). Clearly, when we perform an intra-orbit variation we modify the one-particle density p(x). But when we perform an inter-orbit variation at fixed density, we just modify the orbitals For simplicity, we take these orbitals to be locally-scaled Raffenetti-type functions. The Raffenetti Rno(r) radial orbitals [91] are expanded in terms of set of Is Slater-type orbitals, i.e., orbitals having norbital exponents are defined by aQj = a/31. In the present case we do not consider the orbital coefficients as variational parameters these are obtained by Schmidt orthonormalization of the Raffenetti initial values [91]. Hence, the energy density functional becomes explicitly ... 

Orthonormality is not an inherent property of Slater-type orbitals nor the best fit Gaussian basis sets to these orbitals. This deficiency is compounded in all the kinds of linear combinations we might use to approximate atomic and molecular orbitals. The taking of linear combinations of normalized functions, in general, destroys the normalization. So before considering the concerted transformations of Sections 3.6 to 3.7, it is appropriate to investigate the orthonormality, with respect to the various basis functions that we encountered in Chapter 1. 

In these methods, the ground state must be of a type that can be described by a single Slater determinant of orthonormal spin orbitals. One such type, which is the most common ground state, is a closed-shell system (i.e. all occupied MOs are doubly occupied). We let or 0 > denote the Slater determinant wave function for the ground state, which will usually be made up of HF MOs, although in fact... 

Since the MP perturbation operator involves a two-electron operator, the standard Slater rules for orthonormal spin orbitals can be applied when calculating Koi, V , etc. All matrix elements can be expressed in terms of double-bar integrals (ij ab), which are antisymmetrized two-electron integrals of the general type pq rs) ... 

This boundary condition on the Schrodinger equation may be satisfied provided the approximate wave function is expanded in a set of normalized orbitals. In practice, the orthonormal orbitals from which the determinants are constructed are expanded in a finite set of Gaussian functions (and sometimes Slater-type functions) as discussed in Chapter 6. The asymptotic form of these functions ensures that the wave function is square-integrable. 

In second quantization, the numerical vector-coupling coefficients (the Aff and ) appear as matrix elements of creation and annihilation operators X] and jc> The operator X creates an electron in an orthonormal spin orbital io), where /(j) = /) (j), and (T = a or p. Similarly, operator destroys an electron in the orthonormal spin orbital ia). In quantum chemistry problems in which the number of particles is conserved, the Xj and will always occur in pairs. The role of these operators is easily illustrated by showing their operation on a specific type of CSF, namely a Slater determinant. Thus, as an example, for the determinant... 

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