Slater theory orthonormality

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

The Hartree-Fock model is the simplest, most basic model in ab initio electronic structure theory [28], In this model, the wave function is approximated by a single Slater determinant constructed from a set of orthonormal spin orbitals ... 

The study of orthonormality relationships in basis sets involving more than one Slater function is a very instructive exercise in the development of our understanding of the LCAO-MO theory and continues the discussion on the role of variation parameters. Because these are linear in action they lead to increased computational efficiency, since the integrals between the components need to be calculated only once, which, until recently, was a major consideration. 

It is important, now, to change focus somewhat. We have concentrated on the matching of Herman-Skillman data and the orthonormal functions, which we can generate using Slater functions. This approach is defective for two reasons. Firstly, we have not determined how good are the Herman-Skillman data. Secondly, the whole objective in basis set theory is to provide functions, which lead to simpler integrations, from which can... 

The approximation will be described in Chapter 8 and is based on assuming the wave function in the form of a single Slater determinant built of orthonormal spinorbitals. Car and ParrineUo give their procedure for the density functional theory (DPT), where a single Slater determinant also plays an important role. 

The Slater determinant is the central entity in molecular orbital theory. The exact -electron wave function of a stationary molecule in the Born-Oppenheimer approximation is a 4-dimensional object that depends on the three spatial coordinates and a spin coordinate of the N electrons in the system. This object is of course too complicated for any practical application and is, in first approximation, replaced by a product of N orthonormal 4-dimensional functions that each depend on the coordinates of only one of the electrons in the system. 

According to the ab initio molecular orbital theory methodology, atomic orbitals (set of functions, also called basis sets) combine in a way to form molecnlar orbitals that snrronnd the molecule. The molecular orbital theory considers the molecnlar wave function as an antisymmetiized product of orthonormal spatial molecular orbitals. Then they are constructed as a Slater determinant [56], Essentially, the calculations initially use a basis set, atomic wave functions [57, 58], to constract the molecular orbitals. The first and basic ab initio molecular orbital theory approach to solve the Schrodinger equation is the Hartree-Fock (HF) method [59, 60], Almost all the ab initio methodologies have the same basic numerical approach but they differ in mathematical approximations. As it is clear that finding the exact solution for the Schrodinger equation, for a molecular system, is not possible, various approaches and approximations are used to find the reliable to close-to-accurate solutions [61-68]. 

In the special case where the spin-orbitals are orthonormal and the trial functions are Slater determinants the expressions for the projective reduction coefficients are both simple and limited, given by Slaters rules to be discussed in detail in later chapters in this work. With such a choice there are Hamiltonian matrix elements between functions that differ from each other only in two or fewer orbitals and Mu = 6u- The expressions for these coefficients when the orbitals are not orthogonal involve the overlap integrals S, j between all the orbitals in the functions and there is no limitation on orbital differences between the functions and Mu is not the unit matrix. Every electronic permutation must be considered in their evaluation. For non-orthogonal orbitals it is thus much more difficult to consider systems with more than a few electrons and, because atomic orbitals on different centers are not orthogonal, this difficulty has hindered the development of VB theory in a quantitative manner until very recently. An account of modern VB developments forms a later part of this handbook. Usually LCAO MOs are developed so as to be orthogonal so that given... 

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