Molecular system exact” wave function

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

For hydrogen the exact wave function is known. For helium and lithium, very accurate wave functions have been calculated by including interelectronic distances in the variation functions. For atoms of higher atomic number, the best approach to finding a good wave function lies in first calculating an approximate wave function using the Hartree-Fock procedure, which we shall outline in this section. The Hartree-Fock method is the basis for the use of atomic and molecular orbiteils in many-electron systems. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

These considerations suggest that it might be useful to approximate molecular orbitals as sums of atomic orbitals. Thus for the hydrogen molecule ion hJ, a one-electron system, one could consider a wave function - based on the exact orbitals for the separated atoms - of the form, in... 

Such a Slater determinant, as it is often called, would, in fact, be the correct wave function for a system of noninteracting electrons. Electrons, however, do interact in real molecular systems. In order to obtain a more satisfactory representation, the individual orbitals self-consistent field method, whose main features are as follows, (a) One writes the exact total Hamiltonian for the system with explicit inclusion of electron interactions... 

Regarding the problems of the electronic structure of molecular systems, we notice that in the past, the importance of the qualitative concepts and explanations has been stressed many times. In this context, V.A. Fock [3,4] discussed the (basically metaphysical) problem of interrelation between exact solution and approximate explanation . His point was that any approximation (more precisely, the general form of the trial electron wave function i.e. an Ansatz used for it) sets the system of qualitative concepts (restricted number of variables), which can only be used for interpreting the calculation results and for describing the experiments. A characteristic example... 

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