Many-electron wave functions Slater determinants

This chapter introduces the basic concepts, techniques, and notations of quantum chemistry. We consider the structure of many-electron operators (e.g., the Hamiltonian) and discuss the form of many-electron wave functions (Slater determinants and linear combinations of these determinants). We describe the procedure for evaluating matrix elements of operators between Slater determinants. We introduce the basic ideas of the Hartree-Fock approximation. This allows us to develop the material of this chapter in a form most useful for subsequent chapters where the Hartree-Fock approximation and a variety of more sophisticated approaches, which use the Hartree-Fock method as a starting point, are considered in detail. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

Approximating a many-electron wave function by a finite sum of Slater determinants, e.g. truncating the Cl, CC or MBPT wave function to include only certain excitation types. 

As discussed above, it is impossible to solve equation (1-13) by searching through all acceptable N-electron wave functions. We need to define a suitable subset, which offers a physically reasonable approximation to the exact wave function without being unmanageable in practice. In the Hartree-Fock scheme the simplest, yet physically sound approximation to the complicated many-electron wave function is utilized. It consists of approximating the N-electron wave function by an antisymmetrized product4 of N one-electron wave functions (x ). This product is usually referred to as a Slater determinant, OSD ... 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

Recall that the minimum requirement for a many-electron wave function is that it be written as a suitably antisymmetrized sum of products of one-electron wave functions, that is, as a Slater determinant of MOs [see equation (A.68)] In Chapter 2 and Appendix A, we find that the condition that this be the best possible wave function of this form is that the MOs be eigenfunctions of a one-electron operator, the Fock operator [recall equation (A.42)], from which one can choose the appropriate number of the lowest energy. The Fock operator in restricted form, F( 1) [RHF, the UHF form was given in equation (A.41)], is given by... 

In Hartree-Fock theory, the many-electron wave function is expressed as a single Slater determinant, conveniently abbreviated as... 

Wjl coefficient of the yth Slater determinant in the ith many-electron wave function... 

In the above equations, Wji is the coefficient of the y th Slater determinant in the Zth many-electron wave function and K is the total number of the Slater determinants used for the linear combination. For example, for 4f2 electron configuration, such as ground state configuration of the Pr3"1" ion, two f electrons occupy 14 MOs. Therefore, the number of possible combinations is K = 14 / 2 (14 - 2) = 91. 

In the relativistic DVME method, the interactions among the states represented by the Slater determinants, i.e., the Cl can be analyzed using explicit many-electron wave functions expressed by eq. (23). From the orthonormality of the Slater determinants, the inner product of the /th many-electron wave function with itself can be expanded as... 

In this scheme, the contribution of the jth Slater determinant many-electron wave function lF/ can be simply represented as... 

Since the many-electron wave functions are obtained explicitly as linear combinations of the Slater determinants, the oscillator strengths of an electric dipole transition from the /th state to the yth state can be calculated by... 

We mentioned earlier that the dimensionality of the FCI space is significantly reduced due to spin symmetry. This can be formulated somewhat differently due to the relation existing between the spin and permutation symmetries of the many-electronic wave functions (see [30,42]). Indeed, the wave function of two electrons in two orbitals a and b allows for six different Slater determinants... 

In the DVME method, the many-electron Hamiltonian is diagonalized based on the Cl approach. In this approach, the Zth many-electron wave function Fi is represented as a linear combination of the Slater determinants which are constructed from MOs obtained by the relativistic DV-Xa calculations as ... 

In the MO approach appropriate to outer s and p electrons, the simple formalism does not distinguish between a covalent-ionic band and a metallic band. The use of determinantal (antisymmetrized) wave functions automatically introduces correlations between electrons of parallel spin. Traditionally the many-electron wave function has, at best, been represented by a single Slater determinant of one-electron wave functions (Hartree-Fock approximation), whereas the true wave function would be given by a series of such determi-... 

The many-electron wave function of a molecular system is taken as the antisymmetrized product of (pt, and for closed-shell systems it is convenient to represent it by a Slater determinant. Such an approach is known as the restricted Hartree-Fock (RHF) method and is the most widely used method in chemisorption calculations. Its principal drawback is the neglect of Coulomb electron correlation, which is of crucial importance for adequate treatment of chemical rearrangements with varying numbers of electron pairs. 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

In the second step which is a core part of the DV-ME method, the many-electron wave functions are described as linear combinations of Slater determinants, and all the matrix elements of the many-electron Hamiltonian are calculated, then finally diagonalized to obtain the multiplet energies and many-electron wave functions. 

As the standard ansatz few the many-electron wave function is an antisymmetrized product of one-electron spinors o) it may be written as a Slater determinant or in the language of second quantization (see, for example, Helgaker et al. 2000) as... 

Since the many-electron wave function can be expanded in a linear combination of Slater determinants, its matrix element with a spin-orbit coupling operator of the form of Equation (3.6) can be expressed as a sum of matrix elements of the operator between Slater determinants. For a matrix element between Slater determinants which differ in exactly one spin orbital (i.e. which are singly excited from i — a with respect to each other), the matrix element is... 

As an approximation to the exact many-electron wave function of the real system, we will use the Slater determinant built from the occupied KS MO s [64]. Although it is only an approximation to the exact many-electron wave function of the real system, it seems to be a reasonable one especially if one is interested in calculations of one-electron matrix elements only (in the case of a local multiplicative operator such an approximation yields the exact values of the matrix elements). To describe an excited state corresponding to the transition of an electron from the occupied MO k into the virtual MO a , we will use the many-electron wave function of the excited state in the form of a Slater determinant that differs from the ground state determinant by replacing the occupied MO k by the virtual MO a . 

A many-electron wave function can be constructed from the set of occupied one-electron spinorbitals in the form of the Slater determinant. The molecular orbitals in the LCAO form are determined by solving the Roothaan equations. The MO method is improved by the configuration interaction. For evaluation of the matrix elements of operators over the determinantal functions, the Slater rules are helpful. 

An exact many-electron wave function can be expanded in an infinite series of all the ordered Slater determinants that can be formed from a one-electron basis set. Usually calculations on systems of more than two electrons are made by taking a finite number of such configurations as a linear trial function [configuration-interaction (C.I.) method ]. The expanded form of the ordinary perturbation theory may also be considered an approximate way of doing... 

However, this is not a satisfactory description, especially, for electronic configurations in which there are unpaired electron spins, since this, so-called Hartree product form, does not comply with the anti-symmetry requirement of the Pauli Principle. In general, many-electron wave functions are written as Slater Determinants, which do exhibit the necessary anti-symmetry properties for electron exchange. ... 

Ignoring the general requirement to write the many-electron wave function as a Slater determinant. 

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