Many-electron wave

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

Generalize the solution of Exercise 9-1 to the case of a many-electron wave function [Eq. (9-29)] yielding Pm permutations. 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

To use HyperChem for calculations, you specify the total molecular charge and spin multiplicity (see Charge, Spin, and Excited State on page 119). The calculation selects the appropriate many-electron wave function with the correct number of alpha or beta electrons. You don t need to specify the spin function of each orbital. 

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 ... 

Even though the correct many electron wave function is not available in DFT, we will see in Section 5.3.3 that a related wave function exists, which can often be used for qualitative interpretation. 

The many-electron wave function (40 of any system is a function of the spatial coordinates of all the electrons and of their spins. The two possible values of the spin angular momentum of an electron—spin up and spin down—are described respectively by two spin functions denoted as a(co) and P(co), where co is a spin degree of freedom or spin coordinate . All electrons are identical and therefore indistinguishable from one another. It follows that the interchange of the positions and the spins (spin coordinates) of any two electrons in a system must leave the observable properties of the system unchanged. In particular, the electron density must remain unchanged. In other words, 4 2 must not be altered... 

A many-electron wave function must be antisymmetric... 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

Any many-electron wave function must be antisymmetric to the interchange of the spacial coordinates and spin (collectively referred to as the vector q) of any pair of electrons i and /. 

The superscripts a and P indicate the spin state of the electrons in the many-electron wave-function. Although many biologically important compounds, particularly metallopro-teins46, exist in states with unpaired electrons, our work has not involved the study of open-shell systems. Readers who wish to apply semi-empirical methods in the study of such structures should consult more specialized discussions47,48. In my experience, handling the complications that arise in treating systems with unpaired electrons should probably be left to professional theoreticians ... 

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.)... 

During the last five decades, an alternative way of looking at the quantum theory of atoms, molecules, and solids in terms of the electron density in three-dimensional (3D) space, rather than the many-electron wave function in the multidimensional configuration space, has gained wide acceptance. The reasons for such popularity of the density-based quantum mechanics are the following ... 

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)]. 

Molecular orbital an initio calculations. These calcnlations represent a treatment of electron distribution and electron motion which implies that individual electrons are one-electron functions containing a product of spatial functions called molecular orbitals hi(x,y,z), 4/2(3 ,y,z), and so on. In the simplest version of this theory, a single assignment of electrons to orbitals is made. In turn, the orbitals form a many-electron wave function, 4/, which is the simplest molecular orbital approximation to solve Schrodinger s equation. In practice, the molecular orbitals, 4 1, 4/2,- -are taken as a linear combination of N known one-electron functions 4>i(x,y,z), 4>2(3,y,z) ... 

For elastic scattering, illustrated by Fig. 1.2(a), the initial and final states are the same, that is, n = m. For a system of electrons, represented by a many-electron wave function, we obtain in the approximation that the N electrons are scattering independently ... 

The overall many-electron wave function is formed from the Hartree-Fock orbitals as an antisymmetrized product. If the individual spin orbitals are... 

In this character table, we have included the spectroscopic symbol for the symmetry type. A molecular orbital, in distinction to a full many-electron wave... 

Problem 11-1. Consider three levels of approximation (a) Exact many-electron wave function, (b) Hartree-Fock wave function, (including all electrons), (c) Simple LCAO-MO valence electron wave function. For each of the following molecular properties, would you expect the Hartree-Fock approximation to give a correct prediction (to within 1% in the cases of quantitative predictions) Would you expect the LCAO-MO approximation to give a correct prediction ... 

Spatial extension, as expressed by the expectation value (r), is roughly comparable for 4 f and 5 f wave functions (Figs. 7 and 8). However, the many-electron wave functions resulting from the solution of the relativistic Dirac equation may also be used to calculate a number of physically interesting quantities, i.e. expectation values of observable... 

The exact many-electron wave function for an excited state, kj, f / 0, satisfies orthogonality conditions with respect to other many-electron state including the ground state, ko. For example, for the first excited state with many-electron wave function we have... 

In Eq. 1, is the Hamiltonian operator, is one of the possible wave functions for all of the electrons in a molecule, and E is the energy associated with each of these possible many-electron wave functions. 

Since in Eq. 1 is the wave function for all of the electrons in the molecule, it is simplest to begin trying to find 4/ by assuming that it can be approximated as the product of one-electron wave functions, one wave function for each of the electrons in a molecule. These one-electron wave functions are called orbitals, and they are distinguished from the many-electron wave function by using a lower case psi (v[/) for the former and an upper case psi (th) for the latter. 

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