Wave function, antisymmetric determinant form

Problem 8-9. If a two-particle wave function has the form of a product, f2) =

[Pg.73]

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... 

Although orbital wave functions, such as Hartree-Fock, generalized valence bond, or valence-orbital complete active space self-consistent field wave functions, provide a semi-quantitative description of the electronic structure of molecules, accurate predictions of molecular properties cannot be made without explicit inclusion of the effects of dynamical electron correlation. The accuracy of correlated molecular wave functions is determined by two inter-related expansions the many-electron expansion in terms of antisymmetrized products of molecular orbitals that defines the form of the wave function, and the basis set used to expand the one-electron molecular orbitals. The error associated with the first expansion is the electronic structure method error the error associated with the second expansion is the basis set error. Only by eliminating the basis set error, i.e., by approaching the complete basis set (CBS) limit, can the intrinsic accuracy of the electronic structure method be determined. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

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. 

This type of wave function, which is clearly antisymmetric with respect to exchange of electron 1 and 2, can be also written in a determinant form... 

The antisymmetry principle is also of great importance in understanding the dualism between localised and delocalised descriptions of electronic structure. We shall see that these are just different ways of building up the same total determinantal wave functions.1 This can be developed mathematically from general properties of determinants, but a clearer picture can be formed if we make a detailed study of the antisymmetric wave function for some highly simplified model systems. 

Antisymmetrized wave function (10.8) may be written in the form of a determinant... 

A possible approximation to be used for the cls function can be chosen considering two ideas. In contrast to the directionality and saturability characteristic for organic covalent bonds, those formed by metal ions do not possess these properties. Thus there is no need to invoke the HO formation on the metal ion. At the infinite separation limit, the cls wave function must flow to the antisymmetrized product of the lone pair geminals of eq. (2.61). The latter is in fact a single determinant function with all lone pair HOs doubly filled. With these arguments, we arrive at the conclusion that the single determinant (HFR) wave function is an appropriate form... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

Whilst we are discussing the Pauli principle, it is worthwhile to introduce a further method of expressing the antisymmetric nature of the electronic wave function, namely, the Slater determinant [1], Since both electrons in the ground state of the helium atom occupy the same space orbital, the wave function may be written in the form... 

Consider a model system of four electrons moving in an arbitrary electrostatic field generated by the nuclei in a molecule. For our purposes, it is not necessary to specify the number of these nuclei, their types, or positions only the general form of the electronic wavefunction is of interest. It is convenient to describe the motions of each electron separately by assigning them to one-electron functions, (l),(xi), where Xi is a vector of the coordinates (including spin) of electron 1. In addition, electrons are fermions, so the electronic wave-function must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. A traditional and very useful starting point for such a four-electron wavefunction is the so-called Slater determinant... 

In the last chapter we have seen that a good approximation to the wave function for a system of atoms at a considerable distance from one another is obtained by using single-electron orbital functions wa(l), etc., belonging to the individual atoms, and combining them with the electron-spin functions a and /3 in the form of a determinant such as that of Equation 44-3. Such a function is antisymmetric in the electrons, as required by Pauli s principle, and would be an exact solution of the wave equation for the system if the interactions between the electrons and those between the electrons of one atom and the nuclei of the other atoms could be neglected. Such determinantal... 

The above anticommutation relations for second-quantization operators have been derived using the symmetry properties of one-determinant wave functions with relation to the permutation of the coordinates of particles. Since the second-quantization operators are only defined in the space of antisymmetric wave functions, the reverse statement is true -in second-quantization formalism the permutative symmetry properties of wave functions automatically follow from the anticommutation relations for creation and annihilation operators. We shall write these relations together in the form... 

The variational principle of quantum mechanics allows us to chose the best wave function by minimizing this energy subject to such variations in the form of the wave function that leave it normalized. The Hartree-Fock approximation seeks to chose the most flexible form of the wave function that still puts two electrons (of opposite spins) in one space orbital. Such a wave function that is antisymmetrized under the exchange of any two electrons is obtained by writing it as a so-called Slater determinant . This is an antisymmetrized form of the product (or Hartree) wave function that is, for N electrons,... 

The more accurate Hartree-Fock method approximates the wave function as an antisymmetrized product (Slater determinant or determinants) of one-electron spin-orbitals and finds the best possible forms for the spatial orbitals in the spin-orbitals. Hartree-Fock calculations are usually done by expanding each orbital as a linear combination of basis functions and iteratively solving the Hartree-Fock equations (11.12). The Slater-type orbitals (11.14) are often used as the basis functions in atomic calculations. The difference between the exact nonrelativistic energy and the Hartree-Fock energy is the correlation energy of the atom (or molecule). 

The centrality of the FNA has spawned considerable research into improvement of the approach. The strategies for obtaining better nodes are numerous. Canonical HF orbitals, Kohn-Sham orbitals from density functional theory (DFT), and natural orbitals from post-HF methods have been used. The latter do not necessarily yield better nodes than single configuration wave functions [39-41]. More success has been found with alternative wave function forms that include correlation more directly than sums of Slater determinants. These include antisymmetrized geminal power functions [42,43], valence-bond [44,45] and Pfaffian [46] forms as well as... 

With the various orbital-types described, one can construct a simple many-electron wave function by forming the antisymmetrized product of orbitals. This is conveniently evaluated using the determinant of the Slater matrix, whose elements are the values of each orbital (in rows) evaluated at the coordinates of each electron (in columns). 

The electron-nucleus (e-n) correlation function does not describe electron correlation per se because it is redundant with the orbital expansion of the antisymmetric function. If the correlation function expansion is truncated at Fi and the antisymmetric wave function is optimized with respect to all possible variations of the orbitals, then Fi would be zero everywhere. There remain two strong reasons for including Fi in the correlation function expansion. First, the molecular orbitals are typically expanded in Gaussian basis sets that do not satisly the e-n cusp conditions. The e-n correlation function can satisly the cusp conditions, but F/ influences the electron density in regions beyond the immediate vicinity of the nucleus, so simple methods for determining Fi solely from the cusp conditions may have a detrimental effect on the overall wave function. Careful optimization of a flexible form of is required if the e-n cusp is to be satisfied by the one-body correlation function [115]. 

Thus an arbitrary antisymmetric function of the two variables can be exactly expanded in terms of all unique determinants formed from a complete set of one-variable functions z,( ) - This argument is readily extended to more than two variables, so that the exact wave function for the ground and excited states of our N-electron problem can be written as a linear combination of all possible iV-electron Slater determinants formed from a complete set of spin orbitals /J. 

In this case the mathematical form of the spinorbitals undergoes variation -change n r) as well as tpi2 r) in eq. (8.1) (however you want) to try to lower the mean value of the Hamiltonian as much as possible. The output determinant which provides the minimum mean value of the Hamiltonian is called the Hartree-Fock function. The Hartree-Fock function is an approximation of the true wave function (which satisfies the Schrodinger equation = E ), because the former is indeed the optimal solution, but only among single Slater determinants. The Slater determinant is an antisymmetric function, but an antisymmetric function does not necessarily need to take the shape of a Slater determinant. 

We begin with the ground state of He2. Each separated helium atom has the ground-state configuration 1. This closed-subshell configuration does not have any unpaired electrons to form valence bonds, and the VB wave function is simply the antisymmetrized product of the atomic-orbital functions. In the notation of Eq. (10.47), the He VB ground state wave function is the Slater determinant... 

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