Wave function, Hartree-Fock-type

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

We can discuss the "best uni-configurational wave function by the usual variational method of the Hartree-Fock type. This means making a search for the function W which minimizes the quantity... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

In this type of empirical approximation, the form of the matrix elements Hu has been specified, but it is also possible to specify the form of the wave function and therefore define the Hamiltonian exactly. Thus, if the ground-state function is a Hartree-Fock-type, the Hamiltonian can be evaluated... 

The final level of basis is a very large one that approaches completion so that the molecular orbitals p/ become perfect. With a restricted deter-minantal wave function of the type given in Eq. (3), this is the Hartree-Fock limit. In practice, however, this limit can only be achieved for very small molecules, and it will be difficult to implement in an extended manner. 

In this section we shall introduce the very useful concept of the Fermi vacuum, which makes the evaluation of certain types of matrix elements much easier. As a matter of fact, many (if not most) quantum-chemical considerations and methods are based on the Hartree-Fock single determinantal wave function which serves also as a zeroth-order wave function ( reference state ) in guessing more accurate wave functions as well. For this reason, one is often interested in evaluating expectation values with respect to Hartree-Fock-type wave functions. The evaluation of such expressions will be analyzed below in some detail. 

Derivations of the above type can be made much simpler by introducing the Fermi vacuum. Consider the expectation value of some operator A with a Hartree-Fock-type wave function ... 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

The simplest approximation corresponds to a single-determinant wave function. The best possible approximation of this type is the Hartree-Fock (HF) molecular-orbital determinant. The HF wavefunction is constructed from the minimal number of occupied MOs (i.e., NI2 for an V-eleclron closed-shell system), each approximated as a variational linear combination of the chosen set of basis functions (vide infra). 

The electron densities of the separate atoms are described by Hartree-Fock wave functions approximated by analytic extended (Slater-type) basis sets. 

The description above may seem a little unhelpful since we know that in any interesting system the electrons interact with one another. The many different wave-function-based approaches to solving the Schrodinger equation differ in how these interactions are approximated. To understand the types of approximations that can be used, it is worth looking at the simplest approach, the Hartree-Fock method, in some detail. There are also many similarities between Hartree-Fock calculations and the DFT calculations we have described in the previous sections, so understanding this method is a useful way to view these ideas from a slightly different perspective. 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal-... 

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