The Hartree-Fock Approximation

The basis of this problem has recently been analysed by Buijse and Baerends [33] who studied the electronic structure of the apparently simple d° [Mn04] ion. Given the large HOMO/LUMO gap, there is an a priori expectation that a single determinant HF treatment should be a reasonably 

The Hartree-Fock self-consistent field (SCF) method is the primary tool used in this chapter. It is rooted in the time-independent one-electron Schrodinger equation (in atomic units)  

The effective one-electron operator indicated in brackets includes the kinetic energy operator —and an effective potential energy V ri) taken as an averaged function of ri—the distance of electron 1 from the nucleus. In this approximation, electron 1 

Atomic Charges, Bond Properties, and Molecular Energies, by Sandor Fliszar Copyright 2009 John Wiley Sons, Inc. 

In addition, F contains two bielectronic operators. They describe the interaction between the electron occupying spin orbital i and the other electrons found in the atom. So, for the interaction between electrons 1 and 2 at a distance ri2, we have the Coulomb operator Jj and the exchange operator Kj defined by 

On the other hand, the interaction between that electron, denoted here as electron 1, and electrons 2, 3. n is given by the appropriate sums of Coulomb and exchange integrals, for example, for electron 2 interacting with electron 1  

The Hartree-Fock method assumes that the exact N-electron eigenfunction of the system can be approximated by a single Slater determinant of N orbitals (3.26). Applying the variational principle, one can derive a set of N-coupled equations for the N orbitals. Solutions of these equations enables to determine the Hartree-Fock eigenfunction and energy of the system. 

For molecules, Hartree-Fock approximation is the central starting point for most ab initio quantum chemistry methods. It was then shown by Fock that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, has the same antisymmetric property as the exact solution and hence is a suitable ansatz for applying the variational principle. 

Initially, both the Hartree method and the Hartree-Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to simplify the problem considerably. These approximate methods were (and still are) often used together with the central field approximation, to enforce the condition that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

The molecular orbitals are not something real, they are just models of moving electrons. The notion of molecular orbitals is an essential part of the Hartree-Fock theory and this theory is an approximation of the solution to the electronic Schro-dinger equation. The approximation means that one assumes that each electron feels only the average Coulomb repulsion of all the other electrons. This approximation makes the Hartree-Fock theory much simpler to solve numerically than the original N-body problem. Unfortunately, in many cases iterative procedures based on this approximation diverge rather seriously from the reality and thus give incorrect results. 

The Hartree-Fock approximation [13, 14] plays a central role in the molecular electronic structure theory. In most cases, it provides a qualitatively correct description of the electronic structure of many electron atoms and molecules in their ground electronic state. In addition, it constitutes a basis upon which more accurate methods can be developed. A detailed derivation and discussion of the method can be found in textbooks such as [10, 11]. The Hartree-Fock approximation assumes the simplest possible form for the electronic wavefunction, i.e a single Slater determinant given by Eq. (2.41). Starting from the electronic TISE Eq. (2.5), the Hartree-Fock energy is simply 

The Hartree-Fock approximation relies on the variational principle, which states that any approximate wavefunction has an energy above or equal to the exact ground state energy. This principle has an important consequence for a given system, the wavefunction of the form ofEq. (2.41) that minimizes the energy in Eq.(2.42) is the best possible wavefunction within the single determinant approximation. 

With these definitions, the Slater-Condon rules [12, 15] can be used to obtain an expression for the energy of Eq. (2.42) as a function of the MOs. The contribution of the mono-electronic operator reads 

In this last equation, Jij and Kij denote the Coulomb and exchange integrals, respectively. The Coulomb integral represents the repulsion between the two electronic densities IV i( i) and V y( 2)l -The exchange integral has no classical analogue and is a consequence of the Pauli principle. Using Eqs.(2.50) and (2.51), the Hartree-Fock 

The above equation can be recast in a form where the Coulomb and exchange integrals are replaced by the expectation values of Coulomb and exchange operators Ji and Ki, defined such that 

The total energy with the Hartree-Fock wavefunction is 

Then the single-particle Hartree-Fock equations take the form 

Let us summarize what we have shown so far once N and Vcxt (uniquely determined by ZA and Ra) are known, we can construct H. Through the prescription given in equation (1 -13) we can then - at least in principle - obtain the ground state wave function, which in turn enables the determination of the ground state energy and of all other properties of the 

N and Vext completely and uniquely determine PfJ and E0. We say that the ground 

In this and the following sections we will introduce the Hartree-Fock (HF) approximation and some of the fundamental concepts intimately connected with it, such as exchange, selfinteraction, dynamical and non-dynamical electron correlation. We will meet many of these terms again in our later discussions on related topics in the framework of DFT. The HF 

3 In general there can be more than one function associated with the same energy. If the lowest energy results from n functions, this energy is said to be n-fold degenerate. 

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 %i (x ). This product is usually referred to as a Slater determinant, Osd  

The Hartree-Fock (HF) method, or self consistent field (SCF) method as it is sometimes called, approximates T(l, 2. ) by expressing it solely in terms of functions each of which contains the coordinates of just one electron these functions are called molecular orbitals (MOs). This is an approximation because in reality the position of one electron is always correlated with the positions of the others, so that the function which describes a given electron cannot be independent of the functions describing the other electrons. It is for this reason that the error in the electronic energy in the HF approximation is called the correlation energy. 

Both these laws are satisfied by expressing (1, 2. n) in terms of the MOs with a Slater determinant  

In this determinant 0,(j) symbolizes the th MO as a well-defined function of the coordinates of electron j we say that electron j is occupying the MO Of. If the determinant is multiplied out there will be n terms and each term corresponds to one of the n permutations of the n electrons amongst the n MOs since all permutations are included, every electron is treated equally and the indistinguishability law is satisfied. 

If two electrons are interchanged then, for a Slater determinant, this is equivalent to interchanging two rows and if two rows of a determinant are exchanged it changes sign, hence the antisymmetry law is also satisfied, e.g. if we exchange electrons 1 and 2, we get 

That Y(l, 2. n) necessarily satisfies the Pauli exclusion principle is evident from the fact that if two MOs are absolutely identical (including their spin components), then so are two columns in the Slater determinant and a determinant with two identical columns is zero. 

Within the Hartree method, the electronic spin does not appear explicitly except for the fact that no more than two electrons may go into a single orbital. The existence of the Pauli exclusion principle, however, needs to be accounted for in order to go beyond the Hartree method, and that is what the Hartree-Fock method [120] is all about. We first formulate an arbitrary three-dimensional orbital for electron i by writing it as the product of a purely space-dependent part and a spin function (spinor), a or characterizing spin-up or spin-down electron, for example (pi Xi) = i(ri)iXi here we use x to indicate a variable which includes both space (r) and spin (a). A Hartree-like product wave function between two one-electron wave functions and 2 could then be written as 

Even if no integral parametrizations are introduced and the HF equations are all correctly solved, the method eventually turns out to be theoretically incomplete. Despite the correct treatment of electronic exchange (X) within Hartree-Fock theory, electronic correlation (C) is totally missing. This is easily shown for the case of the H2 molecule in which we use the bonding solution of the H2 molecular ion ( + = cr from Equation (2.15)) to build up an antisymmetrized molecular wave function. This means that we put both electrons (ri and rz) of the H2 molecule into the same ip+ orbital, and Pauli s principle is obeyed by means of the ct/ spinors. Neglecting orbital overlap and any pre-factors, for simplicity, the so-called Hund-Mulliken [124] (another name 

This wave function can be iconized, from left to right, as a mix of the four electronic configurations 

For example, the (f i (ri) pi (t2) summand indicates that both electrons of H2 are found in the Is atomic orbital of the left H atom, whereas the other Is orbital of the right atom is empty, thus H H+. The other icons are self-explanatory. Let us contrast this solution with the valence-bond (VB) solution for H2 which itself rests on the Heitler-London ansatz for the H2 ion. Here one starts from loosely interacting H atoms, which are weakly perturbing each other, such that the atomic orbitals themselves are good approximations for the wave function. In the valence-bond case, the solution for the hydrogen molecule is 

A comparison of Equations (2.78) and (2.79) yields that in both approaches, MO (= Hartree-Fock) and VB, the Pauli exchange has been correctly included. The difference between the two is solely given by their amount of electronic correlation. In the MO approach, the electrons are completely uncorrelated (independent), and they may even go into the same atomic orbital, albeit with different spins, thus producing ionic states (H H+). The MO approach therefore does not take care of the energy penalty due to the Coulomb repulsion between the two electrons (see Section 2.9). Because the electronic Coulomb correlation has been completely ignored, the correlation energy may be defined as the difference between the correct energy and that of the Hartree-Fock solution, that is 

We begin our survey of the standard models of quantum chemistry with a discussion of the Hartiee-Fock model - the simplest wave-function model in ab initio electronic-structure theory. It serves not only as a useful approximation in its own right, but also constitutes a convenient starting point for other, more accurate models of molecular electronic structure. 

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Although it is now somewhat dated, this book provides one of the best treatments of the Hartree-Fock approximation and the basic ideas involved in evaluating the correlation energy. An especially valuable feature of this book is that much attention is given to how these methods are actually implemented. 

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory... 

Another distinguishing aspect of MO methods is the extent to which they deal with electron correlation. The Hartree-Fock approximation does not deal with correlation between individual electrons, and the results are expected to be in error because of this, giving energies above the exact energy. MO methods that include electron correlation have been developed. The calculations are usually done using MoUer-Plesset perturbation theoiy and are designated MP calculations." ... 

In addition, if one goes beyond the Hartree-Fock approximation to something like the configuration interaction approach there is an important sense in which one has gone beyond the picture of a certain number of electrons into a set of orbitals.10 If one insists on picturing this, then rather than just every electron being in eveiy possible orbital... 

It is presumed that this function is optimized in the Hartree-Fock approximation. 

Before discussing the correlation error, we will make some introductory remarks about the Hartree-Fock approximation based on the use of the Slater determinant (Eq. 11.38). We note that, if we... 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

The second-order density matrix is in the Hartree-Fock approximation given by Eqs. 11.44 and 11.53, and we obtain directly... 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

According to Eq. 11.67, the correlation energy is simply defined as the difference between the exact energy and the energy of the Hartree-Fock approximation. Let us repeat this definition in a more precise form ... 

The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. 

We see immediately the connection with the one-electron scheme, but we note that the emphasis is here on the word "complete, whereas, in the Hartree-Fock approximation, one is looking for a finite set of best spin orbitals. 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

A number of techniques have been introduced since 1930 to overcome the problem of the neglect of correlation energy in the Hartree-Fock approximation. 

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

The only term for which no explicit form can be given, i. e the big unknown, is of course Exc- Similarly to what we have done within the Hartree-Fock approximation, we now apply the variational principle and ask what condition must the orbitals cp fulfill in order to minimize this energy expression under the usual constraint of ((p I (pj) = ,j The resulting equations are (for a detailed derivation see Parr and Yang, 1989) ... 

Kitaura K, Morokuma K (1976) A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation. Int J Quantum Chem 10 325... 

Mean-field approximation of quasi-free electrons (the Hartree-Fock approximation). The total wave function is described, in this case, by a single Slater determinant. 

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