The Hartree-Fock Equation

In Hartree-Fock theory, each electron is assigned to a molecular orbital, and the wave function is expressed as a single Slater determinant in terms of the molecular orbitals. For a system with nei electrons the wave function is then given as 

represents a molecular orbital, and xjt designates the spatial and spin coordinates of electron k it and ) 

The molecular orbitals are expressed as linear combinations of atomic orbitals 

The elements Dpx of the density matrix D are computed from the molecular orbital coefficients (assumed to be real) 

The electronic contribution to the Hartree-Fock energy is computed as follows 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the 

This type of constrained minimisation problem can be tackled using the method of Lagrange multipliers. In this approach (see Section 1.10.5 for a brief introduction to Lagrange multipliers) the derivative of the function to be minimised is added to the derivatives of the constraint(s) multiplied by a constant called a Lagrange multiplier. The sum is then set equal to zero. If the Lagrange multiplier for each of the orthonormality conditions is written A , then  

In the Hartree-Fock equations the Lagrange multipliers are actually written —TSy to reflect the fact that they are related to the molecular orbital energies. The equation to be solved is thus  

This expression can be tidied up by introducing three operators that represent the contributions to the energy of the spin orbital Xi in the frozen system  

In the absence of any interelectronic interactions this would be the only operator present, corresponding to the motion of a single electron moving in the field of the bare nuclei. 

These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4) and by Slater2 in 1930 [8], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree s iterative, average-field approach supplemented with electron spin and antisymmetry leads to the Hartree-Fock equations. 

The Hartree wavefunction (above) is a product of one-electron functions called orbitals, or, more precisely, spatial orbitals these are functions of the usual space coordinates x, y, z. The Slater wavefunction is composed, not just of spatial orbitals, but of spin orbitals. A spin orbital ij/ (spin) is the product of a spatial orbital and a spin function, a or / The spin orbitals corresponding to a given spatial orbital are 

2John Slater, bom Oak Park, Illinois, 1900. Ph.D. Harvard, 1923. Professor of physics, Harvard, 1924-1930 MIT 1930-1966 University of Florida at Gainesville, 1966-1976. Author of 14 textbooks, contributed to solid-state physics and quantum chemistry, developed X-alpha method (early density functional theory method). Died Sanibel Island, Florida, 1976. 

The Slater wavefunction differs from the Hartree function not only in being composed of spin orbitals rather than just spatial orbitals, but also in the fact that it is not a simple product of one-electron functions, but rather a determinant (Section 4.3.3) whose elements are these functions. To construct a Slater wavefunction (Slater determinant) for a closed-shell species (the only kind we consider in any detail here), we use each of the occupied spatial orbitals to make two spin orbitals, by multiplying the spatial orbital by a and, separately, by jl. The spin orbitals are then filled with the available electrons. An example should make the procedure clear (Fig. 5.2). Suppose we wish to write a Slater determinant for a four-electron 

11 T of Eq. 5.10. Some authors use the row format for the electrons, others the column format. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

The simplest SCF system is based on Hartree s original model, which treats the many-electron wavefunction as a simple product of one-electron functions fa. Since this is also the rule for combining independent probabilities, this is clearly an independent particle approach, but violates the Pauli principle, because no account is taken of exchange. The next stage beyond this is to write the wavefunction as a properly 

Note that the external, Coulomb and electrostatic repulsion terms are all three multiplicative on fa (as is the Hartree potential). For this reason, they are called local operators. The exchange term contains fa in the kernel of an integral and is referred to as a nonlocal operator. Much of the complexity of solving the Hartree-Fock equations arises from the nonlocality of the exchange term. Various simplifications have therefore been devised so as to replace it by an effective local operator. 

Independent electron models are only an approximation. Any effect not included within a particular independent electron model is called a correlation. Note that electron correlations are defined with respect to a specific model and therefore depend on the model used. Thus exchange forces appear as a Pauli correlation in Hartree s model. The main effect of Pauli correlations is to reduce the probability of electrons with parallel spins approaching each other. Owing to this reduction, each electron seems to be surrounded by a hole or a space devoid of other electrons. 

This is called a Fermi hole and is the first example we encounter of a particle being dressed (i.e. having its properties modified) by many-body forces. Strictly speaking, the Fermi hole differs for each electron, but the interaction can be made local by averaging it over different orbitals, and this is referred to as the Hartree-Slater approximation.  

---

This expression is not orbitally dependent. As such, a solution of the Hartree-Fock equation (equation (Al.3.18) is much easier to implement. Although Slater exchange was not rigorously justified for non-unifonn electron gases, it was quite successfiil in replicating the essential features of atomic and molecular systems as detennined by Hartree-Fock calculations. 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

We now introduce the atomic orbital expansion for the orbitals i/), and substitute for the corresponding spin orbital Xi into the Hartree-Fock equation,/,(l)x,(l) = X (1) ... 

Application of the Hartree-Fock Equations to Molecular Systems... 

The LCAO approximation for the wave functions in the Hartree-Fock equations... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... 

While orbitals may be useful for qualitative understanding of some molecules, it is important to remember that they are merely mathematical functions that represent solutions to the Hartree-Fock equations for a given molecule. Other orbitals exist which will produce the same energy and properties and which may look quite different. There is ultimately no physical reality which can be associated with these images. In short, individual orbitals are mathematical not physical constructs. 

The Hartree-Fock equations have to be solved by the coupled Hartree-Fock method. The following article affords a typical example. 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

When deriving the Hartree-Fock equations it was only required that the variation of the energy with respect to an orbital variation should be zero. This is equivalent to the first derivatives of the energy with respect to the MO expansion coefficients being equal to zero. The Hartree-Fock equations can be solved by an iterative SCF method, and... 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

GombAs, P., Acta Phys. Hung. 4, 187, Erweiterung der Hartree-Fockschen Gleichungen durch die Korrelation der Elektronen/ Extension of the Hartree-Fock equations through the correlation of the electrons. The correlation energy of the alkali metals is estimated with a statistical method. 

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

Tsoucaris, decided to treat by Fourier transformation, not the Schrodinger equation itself, but one of its most popular approximate forms for electron systems, namely the Hartree-Fock equations. The form of these equations was known before, in connection with electron-scattering problems [13], but their advantage for Quantum Chemistry calculations was not yet recognized. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

The Hartree-Fock equations for the /-th element of a set containing occ occupied molecular orbitals i in a closed shell system with n = 2occ electrons are [8]... 

Ehf from equation (1-20) is obviously a functional of the spin orbitals, EHF = E[ XJ]. Thus, the variational freedom in this expression is in the choice of the orbitals. In addition, the constraint that the % remain orthonormal must be satisfied throughout the minimization, which introduces the Lagrangian multipliers e in the resulting equations. These equations (1-24) represent the Hartree-Fock equations, which determine the best spin orbitals, i. e., those (xj for which EHF attains its lowest value (for a detailed derivation see Szabo and Ostlund, 1982)... 

Thus, once we know the various contributions in equation (5-15) we have a grip on the potential Vs which we need to insert into the one-particle equations, which in turn determine the orbitals and hence the ground state density and the ground state energy by employing the energy expression (5-13). It should be noted that Veff already depends on the density (and thus on the orbitals) through the Coulomb term as shown in equation (5-13). Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-electron equations (5-14) also have to be solved iteratively. 

In the literature we may find the procedure for creating localized Hartree-Fock orbitals via an energy minimization based on a Cl procedure employing monoexcitations (see for instance Reference [24]). The scheme starts from a set of given (guess) orbitals and solves iteratively the Hartree-Fock equations via the steps ... 

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

---