Fock equation

If one uses a Slater detemiinant to evaluate the total electronic energy and maintains the orbital nomialization, then the orbitals can be obtained from the following Hartree-Fock equations ... 

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

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. 

Friesner R A 1987 Solution of the Flartree-Fock equations for polyatomic molecules by a pseudospectral method J. Chem. Phys. 86 3522-31... 

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... 

In summary, the Hariree-Fock equation for antisymmetr ized orbitals is written... 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

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

If we assume that S = I (which is not true in general), the matr ix form of the Fock equation can be written... 

Now collect all terms from Problems 9.9.8 through 9.9.10 and show that they add up to the Hartiee-Fock equation... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical 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 first two kinds of terms are called derivative integrals, they are the derivatives of integrals that are well known in molecular structure theory, and they are easy to evaluate. Terms of the third kind pose a problem, and we have to solve a set of equations called the coupled Hartree-Fock equations in order to find them. The coupled Hartree-Fock method is far from new one of the earliest papers is that of Gerratt and Mills. 

The coupled Hartree-Fock equations are then solved (Figure 17.5). 

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. 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

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. 

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