Matrix Hartree-Fock equations

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. 

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 ... 

Most ab initio quantum chemical molecular orbital calculations involve, in some form, the solution of the Hartree-Fock equations. Following Roothaan (13,14) these equations are usually given in a matrix form that for a closed shell molecule takes the deceivingly simple form ... 

L. Cohen and C. Frishberg, Hartree-Fock density matrix equation. J. Chem. Phys. 65,4234 (1976). 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

This last result can be written as a martix product as well, and it is seen that this is now a unitary transformation to the matrix e. We are free to choose U to be whatever we please, and if we choose it to make e diagonal, we can rewrite the Hartree-Fock equations as... 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

Usually the solutions of any version of the Hartree-Fock equations are presented in numerical form, producing the most accurate wave function of the approximation considered. Many details of their solution may be found in [45], However, in many cases, especially for light atoms or ions, it is very common to have analytical radial orbitals, leading then to analytical expressions for matrix elements of physical operators. Unfortunately, as a rule they are slightly less accurate than numerical ones. 

In dealing with the MO-LCAO wave function no additional assumptions concerning the vibronic matrix elements are necessary. The evaluation of the total molecular energy exactly copies the lower sheet of the adiabatic potential. This is a consequence of the well-known fact that the Hartree-Fock equations are equivalent to the statement of the Brillouin theorem the matrix elements of the electronic Hamiltonian between the ground-state and... 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

This is the Hartree-Fock equation for atoms, and F is the Hartree Fock operator. Equation (6.53) can be written in matrix form,... 

Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

We have also examined here the use of approximate solutions of the coupled perturbed Hartree-Fock equations for estimating the Hessian matrix. This Hessian appears to be more accurate than any updated Hessian we have been able to generate during the normal course of an optimization (usually the structure has optimized to within the specified tolerance before the Hessian is very accurate). For semi-empirical methods the use of this approximation in a Newton-like algorithm for minima appears optimal as demonstrated in Table 17. In ab-initio methods searching for minima, the BFGS procedure we describe is the best compromise. 

An approximate or semiempirical Hartree-Fock molecular-orbital method in which electron repulsion integrals and some Hamiltonian matrix elements are approximated and the approximate Hartree-Fock equations solved and integrated to self-consistency... 

It is important that the orbitals r ) that satisfy (5.17) are orthogonal for different eigenvalues We can use them to construct an orthonormal set with which to express the many-electron problem. We have assumed orthogonality in deriving (5.17). We now show that this is consistent by considering (5.17) as an equation for the matrix elements formed for the bra orbital ( from a Schrodinger equation for rj). The set of such equations for N different rj) is called the Hartree—Fock equations. 

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