Roothaans LCAO Hartree-Fock Equation

In the late 1920s and early 1930s, a team led by Hartree [9] formulated a self-consistent-field iterative numerical process to treat atoms. In 1930, Fock [10] noted that the Hartree-SCF method needed a correction due to electron exchange and the combined method was known as the Hartree-Fock SCF method. It was not until 1951 that a molecular form of the LCAO-SCF method was derived by Roothaan [11] as given in Appendix B but we can give a brief outline here. The Roothaan method allows the LCAO to be used for more than one atomic center and so the path was open to treat molecules Now, all we have to do is to carry out the integral for the expectation value of the energy as 

This turns out to be a tremendous problem keeping track of all the terms but it has been done [ 1,3] in many texts which we will only summarize here. Note in particular that we have to integrate over the spin functions as well as the spatial coordinates using simple rules. 

Among the many permutations of the (1, 2,3. n) and the (1,2, 3. n) there will be many integrals in the long summation (there may be thousands of terms) of the general form 

Much of the mathematics for the SCF process was derived for atoms using texts like Condon and Shortly [6] but in 1951 Roothaan [11] solved the problem for molecules using the LCAO philosophy. Instead of using just single orbitals such as l a in our example above for the 

Li atom, the LCAO concept says that (1, 2,..n) is built from many spin orbitals i r, which are linear combinations of basis functions 4 . Thus i r, = The original paper by Roothaan 

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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... 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Substituting into the Hartree-Fock equations Fiji = v. j/ (5.44) the Roothaan-Hall linear combination of basis functions (LCAO) expansions i//, = csi4>s (5.52) for the MO s 1jj gave the Roothaan-Hall equations (Eqs. 5.56), which can be written compactly as FC = SCe (Eqs. 5.57). 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

Quantum chemistry calculations are currently quite common and we recall the excitement this author experienced when first reading Roberts Notes on Molecular Orbital Theory [2] in the early 1960s. Therefore, we include some modern but simple examples here in the hope that the amazement factor is still possible for undergraduates eager to learn up-to-date material. First we can write down the main Hartree-Fock-Roothaan energy operator and at least interpret the various terms. We have used Slater s derivation [1] of the Roothaan LCAO form of the Hartree-Fock equations but prefer Pople s implementation [3] for computer code. First, the one-electron operator... 

Since the first formulation of the MO-LCAO finite basis approach to molecular Hartree-Fock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — are calculated and stored on external storage. The second step then consists of the iterative solution of the Roothaan equations, where the integrals from the first step are read once for every iteration. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The Hartree-Fock and LCAO approximations, taken together and applied to the electronic Schrodinger equation, lead to the Roothaan-Hall equations. ... 

LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Approximates the unknown Hartree-Fock Wavefunctions (Molecular Orbitals) by linear combinations of atom-centered functions (Atomic Orbitals) and leads to the Roothaan-Hall Equations. 

Roothaan-Hall Equations. The set of equations describing the best Hartree-Fock or Single-Determinant Wavefunction within the LCAO Approximation. 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

Use of the LCAO expansion leads to the Hartree-Fock-Roothaan equations Fc = See. Our job is then to find the LCAO coefficients c. This is achieved by transforming the matrix equation to the form of the eigenvalue problem, and to diagonalize the corresponding Hermitian matrix. The canonical molecular orbitals obtained are linear combinations of the atomic orbitals. The lowest-energy orbitals are occupied by electrons, those of higher energy are called virtual and are left empty. 

In this section, we examine the main modifications in the Hartree-Fock Roothaan equations (4.33), it being necessary to take into account the translation symmetry of periodic systems. The first most important difference appears in the LCAO representation of the crystalline crbitals (CO) compared to the molecular orbitals (MO). 

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