Roothaan-Hall equations procedure

Using the Roothaan-Hall Equations to do ab initio Calculations - the SCF Procedure... 

To use the Roothaan-Hall equations we want them in standard eigenvalue-like form so that we can diagonalize the Fock matrix F of Eq. 5.57 to get the coefficients c and the energy levels e, just as we did in connection with the extended Hiickel method (Section 4.4.1). The procedure for diagonalizing F and extracting the c s and e s and is exactly the same as that explained for the extended Hiickel method (although here the cycle is iterative, i.e. repetitive, see below) ... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Let us consider how we might solve the Roothaan-Hall equations and thereby obtain the molecular orbitals. The first point we must note is that the elements of the Fock matrix, which appear on the left-hand side of Equation (2.162), depend on the molecular orbital coefficients c which also appear on the right-hand side of the equation. Thus an iterative procedure is required to find a solution. 

The procedure for determining the electron density and the energy of the system within the DFT method is similar to the approach used in the Hartree-Fock technique. The wavefunction is expressed as an antisymmetric determinant of occupied spin orbitals which are themselves expanded as a set of basis functions. The orbital expansion coefficients are the set of variable parameters with respect to which the DFT energy expression of equation 15 is optimized. The optimization procedure gives rise to the single particle Kohn-Sham equations which are similar, in many respects, to the Roothaan-Hall equations of Hartree-Fock theory. 

We have now succeeded in expressing the Hartree-Fock equations in the AO basis, avoiding any transformation to the MO basis. The pseudo-eigenvalue equations (10.6.16) are called the Roothaan-Hall equations [3,4]. In Exercise 10.4, the Roothaan-Hall SCF procedure is used to calculate the Hartree-Fock wave function for HeH in the STO-3G basis. 

As seen in Section 3.5, the Roothaan-Hall (or Pople-Nesbet for the UHF case) equations must be solved iteratively since the Fock matrix depends on its own solutions. The procedure illustrated in Figure 3.3 involves the following steps. 

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). 

The Hall-Roothaan equations for the case of the helium ground state appear, when equation 5.39 is written out in an appropriate linear combination of functions upon which the variation principle procedure can be applied to return the best energy in a calculation. For example, for the double-zeta basis of Slater functions used in the previous section, we have... 

In order to determine these unknowns the variational minimax principle of chapter 8 is invoked. For this procedure, we may again start from the energy expression of section 10.2 and differentiate it or directly insert the basis set expansion of Eq. (10.3) into the SCF Eqs. (8.185). These options are depicted in Figure 10.2. The resulting Dirac-Hartree-Fock equations in basis set representation are called Dirac-Hartree-Fock-Roothaan equations according to the work by Roothaan [511] and Hall [512] on the nonrelativistic analog. 

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