Roothaan-Hall equations using

Equations (6.123) are similar to the Roothaan-Hall equations used to obtain the Hartree-Fock energy. A full configuration interaction treatment is feasible only for the simplest molecular systems, and therefore much effort has been expended on establishing the best ways to achieve the optimum limited configuration interaction. One... 

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. 

I he Fock matrix is a ff x ff square matrix that is symmetric if real basis functions are used. Tile Roothaan-Hall equations (2.149) can be conveniently written as a matrix equation ... 

Kohn-Sham Equations. The set of equations obtained by applying the Local Density Approximation to a general multi-electron system. An Exchange/Correlation Functional which depends on the electron density has replaced the Exchange Energy expression used in the Hartree-Fock Equations. The Kohn-Sham equations become the Roothaan-Hall Equations if this functional is set equal to the Hartree-Fock Exchange Energy expression. 

We have m x m equations because each of the m spatial MO s i// we used (recall that there is one HF equation for each ip, Eqs. 5.47) is expanded with m basis functions. The Roothaan-Hall equations connect the basis functions (p (contained in the integrals F and S, Eqs. 5.55, above), the coefficients c, and the MO energy levels . Given a basis set energy levels e. The overall electron distribution in the molecule can be calculated from the total wavefunction P, which... 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

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) ... 

Fig. 5.11 STO-1G energy versus bond length r for H-He+. The calculation for r = 0.800 A was done largely by hand (see Section Using the Roothaan-Hall Equations to do Ab initio Calculations - an Example ) the others were done with the program Gaussian 92 [29]...

The overlap matrix. SCF-type semiempirical methods take the overlap matrix as a unit matrix, S = 1, so S vanishes from the Roothaan-Hall equations FC = SCe without the necessity of using an orthogonalizing matrix to transform these equations into standard eigenvalue form FC = Ce so that the Fock matrix can be diagonalized to give the MO coefficients and energy levels (Sections 4.4.3 and 4.4.1 Section 5.2.3.6.2). 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

The derivation of the Roothaan-Hall equations involves some key concepts Slater determinant, Schrodinger equation, explicit Hamiltonian operator, energy minimization, and LCAO. Using these, summarize the steps leading to the Roothaan-Hall equations FC = See. 

A feature common to the semi-empirical methods is that the overlap matrix, S (in Equation (2.225)), is set equal to the identity matrix I. Thus all diagonal elements of the overlap matrix are equal to 1 and all off-diagonal elements are zero. Some of the off-diagonal elements would naturally be zero due to the use of orthogonal basis sets on each atom, but in addition the elements that correspond to the overlap between two atomic orbitals on different atoms are also set to zero. The main implication of this is that the Roothaan-Hall equations are simplified FC = SCE becomes FC = CE and so is immediately in standard matrix form. It is important to note that setting S equal to the identity matrix does not mean that aU overlap integrals are set to zero in the calculation of Fock matrix elements. Indeed, it is important specifically to include some of the overlaps in even the simplest of the semi-empirical models. 

The first consideration is that the total wavefunction and the molecular properties calculated from it should be the same when a transformed basis set is used. We have already encountered this requirement in our discussion of the transformation of the Roothaan-Hall equations to an orthogonal set. To reiterate suppose a molecular orbital is written as a linear combination of atomic orbitals ... 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

The presence of the overlap matrix S in the Roothaan-Hall equations Eq. (2.65) reflects the fact that the basis functions used to expand the orbitals are non-orthogonal. By multiplying from the left by and inserting the unit matrix I =... 

---