Roothaan-Hall equations solving

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. 

Lei 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, u liich appear on the left-hand side of Equation (2.162), depend on the molecular orbital oetficients which also appear on the right-hand side of the equation. Thus an iterative pi oeedure is required to find a solution. 

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 potential (or field) generated by the SCF electron density is identical to that produced by solving for the electron distribution. The Fock matrix, and therefore the total energy, only depends on the occupied MOs. Solving the Roothaan-Hall equations produces a total of Mbasis MOs, i.e. there are A eiec occupied and Mbasis - A dec unoccupied, or virtual, MOs. The virtual orbitals are orthogonal to all the occupied orbitals. 

Since the Fock matrix depends on the density matrix, and therefore on the LCAO coefficients themselves, the Roothaan-Hall equations must in general be solved in an iterative fashion. An initial guess of the density matrix is made (typically a diagonal matrix) and a starting Fock matrix F< > is assembled according to equation (6). This Fock matrix is used in equation (8) to arrive at an approximate set of LCAO coefficients which are then inserted into equation (3) to construct a slightly improved density matrix The whole process is repeated until convergence on P is achieved and an SCF is established. 

Electronic information for a given subsystem is obtained by solving a set of localized Roothaan-Hall equations,... 

By diagonalizing the Fock matrix, the canonical MO coefficient matrix (C) is obtained (see Eq. [7]). However, we have seen in a previous section that almost all elements in the coefficient matrix are significant, which contrasts with the favorable behavior of the one-particle density matrix (P). The density matrix is conventionally constructed from the coefficient matrix by a matrix product (Eq. [8]). Although the Roothaan-Hall equations are useful for small-to medium-sized molecules, it makes no sense to solve first for a nonlocal quantity (C) and generate from this the local quantity (P) in order to compute the Fock matrix or the energy of a molecule. Therefore, the goal is to solve directly for the one-particle density matrix as a local quantity and avoid entirely the use of the molecular orbital coefficient matrix. 

Semi-empirical models begin with the HF and LCAO approximations resulting in the Roothaan-HaU equations (Equations 9-36 through 940). A minimal basis set is used of STO s. The Roothaan-Hall equations are solved in a self-consistent field fashion, however not aU of the integrals are actually solved. In the most severe approximation, there is complete neglect of differential overlap (CNDO). 

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. 

These equations have to be solved iteratively to reach self-consistency. They can be cast in matrix form yielding Roothaan-Hall-like equations (Hall 1951 Roothaan 1951). 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

All quantum chemical calculations are based on the self-consistent field (SCF) method of Hatree and Fock (1928-1930) and the MO theory of Hund, Lennard-Jones, and Mulliken (1927-1929). A method of obtaining SCF orbitals for closed shell systems was developed independently by Roothaan and Hall in 1951. In solving the so-called Roothan equations, ab initio calculations, in contrast to semiempirical treatments, do not use experimental data other than the values of the fundamental physical constants. 

---