The Roothaan-Hall Equations

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall  

F is called the Fock matrix, and it represents the average effects of the field of all the electrons on each orbital. For a closed shell system, its elements are  

The coefficients are summed over the occupied orbitals only, and the factor of two comes from the fact that each orbital holds two electrons. 

Finally, the matrix S from Equation 31 is the overlap matrix, indicating the overlap between orbitals. 

Both the Fock matrix—through the density matrix—and the orbitals depend on the molecular orbital expansion coefficients. Thus, Equation 31 is not linear and must be solved iteratively. The procedure which does so is called the Self-Consistent Field 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

The LCAO approximation for the wave functions in the Hartree-Fock equations 

These are just the secular equations shown in equation set (7-2) with F in place of H and the stacked matrix Eq. (7-6) of eigenvectors in place of a single eigenvector. In matrix notation 

The Roothaan-Hall equation set (9-57) is often written in the notation 

If we assume that S = I (which is not true in general), the matr ix form of the Fock equation can be written 

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Closed-shell Systems and the Roothaan-Hall 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. 

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. 

The Roothaan-Hall equations can now be manipulated as follows. Both sides of Equation (2.162) are pre-multiplied by the matrix... 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

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

Z is the nuclear charge, R-r is the distance between the nucleus and the electron, P is the density matrix (equation 16) and (qv Zo) are two-electron integrals (equation 17). f is an exchange/correlation functional, which depends on the electron density and perhaps as well the gradient of the density. Minimizing E with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations , analogous to the Roothaan-Hall equations (equation 11). 

The first shortcoming of a minimal basis set.. . bias toward atoms with spherical environments. . . may be addressed by providing two sets of valence basis functions ( inner and outer functions). For example, proper linear combinations determined in the solution of the Roothaan-Hall equations allow for the fact that the p orbitals which make up a tight o bond need to be more contracted than the p orbitals which make up a looser k bond. 

This is referred to as the Neglect of Diatomic Differential Overlap or NDDO approximation. It reduces the number of electron-electron interaction terms from 0(N ) in the Roothaan-Hall equations to 0(N ), where N is the total number of basis functions. 

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. 

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. 

Pople-Nesbet Equations. The set of equations describing the best Unrestricted Single Determinant Wavefunction within the LCAO Approximation. These reduce the Roothaan-Hall Equations for Closed Shell (paired electron) systems. 

In the m sets of equations 5.54-1-5.54-m each set itself contains m equations (the subscript of , for example, runs from 1 to m), for a total ofmxm equations. These equations are the Roothaan-Hall version of the Hartree-Fock equations they were obtained by substituting for the MO s j/ in the HF equations a linear combination of basis functions ( s weighted by c s). The Roothaan-Hall equations are usually written more compactly, as... 

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... 

The fact that the Roothaan-Hall equations Eqs. 5.56 are actually a total of mzm equations suggests that they might be expressible as a single matrix equation, since the single matrix equation AB = 0, where A and B are m x m matrices, represents m x m simple equations, one for each element of the product matrix AB (work it out for two 2x2 matrices). A single matrix equation would be easier to work with than m1 equations and might allow us to invoke matrix diagonalization as in the case of the simple and extended Hiickel methods (Sections 4.3.4 and 4.4.1). To subsume the sets of equations 5.54-1-5.54-m, i.e. Eqs. 5.56, into one matrix... 

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. 

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). 

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... 

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 derivation of the Roothaan-Hall equations involves some key concepts Slater determinant, Schrodinger equation, explicit Hamiltonian operator,... 

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). 

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