Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Roothaan's Equations

In order to illustrate the practical application of the Hartree-Fock equation using the basis set expansion method of Roothaan,[15,16] the case of a closed-shell molecule treated via the restricted formalism will be examined. In the case of a closed-shell, N-electron molecule, the restricted Slater determinant contains a total of N/2 spatial orbitals, each of which is occupied by a pair of electrons with opposite spins. The initial step in solving the Hartree-Fock equations is to integrate over the spin coordinates and rewrite the equation in terms of only the spatial orbitals. Remember that the spin functions were never explicitly defined because the effects of spin are taken into account through cancellations due to orthonormality and the Antisymmetry Principle. Therefore, integration over the spin orbitals strips the spin functions from the spatial functions and, in the process, removes all situations unallowed by the spin. Upon completion of the spin integration, the following result is obtained:

$\displaystyle \hat{f} (r_1) \psi_j (r_1)$	=	$\displaystyle \hat{h} (r_1) \psi_j(r_1) + \left[ 2\sum_{a}^{\frac{N}{2}} \int dr_2\psi_a^*(r_2) \frac{1}{{r_1}_2} \psi_a(r_2) \right] \psi_j(r_1)$	(49)
	=	$\displaystyle -\left[ \sum_{a}^{\frac{N}{2}} \int dr_2\psi_a^*(r_2) \frac{1}{{r_1}_2} \psi_j(r_2) \right] \psi_a (r_1) = E_j\psi_j(r_1).$	(50)

From this result, one can extract the spatial form of the closed-shell Fock operator $\hat{f}(r_1)$ by comparing the first two terms. Hence, the spatial form of the closed-shell Fock operator may be written as

$\begin{displaymath}\hat{f} (r_1) = \hat{h} + \sum_{a}^{\frac{N}{2}} 2 \hat{J}_a (r_1) -\hat{K}_a (r_1)\end{displaymath}$

(51)

where $\hat{J}_a (r_1)$ and $\hat{K}_a (r_1)$ are the Coulomb and exchange operators recast in terms of spatial orbitals:

$\displaystyle \hat{J}_a (r_1)$	=	$\displaystyle \int dr_2 {\psi_a}^*(r_2) \frac{1}{{r_1}_2}\psi_a (r_2)$	(52)
$\displaystyle \hat{K}_a (r_1) \psi_b$	=	$\displaystyle \left[ \int dr_2 {\psi_a}^*(r_2)\frac{1}{{r_1}_2} \psi_b (r_2) \right]\psi_a {r_1}.$	(53)

Given the spatial form of the closed-shell Fock operator, the Hartree-Fock equation may be recast, as follows, in terms of spatial orbitals:

$\begin{displaymath}\hat{f}(r_1) \psi_j (r_1) = E_j \psi_j (r_1).\end{displaymath}$

(54)

Moreover, by carrying through the spin integrations described above, the ground state energy of a closed-shell molecule may likewise be recast in terms of the spatial orbitals:

$\begin{displaymath}E_0 = 2 \sum_a \langle \psi_a \vert \hat{h} \vert \psi_a \ran...... \rangle - \langle \psi_a \psi_b\vert \psi_b \psi_a \rangle.\end{displaymath}$

(55)

For the closed shell case, the sole criterion required to obtain a minimum expectation value for the total energy, the Hartree-Fock equation, may be expressed entirely as a function of the spatial orbitals, resulting in the integro-differential equation (1.54). Like the Schrödinger equation, this is an eigenvalue equation that must be solved for the orbital energy, $\epsilon_i$ , and the wave function, $\psi_i$ . The subscript on the wave function serves to indicate that it is only one of several spatial orbitals required to construct a complete Slater determinant and accurately model a closed-shell molecule. In a closed-shell molecule every spatial orbital is occupied by a pair of electrons with opposite spins; therefore, to form a complete Slater determinant description one must solve a Hartree-Fock equation for each doubly occupied spatial orbital. The contribution of Roothaan[15,16] to the Hartree-Fock method was to determine how to simultaneously solve these equations by writing the spatial orbitals as a linear combination of a set of basis functions. Roothaan's method is elegant in its simplicity because the differential equation is reduced to a set of algebraic equations that may be solved using matrix methods which can, in turn, be efficiently implemented on modern computers. The first step in the reduction is to expand the unknown spatial orbital $\psi_i$ in terms of a generic basis set,

$\begin{displaymath}\psi_i = \sum_{v=1}^{K} {C_v}_i \phi_v,\end{displaymath}$

(56)

and substitute this expansion into the Hartree-Fock equation:

$\begin{displaymath}f(r_1) \sum_{v}^K {C_v}_i \phi_v(r_1) = E_i \sum_{v}^K {C_v}_i\phi_v (r_1).\end{displaymath}$

(57)

By multiplying equation (1.57) through on the left by ${\phi_u}^*(r_1)$ and then integrating, one obtains

$\begin{displaymath}\sum_{v}^K {C_v}_i \int dr_1 {\phi_u}^*(r_1) \hat{f}(r_1) \ph...... E_i \sum_{v}^H {C_v}_i \int dr_1 {\phi_u}^* (r_1) \phi_v(r_1)\end{displaymath}$

(58)

which may be recognized as a matrix equation after two definitions have been made. Starting with the right-hand side, the first definition needed is that of the Fock matrix. The Fock matrix is defined to have elements

F_uv	=	$\displaystyle \int dr_1 {\phi_u}^* (r_1) \hat{f}(r_1) \phi_v (r_1)$	(59)
	=	$\displaystyle \int dr_1 {\phi_u}^* (r_1) \left[ h(r_1) + \sum_a (2 \hat{J_a}(r_1) - \hat{K_a}(r_1)) \right] \phi_v(r_1)$
	=	$\displaystyle \int dr_1 \phi_u^*(1) h(1) \phi_v(1) +$
		$\displaystyle \sum_a \left[ 2 \int dr_1 \phi_u^(1) \hat{J_a}(1) \phi_v(1)- \int dr_1 \phi_u^(1) \hat{K_a}(1) \phi_v(1) \right]$	(60)
	=	$\displaystyle h_{uv} + \sum_a 2(uv\vert aa) - (ua\vert av)$	(61)
	=	$\displaystyle h_{uv} + \sum_a \left [\sum_{\rho\sigma} C_\rho^a C_\sigma^a2(uv\vert\rho\sigma) - (u \rho\vert\sigma v) \right]$	(62)
	=	$\displaystyle h_{uv} + \sum_{\rho\sigma} (\sum_a C_\rho^a C_\sigma^a)\left[ 2(uv\vert\rho\sigma) - (u \rho\vert\sigma v) \right]$	(63)
	=	$\displaystyle h_{uv} + \sum_{\rho\sigma} D_{\rho\sigma}\left[ 2(uv\vert\rho\sigma) - (u \rho\vert\sigma v) \right]$	(64)

and is the matrix representation of the Fock operator in the generic basis set. The quantity $D_{\rho\sigma}$ is known as either the density matrix or charge-density bond-order matrix because it recurs in the determination of these properties. The second definition needed is that of the overlap matrix, which has elements

$\begin{displaymath}{S_u}_v = \int dr_1 {\phi_u}^* (r_1) \phi_v(r_1)\end{displaymath}$

(65)

and represents the overlap between $\phi_u$ and $\phi_v$ as a positive or negative fraction between zero and one. The closer the overlap is to one, the closer the two basis functions approach linear dependence. The overall sign depends on the spatial orientation and signs of the individual basis functions. Given these definitions, the integrated Hartree-Fock equation can be written as

$\begin{displaymath}\sum_v {F_u}_v {C_v}_i = \epsilon_i \sum_{v} {S_u}_v {C_v}_i,\end{displaymath}$

(66)

which can be written more compactly as the standard matrix equation

$\begin{displaymath}{\bf FC} = {\bf SCE}.\end{displaymath}$

(67)

Finally, in order to simplify the equations to a computationally efficient form, one performs the orthogonalizing transformation

$\begin{displaymath}{\bf F^{T}} = {\bf S^{\frac{1}{2}} F S^{\frac{1}{2}}}\end{displaymath}$

(68)

which reduces equation (1.65) to an eigenvalue equation:

$\displaystyle {\bf F^{T} C^{\dagger}}$	=	$\displaystyle {\bf C^{\dagger} E},$	(69)
$\displaystyle {\rm with} \; {\bf C^{\dagger}}$	=	$\displaystyle {\bf S^{\frac{1}{2}} C}.$	(70)

From the mathematics of linear algebra, this eigenvalue equation may be solved by diagonalizing the transformed Fock matrix ${\bf F^T}$ . Many efficient programs exist that can diagonalize a matrix such as ${\bf F^T}$ and return the eigenvalues, E, and eigenvectors, ${\bf C^{\dagger}}$ . The final step is to reverse the transformation (1.68) and obtain the optimal set of LCAO-MO coefficients.

This page maintained by Brian C. Hoffman