18.5: The LCAO-MO Expansion

The HF equations \(F \phi_i = \epsilon_i \phi_i\) comprise a set of integro-differential equations; their differential nature arises from the kinetic energy operator in h, and the coulomb and exchange operators provide their integral nature. The solutions of these equations must be achieved iteratively because the \(J_i \text{ and } K_i\) operators in F depend on the orbitals \(\phi_i\) which are to be solved for. Typical iterative schemes begin with a 'guess' for those \(\phi_i \text{ which appear in } \Psi\), which then allows F to be formed. Solutions to \(F \phi_i = \epsilon_i \phi_i\) are then found, and those \(\phi_i\) which possess the space and spin symmetry of the occupied orbitals of \(\Psi\) and which have the proper energies and nodal character are used to generate a new F operator (i.e., new \(J_i \text{ and } K_i\) operators). The new \(\hat{F}\) operator then gives new \(\phi_i \text{ and } \epsilon_i\) via solution of the new equations:

\[\hat{F} \phi_i = \epsilon_i \phi_i. \label{EQ1} \]

This iterative process is continued until the \(\phi_i \text{ and } \epsilon_i\) do not vary significantly from one iteration to the next, at which time one says that the process has converged. This iterative procedure is referred to as the Hartree-Fock self-consistent field (SCF) procedure because iteration eventually leads to coulomb and exchange potential fields that are consistent from iteration to iteration.

In practice, solution of Equation \(\ref{EQ1}\) as an integro-differential equation can be carried out only for atoms (C. Froese-Fischer, Comp. Phys. Commun. 1, 152 (1970)) and linear molecules (P. A. Christiansen and E. A. McCullough, J. Chem. Phys. 67 , 1877 (1977)) for which the angular parts of the \(\phi_i\) can be exactly separated from the radial because of the axial- or full- rotation group symmetry (e.g., \(\phi_i = Y_{l,m} R_{n,l} \text{(r) for an atom and } \phi_i = e^{im\phi} R_{n,l,m} (r,\theta )\) for a linear molecule). In such special cases, \(F \phi_i = \epsilon_i \phi_i\) gives rise to a set of coupled equations for the \(R_{n,l}\text{(r) or R}_{n,l,m}\)(r,q) which can and have been solved. However, for non-linear molecules, the HF equations have not yet been solved in such a manner because of the three-dimensional nature of the \(\phi_i\) and of the potential terms in F.

In the most commonly employed procedures used to solve the HF equations for non-linear molecules, the \(\phi_i\) are expanded in a basis of functions \(\chi_m\) according to the LCAO-MO procedure:

\[ \phi_i = \sum\limits_\mu C_{\mu ,i}\chi_\mu . \nonumber \]

Doing so then reduces F \(\phi_i = \epsilon_i \phi_i\) to a matrix eigenvalue-type equation of the form:

\[ \sum\limits_\nu F_{\mu ,\nu}C_{\nu ,i} = \epsilon_i \sum\limits_\nu \textbf{S}_{\mu ,\nu}C_{\nu ,i}, \nonumber \]

where \(\textbf{S}_{\mu ,\nu} = \langle \chi_\mu \big| \chi_\nu \rangle\) is the overlap matrix among the atomic orbitals (aos) and

\[ F_{\mu ,\nu} = \langle \chi_\mu \big| h \big| \chi_\nu \rangle + \sum\limits_{\delta ,\kappa} \left[ \gamma_{\delta ,\kappa} \langle \chi_\mu \chi_\delta \big| g \big| \chi_\nu \chi_\kappa \rangle -\gamma_{\delta ,\kappa}^{ex} \langle \chi_\mu \chi_\delta \big| g \big| \chi_\kappa \chi_\nu \rangle \right] \nonumber \]

is the matrix representation of the Fock operator in the ao basis. The coulomb and exchange-density matrix elements in the ao basis are:

\[ \gamma_{\delta ,\kappa} = \sum\limits_{i\text{(occupied)}}C_{\delta ,i}C_{\kappa ,i} \text{, and} \nonumber \]

\[ \gamma_{\delta ,\kappa}^{ex} = \sum\limits_{i\text{(occ., and same spin)}}C_{\delta ,i}C_{\kappa ,i}, \nonumber \]

where the sum in \(\gamma_{\delta ,\kappa}^{ex}\) runs over those occupied spin-orbitals whose \(m_s\) value is equal to that for which the Fock matrix is being formed (for a closed-shell species, \(\gamma_{\delta ,\kappa}^{ex} = \frac{1}{2} \gamma_{\delta ,\kappa}\)).

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution; the \(F_{\mu ,\nu}\) matrix elements depend on the \(C_{\nu ,i}\) LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations:

\[ \sum\limits_{\nu} F_{\mu ,\nu}C_{\nu ,i} = \epsilon_i \sum\limits_\nu \textbf{S}_{\mu ,\nu}C_{\nu ,i}. \label{Roothan} \]

One should also note that, just as \(F \phi_i = \epsilon_i \phi_j\) possesses a complete set of eigenfunctions, the matrix \(F_{\mu ,\nu}\), whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues \(\epsilon_i\) and M eigenvectors whose elements are the \(C_{\nu ,i}.\) Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with \(C_{\nu ,i}\) coefficients obtained via solution of Equations \(\ref{Roothan}\).