# 10.8: The Self-Consistent Field and the Hartree-Fock Limit

- Page ID
- 1976

In a modern *ab initio *electronic structure calculation on a closed shell molecule, the electronic Hamiltonian is used with a single determinant wavefunction. This wavefunction, \(\psi\), is constructed from molecular orbitals, \(\psi\) that are written as linear combinations of contracted Gaussian basis functions, \(\varphi\)

\[\varphi _j = \sum \limits _k c_{jk} \psi _k \label {10.69}\]

The contracted Gaussian functions are composed from primitive Gaussian functions to match Slater-type orbitals (STOs). The exponential parameters in the STOs are optimized by calculations on small molecules using the nonlinear variational method and then those values are used with other molecules. The problem is to calculate the electronic energy from

\[ E = \dfrac {\int \psi ^* \hat {H} \psi d \tau }{\int \psi ^* \psi d \tau} \label {10.70}\]

and find the optimum coefficients \(c_{jk}\) for each molecular orbital in Equation \(\ref{10.69}\) by using the Self Consistent Field Method and the Linear Variational Method to minimize the energy as was described in the previous chapter for the case of atoms.

To obtain the total energy of the molecule, we need to add the internuclear repulsion to the electronic energy calculated by this procedure. The total energy of the molecule can be calculated for different geometries (i.e. bond lengths and angles) to find the minimum energy configuration. Also, the total energies of possible transition states can be calculated to find the lowest energy pathway to products in chemical reactions.

\[ V_{rs} = \sum \limits _{r=1}^{N-1} \sum \limits _{s=r+1}^{N} \dfrac {Z_r Z_s}{r_{rs}} \label {10.71}\]

As we improve the basis set used in calculations by adding more and better functions, we expect to get better and better energies. The variational principle says an approximate energy is an upper bound to the exact energy, so the lowest energy that we calculate is the most accurate. At some point, the improvements in the energy will be very slight. This limiting energy is the lowest that can be obtained with a single determinant wavefunction. This limit is called the *Hartree-Fock limit*, the energy is the *Hartree-Fock energy*, the molecular orbitals producing this limit are called *Hartree-Fock orbitals*, and the determinant is the *Hartree-Fock wavefunction*.

## Contributors

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")