5.1. Post-Hartree Fock Methods: Theory Recap

This section give a brief summary of the theory behind two Post-Hartree-Fock methods: Configuration Interaction (CI) and Møller Plesset Perturbation Theory (MPn, where \(n\) indicates the order of MP theory).

5.1.1. Configuration Interaction Theory

Of all the ab initio methods, CI is probably the easiest to understand, and perhaps one of the hardest to implement efficiently on a computer! The basic idea of CI is to improve the HF solution by increasing the space of all possible many-electron wavefunctions from a single Slater determinant (as in HF theory), to a set of many, in principle infinite, Slater determinants.

An arbitrary \(N\)-electron wavefunction \(\left|\Psi_j\right>\) can be expanded as a linear combination of \(N\)-electron basis functions \(\left|\Phi_i\right>\)

\[\begin{aligned} \left|\Psi_j\right>=\sum_{i=1}^M c_{ij}\left|\Phi_j\right> \end{aligned} \]

In the case of HF theory, this expansion had a single element (\(M=1\)), i.e. a single Slater determinant made of the occupied HF one-electron orbitals. In general, there are \(M\) possible \(N\)-electron basis functions to be used, where a complete basis is reached if \(M\) is infinite.

To better the idea behind CI theory, we can express the \(N\)-electron basis function as an expansion of “excitations” from the HF “reference” determinant:

\[\begin{aligned} \left| \Psi_j \right> = c_0 \left|\Phi_0 \right> +\sum_{ra} c_{a}^{r}\left|\Phi_{a}^{r}\right>+\sum_{a<b,r<s} c_{ab}^{rs}\left|\Phi_{ab}^{rs}\right> + \dots \end{aligned}\]

where \(\left|\Phi_{a}^{r}\right>\) indicates the Slater determinants formed by replacing the spin-orbital \(a\) in \(\left|\Phi_0\right>\) with the spin orital \(r\), and so on. Every \(N\)-electron Slater determinant can be described by the set of \(N\) spin orbitals from which it is formed, and this set is often referred to as configuration. Thus, the configuration interaction methods is nothing more than the matrix solution of the time-independent non-relativistic Schrödinger equation, using the (truncated) expression for \(\left|\Psi_j\right>\), as expressed above.

In practice, the CI expansion is typically truncated according to excitation level for computational tractability. CIS will include all excitations where one electron is promoted from an occupied orbital to an unoccupied one. Similarly, CISD will include all single and double excitations, and so on for CISDT, CISDTQ, etc. If all possible \(N\)-electrons excitations are taken into account, the porocedure is called full CI. If the one-electron basis is also complete (never in practice since we are using a computer, but it may be in theory), we have complete CI.

In practice, Full CI has been performed for very small molecules for benchmarks studies. CIS is sometimes used for approximate excited state calculations, while otherwise CI method is not often used because it is too expensive and other methods give results of comparable quality at lower cost.

5.1.2. Møller-Plesset Perturbation Theory

Perturbation methods assume that the problem under investigation only differs slightly from a problem which has already been solved, exactly or approximately. In quantum mechanics, perturbation methods can be used to add corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT).

According to Rayleigh-Schrödinger perturbation theory, an instantaneous perturbation of a system described by a Hamiltonian \(\hat{\mathrm{H}}_0\) is described by the perturbed Hamiltonian

\[\begin{aligned} \hat{\mathrm{H}} = \hat{\mathrm{H}}_0 + \lambda V.\end{aligned}\]

For a sufficiently weak perturbation, the eigenstates and eigenvalues may then be expanded in a power series:

\[\begin{split}\begin{aligned} & \bra{\Psi} = \bra{\Psi^{(0)}} + \lambda\bra{\Psi^{(1)}} + \lambda^2 \bra{\Psi^{(2)}} + \dots\\ & E_\Psi = E_\Psi^{(0)} + \lambda E_\Psi^{(1)} + \lambda^2 E_\Psi^{(2)} + \dots\end{aligned}\end{split}\]

The perturbing operator \(\hat{\mathrm{V}}\) which is introduced in Møller-Plesset perturbation theory is the difference between the true ground state Hamiltonian and the Hartree-Fock Hamiltonian; hence, the perturbation may be written in terms of excited Slater determinants (which also imposes orthonormality). By carrying out the expansion, collecting terms of the same order and applying the Slater-Condon rules (see chapter 5.4 of the course script), one concludes that only doubly-excited Slater determinants can contribute to the second- and third-order term. Only the fourth-order energy will include up to quadruply excited determinants. By truncating the expansion at second order, one arrives at the MP2 expression for the energy, where electron correlation is now included as a perturbation to the uncorrelated Hartree-Fock wavefunction:

\[\begin{aligned} E_0^{(2)} = \frac{1}{4} \sum_a \sum_b \sum_r \sum_s \frac{\left|\Bra{ab}\Ket{rs}\right|^2}{\epsilon_a+\epsilon_b-\epsilon_r-\epsilon_s},\end{aligned}\]

where \(a,b\) denote occupied, \(r,s\) denote virtual orbitals and \(\epsilon\) are the respective energy eigenvalues.

The total energy is given by the perturbative contribution and the Hartree-Fock energy:

\[\begin{aligned} E_0^{total} = E_0^{HF} + \sum_{n=1} E_0^{(n)}.\end{aligned}\]

(We note en passant that the first-order Møller Plesset energy is nothing but the expression for the total Hartree Fock energy, such that the zero-order term is given by a sum over orbital eigenvalues.) The MP approach is often accurate enough; however, it is not variational, i.e. the resulting MPn energy may be lower than the true ground-state energy.