Møller-Plesset Perturbation Theory

Møller-Plesset Perturbation theory [46,47] treats the exact Hamiltonian, $\hat{H}$, as a small perturbation from the HF Hamiltonian, $\hat{H}_{0}$ -- the sum of the one-electron Fock operators defined by equation (1.34). That is,
$$\begin{align}\hat{H} = \hat{H}_{0} + \lambda \hat{V}. \end{align}$$
If we expand the exact energy and wavefunction in terms of the perturbation
 $$\begin{align} \begin{split} \Psi_{i} = \Psi^{(0)}_{i} + \lambda \Psi^{(1)}_{i} +... ...lambda E^{(1)}_{i} + \lambda^{2} E^{(2)}_{i} + \, \ldots, \end{split}\end{align}$$
where $\Psi^{(n)}_{i}$ is the n-th state HF wavefunction. Substituting equations (1.65) into the Schrödinger equation and collating the powers of $\lambda$ gives the equations:

 

$$\hat{H}_{0}\Psi^{(0)}_{i}$$ = E⁽⁰⁾_i (5.5)

$$\hat{H}_{0}\Psi^{(1)}_{i} + \hat{V}\Psi^{(0)}_{i} E^{(0)}_{i}\Psi^{(1)}_{i} + E^{(1)}_{i}\Psi^{(0)}_{i}$$ = (5.6)

$$\hat{H}_{0}\Psi^{(2)}_{i}+\hat{V}\Psi^{(1)}_{i} E^{(0)}_{i}\Psi^{(2)}_{i}+E^{(1)}_{i}\Psi^{(1)}_{i}+E^{(2)}_{i}\Psi^{(0)}_{i}$$ = (5.7)

and so on. Multiplying each of these equations on the left by $\Psi_{0}$ and integrating over all space yields expressions for E⁽ⁿ⁾ in terms of $\hat{V}$ and $\Psi^{(n-1)}$:

$$\langle \Psi^{(0)}_{i} \vert \hat{H}_{0} \vert \Psi^{(0)}_{i} \rangle$$ E⁽⁰⁾_i = (5.8)

$$\langle \Psi^{(0)}_{i} \vert \hat{V} \vert \Psi^{(0)}_{i} \rangle$$ E⁽¹⁾_i = (5.9)

$$\langle \Psi^{(0)}_{i} \vert \hat{V} \vert \Psi^{(1)}_{i} \rangle$$ E⁽²⁾_i = (5.10)

$$\langle \Psi^{(0)}_{i} \vert \hat{V} \vert \Psi^{(2)}_{i} \rangle$$ E⁽³⁾_i = (5.11)

and so on. From this it can be seen that the HF energy is the sum of E⁽⁰⁾₀ and E⁽¹⁾₀.

By using the expansion
$$\begin{align}\Psi^{(1)}_{i} = \sum_{n} c^{(1)}_{n} \Psi^{(0)}_{n} \end{align}$$
in equation (1.67) and rearranging, an expression for the coefficients can be found:
$$\begin{align}c^{(1)}_{n} = - \frac{\langle \Psi^{(0)}_{n} \vert \hat{V} \vert \Psi^{(0)}_{0} \rangle}{E^{(0)}_{n}-E^{(0)}_{0}}. \end{align}$$
Inserting this expansion into the second-order energy expression gives a readily computable formula for the second-order Møller-Plesset (MP2) energy
 $$\begin{align} E^{(2)}_{0} = \frac{1}{4}\sum_{ijab}\frac{((ia\vert jb)-(ib\vert ja))^{2}}{\epsilon_{a}+\epsilon_{b}-\epsilon_{i}-\epsilon_{j}}. \end{align}$$

The MPn energies are size consistent, but not variational. Size consistency can be seen by considering the MP2 energy for two widely separated systems A and B. The energy expression will only be non-zero if the orbitals $\psi_{i},\psi_{j},\psi_{a},\psi_{b}$ are all on A, or all on B. Thus there are no cross-correlation terms.

The computational cost scaling of the MPn energy is O(N⁽ⁿ⁺³⁾). For MP2 this arises from the need to transform the integrals over atomic orbitals into integrals over molecular orbitals. MP2 is a relatively cheap form of correlation, yet the higher orders become comparitively very expensive, especially considering that a CC or QCI calculation may be more accurate.

Perturbation theory relies on the starting wavefunction being close to the exact wavefunction. When this is the case, convergence of the MP series is rapid. However, when bonds are stretched the MP series becomes oscillatory. Also, if a UHF wavefunction with high spin is used, convergence can be extremely slow [28,48]. Recent results have suggested that with large basis sets divergence can occur even for systems where HF is a good starting point [49]. For all these reasons it is expected that MP theory will become less popular.