What are the types of PT applied to electronic structure methods?

Question

The motivation for this question is similar in nature to the series of questions on different methods :

• What are the types of SCF?
• What are the types of MCSCF?
• What are the types of Quantum Monte Carlo?
• What are the types of ab initio Molecular Dynamics?
• What are the types of DFT?
• What are the types of DMRG?

The answers to these questions were really insightful, so I am looking for an answer that briefly describes the different variants of Perturbation Theory (PT), and the context in which they are applied on different electronic structure methods.

Rayleigh-Schrodinger perturbation theory (RSPT) is the most common form that is used, but there's also the Lennard-Jones-Brillouin-Wigner perturbation theory (BWPT). In fact, RSPT could be derived from BWPT. Again, depending the type of partitioning of the Hamiltonian, there are two variants: Moller-Plesset and Epstein-Nesbet. Then there are also many-body and diagrammatic PT. These methods have also been applied with both multi and single reference schemes as well as multi-configurational methods.

Here are a few examples of such methods:

• Second order corrections to ground state energies(the standard MPn methods)
• Perturbative corrections to CI (see for example CIS(D) by Martin Head-Gordon et al)
• CASPT2 (multireference second-order PT)

Answer 1

Moller-Plesset (MPn)

Moller-Plesset perturbation theory combines the Rayleigh-Schrodinger style of perturbation expansion with a particular partitioning of the molecular Hamiltonian in order to compute the correlation energy (and/or perturbed wavefunctions).

We express the Hamiltonian $H$ as an unperturbed part $H_0$ and a perturbation $V$. For the unperturbed part, we choose a sum of Fock operators for each electron: $$H_0=\sum_i^Nf(i) = \sum_i^N h(i)+v^\text{HF}(i)$$ This is a convenient choice because the HF wavefunction is an exact eigenfunction of this Hamiltonian, $E_0^{(0)}=\sum_i^N \epsilon_i$.

This would correspond to MP0, which is never used on its own because it's a very poor prediction for the energy that can be easily (at least conceptually) improved on. To do this, we express the wavefunction and energy as perturbative expansions:

$$H=H_0+\lambda V$$ $$\Psi_0=\Psi_0^{(0)}+\lambda \Psi_0^{(1)}+\lambda^2\Psi_0^{(2)}+...$$ $$E_0=E_0^{(0)}+\lambda E_0^{(1)}+\lambda^2E_0^{(2)}+...$$

By plugging these expressions into the Schrodinger equation and separating by order (i.e how many factors of $\lambda$ are in each term), we get a series of energy/wavefunction contributions that ideally converge to the true values for an infinite order expansion (practically, we are hoping for very good convergence with just a few terms included).

This leads to a general form for the correction terms to the energy: $$E_0^{(n)}=\langle\Psi_0^{(0)}|V|\Psi_0^{(n-1)}\rangle$$ The general form for the wavefunction corrections is more involved, but the form for a given order is not conceptually difficult to obtain.

Now we introduce the particular perturbation for MP $$V=\sum_{i<j}r_{ij}^{-1}- \sum_iv^\text{HF}(i)$$

With $V$ we can explicitly compute a few orders of energy corrections and we call the resulting energy summing up to that order the MPn energy. MP1 is actually just the Hartree-Fock energy, so it takes until MP2 before we get a nontrivial result for our effort.

MPn methods are potentially useful whenever you need to include correlation effects. On the other hand, MPn calculations scale as $O(N^{3+n})$, making higher order expansions impractical for most systems (and often worse accuracy than comparably scaling coupled cluster methods). MP2 sees the most widespread use due to having a good balance of cost and accuracy. While DFT has started to take some of the work formerly reserved, MP2 still sees use both as an individual method and as a correction to DFT in double hybrid functionals.