Møller–Plesset perturbation theory

Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.^[1]

Rayleigh–Schrödinger perturbation theory[]

The MP perturbation theory is a special case of RS perturbation theory. In RS theory one considers an unperturbed Hamiltonian operator ${\displaystyle {\hat {H}}_{0}}$, to which a small (often external) perturbation ${\displaystyle {\hat {V}}}$ is added:

${\displaystyle {\hat {H}}={\hat {H}}_{0}+\lambda {\hat {V}}.}$

Here, λ is an arbitrary real parameter that controls the size of the perturbation. In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The perturbation is the correlation potential. In RS-PT the perturbed wave function and perturbed energy are expressed as a power series in λ:

${\displaystyle \Psi =\lim _{m\to \infty }\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)},}$

${\displaystyle E=\lim _{m\to \infty }\sum _{i=0}^{m}\lambda ^{i}E^{(i)}.}$

Substitution of these series into the time-independent Schrödinger equation gives a new equation as ${\displaystyle m\to \infty }$:

${\displaystyle \left({\hat {H}}_{0}+\lambda V\right)\left(\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)}\right)=\left(\sum _{i=0}^{m}\lambda ^{i}E^{(i)}\right)\left(\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)}\right).}$

Equating the factors of ${\displaystyle \lambda ^{k}}$ in this equation gives a kth-order perturbation equation, where k = 0, 1, 2, ..., m. See perturbation theory for more details.

Møller–Plesset perturbation[]

Original formulation[]

The MP-energy corrections are obtained from Rayleigh–Schrödinger (RS) perturbation theory with the unperturbed Hamiltonian defined as the shifted Fock operator,

${\displaystyle {\hat {H}}_{0}\equiv {\hat {F}}+\langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle }$

and the perturbation defined as the correlation potential,

${\displaystyle {\hat {V}}\equiv {\hat {H}}-{\hat {H}}_{0}={\hat {H}}-\left({\hat {F}}+\langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle \right),}$

where the normalized Slater determinant Φ₀ is the lowest eigenstate of the Fock operator:

${\displaystyle {\hat {F}}\Phi _{0}\equiv \sum _{k=1}^{N}{\hat {f}}(k)\Phi _{0}=2\sum _{i=1}^{N/2}\varepsilon _{i}\Phi _{0}.}$

Here N is the number of electrons in the molecule under consideration (a factor of 2 in the energy arises from the fact that each orbital is occupied by a pair of electrons with opposite spin), ${\displaystyle {\hat {H}}}$ is the usual electronic Hamiltonian, ${\displaystyle {\hat {f}}(k)}$ is the one-electron Fock operator, and ε_i is the orbital energy belonging to the doubly occupied spatial orbital φ_i.

Since the Slater determinant Φ₀ is an eigenstate of ${\displaystyle {\hat {F}}}$, it follows readily that

${\displaystyle {\hat {F}}\Phi _{0}-\langle \Phi _{0}|{\hat {F}}|\Phi _{0}\rangle \Phi _{0}=0\implies {\hat {H}}_{0}\Phi _{0}=\langle \Phi _{0}|{\hat {H}}|\Phi _{0}\rangle \Phi _{0},}$

i.e. the zeroth-order energy is the expectation value of ${\displaystyle {\hat {H}}}$ with respect to Φ₀, the Hartree-Fock energy. Similarly, it can be seen that in this formulation the MP1 energy

${\displaystyle E_{\text{MP1}}\equiv \langle \Phi _{0}|{\hat {V}}|\Phi _{0}\rangle =0}$.

Hence, the first meaningful correction appears at MP2 energy.

In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-PT formula is written in a basis of doubly excited Slater determinants. (Singly excited Slater determinants do not contribute because of the Brillouin theorem). After application of the Slater–Condon rules for the simplification of N-electron matrix elements with Slater determinants in bra and ket and integrating out spin, it becomes

${\displaystyle {\begin{aligned}E_{\text{MP2}}&=2\sum _{i,j,a,b}{\frac {\langle \varphi _{i}\varphi _{j}|{\hat {\tilde {v}}}|\varphi _{a}\varphi _{b}\rangle \langle \varphi _{a}\varphi _{b}|{\hat {\tilde {v}}}|\varphi _{i}\varphi _{j}\rangle }{\varepsilon _{i}+\varepsilon _{j}-\varepsilon _{a}-\varepsilon _{b}}}-\sum _{i,j,a,b}{\frac {\langle \varphi _{i}\varphi _{j}|{\hat {\tilde {v}}}|\varphi _{a}\varphi _{b}\rangle \langle \varphi _{a}\varphi _{b}|{\hat {\tilde {v}}}|\varphi _{j}\varphi _{i}\rangle }{\varepsilon _{i}+\varepsilon _{j}-\varepsilon _{a}-\varepsilon _{b}}}\\\end{aligned}}}$

where