Møller–Plesset perturbation theory

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 𝜑_i and 𝜑_j are canonical occupied orbitals and 𝜑_a and 𝜑_b are virtual (or unoccupied) orbitals. The quantities ε_i, ε_j, ε_a, and ε_b are the corresponding orbital energies. Clearly, through second-order in the correlation potential, the total electronic energy is given by the Hartree–Fock energy plus second-order MP correction: E ≈ E_HF + E_MP2. The solution of the zeroth-order MP equation (which by definition is the Hartree–Fock equation) gives the Hartree–Fock energy. The first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy.

Equivalent expressions are obtained by a slightly different partitioning of the Hamiltonian, which results in a different division of energy terms over zeroth- and first-order contributions, while for second- and higher-order energy corrections the two partitionings give identical results. The formulation is commonly used by chemists, who are now large users of these methods.[2] This difference is due to the fact, well known in Hartree–Fock theory, that

${\displaystyle \langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle \neq 0\qquad \Longleftrightarrow \qquad E_{\text{HF}}\neq 2\sum _{i=1}^{N/2}\varepsilon _{i}.}$

(The Hartree–Fock energy is not equal to the sum of occupied-orbital energies). In the alternative partitioning, one defines

${\displaystyle {\hat {H}}_{0}\equiv {\hat {F}},\qquad {\hat {V}}\equiv {\hat {H}}-{\hat {F}}.}$

Clearly, in this partitioning,

${\displaystyle E_{\text{MP0}}=2\sum _{i=1}^{N/2}\varepsilon _{i},\qquad E_{\text{MP1}}=E_{\text{HF}}-2\sum _{i=1}^{N/2}\varepsilon _{i}.}$

Obviously, with this alternative formulation, the Møller–Plesset theorem does not hold in the literal sense that E_MP1 ≠ 0. The solution of the zeroth-order MP equation is the sum of orbital energies. The zeroth plus first-order correction yields the Hartree–Fock energy. As with the original formulation, the first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy. To reiterate, the second- and higher-order corrections are the same in both formulations.