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.
is added: Here, λ is an arbitrary real parameter that controls the size of the perturbation.
In RS-PT the perturbed wave function and perturbed energy are expressed as a power series in λ: Substitution of these series into the time-independent Schrödinger equation gives a new equation as
The MP-energy corrections are obtained from Rayleigh–Schrödinger (RS) perturbation theory with the unperturbed Hamiltonian defined as the shifted Fock operator, and the perturbation defined as the correlation potential, where the normalized Slater determinant Φ0 is the lowest eigenstate of the Fock operator: 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),
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 where 𝜑i and 𝜑j are canonical occupied orbitals and 𝜑a and 𝜑b are virtual (or unoccupied) orbitals.
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.
As with the original formulation, the first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy.
Convergence can be slow, rapid, oscillatory, regular, highly erratic or simply non-existent, depending on the precise chemical system or basis set.
[9][10] The eigenvalues of the response density matrix (which are the occupation numbers of the MP2 natural orbitals) can therefore be greater than 2 or negative.
[11] Additionally, various important molecular properties calculated at MP3 and MP4 level are no better than their MP2 counterparts, even for small molecules.
[12] For open shell molecules, MPn-theory can directly be applied only to unrestricted Hartree–Fock reference functions (since UHF states are not in general eigenvectors of the Fock operator).
However, the resulting energies often suffer from severe spin contamination, leading to large errors.
Some of the ROHF based MP2-like theories suffer from spin-contamination in their perturbed density and energies beyond second-order.
Multi-configurational self-consistent field (MCSCF) methods use several determinants and can be used for the unperturbed operator, although not uniquely, so many methods, such as complete active space perturbation theory (CASPT2),[25] and Multi-Configuration Quasi-Degenerate Perturbation Theory (MCQDPT),[26][27] have been developed.