Møller–Plesset perturbation theory

Møller–Plesset perturbation theory
Name	Møller–Plesset perturbation theory
Class	Post-Hartree–Fock method
Authors	Christian Møller, Milton S. Plesset
Year	1934
Publisher	Physical Review

Møller–Plesset perturbation theory

Møller–Plesset perturbation theory is a cornerstone post-Hartree–Fock method in computational chemistry for approximating electron correlation energy. Developed by Christian Møller and Milton S. Plesset in a seminal 1934 paper in Physical Review, it applies Rayleigh–Schrödinger perturbation theory to the Hartree–Fock wavefunction. The approach systematically improves upon the mean-field description of Hartree–Fock theory by treating electron correlation as a perturbation to the Fock operator.

Theoretical foundation

The theoretical foundation rests on partitioning the full molecular Hamiltonian of the Schrödinger equation into a solvable zeroth-order part and a perturbation. The unperturbed Hamiltonian is chosen as the sum of the one-electron Fock operators, with its eigenfunctions being the Slater determinants from a Hartree–Fock calculation. The perturbation is then defined as the difference between the exact electron-electron repulsion operator and the Hartree–Fock potential, which accounts for the mean-field approximation of electron interaction. This framework ensures the Hartree–Fock energy is the sum of the zeroth- and first-order energy corrections, making the first non-zero correlation energy contribution appear at the second order. The method is formally derived from many-body perturbation theory within the field of quantum chemistry.

Formulation and equations

In the standard formulation, the wavefunction and energy are expanded as a power series in a coupling parameter. The zeroth-order wavefunction is typically the Hartree–Fock reference wavefunction, denoted |Ψ⁽⁰⁾⟩. The first-order correction to the wavefunction involves a sum over all possible excited Slater determinants, generated by promoting electrons from occupied to virtual molecular orbitals. The second-order energy correction, which is the first to include electron correlation, is given by a well-known sum-over-states formula involving double excitations. This expression depends on two-electron repulsion integrals, often computed in the molecular orbital basis obtained from programs like GAUSSIAN or GAMESS (US).

Orders of perturbation (MP2, MP3, MP4, etc.)

The method is conventionally identified by the highest order of energy correction included. MP2 (second order) is the most widely used due to its favorable balance of cost and accuracy, capturing a significant portion of dynamic correlation. MP3 (third order) includes contributions from single excitations and double excitations but offers only marginal improvement over MP2 for most systems. MP4 (fourth order) incorporates triple excitations and quadruple excitations, providing greater accuracy for properties like molecular geometry but at substantially higher computational cost. Higher orders like MP5 and beyond are rarely used in practice due to prohibitive expense and issues with convergence of the perturbation series, which can be divergent for systems with significant static correlation.

Applications and limitations

The primary application is the calculation of accurate energies, geometries, and frequencies for molecules where Hartree–Fock theory is qualitatively correct. It is extensively used for studying intermolecular forces in complexes like the water dimer, reaction energies, and spectroscopic constants. A major limitation is its inability to properly describe systems with multireference character, such as transition metal complexes, diradicals, or during bond dissociation, where the Hartree–Fock reference is a poor starting point. The method is also not variational, meaning calculated energies can be below the true ground state energy.

Computational cost and scaling

The computational cost, characterized by its scaling with the number of basis functions (N), is a critical practical factor. The MP2 method scales formally as O(N⁵), due to the transformation of two-electron integrals from the atomic orbital basis to the molecular orbital basis and the subsequent energy summation. MP3 scales as O(N⁶), while MP4 scales as O(N⁷), primarily due to the inclusion of triple excitations. These steep scaling laws limit the application of higher-order methods to relatively small molecules. However, efficient implementations using density fitting or local correlation approximations in software like ORCA or Molpro can reduce the effective scaling of MP2 to near O(N⁴) or O(N³).

Connection to other methods

The theory is deeply connected to other quantum chemistry methodologies. It is a specific application of many-body perturbation theory within the Møller–Plesset partitioning scheme. MP2 can be seen as an approximation to the coupled cluster method, specifically to CCD (coupled cluster doubles). Conversely, coupled cluster methods like CCSD(T) often outperform even high-order MP calculations. The relationship with density functional theory is more distant, though some hybrid functionals incorporate a fraction of MP2-like correlation. It also serves as a benchmark for more approximate methods and is a component in some composite methods like the Gaussian-n theories developed by John Pople and his group.