Møller–Plesset perturbation theory
Møller–Plesset perturbation theory (MPPT) is a post-Hartree–Fock ab initio method in quantum chemistry that applies Rayleigh–Schrödinger perturbation theory to account for electron correlation energy beyond the mean-field approximation of Hartree–Fock theory. It treats the Hartree–Fock Hamiltonian as the unperturbed zeroth-order operator and the difference between the exact Hamiltonian and this Fock operator as the perturbation, yielding a perturbative series for the correlation energy denoted as MPn, where n indicates the order of approximation.[1] The second-order term (MP2) is the most widely used, incorporating pairwise electron correlation through double excitations and recovering approximately 90% of the total correlation energy for many systems.[2] Originally proposed by Christian Møller and Milton S. Plesset in 1934 as an approximation for many-electron systems, MPPT remained largely overlooked for decades until its revival in the 1970s through connections to many-body perturbation theory.[3] During the 1970s and 1980s, implementations up to fifth order (MP5) were developed, but higher-order terms often exhibited oscillatory convergence or divergence, limiting practical use to MP2 and occasionally MP3 or MP4.[3] The theory's canonical formulation scales steeply with system size—fifth power for MP2—though linear-scaling variants, such as local MP2 (LMP2) and resolution-of-the-identity MP2 (RI-MP2), have enabled applications to large molecules with thousands of atoms since the 1990s.[3][4] MPPT plays a central role in computational chemistry for predicting molecular energies, geometries, and properties, often serving as a benchmark or component in hybrid methods like coupled-cluster theory with perturbative triples (CCSD(T)).[1] It excels in describing dynamic electron correlation in closed-shell systems but struggles with strong correlation or open-shell cases without modifications, such as spin-component scaling or orbital optimization.[3] Despite limitations like basis set superposition error sensitivity and size-consistency issues in non-canonical forms, MP2 remains a cornerstone for thermochemistry, non-covalent interactions, and excited-state calculations due to its balance of accuracy and computational efficiency.[3]Theoretical Background
Rayleigh–Schrödinger Perturbation Theory
Rayleigh–Schrödinger perturbation theory (RSPT) provides a systematic framework for approximating the eigenvalues and eigenfunctions of a quantum mechanical Hamiltonian that consists of a solvable unperturbed part and a small perturbation.[5] It is particularly applicable to the time-independent Schrödinger equation, where the total Hamiltonian is partitioned as H = H_0 + \lambda V, with H_0 being the unperturbed Hamiltonian whose spectrum is known, V the perturbation, and \lambda a small dimensionless parameter that scales the strength of the perturbation. The theory assumes a non-degenerate unperturbed ground state, meaning the eigenvalue E_0^{(0)} of interest has no other states at the same energy, and that \lambda is sufficiently small relative to the gaps between E_0^{(0)} and other unperturbed energies to ensure the series converges.[5] The energy E(\lambda) and wavefunction \Psi(\lambda) are expanded as power series:E(\lambda) = E_0 + \lambda E^{(1)} + \lambda^2 E^{(2)} + \cdots + \lambda^n E^{(n)} + \cdots,
\Psi(\lambda) = \Psi_0 + \lambda \Psi^{(1)} + \lambda^2 \Psi^{(2)} + \cdots + \lambda^n \Psi^{(n)} + \cdots,
where H_0 \Psi_0 = E_0 \Psi_0 and the corrections are determined order by order.[5] The nth-order energy correction follows the recursive formula
E^{(n)} = \langle \Psi_0 | V | \Psi^{(n-1)} \rangle - \sum_{k=1}^{n-1} E^{(k)} \langle \Psi_0 | \Psi^{(n-k)} \rangle,
with the wavefunctions normalized such that \langle \Psi_0 | \Psi_0 \rangle = 1 and \langle \Psi_0 | \Psi^{(m)} \rangle = 0 for m \geq 1.[5] This yields the first-order correction E^{(1)} = \langle \Psi_0 | V | \Psi_0 \rangle and higher orders involving projections onto the unperturbed excited states.[5] The origins of RSPT trace back to Lord Rayleigh's work in the 1870s on classical problems of vibrating strings with small density variations, which introduced the perturbative expansion technique.[6] Erwin Schrödinger formalized its quantum mechanical version in 1926, applying it to the eigenvalue problem in wave mechanics shortly after developing his equation. In quantum chemistry, RSPT serves as the foundational perturbation scheme for methods like Møller–Plesset theory, where the Hartree–Fock approximation typically defines the unperturbed reference.[5] A classic illustration of RSPT's application and convergence is the one-dimensional quartic anharmonic oscillator, with Hamiltonian H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} m \omega^2 x^2 + \lambda x^4, treating the unperturbed part as the harmonic oscillator (\lambda = 0).[7] The ground-state energy corrections alternate in sign and decrease in magnitude for small \lambda, with even orders contributing positively and odd orders negatively, demonstrating asymptotic convergence up to a radius determined by the perturbation strength; for instance, at \lambda / (\hbar \omega)^{3/2} = 0.1, the series up to fourth order recovers over 99% of the exact shifted energy.[7] This example highlights how RSPT systematically improves accuracy for weakly anharmonic systems.[8]