Møller-Plesset perturbation theory

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Møller-Plesset perturbation theory (MP) is one of several quantum chemistry post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It improves on the Hartree-Fock method by adding electron correlation effects using perturbation theory, usually to second (MP2), third (MP3) or forth (MP4) order.

The unperturbed Hartree-Fock Hamiltonian operator \hat{H}_{0} is extended by adding a small perturbation V:

\hat{H} = \hat{H}_{0} + \lambda \hat V,

where λ is an arbitrary parameter. The Hartree-Fock wave function is an approximate eigenfunction of the exact Hamiltonian operator but is an exact eigenfunction of an operator which is the sum of individual Fock operators \hat{H}_{0}. The perturbation is the difference between this operator and the exact Hamiltonian. If the perturbation is sufficiently small, then the resulting wavefunction and energy can be expressed as a power series in λ:

\Psi = \lim_{n \to \infty} \sum_{i}^{n} \lambda^{i} \Psi^{(i)},
E = \lim_{n \to \infty} \sum_{i}^{n} \lambda^{i} E^{(i)}.

Substitution of these series into the time-independent Schrödinger equation gives a new equation:

\left( \hat{H}_{0} + \lambda V \right) \left( \sum_{i}^{n} \lambda^{i} \Psi^{(i)} \right) = \left( \sum_{i}^{n} \lambda^{i} E^{(i)} \right) \left( \sum_{i}^{n} \lambda^{i} \Psi^{(i)} \right).

The solution of this equation to zero order gives an energy which is the sum of the orbital energies for the electrons. The solution to first order (n = 1) corrects this energy and gives the Hartree-Fock energy and wave function. To go beyond the Hartree-Fock treatment it is necessary to go beyond first order. Second (MP2), third (MP3), and fourth (MP4) order Møller-Plesset calculations are standard levels used in calculating small systems and are implemented in many computational chemistry codes. Higher level MP level calculations are possible in some codes, however, they are rarely used.

Systematic studies of MP perturbation theory have shown that it is not necessarily a convergent theory at high orders. The convergence properties can be slow, rapid, oscillatory, regular or highly erratic, depending on the precise chemical system or basis set.1

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