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MøllerPlesset perturbation theory
MøllerPlesset perturbation theory (MP) is one of several quantum chemistry postHartreeFock ab initio methods in the field of computational chemistry. It improves on the HartreeFock method by adding electron correlation effects by means of RayleighSchrödinger perturbation theory (RSPT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934.^{[1]} Additional recommended knowledge
RayleighSchrödinger perturbation theoryThe MPtheory is a special application of RSPT. In RSPT one considers an unperturbed Hamiltonian operator to which is added a small (often external) perturbation :
where λ is an arbitrary real parameter. In MPtheory the zerothorder wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The perturbation is the correlation potential. In RSPT the perturbed wave function and perturbed energy are expressed as a power series in λ:
Substitution of these series into the timeindependent Schrödinger equation gives a new equation: ()
Equating the factors of λ^{k} in this equation gives an kthorder perturbation equation, where k=0,1,2, ..., n. MøllerPlesset perturbationOriginal formulationThe MPenergy corrections are obtained from RayleighSchrödinger (RS) perturbation theory with the perturbation (correlation potential): where the normalized Slater determinant Φ_{0} is the lowest eigenfunction of the Fock operator Here N is the number of electrons of the molecule under consideration, H is the usual electronic Hamiltonian, f(1) is the oneelectron Fock operator, and ε_{i} is the orbital energy belonging to the doubly occupied spatial orbital φ_{i}. The shifted Fock operator serves as the unperturbed (zerothorder) operator. The Slater determinant Φ_{0} being an eigenfunction of F, it follows readily that so that the zerothorder energy is the expectation value of H with respect to Φ_{0}, i.e., the HartreeFock energy: Since the firstorder MP energy is obviously zero, the lowestorder MP correlation energy appears in second order. This result is the MøllerPlesset theorem:^{[1]} the correlation potential does not contribute in firstorder to the exact electronic energy. In order to obtain the MP2 formula for a closedshell molecule, the second order RSPT formula is written on basis of doublyexcited Slater determinants. (Singlyexcited Slater determinants do not contribute because of the Brillouin theorem). After application of the SlaterCondon rules for the simplification of Nelectron matrix elements with Slater determinants in bra and ket and integrating out spin, it becomes where φ_{i} and φ_{j} are canonical occupied orbitals and φ_{a} and φ_{b} are canonical virtual orbitals. The quantities ε_{i}, ε_{j}, ε_{a}, and ε_{b} are the corresponding orbital energies. Clearly, through secondorder in the correlation potential, the total electronic energy is given by the HartreeFock energy plus secondorder MP correction: E ≈ E_{HF} + E_{MP2}. The solution of the zerothorder MP equation (which by definition is the HartreeFock equation) gives the HartreeFock energy. The first nonvanishing perturbation correction beyond the HartreeFock treatment is the secondorder energy. Alternative formulationEquivalent expressions are obtained by a slightly different partitioning of the Hamiltonian, which results in a different division of energy terms over zeroth and firstorder contributions, while for second and higherorder 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, wellknown in HartreeFock theory, that (The HartreeFock energy is not equal to the sum of occupiedorbital energies). In the alternative partitioning one defines, Clearly in this partioning, Obviously, the MøllerPlesset theorem does not hold in the sense that E_{MP1} ≠ 0. The solution of the zerothorder MP equation is the sum of orbital energies. The zeroth plus first order correction yields the HartreeFock energy. As with the original formulation, the first nonvanishing perturbation correction beyond the HartreeFock treatment is the secondorder energy. We reiterate that the second and higherorder corrections are the same in both formulations. Use of MøllerPlesset perturbation methodsSecond (MP2), third (MP3), and fourth (MP4) order MøllerPlesset calculations are standard levels used in calculating small systems and are implemented in many computational chemistry codes. Higher level MP calculations, generally only MP5, are possible in some codes. However, they are rarely used because of their costs. 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, highly erratic or simply nonexistent, depending on the precise chemical system or basis set.^{[3]} Additionally, various important molecular properties calculated at MP3 and MP4 level are in no way better than their MP2 counterparts, even for small molecules.^{[4]} For open shell molecules, MPntheory can directly be applied only to unrestricted HartreeFock reference functions (since RHF states are not in general eigenvectors of the Fock operator). However, the resulting energies often suffer from severe spin contamination, leading to very wrong results. A much better alternative is to use one of the MP2like methods based on restricted HartreeFock references. These methods, HartreeFock, unrestricted HartreeFock and restricted HartreeFock use a single determinant wave function. Multiconfigurational selfconsistent field methods use several determinants and can be used for the unperturbed operator, although not in a unique way so many methods, such as Complete Active Space Perturbation Theory (CASPT2) have been developed. See also
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "MøllerPlesset_perturbation_theory". A list of authors is available in Wikipedia. 