Møller-Plesset perturbation theory

Møller-Plesset perturbation theory (MP) is one of several quantum chemistry post-Hartree-Fock ab initio methods in the field of computational chemistry. 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.Its main idea was published as early as 1934.cite journal
author = Møller C., Plesset M.S.
year = 1934
title = Note on an Approximation Treatment for Many-Electron Systems
journal = Phys. Rev. | volume = 46 | pages = 618–622
doi = 10.1103/PhysRev.46.618
url = http://link.aps.org/abstract/PR/v46/p618
note = This article contains several minor, albeit annoying problems in the mathematics as published. For a concise derivation of MP perturbation theory to n^th order, see any good quantum mechanics textbook.
format = abstract ]

Rayleigh-Schrödinger perturbation theory

See perturbation theory.The MP-theory is a special application of RS-PT. In RS-PT one considers an unperturbed Hamiltonian operator $hat\{H\}\_\{0\}$ to which is added a small (often external) perturbation $hat\{V\}$:

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

where λ is an arbitrary real parameter. In MP-theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The perturbation is the correlation potential.

In RS-PT the perturbed wave function and perturbed energy are expressed as a power series in λ:

:$Psi\; =\; lim\_\{n\; o\; infty\}\; sum\_\{i=0\}^\{n\}\; lambda^\{i\}\; Psi^\{(i)\}$,:$E\; =\; lim\_\{n\; o\; infty\}\; sum\_\{i=0\}^\{n\}\; lambda^\{i\}\; E^\{(i)\}$.

Substitution of these series into the time-independent Schrödinger equation gives a new equation: ($n\; o\; infty$)

:$left(\; hat\{H\}\_\{0\}\; +\; lambda\; V\; ight)\; left(\; sum\_\{i=0\}^\{n\}\; lambda^\{i\}\; Psi^\{(i)\}\; ight)\; =\; left(\; sum\_\{i=0\}^\{n\}\; lambda^\{i\}\; E^\{(i)\}\; ight)\; left(\; sum\_\{i=0\}^\{n\}\; lambda^\{i\}\; Psi^\{(i)\}\; ight)$.

Equating the factors of $lambda^k$ in this equation gives an "k"th-order perturbation equation, where "k"=0,1,2, ..., n. See perturbation theory for more details.

Møller-Plesset perturbation

Original formulation

The MP-energy corrections are obtained from Rayleigh-Schrödinger (RS) perturbation theory with the perturbation("correlation potential"):

:$hat\{V\}equiv\; H-F-\; langlePhi\_0|H-F|Phi\_0\; angle,$

where the normalized Slater determinant Φ₀ is the lowest eigenfunction of the
Fock operator

:$FPhi\_0\; equivleft(\; sum\_\{k=1\}^\{N\}\; f(k)\; ight)\; Phi\_0\; =\; 2left(sum\_\{i=1\}^\{N/2\}varepsilon\_i\; ight)Phi\_0.$ Here "N" is the number of electrons of the molecule under consideration, "H" is the usual electronic Hamiltonian, $f(1)$ is the one-electron Fock operator, and ε_"i" is the orbital energy belonging to the doubly occupied spatial orbital φ_"i". The shifted Fock operator:$hat\{H\}\_\{0\}\; equiv\; F+langlePhi\_0|\; H-F\; |\; Phi\_0\; angle$ serves as the unperturbed (zeroth-order) operator. The Slater determinant Φ₀ being an eigenfunction of "F", it followsreadily that:$F\; Phi\_0\; -\; langle\; Phi\_0|\; F\; |\; Phi\_0\; angle\; Phi\_0\; =\; 0\; Longrightarrow\; hat\{H\}\_\{0\}\; Phi\_0\; =\; langle\; Phi\_0|\; H\; |\; Phi\_0\; angle\; Phi\_0,$so that the zeroth-order energy is the expectation value of "H" with respect to Φ₀, "i.e.," the Hartree-Fock energy::$E\_\{mathrm\{MP0equiv\; E\_\{mathrm\{HF\; =\; langlePhi\_0|H|Phi\_0\; angle.$

Since the first-order MP energy:$E\_\{mathrm\{MP1\; equiv\; langlePhi\_0|hat\{V\}|Phi\_0\; angle\; =\; 0$ is obviously zero, the lowest-order MP correlation energy appears in second order.This result is the Møller-Plesset theorem: "the correlation potential does not contribute in first-order to the exact electronic energy."

In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-PT formula is written on basis of doubly-excited Slater determinants. (Singly-excited Slater determinants do not contribute because of the Brillouin theorem).After application of the Slater-Condon rules for the simplification of "N"-electron matrixelements with Slater determinants in bra and ket and integrating out spin, it becomes

:$E\_\{mathrm\{MP2\; =sum\_\{i,j,a,b\}langlevarphi\_i(1)varphi\_j(2)|r\_\{12\}^\{-1\}|varphi\_a(1)varphi\_b(2)\; angle$:::$imes\; frac\{2langlevarphi\_a(1)varphi\_b(2)|r\_\{12\}^\{-1\}|varphi\_i(1)varphi\_j(2)\; angle-langlevarphi\_a(1)varphi\_b(2)|r\_\{12\}^\{-1\}|varphi\_j(1)varphi\_i(2)\; angle\}\{varepsilon\_i\; +varepsilon\_j-varepsilon\_a-varepsilon\_b\},$

where φ_"i" and φ_"j" are canonical occupied orbitals andφ_"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 second-order in the correlation potential, the total electronic energy is given by the Hartree-Fock energy plus second-order MP correction: "E" ≈ "E"_HF + "E"_MP2. The solution of the zeroth-order MP equation (which by definition is the Hartree-Fock equation) gives the Hartree-Fock energy. The first non-vanishing perturbation correction beyond the Hartree-Fock treatment is the second-order energy.

Alternative formulation

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. The formulation is commonly used by chemists, who are now large users of these methods. [ See all volumes in the "Further reading" section. ] This difference is due to the fact, well-known in Hartree-Fock theory, that:$langle\; Phi\_0\; |\; H-\; F\; |\; Phi\_0\; angle\; e\; 0\; quad\; Longleftrightarrow\; quadE\_\{mathrm\{HF\; e\; 2\; sum\_\{i=1\}^\{N/2\}\; varepsilon\_i.$(The Hartree-Fock energy is not equal to the sum of occupied-orbital energies).In the alternative partitioning one defines,:$hat\{H\}\_0\; equiv\; F,\; qquad\; hat\{V\}\; equiv\; H-F.$Clearly in this partioning,:$E\_\{mathrm\{MP0\; =\; 2\; sum\_\{i=1\}^\{N/2\}\; varepsilon\_i,\; qquad\; E\_\{mathrm\{MP1\; =\; E\_\{mathrm\{HF-\; 2\; sum\_\{i=1\}^\{N/2\}\; varepsilon\_i\; .$Obviously, the Møller-Plesset theorem does not hold in the sense that "E"_MP1 ≠ 0. The solution of the zeroth-order MP equation is the sum of orbital energies. The zeroth plus first order correction yields the Hartree-Fock energy. As with the original formulation, the first non-vanishing perturbation correction beyond the Hartree-Fock treatment is the second-order energy. We reiterate that the second- and higher-order corrections are the same in both formulations.

Use of Møller-Plesset perturbation methods

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 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 non-existent, depending on the precise chemical system or basis set. [cite journal
author = Leininger M.L., Allen W.D., Schaefer H.F., Sherrill C.D.
year = 2000
title = Is Moller-Plesset perturbation theory a convergent ab initio method?
journal = J. Chem. Phys.
volume = 112
issue = 21
pages = 9213–9222
doi = 10.1063/1.481764 ] Additionally, various important molecular properties calculated at MP3 and MP4 level are in no way better than their MP2 counterparts, even for small molecules. [ cite book | last = Helgaker | first = Trygve | authorlink = Trygve Helgaker
coauthors = Poul Jorgensen and Jeppe Olsen
title = Molecular Electronic Structure Theory | publisher = Wiley | date = 2000
location = | pages = | isbn = 978-0471967552 ]

For open shell molecules, MPn-theory can directly be applied only to unrestricted Hartree-Fock 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 MP2-like methods based on restricted Hartree-Fock references.

These methods, Hartree-Fock, unrestricted Hartree-Fock and restricted Hartree-Fock use a single determinant wave function. Multi-configurational self-consistent 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

* Electron correlation
* Perturbation theory (quantum mechanics)
* Post-Hartree-Fock
* Quantum chemistry computer programs

References

Further reading

* cite book
last = Cramer | first = Christopher J.
title = Essentials of Computational Chemistry
publisher = John Wiley & Sons, Ltd. | date = 2002 | location = Chichester
pages = 207 - 211 | isbn = 0-471-48552-7

* cite book
last = Foresman | first = James B. | coauthors = Æleen Frisch
title = Exploring Chemistry with Electronic Structure Methods
publisher = Gaussian Inc. | date = 1996 | location = Pittsburgh, PA
pages =267 - 271 | isbn = 0-9636769-4-6

* cite book
last = Leach | first = Andrew R.
title = Molecular Modelling
publisher = Longman | date = 1996 | location = Harlow
pages = 83 - 85 | isbn = 0-582-23933-8

* cite book
last = Levine | first = Ira N.
title = Quantum Chemistry
publisher = Prentice Hall | date = 1991 | location = Englewood Cliffs, New jersey
pages = 511 - 515 | isbn = 0-205-12770-3

* cite book
last = Szabo | first = Attila | coauthors = Neil S. Ostlund
title = Modern Quantum Chemistry
publisher = Dover Publications, Inc | date = 1996 | location = Mineola, New York
pages = 350 - 353 | isbn = 0-486-69186-1