Perturbation theory of Møller-Plesset

The perturbation theory of Møller-Plesset (MP) is one of many the methods post-Hartree-Fock Ab.initio in quantum Chimie applied within the framework of the numerical Chimie. It improves the Méthode Hartree-Fock while bringing the effects of electronic Corrélation there by means of the perturbation theory of Rayleigh-Schrödinger (RS-PT) to second (MP2), third (MP3) or fourth (MP4) order usually. The principal idea of this method was published since 1934.

Perturbation theory of Rayleigh-Schrödinger

Theory MP is a particular application of the perturbation theory of Rayleigh-Schrödinger (in English Rayleigh-Schrödinger disturbance theory , RS-PT). In the RS-PT, one considers a nondisturbed Hamiltonian operator $\backslash \; hat\; \{H\}\; \_\; \{0\}$ to which a small disturbance is added (sometimes external) $\backslash \; hat\; \{V\}$:

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

where λ is a real arbitrary parameter. In theory MP, the function of wave of order 0 is an exact clean function of the Opérateur of Fock, which is used then as nondisturbed operator. The disturbance is the potential of corrélation.
In the RS-PT, the function of disturbed wave and disturbed energy are expressed in the form of whole series of λ:

$\backslash \; Psi\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \backslash \; lambda^\; \{I\}\; \backslash \; Psi^\; \{(I)\}$,

$E\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \backslash \; lambda^\; \{I\}\; E^\; \{(I)\}$.

The inroduction of these series in the equation of Schrödinger dependant on time gives a new equation:

$\backslash \; left\; (\backslash \; hat\; \{H\}\; \_\; \{0\}\; +\; \backslash \; lambda\; V\; \backslash \; right)\; \backslash \; left\; (\backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \backslash \; lambda^\; \{I\}\; \backslash \; Psi^\; \{(I)\}\; \backslash \; right)\; =\; \backslash \; left\; (\backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \backslash \; lambda^\; \{I\}\; E^\; \{(I)\}\; \backslash \; right)\; \backslash \; left\; (\backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \backslash \; lambda^\; \{I\}\; \backslash \; Psi^\; \{(I)\}\; \backslash \; right)$.

The equalization of the factors of the $\backslash \; lambda^k$ in this equation gives the equation of the disturbance of order K , where K =0,1,2,…, N.

Disturbance of Møller-Plesset

Initial formulation

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

$\backslash \; hat\; \{V\}\; \backslash \; equiv\; HF\; \backslash \; langle\; \backslash \; Phi\_0|HF|\backslash \; Phi\_0\; \backslash \; rangle,$

in which the Determinant of Slater standardized Φ ₀ is the clean function lowest of the Opérateur of Fock:

$F\; \backslash \; Phi\_0\; \backslash \; equiv\; \backslash \; left\; (\backslash \; sum\_\; \{k=1\}\; ^\; \{NR\}\; F\; (K)\; \backslash \; right)\; \backslash \; Phi\_0\; =\; 2\; \backslash \; left\; (\backslash \; sum\_\; \{i=1\}\; ^\; \{N/2\}\; \backslash \; varepsilon\_i\; \backslash \; right)\; \backslash \; Phi\_0.$

Here, NR is the number of electrons of the molecule considered, H is the usual electric Hamiltonian, $f\; (1)$ is the operator of Fock mono-electronics, and ε _I is energy orbitalaire pertaining to orbital space the doubling occupied φ _I . The operator of Fock shifted

$\backslash \; hat\; \{H\}\; \_\; \{0\}\; \backslash \; equiv\; F+\; \backslash \; langle\; \backslash \; Phi\_0|\; HF\; |\; \backslash Phi\_0\; \backslash rangle$

is used as nondisturbed operator (order zero).

The determinant of Slater Φ ₀ being a clean function of F , it follows that:

$F\; \backslash \; Phi\_0\; -\; \backslash \; langle\; \backslash \; Phi\_0|\; F\; |\; \backslash \; Phi\_0\; \backslash \; rangle\; \backslash \; Phi\_0\; =\; 0\; \backslash \; Longrightarrow$

\ hat {H} _ {0} \ Phi_0 = \ langle \ Phi_0| H | \Phi_0\rangle \Phi_0, thus that energy with order zero is the awaited value of H according to Φ ₀, i.e. Hartree-Fock energy:

$$

E_ {\ mathrm {MP0}} \ equiv E_ {\ mathrm {HF}} = \ langle \ Phi_0|H|\Phi_0\rangle.

Like energy MP of first order:

$$

E_{\mathrm{MP1}} \equiv \langle\Phi_0|\ hat {V}|\ Phi_0 \ rangle = 0 is of course null, the energy of correlation MP appears in the term of second order. This result is the theorem of Møller-Plesset' . This difference is due to the fact, well-known in Hartree-Fock theory, that:

$\backslash langle\; \backslash Phi\_0\; |\; H\; F\; |\; \backslash \; Phi\_0\; \backslash \; rangle\; \backslash \; 0\; \backslash \; quad\; \backslash \; Longleftrightarrow\; \backslash \; quad$

E_ {\ mathrm {HF}} \ 2 \ sum_ {i=1} ^ {N/2} \ varepsilon_i. (energy Hartree-Fock is not equal to the sum of energies of orbital occupied). In alternative separation, one defines:

$\backslash \; hat\; \{H\}\; \_0\; \backslash \; equiv\; F,\; \backslash \; qquad\; \backslash \; hat\; \{V\}\; \backslash \; equiv\; HF.$

One has in an obvious way, in this separation:

$$

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. The theorem of Møller-Plesset does not abound in the direction where E _MP1 ≠ 0. The solution with equation MP of order zero is the sum of energies orbitalaires. The correction of order zero plus one gives Hartree-Fock energy. As in the initial formulation, the first term of disturbance which is not cancelled beyond the Hartree-Fock treatment is the energy of second order, which is, let us recall it, the same one in the two formulations.

Use of the methods of Møller-Plesset disturbance

Møller-Plesset calculations of second (MP2), third (MP3) and fourth order (MP4) are the standard levels used for small systems and are implemented in many codes of numerical chemistry. More raised calculations MP of level, in general only of the MP5, are available in certain codes. However, they are seldom used because of their coûts.
Systematic studies on the perturbation theory MP showed that it is not inevitably convergent with high orders. The properties of convergence can be slow, fast, oscillating, regular, highly erratic or simply non-existent, according to the studied chemical system or bases it used. Moreover, various important molecular properties calculated on level MP3 and MP4 are not in any way better than their equivalents MP2, even for small systems. For the molecules with opened layers, theory MP of order N can be applied directly only to the functions of reference the nonrestricted Méthode of Hartree-Fock (the states of the restricted Hartree-Fock method are not general clean vectors of the operator of Fock). However, resulting energies suffer of severe a Contamination of spin, leading to completely erroneous results. A better alternative is to use one of the methods similar to method MP2 based on references Hartree-Fock restricted for open layer.
These methods, Hartree-Fock, Hartree-Fock not-restricted and Hartree-Fock restricted use a function of wave to simple determinant. The methods with multi-configurational self-coherent Champ use several determinants and can be used for the not-disturbed operator, although there is not only one method of application, like the space Perturbation theory supplements ( Complete Activates Space Perturbation Theory , CASPT2).

See too

• electronic Correlation
• Method post-Hartree-Fock
• Perturbation theory (quantum chemistry)

References

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