Third-order contributions to the correlation energy

On taking the Rayleigh-Schrodinger perturbation expansion for the energy beyond second-order, we find that so-called renormalization terms appear. In third-order Rayleigh-Schrodinger perturbation theory, the energy coefficient can be written in the form 

The first term in this expression is called the direct term whilst the second term is called the renormalization term. If the perturbation expansion is developed with respect to a reference function constructed from canonical Hartree-Fock orbitals, then the renormalization term is equal to zero  

The expression for the third-order energy coefficient given in eq. (3.181)  

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Fig. 5. Hugenholtz diagrams for the third order contribution to the correlation energy in the ground state...

The third-order contribution to the correlation energy for the HeH of Problem 2.1 can now be determined, using the HF orbital energies and the one- and two-electron integrals in the HF basis that is determined there. 

Triple-excitations and Quadruple-excitations.—Only double excitations contribute to the correlation energy and other properties through third-order triple-, quadruple and higher-order excitations are usually neglected. Triple- and quadruple-excitations first arise in the fourth-order of the perturbation series. 

When this term is evaluated to lowest nonvanishing order (third order), it leads to the following contribution to the correlation energy ... 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

Traditionally, the G3 energy is written in terms of corrections (basis set extensions and correlation energy contributions) to the MP4/d energy. Alternatively, the G3 energy can be specified in terms of HF and perturbation energy components. Denoting the second-, third-, and fourth-order contributions from perturbation theory by E2, E3, and E4, respectively, and the contributions beyond fourth order in a QCISD(T) calculation by EAqci, the G3 energy can be expressed as... 

The dispersion interaction in the third-order perturbation theory contributes to the three-body non-additivity and is called the Axilrod-Teller energy. The term represents a correlation effect. Note that the effect is negative for three bodies in a linear configuration. 

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