Fourth Order Energy Components

5 Higher Order Correlation Energy Components. - 2.5.1 Fourth order energy components. - The general fourth order term for the correlation energy expansion of the closed-shell system described in zero order by a single determinant is 

The numerator is a product of four matrix elements. Specifically, it takes the form 

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The fourth order energy component takes the form... 

Table 6 Fourth order energy components involving singly excited intermediate states...

When the canonical Hartree-Fock orbitals are used to construct the reference function with respect to which the perturbation expansion is made, only one non-vanishing renormalization term occurs in the fourth-order energy component. The fourth-order energy can be written as... 

AT is a constant, fi and v denote compound indices representing a maximum of four actual indices. The intermediates and gp differ only in that in the latter there is an additional denominator factor. A number of calculations of the quadruple-excitation component of the fourth-order energy have been reported.131-140... 

The triple-excitation fourth-order energy, in contrast to the quadruple-excitation component, arises from connected wave function diagrams. The algorithm required to evaluate this energy component is considerably less tractable than that for the quadruple-excitation energy, depending on 7, where n is the number of basis functions. The triple-excitation diagrams can be written in terms of the intermediates. 

When the single determinant many-electron functions are constructed from canonical Hartree-Fock orbitals, the excited functions, and , are doubly excited with respect to the reference function . The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form... 

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... 

Table VII Fourth-order linked-diagram triple-excitation and quadruple excitation components of the electron correlation energy in a number of atoms and small molecules...

The fourth-order quadruple-excitation energy diagrams arise from the disconnected wave function diagrams shown in Figure 9. This observation leads to considerable simplification in the evaluation of this energy component. For example, using the identity... 

Rendell et al. compared three previously reported algorithms to the fourth-order triple excitation energy component in MBPT." The authors investigated the implementation of these algorithms on current Intel distributed-memory parallel computers. The algorithms had been developed for shared-... 

The inclusion of higher components of the correlation energy (not included in up to the fourth order of perturbation theory, MP4) was performed with the CCSD(T) method. This method is known to be very reliable for systems containing transition metals, while some similar approaches like QCISD(T) failed.83,84... 

The data points are fitted in a least-square sense to a fourth degree polynomial, and the properties thereby obtained are presented in Table 3. Since the atom possesses spherical symmetry there is only a single independent component of the a-tensor as well as the y-tensor. The curvature of the energy, or the polarizability, at the SCF level differs by less than 5% compared to the FCI result, and the MP2 value captures slightly more than half of the correlation effect. Electron correlation plays a more important role in the determination of the fourth-order property y. Again the MP2 method captures slightly more than half of the total contribution, which amounts to 21% at the FCI level of theory. The trends we have seen here in the example of the helium atom are more or less representative for closed-shell molecules in general. 

Relative importance of the fourth-order triple-exdtation and quadruple-excitation components of the correlation energy as a function of basis set quality for the ground state of the water molecule."... 

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