Second order energy expression

The theoretical understanding of the interaction between molecules at distances where the overlap is negligible has been well established for some years. The application of perturbation theory is relatively straightforward, and the recent work in this area has consisted in the main of the application of well-known techniques. In the case of neutral molecules, the first non-zero terms appear in the second order of perturbation, so that some method of obtaining the first-order wavefunction, or of approximating the infinite sum in the traditional form of the second-order energy expression, is needed. 

The linear response function in Eq. (11) has the same structure as the second-order energy expression in Eq. (2) and we note that for A = V and wj = 0 they are identical, except for a factor of two. Similarly, Eq. (10) defines the quadratic response function... 

Explicit expressions corresponding to the four diagrams shown in Figure 4 can be written down by following the rules given in the previous section. The second-order energy expression has the form... 

From the theoretical point of view, the electrophilicity concept has been recently discussed in terms of global reactivity indexes defined for the ground states of atoms and molecules by Roy et al.18 19. In the context of the conceptual density functional theory (DFT), a global electrophilicity index defined in terms of the electronic chemical potential and the global hardness was proposed by Maynard et al.20 in their study of reactivity of the HIV-1 nucleocapsid protein p7 zinc finger domains. Recently, Parr, Szentp ly and Liu proposed a formal derivation of the electrophilicity, co, from a second-order energy expression developed in terms of the variation in the number of electrons.21... 

According to this definition the second-order energy expressions in Eqs (38) and (55) are identical for a given set c,T. The minimization of the energy expectation value (Eq. (55)) with respect to the c yields the eigenvalue equation... 

The representation of the K operator using the elements of the k vector of Eq. (112) is most useful in this representation. The other terms of Eq. (139) may also be factored as in Eq. (141) to give the following set of equivalent matrix expressions of the truncated second-order energy expression ... 

The energy expression in (49) is perturbative in the sense that we make use of a U matrix optimized with respect to the second-order energy expression of (37). We refer to this method as CV(oo) — DFT [145]. We shall now seek ways in which to find vectors f7 2,) that optimize F,xs [p0 + Ap(°°)(t/)]. To this end, we can start with... 

This is for the UHF optimum single determinant, the closed-shell case would be expressed in a basis of spatial functions of half the dimension and multiplied by a factor of 2 overall with a corresponding change in the definition of the elements of G. The same is true for the second-order energy expression considered shortly. 

Using the notation already established in eqn ( 26.11), the second-order energy expression when there is no perturbation is... 

The second-order energy expression from Eq. (3.11) reduces, using Q j = 0, to... 

The first term in the square brackets results in the second-order energy expression [Eq. (4.21)] when used in Eq. (4.16). 

With this perturbative correction to the wave function, we can rewrite the second-order energy expression in the familiar form as [353]... 

If the zeroth-order solution of the problem provides a good approximation, one expects that low-order PT contributions will be sufficient to consider. In this case, not all the terms of the perturbation operator W will contribute to the energy formula. For example, only two-bond terms will enter the second-order energy expression. 

We may make use of the variation conditions and the normalization condition to reduce the second-order energy expression. Provided is restricted to the variational space of we may write... 

The variation equation for the second-order wave function comes from the fourth-order energy. If we were to make the variation of the second-order wave function in the second-order energy expression we would only get the zeroth-order equation again, so we need a functional that is quadratic in the second-order wave function. Variation of the fourth-order energy subject to the normalization conditions gives the equation... 

In fact, the quantum mechanical relationships for magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling arrived at from the derivatives of the second-order energy expressed in flie form (7.26)-(7.28) are the same as those obtained via RSPT [2-4],... 

Here we have replaced the fluctuation potential (14.2.5) by the full Hamiltonian and introduced the first-order wave function in the form (14.2.21). As a final rearrangement of the second-order energy expression, we introduce a commutator between the Hamiltonian and the cluster operator... 

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