Second-order energy correction

The second-order energy corrections have the form (B.8) with... 

The second-order energy correction is expressed as follows ... 

When this result is used in the earlier expression for the second-order energy correction, one obtains ... 

The second-order energy correction can be evaluated in like fashion by noting that... 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

Let us look at the expression for the second-order energy correction, eq. (4.38). This involves matrix elements of the perturbation operator between the HF reference and all possible excited states. Since the perturbation is a two-electron operator, all matrix elements involving triple, quadruple etc. excitations are zero. When canonical HF... 

The only nonvanishing matrix elements of x3 are those with j = n 1 and j = n 3. This result is obtained by repeated application of Eq. (40), as before. Thus, there are four terms that the cubic potential constant contributes to the second-order energy correction, Eq, (35). The final result can be written as... 

First- and second-order energy corrections then are ... 

The energy correction (4.52) takes account of the effect of the terms (bq2 + dq4) in the perturbation (4.36). However, since the first-order correction due to the terms (aq + cq3) vanished, we must go to the second-order energy correction to take these terms into account. The contribution of (aq + cq1) to (2) will turn out to be of the same order of magnitude as the contribution of (bq2 + dq4) to (,). We have the correct zeroth-order wave functions (4.38), and we can use (1.202) for the second-order energy correction for state n ... 

As for the denominator of Eq. (7.40), from inspection of Eq. (7.43), a(0) for each doubly excited determinant will differ from that for the ground state only by including in the sum the energies of the virtual orbitals into which excitation has occurred and excluding the energies of the two orbitals from which excitation has taken place. Thus, the full expression for the second-order energy correction is... 

A large number of spin-orbit properties can now be derived from the response functions. From the linear response function we can deduce the second-order energy correction due to SOC (see section 4.1),... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy corrections. 

Perturbation theory corrections to variational energies have also been considered recently by Mitrushenkov and Dmitriev (1995),216 who express the second-order energy correction as... 

Applying standard Rayleigh-Schrodinger perturbation theory, the first-order wave function and second-order energy correction F 2 are... 

The orbitals and the orbital energies ej are first-order corrections. These orbitals can interact with other unperturbed orbitals (/ 1,2, n) to give rise to second-order energy corrections (i.e. those similar to the last term of Eq. (16)). 

The second-order energy correction arising from the excited states (out of the active space) is given by the standard formula of perturbation theory... 

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