Second-order perturbation coefficient

Obviously, the second-order perturbed coefficients Cf can be eliminated from this by a technique exactly analogous to the first derivative case. In order to do this, the coefficients Cf are formally expressed from the second-order response equations... 

The second-order perturbed coefficients in the original third derivative equation (19), and those introduced by the Handy-Schaefer device, appear in linear combinations like... 

The most convenient procedure for attaining the minimum of the second order perturbation expression of the energy, so as to generate the optimized virtual orbitals, depends on the kind of problem being studied. In the case of intermolecular interactions, convergence is quite easy with just a gradient-based procedure. The minimization scheme can be recast in such a way that the coefficients of the improved virtual orbitals can be obtained, at each step, by a resolution of a linear system of NA+NB equations. Specifically. 

The coefficients of configurations not contained in c(, can also be estimated in second-order perturbation theory, using the Ak method [22]. 

The presence of the viscous damping term results in a second-order perturbation of the wave velocity and a first-order contribution to the attenuatitm. Since for most materials a < k, Equation 2.22 enables solution for the attenuation coefficient a ... 

B and C are Racah parameters. A representation of these energies vs. A is given in Figure 19. Using second-order perturbation theory the following coefficients are obtained. 

Figure 4.7 Comparison of first (dashed line), second order (thin solid line) and full HMO calculations (thick solid line) for an inductive perturbation of ethylene. The radius of the circles indicates the size of the MO coefficients of and on atoms 1 and 2, which remain unchanged upon first order perturbation. The MOs obtained by second order perturbation or by full HMO are concentrated more on the perturbed atom 1 in the bonding orbital...

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

A straightforward hut approximate application of second-order perturbation theory by R. Eisenschitz and F. London gave the value 6.47 for this coefficient [Z. f. Phys. 60, 491 (1930)]. The first attack on this problem was made by S. C. Wang, Phys. Z. 28, 663 (1927). The value found by him for the coefficient, = 8.68, must be in error (as first pointed out... 

In principle, this hyperconjugation model (scheme 43) still represents a first-order perturbation it is based on the ESR spectroscopically proven, nearly constant phenyl ring spin population in all related radical cations, i.e. constant squared coefficients c2 (scheme 35), and substitutes the perturbation Sax by the angle-dependent contributions dcx. For a second-order perturbation example, which introduces interactions to additional substituent orbitals (scheme 35), reference is made to the silyl and methyl acetylenes121 (Section IV.E). 

The terms on the first line of Eq. (81) describe single and double excitations of the closed core, while those on the second line describe single and double excitations of the atom where the valence orbital is also excited. Substituting Eq. (81) into the Schrodinger equation one obtains a set of coupled equations for the expansion coefficients that can be found in Ref. [44]. The first and second iterations of the equations for the expansion coefficients leads to results that are identical to first- and second-order perturbation theory. In third-order perturbation theory, terms associated with triple excitations contribute to the energy. These terms have no counterpart in the iterative solution to the equations under consideration. 

Subsequently, Kozlowski et al. [24] also revisited the Cope rearrangement with inclusion of dynamic correlation between the active and inactive electrons. However, they used Davidson s own version of multi-reference, second-order perturbation theory [25], which allows the coefficients of the configurations in the CASSCE wave function to be recalculated after inclusion of dynamic electron correlation. Kozlowski et al. found that the addition of dynamic correlation to the (6/6)CASSCE wave function for the Cope TS causes the weight of the RHE configuration to increase at the expense of the pair conhgurations that are necessary to describe the two diradical extremes in Eig. 30.1. Thus, without the inclusion of dynamic electron correlation in the wave function, (6/6) CASSCF overestimates the diradical character of the C2 wave function [24]. 

The expansion of potentials in spherical harmonics and the resulting contribution to thermodynamic properties were initially presented by Pople (12) and more recently developed in further studies (13,14). The equation produced for the excess over hard-sphere properties shows that the first-order perturbation coefficient (1/kT) involves only the symmetric portion of all potentials. The asymmetric contributions first appear in the coefficient of the second-order (1/kT)2 term where they are weighted... 

---