Second-order perturbation equation

Starting from the second-order perturbation equation (4.32), analogous formulas can be generated for the second-order corrections. Using intermediate normalization... 

As a simple introduction to PMO theory suppose we consider the bond formation between the two-atom, two-orbital system shown in Figure 4.8. The energy gained on forming the bond A—B is given by the second-order perturbation equation... 

To illustrate the accuracy of first- (Equation 4.13) and second-order (Equation 4.15) calculations, their results are compared with the exact solution (Equation 4.17) in Figure 4.7. The perturbation da j is varied from 0 to 3/8. The parameters for N (ethylenimine) and O (formaldehyde) are 5aM//3 = 0.5 and 1.0, respectively. If the term (5aM//8)2 in the discriminant of Equation 4.17 is much smaller than 4 and can be neglected altogether, then Equation 4.17 is equal to the result of first-order perturbation theory (Equation 4.13). If (5aM//8)2 is smaller than 4 but non-negligible, the square root of Equation 4.17 can be expanded, (1 + xf2 1 + 1/2x+. .., and we have the result of second-order perturbation (Equation 4.15). Thus, the results of first-order perturbation are adequate for even a substantial perturbation such as the replacement of CH2 by O in formaldehyde. 

We use the notation of the previous chapter (Section 2.5). Since the bonding MOs in an even AH are all filled and the antibonding ones are empty, any degeneracy between R and S involves pairs of filled MOs or pairs of empty MOs. The n energy of union A rs is then a sum of second-order perturbations [equation (2.19)], involving interactions between filled MOs of R and empty MOs of S, and between filled MOs of S and empty MOs of R i.e.. 

The problem raised at the end of the last section, that the exact solutions to the second-order perturbation equations are not known except for very simple cases, is not the only problem in the application of direct perturbation theory. The major problem is that the exact solutions of the zeroth-order problem, that is, of the Schrodinger equation, are not known except for the same simple cases. As a consequence, the zeroth-order wave function is not the eigenfunction of the zeroth-order Hamiltonian and the perturbation... 

An expression for the second-order wave function can be obtained from the second-order perturbation equation (1.29) and then substituted in expression (1.40). 

Fortunately, for non-integer quadnipolar nuclei for the central transition = 0 and the dominant perturbation is second order only (equation Bl.12.8) which gives a characteristic lineshape (figure B1.12.1(cB for axial synnnetry) ... 

To obtain the force constant for constructing the equation of motion of the nuclear motion in the second-order perturbation, we need to know about the excited states, too. With the minimal basis set, the only excited-state spatial orbital for one electron is... 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

From second-order perturbation theory (Section 4.8) the following equation for the change in energy can be derived. ... 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

Moreover, if the wave function + Xxp P is used as a trial function 0, then the quantity W from equation (9.2) is equal to the second-order energy determined by perturbation theory. Any trial function 0 with parameters which reduces to -h 20o for some set of parameter values yields an approximate energy W from equation (9.2) which is no less accurate than the second-order perturbation value. 

In addition, it can be shown that second-order vibronic perturbation will make possible some intersystem crossing to the 3B3u(n, tt ) state. However, this second-order perturbation should be much less important than the first-order spin-orbit perturbation.(19) This will produce the unequal population of the spin states shown in Figure 6.1. In the absence of sir the ratios of population densities n are given by the following equations ... 

Some transition ions have central hyperfine splittings somewhat greater than this value, for example, for copper one typically finds Az values in the range 30-200 gauss, and so in these systems the perturbation is not so small, and one has to develop so-called second-order corrections to the analytical expression in Equation 5.12 or 5.13 that is valid only for very small perturbations. The second-order perturbation result (Hagen 1982a) for central hyperfine splitting is ... 

A more accurate description is obtained by including other additional terms in the Hamiltonian. The first group of these additional terms represents the mutual magnetic interactions which are provided by the Breit equation. The second group of additional terms are known as effective interactions and represent, to second order perturbation treatment, interaction with distant configurations . These weak interactions will not be considered here. 

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

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