Perturbation theory second-order effects

Long-range perturbation theory—second-order effects... 

Thus far, we have investigated the various contributions to the effective Hamiltonian for a diatomic molecule in a particular electronic state which arise from the spin-orbit and rotational kinetic energy terms treated up to second order in degenerate perturbation theory. Higher-order effects of such mixing will also contribute and we now consider some of their characteristics. 

Our analysis thus far has assumed that solution of the spin Hamiltonian to first order in perturbation theory will suffice. This is often adequate, especially for spectra of organic radicals, but when coupling constants are large (greater than about 20 gauss) or when line widths are small (so that line positions can be very accurately measured) second-order effects become important. As we see from... 

The last important contribution to intermolecular energies that will be mentioned here, the dispersion energy (dEnis). is not accessible in H. F. calculations. In our simplified picture of second-order effects in the perturbation theory (Fig. 2), d mS was obtained by correlated double excitations in both subsystems A and B, for which a variational wave function consisting of a single Slater determinant cannot account. An empirical estimate of the dispersion energy in Li+...OH2 based upon the well-known London formula (see e.g. 107)) gave a... 

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

In the early history of high resolution NMR, the theory was developed by use of perturbation theory. First-order perturbation theory was able to explain certain spectra, but second-order perturbation theory was needed for other cases, including the AB system. Spectra amenable to a first-order perturbation treatment give very simple spectral patterns ( first-order spectra), as described in this section. More complex spectra are said to arise from second-order effects. ... 

The second-order effects comprise three contributions. The second-order Zee-man and hyperfine interactions involve obvious extensions of second-order Rayleigh-Schrodinger perturbation theory using the appropriate operators. If we introduce an arbitrary gauge origin Ra, and the variable tq — r — Rq, the second-order term involving both Zeeman and hyperfine operators is... 

The most precise measurements of the fine-structure parameters D and E have in fact been carried out using zero-field resonance. Figure 7.6 shows the three zero-field transitions in the Ti state of naphthalene molecules in a biphenyl crystal at T = 83 K. In these experiments, the absorption of the microwaves was detected as a function of their frequency [5]. The lines are inhomogeneously broadened and nevertheless only about 1 MHz wide. Owing to the small hnewidth of the zero-field resonances, the fine-structure constants can be determined with a high precision. This small inhomogeneous broadening is due to the hyperfine interaction with the nuclear spins of the protons (see e.g. [M2] and [M5]). For triplet states in zero field, the hyperfine structure vanishes to first order in perturbation theory, since the expectation value of the electronic spins vanishes in all three zero-field components (cf Sect. 7.2). The hyperfine structure of the zero-field resonances is therefore a second-order effect [5]. 

Theoretical work has focused on two aspects of the problem perturbation theory and other calculations of the transition probabilities in the material, and the effect on the transition probability of the statistical nature of the radiation field. Makinson and Buckingham [7,29] were the first to predict the second-order effect and calculate its magnitude based on a surface model of photoemission this work was expanded by Smith [7.30], Bowers [7.31], and Adawi [7.32], The analogous volume calculation was performed by Bloch [7.33] and later corrected by Teich and Wolga [7.24, 25],... 

So by symmetry the response must be an even function of one looks first for leading terms in E. in the response theory perturbation treatment. These will evidently result from first order effects of sC polarizability torques involving P2(cos9) only but for polar molecules there can also be second order effects of torques on the permanent dipoles involving Pj(cosd) - cosd as well as 2. It is here that complications of non-linearity develop as the effect of the operator d/dp on the first order perturbation fj(t) and hence from equation (10) on F Ct) must be considered unless F] (t) - F] (0) which is independent of p. ... 

Accuracy of the SLG approximation can be improved by perturbation theory. Second quantization provides us a powerful tool in developing a many-body theory suitable to derive interbond delocalization and correlation effects. The first question concerns the partitioning of the Hamiltonian to a zeroth-order part and perturbation. LFsing a straightforward generalization of the Moller-Plesset (1934) partitioning, the zeroth-order Hamiltonian is chosen as the sum of the effective intrabond Hamiltonians ... 

In discussing second-order effects, which arise jointly from two perturbations, two forms of the perturbation theory of Section 11.4 are open to us. It will often be convenient, as in Sections 11.5 and 11.6, to use the form in which one perturbation is applied first and the second is used as a probe to study its effect. In this way, it is possible to gain valuable insight into the origin of various types of coupling. At the same time, the approach leads naturally towards the theory of linear response, taken up in the next chapter. Again, we consider one-by-one a few typical and important examples. 

When X or M is zero, the first-order effect vanishes, and only a quadratic effect is found, as for a linear molecule. The second-order effect calculated for a linear molecule also applies to a symmetric top when X is zero. Standard second-order perturbation theory gives for the Stark correction of the 7, M level. 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

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