Second order contribution to the

Note that, just like for the first-order expression in Equation 5.12 also the second-order expression in Equation 5.18 applies to field-swept spectra, and a different expression found in EPR textbooks (Pake and Estle 1973) applies to frequency-swept spectra. The effect of including a second-order contribution to the central hyperfine splitting is illustrated in Figure 5.7 on the spectrum of a not uncommon contaminant of metalloprotein preparations Cu(II) ion coordinated by nitrogens of tris-hydroxy-ethyl aminomethane or Tris buffer. 

The second order contribution to the transition frequency of a single nucleus (central ion or ligand) with spin I is given by... 

Contents Introduction. - Symmetry An Excursion Through its Formal Apparatus. - Symmetry-Adapted Perturbation Theory A General Approach. - Why Symmetry-Adapted Perturbation Theories are Needed - Symmetry-Adapted Perturbation Theories at Low Orders From Ht to the General Case. - The Calculation of the 1-st Order Interaction Energy. - The Second-Order Contribution to the Interaction Energy. -Epilogue. - Appendix A. - Appendix B. -Appendix C. - Appendix D. - References. 

Calculation of the Pauli form factor contribution follows closely the one which was performed in order a Za), the only difference being that we have to employ the second order contribution to the Pauli form factor (see Fig. 3.3) calculated a long time ago in [26, 27, 28] (the result of the first calculation [26] turned out to be in error)... 

This term is important only in that it gives a second-order contribution to the hyperfine interaction by allowing the nuclear spin and electron spin to couple indirectly through the orbital momentum. 

Polarizability.—The hamiltonian for a molecule in a uniform electric field is given by H—ft° +JT(1), where i/(1)= — paF, F being the electric field vector. Developing a normal Rayleigh-Schrodinger perturbation scheme, the second-order contribution to the energy is... 

B. Second Order Contribution to the Correlation Energy. An Example... 

We use the second quantization formalism to express the second order contribution to the correlation energy of the closed-shell ground state. By substituting the expression (67) for W in Eq. (72) we obtain... 

Fig. 3. Goldstone diagrams for the second order contribution to the correlation energy in the ground state...

Hartree potential, (d) the Fock non-local exchange potential, (e-g) second-order contributions to the self-energy Xj(E) (e) direct (optical potential), (f) exchange and (g) Fermi sea correlation... 

The rank k can take values 0, 1 and 2 by the triangle rule. Of these, the scalar term with k = 0 has no A dependence and hence does not affect the relative positions of the ro-vibrational energy levels. It just makes a small contribution to the electronic energy of the state r], A). The first-rank term produces a second-order contribution to the spin orbit interaction because it is directly proportional to the quantum number A from the 3-j symbol in the first line of (7.119). The contribution to the spin-orbit parameter A(R) which arises in this way is given (in cm-1) by... 

In a similar manner, there is a second-order contribution to the spin-rotation parameter which arises from the cross-term between the spin orbit coupling and the... 

In other words, each of the parameters is the sum of a first-order and a second-order contribution. We have met equation (7.126) for the effective rotational constant operator before, in an earlier section, where we pointed out that the second-order contribution Ba> is very much smaller than BiV) and that these two contributions have a different reducedmass dependence. It is importantto realise thatthis is not generally true. Indeed, except for molecules with very light atoms such as H2, the second-order contribution to the spin-rotation parameter is usually very much larger in magnitude than the first-order contribution. The same is also often true for the spin-spin coupling parameter /.. The reduced mass dependences of the two contributions to the spin-rotation parameter y are different from each other and quite complicated. However, Brown and Watson [17] were able to show the rather remarkable result that when one takes the first- and second-order contributions together as in equation (7.127), the reduced mass dependence of the resultant parameter y(R) is simply /u-1. 

The cross term between X and Xr<)t can be treated in exactly the same way The result is a second-order contribution to the effective Zeeman Hamiltonian of the form... 

The effects of the off-diagonal terms when folded-in by perturbation theory are of two types. They can either produce operators of the same form as those which already exist in the Hamiltonian constructed from the Azl = 0 matrix elements (the zeroth-order Hamiltonian), or they can have a completely novel form. A good example of the former type is the second-order contribution to the rotational constant which arises from admixture of excited and A states,... 

For certain macroscopic nonlinear parameters the tensor notation can be simplified due to the intrinsic symmetry of the experiment, e.g., second-harmonic generation and the linear electro-optic effect. Let us first consider SHG. The second-order contribution to the polarization is given by Eq. (9). 

Table 6 Anisotropy of the first and second-order contributions to the interaction energy of HeH2 at R = 6.5 bohr. The B44 basis set was used. Energies are in //hartree.

The second-order contribution to the EP given above is, of course, identical to the one derived in Eq. (6.79). 

Start with a density that is so slowly varying that the second-order gradient expansions are valid for Ex (Eq. (16)) and for Exc (Eq. (20)). This requires that the reduced density gradients on both length scales of Section 5 must be small. Thus p and q of Eqs. (10) and (11) have magnitudes much less than 1, and so do pc = k.F/ksf p and = kp/ksfq- Moreover, the second-order contribution to the correlation energy... 

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