Third-Order Contributions

The Judd-Ofelt model of the/ —s- / transitions extended by the third-order contributions to the transition amplitude is based on the double perturbation theory applied for the following Hamiltonian, 

In this perturbing operator, uhf is the centro-symmetric Hartree-Fock (H-F) potential. Similarly as in the case of the crystal field potential included as a perturbation, also here the inter-shell interactions via the Vcon- are taken into account, thus the same partitioning of space in equation (10.26). However, for a certain order of three operators, D g Vcryst, and Vcorr in the final third-order contributions, it is also possible to take into account the interactions via the perturbing operator within the Q-space. This is why additional operators, QVcrystQ and QVQ, are included in the Hamiltonian of equation (10.26). 

Following the standard procedure of double perturbation theory, the transition amplitude is now determined by the following contributions. 

The terms associated with A define the second-order contributions, and they lead to the Judd-Ofelt theory this part is known as presented in the previous section. The terms 

In general the third-order contributions to the transition amplitude that originate from the inter-shell interactions have the usual perturbing form, now with two energy denominators. 

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The third order contribution to the amplitude that the vacuum remain a vacuum under the influence of the external field is exhibited by the two diagrams of Fig. 10-7, and is given by... 

The quadrupolar effects of order higher than two (7) are usually assumed to be negligible, especially at high magnetic fields. However, once the first- and second-order effects are removed, the measurement of third-order contributions becomes realistic. It can be easily shown that, similar to the first-order case, the CT and all symmetric MQ transitions (q = 0) are free of the third-order contribution, which thus can be safely ignored in DAS, DOR, and MQMAS experiments [161,162]. This is not the case for transitions between non-symmetric spin states, such as the STs. Indeed, numerical simulations of the third-order effect have explained the spectral features that have been observed in 27A1 STMAS spectra of andalusite mineral [161]. 

For calculation of the Pauli form factor contribution to the Lamb shift the third order contribution to the Pauli form factor (Fig. 3.5), calculated numerically in [33], and analytically in [34] is used... 

Fig. 5. Hugenholtz diagrams for the third order contribution to the correlation energy in the ground state...

Figure 7.2. Diagrammatic representation of some third-order contributions to the matrix elements 2 = l/2 3Q,ff f2 = 3/2) ina4 state. The three top diagrams illustrate the admixture of 4n states, whilst the lower diagram shows the admixture of a 2 state. 

The second term will be neglected in the matrix element, since it corresponds to a third order contribution to the energy. We thus have ... 

As an example the case of the third order contribution to frequency doubling, where the input field consists of a static field plus a field at frequency, co, is considered as in the EFISH experiment. Both the fields are applied along the same axis... 

The first step is the formation of H-bonded intermediate 49, in which Ccarbene takes on substantial cationic character. Next, termolecular attack by the amine in the presence of Y provides tetrahedral intermediate 50, which then breaks down into products. The reaction is sensitive to steric hindrance, with ammonia and primary amines reacting rapidly (several orders of magnitude faster than aminolysis of carboxylic acid esters) and secondary amines reacting much more sluggishly. The actual kinetic order associated with the amine is also a function of the solvent. Aprotic solvents such as hexane require a rate law with a third-order contribution from the amine pro tic solvents such as methanol show a mixed first- and second-order contribution from the amine. 

The third order contribution improves the result for both MP and EN schemes. Thus, in the MP case, is small and negative, while the EN value is positive. Both MP3 and EN3 results thus recover 96-98% of the exact correlation energy. As noted at the dose of the preceding... 

The EDM ionization potential calculations discussed in Sections III.A and III.B include some numerically important third-order terms while neglecting other third-order terms whose importance is suggested by means of the various comparisons. In this section we investigate these remaining third-order contributions in EOM IP calculations. 

Three types of third-order contribution are omitted in the calculations in Sections III.A and III.B. Two of these arise because of single excitations from the SCF determinant in the ground-state wave function 0>. These singly excited configurations (SEC) in 0> produce third-order terms in A and second-order ones in... 

The terms are present because an orthogonalized operator basis set is not employed in these calculations. The last row of Table III presents results for calculations that include both the third-order contributions to the partitioned EOM equation due to single excited configurations and use an operator basis in which the 3-block operators are Schmidt-orthogonalized to the 1-block (therefore, D - =0) through second order. These results differ from the results using the nonorthogonal operator basis by only 0.0 to 0.01 eV for each symmetry. 

The only remaining third-order contribution to the partitioned EOM equation (37) arises from the second term in the Born expansion of (Agg —, (39). This term has the form A Mq M Mq A, where... 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted... 

The third-order contribution to the correlation energy for the HeH of Problem 2.1 can now be determined, using the HF orbital energies and the one- and two-electron integrals in the HF basis that is determined there. 

Compare the third-order contribution with the second-order contributions determined in Problem 3.1 and the full Cl correlation energy. 

The third-order contributions to the self-energy can also readily be calculated. These terms lead to fivefold summations and are more cumbersome to compute. ... 

Diagram the third-order contributions to the electron propagator selfenergy listed in Appendix D. 

Several improvements to the original Kuo-Brown approach can be found in the literature [51], using either the partial summations of the folded diagrams as discussed above, or by including renormalized particle-hole interactions as advocated by the random-phase approximation or studying the perturbation expansion order by order in the interaction. Following the latter philosophy, Barrett and Kirson [52] showed that third-order contributions played a sizeable role. Typical examples of third-order topologies which play an important role are shown in Fig. 7. 

Fig. 7. Significant third-order contributions to the effective interaction.

For the HO choice, one observes a clearly nonconvergent behavior when going from second to third order in the perturbation expansion for the lowest J = 0 and J = 2 states. Actually, as can be seen from the above figure and as we will show in Section 4, third-order contributions are sizable, a change of 1.8 MeV for the lowest J = 0 state, and care should therefore be exercised when approximating either the perturbation expansion or the Q-ho to a given order. 

In this, and the subsequent figure, we have omitted states which represent possible intruder states in the low-lying spectra. Inclusion of such states may bring in an additional attraction of 0.1-0.5 MeV [42], thereby degrading the agreement with the data. However, as we will show below, third-order contributions may be large, and their sign is not a priori known. 

In order to understand the importance of the third-order contributions, we list in Table 4 the contributions from the two-body diagrams up to third-order in the interaction for the Bonn B potential. No folded diagrams are included in our third-order diagrams. For the JT = 01 configurations only certain nondiagonal third-order contributions are sizable, and one could therefore be tempted to approximate the Q-box with second-order diagrams only. Inspection of Table 4 tells us, however, that in the JT = 10 channel, third-order contributions may be large, actually of the size of the bare G-matrix or the total second-order contributions ... 

From Table 5 we see that the Bonn A potential is the one which gives the strongest binding in the ground state, irrespective of the approximation used for the 0-box. An LS calculation with a second-order 0-box results in the best reproduction of the data, when employing the Bonn A potential. The problem of the JT = 10 state we discussed in connection with second-order perturbation theory has vanished. However, in order to be consistent, third-order contributions have to be included. In this case, potential A introduces too much binding for the JT = 10 state, in line with our previous comments on result for the sd-shell. If one were to account for so-called intruder states, potential B is seemingly the most appropriate candidate for nuclear structure... 

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