Skeleton integral approach

Skeleton Integral Approach to Calculations of Radiative Corrections... 

The correction of order a Za) induced by the polarization operator insertions in the external photon lines in Fig. 3.10 was obtained in [40, 41, 42] and may again be calculated in the skeleton integral approach. We will use the simplicity of the one-loop polarization operator, and perform this calculation in more detail in order to illustrate the general considerations above. For calculation of the respective contribution one has to insert the polarization operator in the skeleton integrand in (3.33)... 

Calculation of the radiative-recoil correction generated by the one-loop polarization insertions in the exchanged photon lines in Fig. 5.2 follows the same path as calculation of the correction induced by the insertions in the electron line. The respective correction was independently calculated analytically both in the skeleton integral approach [8] and with the help of the Braun formula... 

Technically the lower order contributions to HFS are produced by the constant terms in the low-frequency asymptotic expansion of the electron factor. These lower order contributions are connected with integration over external photon momenta of the characteristic atomic scale mZa and the approximation based on the skeleton integrals in (9.9) is inadequate for their calculation. In the skeleton integral approach these previous order contributions arise as the infrared divergences induced by the low-frequency terms in the electron factors. We subtract leading low-frequency terms in the low-frequency as Tnp-totic expansions of the electron factors, when necessary, and thus get rid of the previous order contributions. 

Calculation of the respective radiative-recoil correction of order a Za) m/M)EF in the skeleton integral approach is quite straightforward and may readily be done. However, numerically the correction in (11.24) is smaller than the uncertainty of the Zemach correction, and calculation of corrections to this result does not seem to be an urgent task. 

Radiative-recoil corrections of order a Za) m/M)Ep are similar to the radiative corrections to the Zemach contribution, and in principle admit a straightforward calculation in the framework of the skeleton integral approach. Leading logarithmic contributions of this order were considered in [6, 7]. The logarithmic estimate in [7] gives... 

The crucial property of the integrand in Eq. (10.16), which facilitates calculation, is that the denominator admits expansion in the small parameter /i prior to momentum integration. This is true due to the inequality j 2 2 2 which is valid according to the definitions of the functions a and b. In this way, we may easily reproduce the nonrecoil skeleton integral in (9.9), and obtain once again the nonrecoil corrections induced by the radiative insertions in the electron line [32, 33, 34]. This approach admits also an analytic calculation of the radiative-recoil corrections of the first order in the mass ratio. 

Within our approach the entire molecnlar symmetry is exploited to increase the efficiency of the code in every step of the calcnlation. For a molecule belonging to a group G of order G, only 7v /(8 G ) symmetry-distinct two-electron integrals over a basis set of J f Ganssian atomic fnnctions are calcnlated and processed at each iteration within SCF, first- and second-order CHF procednres. A skeleton Conlomb repulsion matrix is obtained by processing the non-rednndant list of nniqne two-electron integrals, then the actual repulsion matrices G , a < /7, are obtained via the equation... 

Fig. 14 A sketch of the charge dependence of the distance of closest approach of solvent molecules to the skeleton, a, the center of mass of the excess charge distribution, z , and the corresponding plots for the integral capacitance of the compact layer, Kh- In the case of soft landing of the molecule onto the metal skeleton edge (dashed curves), the spike in the capacitance transforms into a hump.

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