Zero-multipole method

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

Five components (Q = —2,—1,0,1,2) of the multipole moment pq of rank K = 2 form the tensor which characterizes alignment. The form of the probability density in the case where only p and pq are non-zero is presented in Fig. 2.3(c,d,e). The component p characterizes longitudinal alignment, whilst components p x, p 2 characterize transversal alignment. The method of transforming pq on turning the coordinate system is analyzed in Appendix D. 

An electric multipole moment of order n in the form (40) has quite generally (2 + 1) independent components, and their number undergoes a further reduction for various molecular symmetries (Table 2). Buckingham, by methods of group theory, calculated the number of independent tensor components of multipoles (40) from n = 1 up to = 6 for 35 point-group symmetries. In Table 3, we give their numbers for the first four moments as well as the number of non-zero components. 

The non-zero tensor components of multipole moments have been determined specifically for the tetrahedraland octahedral symmetries, beside the axial symmetry for which we have the general formula (40a). Lately, Kielich and Zawodny, resorting to methods of group theory, have calculated and tabulated all non-zero and independent tensor components of electric dipole, quadrupole, octupole, and hexadecapole moments for 51 point groups (Tables 4—7). 

Table 8.4. State multipoles at Eo=350 eV, 6=3° for the n=2 excitation of hydrogen using positive, negative and zero electric fields. The experimental data are due to Williams and Heck (1988). Errors in the final significant figures are given in parentheses. Calculations are CCO, method of Bray, Madison and McCarthy (1990) and vWW, van Wyngaarden and Walters (1986)...

The potential energy terms V bring in the geometry of the molecule they are defined by (IIIB-1, 10—13). We have two methods of approximating these terms. We can expand F in a multipole expansion in (1/2 ,), the distances between groups. The first non-zero term is the dipole-dipole term which can be written as (Kirkwood, 1937)... 

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