Multipole electric, calculation

Yu.N. Kalugina, V.N. Cherepanov, Multipole electric moments and higher polarizabilities of molecules the methodology and some results of ab initio calculations. Atmos. Oceanic Opt 28(5), 406 14 (2015)... 

In the Buckingham-Fowler model, each monomer electric charge distribution is described by a set of point multipoles (charges, dipoles and quadru-poles) located on the atoms and, sometimes, additionally at bond midpoints. The values of the point multipoles are determined by the so-called distributed multipole analysis (DMA) of an ab initio wavefunction. This multicentric representation of the charge distribution shows superior convergence behaviour to the one-centre molecular multipoles when calculating the electrostatic potential around a molecule. 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Now, the quadrupole moment can next be calculated by differentiating the potential to get the electric field due to the dipole moment. The reader can now see that an infinite series can be thus generated. The total electric field is simply the sum of all the individual multipole contributions, given by... 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Although 1 is one of the best investigated molecules, there is, apart from data concerning its electron density distribution, very little information available on its one-electron properties. In principle, accurate data could be obtained by correlation-corrected ab initio methods, but almost nothing has been done in this direction, which of course has to do with the fact that experimental data on one-electron properties of 1 are also rare, and therefore, it is difficult to assess the accuracy and usefulness of calculated one-electron properties such as higher multipole moments, electric field gradients, etc. 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

Proceeding as above, the general expression (58) can be used to calculate other particular cases involving higher multipoles, on resorting to Tables 4—7, where the multipole elements are listed. Interactions between magnetic multipoles are also the subject of discussion. The theory of electrostatic interactions for electric multipoles has been dealt with in various approaches by Frenkel, Pople, Jansen, and others, as well as by Gray. The above presentation follows the concise uniform treatment of Kielich. > > ... 

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). 

Energy of Induced Multipole Interaction.—Interaction between a Multipolar System and External Fields. We calculated above the potential energy of electrostatic interaction between a multipole system and external fields, or thefirst-orderenergyduetothefirstpowerofthefield. Besides that energy, which took account only of the reoriratation of permanent multipoles, we have to take into consideration contributions due to the drcum-stance that an external electric field induces higher-order multipole moments given by the expressions (72) and (79). 

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