Multipole Moment Method

An alternative approach to the evaluation of fhe resulfs obfained in the previous section, and one that provides additional physical insight into the manifestation of the process, is through the coupling of induced multipole moments [51-53]. This method also enables results for energy shifts to be obtained with considerably reduced computational effort, as will be shown in what follows. 

Consider a collection of charged particles forming an atom or molecule. A key attribute of such a body is that it is polarizable. Application of an electromagnetic field induces mulfipole moments in the system. The first few terms of the electric response, resulting in an electric dipole moment being induced, is given by the expansion 

An expression for fhe inferaction energy wriften explicitly in terms of fhe response tensors is easily derived on substituting Eq. (41) into Eq. (42). This 

It is seen from Eq. (44) that for the radiation field, the expectation value is taken over the spatial correlation function, which is easily found to be 

Inserting the right-hand side of Eq. (45) into Eq. (44) gives 

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The inclusion of coherent states of the radiation field in the formalism describing opfically induced forces is most conveniently carried out within the induced multipole moment method delineated in Section 5. Instead of number sfafes n(fc, 2.)) = (k)), coherent states a= =) are defined... 

Molecular dipole moments are often used as descriptors in QPSR models. They are calculated reliably by most quantum mechanical techniques, not least because they are part of the parameterization data for semi-empirical MO techniques. Higher multipole moments are especially easily available from semi-empirical calculations using the natural atomic orbital-point charge (NAO-PC) technique [40], but can also be calculated rehably using ab-initio or DFT methods. They have been used for some QSPR models. 

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. 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

Many authors [8-10] have demonstrated that the CP method undercorrects the BSSE. Moreover, Karlstrom and Sadlej [11] pointed out that addition of the partner orbitals to the basis set of a molecule not only lowers its energy, in accordance with the variation principle, but also affects the monomer properties (multipole moments and polarizabilities). Latajka and Scheiner [12] found that in a model ion-neutral system such as Li" -OH2, this secondary BSSE can be comparable in magnitude to the primary effect at both SCF and MP2 levels. The same authors also underlined the strong anisotropy of secondary error [13]. 

There have been many attempts to formulate a procedure to avoid it and both a posteriori and a priori schemes are available. The counterpoise approach (CP) (Boys and Bemardi, 1970) and related methods are the most conunon a posteriori procedures. Although this technique represents the most frequently employed a posteriori procedure to estimate this error, several authors have emphasised that the method introduced by Boys and Bemardi does not allow a clear and precise determination of the BSSE. The addition of the partner s functions introduces the "secondary superposition error" a spurious electrostatic contribution due to the modification of the multipole moments and polarizabilities of the monomers. This is particularly important in the case of anisotropic potentials where these errors can contribute to alter the shape of the PES and the resulting physical picture (Xantheas, 1996 and Simon et al., 1996). 

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. 

Recently, Sokalski et al. presented distributed point charge models (PCM) for some small molecules, which were derived from cumulative atomic multipole moments (CAM Ms) or from cumulative multicenter multipole moments (CMMMs) [89,90] (see Sect. 3.2). For this method the starting point can be any atomic charge system. In their procedure only analytical formulas are used,... 

In this case, and perhaps for all robust fits, if the fit is robust then its LCAO coefficients can be determined by variation of the energy. In that case the fit is said to be variational. Quantum chemists are beginning to use variational fits, but they do not yet include robust energies, in a method that they call resolution of the identity [14,15]. Equation (6), with pab replaced by PlM where L and M are the usual multipole-moment quantum numbers, can also be used to remove the first order error from fast-multipole methods [16]. 

The expansion coefficients Pq are called polarization moments or multipole moments. The expansion (2.14) may also be carried out by slightly alternative methods which are presented in Appendix D and differ from the above one by the normalization and by the phase of the complex coefficients Pq. The normalization used in (2.14) agrees with [19]. Considering the formula (B.2) from Appendix B of the complex conjugation for the spherical function Ykq(0, [Pg.30]

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. 

In order to describe a signal by this method we will first use the classical approach. At the beginning we will ascertain how either probability density Pb(9, multipole moments ipq of the excited state 6, entering into the fluorescence intensity expressions (2.23) or (2.24), are connected to the corresponding magnitudes pa(9, ground state a. The respective kinetic balance equation for probability density and its stationary solution, assuming that the conditions supposed to hold in Eq. (3.4) are in force, is very simple indeed ... 

Thus, we have attempted to give, in the present appendix, an idea of the various methods of determining classical and quantum mechanical polarization moments and some related coefficients. We have considered only those methods which are most frequently used in atomic, molecular and chemical physics. An analysis of a great variety of different approaches creates the impression that sometimes the authors of one or other investigation find it easier to introduce new definitions of their own multipole moments, rather than find a way in the rather muddled system of previously used ones. This situation complicates comparison between the results obtained by various authors considerably. We hope that the material contained in the present appendix might, to some extent, simplify such a comparison. 

Also as in the case of helium, asymptotic expansion methods can be applied to the Rydberg states of lithium and compared with high precision measurements [73,74]. This case is more difficult because the Li+ core is a nonhydrogenic two-electron ion for which the multipole moments cannot be calculated analytically, and variational basis set methods must be used instead. However, the method is in principle capable of the same high accuracy as for helium. 

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