Monopole expansion

The dipole approximation is valid only for point dipoles, i.e. when the donor-acceptor separation is much larger than the molecular dimensions. At short distances or when the dipole moments are large, it should be replaced by a monopole-monopole expansion. Higher multipole terms should also be included in the calculations. 

An alternative method for obtaining atomic charge is to fit this elearostatic potential to a series of point charges centered on the atomic nuclei. This monopole expansion VM(r) is given by Eq. [53]. 

Figure 4.7. Shape of the molecular electrostatic potential in the outer space of digitoxigenin when occupying the probable receptor-bound conformation. Using the co-ordinates derived from X-ray crystal structure analysis, the equipotential energy contours (expressed in kJtmol) are calculated with the use of an optimized monopol expansion. The energy contours refer to the plane laid across carbon atoms 6, 8, and 9 involved in forming rings B and C. Reproduced from [128],...

This said, the number of the local expansions, their location, the number of expansion terms for each centre, and the method to get the numerical coefficients for each expansion term must be defined. There are no formal constraints, and the strategy should be selected on the basis of its efficacy computer time and precision. A detailed and clear exposition of the problems involved in, and of the options open to the definitions of local expansions has been recently done by F. Vigne-Maeder and the late P. Claverie [72]. We shall follow in part this exposition, giving more emphasis, at the end, to the use of atomic monopole expansions (i.e. atomic charges) and to mixed representations, which represent, in our opinion, the most versatile method for chemical reactivity problems. 

Note that, for spherical overlap distributions centred at P and Q, (9.13.24) and (9.13.25) are the only nonzero multipole moments and the monopole expansion (9.13.27) then repcsents an exact expression for the two-electron integral (assuming disjoint charge distributions) - see also the discussion in Section 9.12.3. In the same mannra-, the multipole expansion (9.13.19) terminates exactly after a finite number of terms whenever the charge distributions of the electrons are one-centre functions, whose centres are chosen as origins of the multipole expansions. In general, however, the bipolar multipole expansion does not terminate and the expansion is then truncated when the remainder is sufficiently small as discussed in Section 9.13.2. 

The usual multipole expansion is inappropriate for the treatment of London energies associated with the interatomic charge oscillation. However, London23 has given an appropriate method, known as the monopole method. Coulson and Davies7 and Haugh and Hirschfelder16 have applied the monopole method to the inter-... 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

On the carbons, nitrogens and oxygens expansions up to octapole level were introduced, whereas the expansions were limited to quadrupole level for the hydrogen atoms. All atoms were given a k expansion/contraction parameter for the spherical monopole term, and all atoms except the hydrogens were given k parameters to expand or contract the non-spherical poles. The k and k values on 0(1) and 0(3), on N(l) and N(3), on C(l) and C(3) and on 0(21) and 0(22) were constrained to be equal. 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

Only the spherical and dipolar density terms contribute to the integral on the right. Assuming, for simplicity, that the deformation is represented by the valence-shell distortion (i.e., the second monopole in the aspherical atom expansion is not used), we have, with density functions p normalized to 1, for each atom ... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

Yu. A. Simonov, Cluster Expansion, Non-Abelian Stokes Theorem and Magnetic Monopoles,... 

Representation of the density n(r) [or, effectively, the electrostatic potential — 0(r)] near any one of the sinks as an expansion in the monopole and dipole contribution only [as in eqn. (230c)] is generally, unsatisfactory. This is precisely the region where the higher multipole moments make their greatest contribution. However, the situation can be improved considerably. Felderhof and Deutch [25] suggested that the physical size of the sinks and dipoles be reduced from R to effectively zero, but that the magnitude of all the monopoles and dipoles, p/, are maintained, by the definition... 

Charges can be obtained at different level of moments such as monopole (s = 1), dipole (s = 3) and quadrupole (s = 9). Torsion energy barriers for the HS-SH molecule calculated by several methods can be seen in Fig. 9 [90]. For the PCM model of this molecule the number of expansion centers is six (c = 6) beside the atomic centers, one center per S-H bond is further included. It can be seen that the PCM result is very close to the CMMM one and the PCM charges can be used for calculating intramolecular electrostatic interactions as well. 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

We shall now analyze the self-energy 2j(E) of the hole in a way that connects the physical picture of the relaxation process with the more technical pictures based on expansions in terms of one-electron basis orbitals having the full symmetry of the system, involving concepts like monopole relaxation and shake-up, fluctuation and correlation. To this end, let us divide the self-energy -Xj(E) into two parts corresponding to relaxation (R) (Figs. 9e, f) and Fermi sea (ground-state) correlation (C) (Fig. 9g)... 

The static monopole relaxation diagram clearly describes the response to a delocalized core hole while the dynamic relaxation (fluctuation) process describes the response to the dipole moment of the hole. Together, the two diagrams (Figs. 38c,e) describe the response of the system to a hole localized to an atomic core orbital on either nucleus41, and individually they represent a multipole expansion of the hole. 

The equilibrium state is determined by a minimization of the free energy. The total interaction in colloidal systems is mainly determined by electrostatic interactions and can be divided in three components. The first component occurs at interactions between net charged molecules or molecules with asymmetric charged distribution. These charge distributions can often be described by multipole expansions, i.e., a combination of monopole, dipole, quadrupole, etc., and is a fruitful approach if each multipole expansion can be described by one or two terms. The interaction is given by the sum of interactions between the terms, where the first contribution is the interaction between ions and is given by Coulomb s law and is the main contribution in systems with... 

As emphasized, the Born—Kirkwood—Onsager (BKO) approach includes only the solute s monopole and dipole interaction with the continuum. That is, the full classical multipolar expansion of the total solute charge distribution is truncated at the dipole term. This simplification of the electronic distribution fails most visibly for neutral molecules whose dipole moments vanish as a result of symmetry. A distributed monopole or distributed dipole model is more... 

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