Multipole coefficients

At each stage of the refinement of a new set of parameters, the hat matrix diagonal elements were calculated in order to detect the influential observations following the criterium of Velleman and Welsh [8,9]. The inspection of the residues of such reflections revealed those which are aberrant but progressively, these aberrations disappeared when the pseudo-atoms model was used (introduction of multipoler coefficients). This fact confirms that the determination of the phases in acentric structures is improved by sophisticated models like the multipole density model. 

For the static density, the zero term in the potential can be expressed in terms of the multipole coefficients of the aspherical-atom formalism. Substituting for atom j at... 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

It is assumed here that 5t (r) = iftpp (r). (For atomic Hartree-Fock radial orbitals, the x-ray scattering by and iftpp are almost the same.19) Notice in Eq. (15) that there are only three multipole coefficients but four different orbital product coefficients. The spherically symmetrical component in the 2pa2pa products can not be distinguished from (2s)2 with the radial functions that have been used to date. This points out the ambiguity of so-called if/a charge ratios that have been recently reported.21 Note from Eq. (15) that for a ir/o ratio,... 

In general, the multipole coefficients of an arbitrary charge distribution p(r) can be obtained as... 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

As electric fields and potential of molecules can be generated upon distributed p, the second order energies schemes of the SIBFA approach can be directly fueled by the density fitted coefficients. To conclude, an important asset of the GEM approach is the possibility of generating a general framework to perform Periodic Boundary Conditions (PBC) simulations. Indeed, such process can be used for second generation APMM such as SIBFA since PBC methodology has been shown to be a key issue in polarizable molecular dynamics with the efficient PBC implementation [60] of the multipole based AMOEBA force field [61]. 

The uniaxial orientation parameter is the most simple way to characterize preferred orientation. It is simple, because it is only a number - in fact, for = a is the first non-trivial expansion coefficient in a multipole expansion of the normalized... 

As are the other multipole-expansion coefficients, the uniaxial orientation parameter is computed from Eq. (9.6). For materials with fiber symmetry the relation simplifies4 and... 

In the literature slightly different definitions of the multipole expansion are found, depending on how the pre-factor (2( — l)/2 is distributed between expansion equation and the definition of die coefficients. Cf. (Ward [251], eq. 5.2)... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

The coefficients of the multipole expansion are computed from Eq. (9.8), and after analogous expansions of both the intensity of the perfectly oriented structural entity (i0pt, be), and of the measured intensity (/, ce), Ruland [253] obtains a set of algebraic equations among the expansion coefficients,... 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The accuracy of the electrostatic moments based on the multipole parameters is a function of the errors in both the population coefficients Tvai and the atomic parameters Pimp- Let M represent the m x m variance-covariance matrix for these parameters, as in chapter 4. Let D be the derivative matrix with elements... 

The similar accuracies of different well-parameterized continuum models implies that they will also perform similarly for the computation of partition coefficients, and that has proven to be the case in most studies to date (see, for example, Bordner, Cavasotto, and Abagyan 2002 and Curutchet et al. 2003b). In Table 11.4 the previously presented SMx results for the chloroform/water partitioning of die methylated canonical nucleic acid bases are compared to results from die MST-ST/HF/6-31G method, and also to purely electrostatic results obtained using a multipole expansion SCRF method. As the latter does not include any accounting for non-electrostatic effects, its performance is significantly degraded compared to the other two. 

Here, the subscript (c) is short for the set of expansion parameters (c) = (2i, 22, A, L, oi, u2) r, is the vibrational coordinate of the molecule i R is the separation between the centers of mass of the molecules the Q, are the orientations (Euler angles a, jS y,) of molecule i Q specifies the direction of the separation / the C(2i22A M[M2Ma), etc., are Clebsch-Gordan coefficients the DxMt) are Wigner rotation matrices. The expansion coefficients A(C) = A2i22Al u1u2(ri,r2, R) are independent of the coordinate system these will be referred to as multipole-induced or overlap-induced dipole components - whichever the case may be. 

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]

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