Atomic multipole parameter

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... 

While the calculation of the electrostatic functions from the multipole parameters parallels that of the calculation of the atomic electrostatic moments, there is an... 

The EFG data from the multipole parameters are, in principle, for the static crystal while the spectroscopic data are affected by vibrations. There may therefore by a systematic difference between the two sets of values, which is evident for a number of hydrogen-bonded hydroxyl groups and water molecules studied by Tegenfeldt and Hermansson (1985), but is not apparent in the data in Table 8.3. The EFG values for H atoms in hydrogen-bonds is further discussed in chapter 12. 

Two advanced techniques have been proposed and applied to some crystal structures (Section IV,C), in which aspherical distributions of valence electrons around an atom are directly taken into account in the least-squares calculations. Aspherical atomic form factors are introduced in the least-squares refinement in the first method (29, 38, 80) and multipole parameters describing the aspherical valence distributions are used in the second method (31, 34, 46). 

Because electron density is a local property, electron density studies of the peptide-like molecules show that the nonspherical part of the deformation density (i.e the P]m parameters of Eq. 8) remain essentially the same for a given atom in the same environment (the peptide residue, a phenyl ring, a methyl group...) [29], The same observation was made for porphyrin ligands [30] and by Brock, Dunitz, and Hirshfeld [37] for naphthalene and anthracene type molecules. All these observations suggest that the multipole parameters are highly transferrable from one atom to a chemically similar atom in different molecules and crystals. A key question is is it possible to determine for each chemical type of a given atom a small set of pseudoatom multipole parameters, and can such parameters be used to calculate electrostatic properties of new molecules To answer this question [29], two accurate but low resolution X-ray data sets (sin 0/Xmax = 0.65 A-1) were... 

As described by Stevens [35] and Holladay, Leung, and Coppens [36], d orbital occupancies of the metal atom can be derived from the multipole parameters assuming that the overlap density and the asphericity of any 4p orbital density are small. For first row transition metals, the d orbital expansion and the overlap density between d orbitals and the ligands are small then, the asphericity of the electron density of the metal atom is mainly due to the d orbital occupancies. From Eq. 8, we can write ... 

Determination of the electrostatic parameters is among the more difficult problems in the derivation of reliable force fields. X-Ray studies of crystal structures can provide us with information regarding atomic multipoles how-... 

The table shows that the M + D + Q atomic multipole expansion is sufficient for most purposes. The largest rms value is just 0.03 kj/mol. So the curve fitting exercise is successful when carried to atomic quadrupole level, satisfying mathematical requirements of accuracy and convergence. But what about chemical reasonableness and transferability. Judged by these criteria, the M a- D -I- Q model is poor. As a practical matter, it may be necessary to always evaluate M + D -I- Q parameters for the specific molecule being considered and not expect the parameter values to have any obvious intuitive chemical meaning. 

It is has been known that the atomic multipole moments for atoms in AMOEBA model can be calculated through quantum mechanics method and Stone s distributed multipole analysis [61]. Thus, it is straightforward to obtain the parameters of electric multipole potentials based on the distributed multipole analysis after the EMP sites of Gay-Berne particles are decided or directly from AMOEBA force field. However, the EMP parameters of Gay-Berne particles need to be optimized because they are derived based on the gas-phase ab initio quantum mechanics. One possible solution would be to match GBEMP and AMOEBA results for the electrostatic energies between CG particles and water molecules, or between CG particle dimers, at various separations and/or in different orientations. 

The actual values of the atomic multipole moments depend on the coefficients of the hybrid orbitals in the individual bond orbitals (bond polarities), on the orientation of the hybrids and on the degree of hybridization. Since these latter two parameters depend significantly on the geometrical arrangement of the atoms [47], the bond increment method may be an adequate tool for constructing the zeroth order wave function, which nevertheless describes the main trends in the conformational and geometrical dependence of the atomic charge distributions and consequently of the electrostatic potential. 

More promising is to describe the deformation electron density by a series of spherical harmonic density functions (multipoles), which can be included into least-squares refinement. The inner (core) electron shells of an atom are presumed and the k parameter, which describes the isotropic expansion (ic <1) or contraction (/c > 1) of the valence shell as a whole. Multipole parameters of higher orders describe deviations of the electron density from spherical symmetry. They can be related to the products of atomic... 

It is evident that the computational results of the method critically depend on the quality of the parameterization and in particular on the point polarizabilities usually associated with atoms. The atomic polarizabilily parameters are generally obtained by fitting to either experimental or QM molecular polarizabilities or QM electrostatic potentials. The methods can also be divided into two groups additive and interactive models, depending on the level of interactions permitted between induced dipoles [20]. In the additive approach, polarizable sites are allowed to respond to an external electric field but not to permanent and induced multipoles on other sites within a molecule. In nonadditive, also called interactive, polarization models, instead, each of a molecule s polarizable sites is allowed to respond to an external electric field not only from other molecules but from other sites within the same molecule. Consequently, aU interacting sites polarize themselves. Under certain conditions, two inducible dipoles at short distances can cause a polarization... 

With only s- and p-functions present, the two-centre two-electron integrals can be modelled by multipoles up to order 2 (quadrupoles), however, with d-functions present multipoles up to order 4 must be included. In MNDO/d all multipoles beyond order 2 are neglected. The resulting MNDO/d method typically employs 15 parameters per atom, and it currently contains parameters for the following elements (beyond those already present in MNDO) Na, Mg, Al, Si, P, S, Cl, Br, 1, Zn, Cd and Hg. 

Thermal parameters of conventional independent-atom refinements using BLFLS [8] were applied as starting values for full multipole refinements, which were performed with VALRAY [10]. Both data sets were successfully refined. The results were compared to those published by Kirfel and Eichhom [7], and good agreement was found. 

Electron population parameters of inner monopoles were constrained to be equal for all 40 non-H atoms. Single exponentials r exp(-ar) were adopted as radial functions for the higher multipoles, with n = 2, 2, 3 respectively for dipole, quadrupole, and octopole of the species C, N and 0, and n = 4, 4, 4 for the same multipoles of the S atom. A radial scaling parameter k, to shape the outer shell monopoles, and the exponential parameter a of all non-H atomic species were also refined. H atoms were initially given scattering factors taken from the H2 molecule [15] and polarised in the direction of the atom to which they are bonded. 

In the final stages of the refinement the positional parameters of the H atoms were kept fixed, and these atoms too were described with multipoles, up to the dipole level. For both poles of the H pseudoatoms the radial functions were again single exponentials, with n = 0, 1 for monopole and dipole respectively, and the a value was 2.48 bohr1. 

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. 

Where p (r) is the electron density of each pseudo atom, Pcore(r) and pvai ( r) are the core and spherical densities of the valence electron shells, Pvai and Pim (multipoles) describe the electron shell occupations, k and k denote the spherical deformation and y (r/r) is a geometrical function. The parameters K, k , Pvai and Pim are refined during adjustment of the experimental and models structure amplitudes. 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

Many molecules contain chemically equivalent atoms, which, though in a different crystal environment, have, to a good approximation, the same electron distribution. Such atoms may be linked, provided equivalent local coordinate systems are used in defining the multipoles. In particular, for the weakly scattering hydrogen atoms, abundant in most organic molecules, this procedure can lead to more precisely determined population parameters. 

---