Atomic monopoles

Sokalski, W. A., D. A. Keller, R. L. Ornstein, and R. Rein. 1993. Multipole Correction of Atomic Monopole Models of Molecular Charge Distribution. I. Peptides. J. Comput. Chem. 14, 970-976. 

Before the data of Table 11 are shortly discussed, one has to stress that the partial charges derived from adiabatic infrared intensities are not related to Mulliken charges, virial charges or most other atomic charges used in ab initio theory. The partial charges p are effective charges which in addition to the atomic monopole contribution, cover the atomic dipole contribution as well. They are... 

We conclude that the adiabatic mode intensities and effective charges derived from them are the localized counterparts of those effective charges derived from measured intensities. They should be more appropriate for the description of the properties of individual bonds. In particular, they should lead to chemically more meaningful effective charges where future work has to show how effective charges, atomic monopole and dipole contributions, and the charge flux are related. 

The final term in Eq. [11] is the Coulombic or electrostatic potential energy term and can be represented as the interaction of bond dipoles or atomic monopoles. With the latter,... 

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. 

In a more recent study, Hillier placed two water molecules around the oxygen of allyl vinyl ether and its transition state for formation of 4-pentenal in an MP2/RHF/6-31G calculation [56]. When the SCRF model was used, no net decrease in activation free energy was obtained at the /= 1 level (atomic monopoles) and lack of convergence accompanied attempts to use higher terms in the multipole expansion. However, the PCM model provided a net energy decrease of 4.3kcalmol, which corresponds favorably to experiment. Somewhat disconcerting, however, were the calculated kinetic isotope effects = 1.149 and =0.919), which differed from the exper-... 

Volumen and Hydratationswarme der lonen. Zeitschrift filr Physik 1 45-48. aan C M and K B Wiberg 1990. Determining Atom-Centred Monopoles from Molecular Electro-itic Potentials. The Need for High Sampling Density in Formamide Conformational Analysis. imal of Computational Chemistry 11 361-373. 

The above potential describes the monopole-monopole interactions of atomic charges Q and Qj a distance Ry apart. Normally these charge interactions are computed only for nonbonded atoms and once again the interactions might be treated differently from the more normal nonbonded interactions (1-5 relationship or more). The dielectric constant 8 used in the calculation is sometimes scaled or made distance-dependent, as described in the next section. 

Each two-electron integral is the sum of all the terms arising from the charge distribution representative of the first pair of atomic orbitals interacting with the charge distribution representative of the second pair of atomic orbitals. Thus in the simplest case, the (ssiss) interaction is represented by the repulsion of two monopoles, while a (pj pjjlp jjp jj), a much more complicated interaction,... 

CM Breneman, KB Wiberg. Determining atom-centered monopoles from molecular electrostatic potentials. The need for high sampling density m formamide conformational analysis. J Comput Chem 11 361-373, 1990. 

Fig. 11. Peptide moiety indicating the monopoles (net charges) on each of the atoms 54 551. This demonstrates that, on the basis of size, charge and accessibility, a polypeptide could more effectively provide selectivity for cations...

London23 has treated the case of the attractive force between anisotropic molecules on the dipole-dipole interaction basis as well as on the monopole basis mentioned above. The small anisotropy found for the chlorine atom makes the dipole-dipole formulation appropriate. For the symmetrical orientation in the Cl2 molecule the London formula is... 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

Examination of the multipole populations gives no indication of the discrepancy observed in the model maps, all populations from parallel refinements agreeing to within two esd s (Table 5). The one striking exception is the monopole population (P,) for carbon. This must be a simple difference in the partitioning of the charge density between atom centers in the model as there is no discernible difference in the model maps around the carbon position. 

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]. 

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. 

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. 

Neutrino Mass, Dark Matter, Gravitational Waves, Condensation of Atoms and Monopoles, Light Cone Quantization... 

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 ... 

It has been recently shown [12] that the ELF topological analysis can also be used in the framework of a distributed moments analysis as was done for Atoms in Molecules (AIM) by Popelier and Bader [32, 33], That way, the Mo( 2) monopole term corresponds to the opposite of the population (denoted N) ... 

In SIBFA, electrostatics is computed upon using distributed multipoles (monopoles, dipoles, quadupole) located on atoms and bond midpoints as ... 

Ry is the distance between centers i and j. i is the monopole associated to center i. E and T are empirical parameters associated to each atom types as rVdw are the atom effective radii. 

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