Atomic Monopole Models

A class of improved population analysis schemes has been designed to reproduce the total dipole moment of the molecule when calculated by point charges. Such a dipole moment conserving procedure was proposed e.g. by Jug [99] and by Thole and van Duijnen [100]. A more general multipole fitted scheme has also been derived [101]. A slightly different approach is to determine potential derived atomic charges which are fitted to reproduce the values of the electrostatic potential outside the van der Waals envelope of the molecule [102, 103]. 

In order to appreciate the differences in the atomic charges obtained for the same amide fragment (NHCOC H ) of peptides and proteins a number of point charge sets were compared by Rullman [104] (Table 1). The charges used in various empirical force fields come from different quantum chemical calculations. For example, the ECEPP [105] and CHARMM [106] charges were determined on the basis of CNDO/2 studies, while the AMBER [107] and OPES [108] charges came from approximately scaled ab initio SCF calculations. Rullman and van Duijnen [109] used the dipole moment conserving population analysis [100] for the HF SCF wave function with the Mehler-Paul basis set [110]. 

It is important to note that an uncertainty of 0.02 atomic units in the atomic charge may correspond to electrostatic energy errors in the order of 4kJ/mol which may be chemically significant [104]. 

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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. 

Surprisingly, the atomic dipole model (D) is nearly as good as the atomic charge (M) model. If only dipoles are used, it may be better to locate them in the bonds. The table shows that the bond dipole model (BD) gives a better representation of the electric potential for most of the molecules considered. Of course, at least one monopole is needed if the molecule is an ion and therefore has a net charge. 

Donkersloot and Walmsley (1970) discussed calculations for the q = 0 lattice modes of a-N. They used two potential models. First these authors assumed an atom-atom potential with 6-12 terms similar to that of Kuan, Warshel, and Schnepp (1969), but they adopted a more restrictive procedure for the evaluation of the parameters. Second, Donkersloot and Walmsley assumed an explicit charge distribution (monopole model), which was compatible with the experimental value of the molecular quadrupole moment. Neither of these models reproduced the librational mode frequencies satisfactorily. 

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-... 

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... 

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]. 

The generalized Bom model (GBM) can be regarded as a special case of the preceding procedures the reaction field is expressed in terms of a multi-center monopole representation of the solute molecule, using the Bom formula, Eq. (32).l3 ,6, 3 "5 The centers are the atomic nuclei. The results are quite sensitive to the method used to calculate the atomic charges Cramer and Truhlar, who have applied the GBM approach extensively13,16,107 use their Class IV charges for this purpose.16,107,116 Various techniques have been utilized to determine the radii.16,95,101,107... 

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]. 

These polarity descriptors combine charge and geometry. Dipole moments are used to model dipole-monopole, dipole-dipole, dipole-induced dipole, and other interactions. Both molecular dipole (fi) as well as bond dipole moments may be defined for neutral molecules. A bond dipole moment due to atoms k and / separated by distance, rki, can be defined as (]i-The topographic electronic index defined in Eq. [12] is another measure (index) of polarity.The sum extends over the number of bonded atoms, N. ... 

Electrostatics. The most difficult aspect of molecular mechanics is electrostatics (35-38). In most force fields, the electronic distribution surrounding each atom is treated as a monopole with a simple coulombic term for the interaction. The effect of the surrounding medium is generally treated with a continuum model by use of a dielectric constant. More... 

Figure 3.12. Different approaches to localization of chaigeused in electrostatic models, (a) Atom-centered monopole (b) atom-centered dipole and (c) atom-centered quadrapole.

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