Distribution Calculations Alternative Approaches

European Molecular Biology Laboratory, Heidelberg, Germany Christophe Chipot 

DMA = distributed multipole analysis ESP = electrostatic potential GAPT = generalized atomic polar tensor NAP = natural atomic population NMB = natural minimal basis NRB = natural Rydberg basis OMM = overlap multipole moments PEOE = partial equalization of orbital electronegativities RESP = restrained electrostatic potential. 

A routine way to obtain atom-centered charge distributions is provided by the Mulliken population analysis, available in most ab initio and semiempirical quantum chemistry packages. In this approach, the net charge borne by atom i, of a given molecule, takes the form  

The problem of negative atomic populations, sometimes encountered with Mulliken population analysis, often results from the use of a non-orthogonal basis set. This difficulty can be circumvented by transforming, in a symmetric manner, all atomic orbitals into an orthogonal basis set. With such a basis transformation, the net chaige borne by atom i can be expressed as  

This is precisely the leading idea of Lowdin population analysis. It can be observed that the mathematical use of the trace of P S, i.e., Tr(P S) = Tr(P S - S ) = Tr(S P S - ), yields an infinite number of possibilities, among which the approaches of Mulliken and Ldwdin are particular cases. In 

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Charge Distribution Calculations Alternative Approaches Force Fields A General Discussion Linear Free Energy Relationships (LFER) Population Analyses for Semiempiri-cal Methods Quantitative Structure-Property Relationships... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

Significant progress in the optimization of VDW parameters was associated with the development of the OPLS force field [53]. In those efforts the approach of using Monte Carlo calculations on pure solvents to compute heats of vaporization and molecular volumes and then using that information to refine the VDW parameters was first developed and applied. Subsequently, developers of other force fields have used this same approach for optimization of biomolecular force fields [20,21]. Van der Waals parameters may also be optimized based on calculated heats of sublimation of crystals [68], as has been done for the optimization of some of the VDW parameters in the nucleic acid bases [18]. Alternative approaches to optimizing VDW parameters have been based primarily on the use of QM data. Quantum mechanical data contains detailed information on the electron distribution around a molecule, which, in principle, should be useful for the optimization of VDW... 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

An alternate approach for determining particle density involves fraction collection followed by the application of quasi-elastic light scattering (QEL) to the fraction (19). The QEL technique is particularly effective in providing a mean particle diameter d for narrow particle distributions. With this d and the X obtained from V (the volume at which the fraction was collected), Ap can be calculated directly from Equation 3. 

Mikheikin et al. (11) have formulated an alternative approach where terminal valencies are saturated by monovalent atoms whose quantum-chemical parameters (the shape of AO, electronegativity, etc.) are specially adjusted for the better reproduction of given characteristics of the electron structure of the solid (the stoichiometry of the charge distribution, the band gap, the valence band structure, some experimental properties of the surface groups, etc.). Such atoms were termed pseudo-atoms and the procedure itself was called the method of a cluster with terminal pseudo-atoms (CTP). The corresponding scheme of quantum-chemical calculations was realized within the frames of CNDO/BW (77), MINDO/3 (13), and CNDO/2 (30) as well as within the scope of the nonempirical approach (16). 

The only alternative approaches for evaluating Sc, known to the authors, are based on pull-out and fragment length measurements.46 Both quantities depend on Sc and m, as well as r. Consequently, if r is known, Sc can be determined. For example, m can be evaluated by fitting the distribution of fiber pull-out lengths to the calculated function. Then, Sc can be obtained for the mean value, h, using Eqn. (12). This approach has not been extensively used and checked. 

A)jS, whether sampled from probability distribution functions or calculated by regression equations or surface-complexation models, can be used in many contaminant transport models. Alternate forms of the retardation factor equation that use a (Equation (3)) and are appropriate for porous media, fractured porous media, or discrete fractures have been used to calculate contaminant velocity and discharge (e.g., Erickson, 1983 Neretnieks and Rasmuson, 1984). An alternative approach couples chemical speciation calculations... 

An alternative approach to that described above is a direct use of the retention data to calculate the adsorption energy distribution [138-146]. This approach was initiated in 1974 by Rudzinski et al. [138], who showed that the energy distribution function can be expressed as a series, which contains derivatives of the retention volume with respect to the equilibrium pressure. According to this formulation, the retention volume plotted as a function of the adsorption energy is the first-order approximation of the energy distribution. However, first derivative of the retention volume is the second-order approximation of this distribution. This approach was later refined and used to calculate energy distributions for different porous solids [143]. 

If the NDF is a function of the particle velocity then the solution of the GPBE provides the modeler with the essential information for calculating the real-space advection term. This approach is used whenever the particle Stokes number is not small, and will result in the development of a particle-velocity distribution. More details on this topic can be found in Chapter 8. An alternative approach consists of integrating the NDF with respect to the particle velocity. Let us consider, for example, a generic NDF n(t,x, p, p), which is a function of the time t, space x, particle velocity Vp, and internal coordinates p. By integrating out the particle velocity the following NDF is obtained ... 

Many attempts were made to explain lipophilicity by related properties, e.g. by solubility, solvent-accessible surface, and charge distributions calculated from semiempirical methods ([190, 272 — 284] and references cited therein) while some of the results allow a better understanding of the intrinsic nature of lipophilicity, none of these alternative approaches have led to a reliable log P prediction system so far. 

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