Atomic multipoles

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

It is interesting to note that the expression for atomic multipoles in Eq. 6.21 can be obtained by defining the atomic charge density p as 

This definition of the atomic charge density is consistent with general physical considerations, for example it gives the total electron density when summed for all atoms  

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In the AMOEBA force field the permanent atomic multipoles are determined from QM calculations [40], The prescription considers that the resulting multipoles on the atoms result from two components, the permanent and the induced moments ... 

The scaling factor Sj can take any value between 0 and 1 and is applied to site j. The superscripts p and m indicate permanent and mutual induction, respectively. Equation (9-19) can be solved iteratively using similar procedures to those used to solve Eq. (9-3). The formal permanent moments can be calculated by subtracting induced moments from moments from ab initio calculations. For any conformation of a given compound the atomic multipoles can be determined from Distributed Multipole Analysis (DMA) [51]. 

Ren PY, Ponder JW (2003) Polarizable atomic multipole water model for molecular mechanics simulation. J Phys Chem B 107(24) 5933-5947... 

Jiao D, King C, Grossfield A, Darden TA, Ren PY (2006) Simulation of Ca2+ and Mg2+ solvation using polarizable atomic multipole potential. J Phys Chem B 110(37) 18553—18559... 

Williams, D. E. 1988. Representation of the Molecular Electrostatic Potential by Atomic Multipole and Bond Dipole Models. J. Comp. Chem. 9, 745. 

Sokalski, W. A., K. Maruszewski, P. C. Hariharan, and J. J. Kaufman. 1989. Library of Cumulative Atomic Multipole Moments II. Neutral and Charged Amino Acids. Int. J. Quantum Chem. Quantum Biol. Symp. 16,119-164. 

Recently, Sokalski et al. presented distributed point charge models (PCM) for some small molecules, which were derived from cumulative atomic multipole moments (CAM Ms) or from cumulative multicenter multipole moments (CMMMs) [89,90] (see Sect. 3.2). For this method the starting point can be any atomic charge system. In their procedure only analytical formulas are used,... 

These atomic contributions depend on the choice of the coordinate origin. A space invariant form can be obtained using the cumulative approach [93, 94]. The space invariant cumulative atomic multipole moments (CAMMs) do not contain contributions from lower moments. The definition of the CAMMs is as follows ... 

The expression for the electrostatic energy given above can be further subdivided to give a sum of interactions between atomic multipoles, which is in turn summed over all possible pairs of interacting atoms. 

Atomic multipoles are estimated by fitting the atomic multipole expansion to the detailed features of the ground-state wave function obtained from ab initio quantum mechanical calculations. Rein (1975) reviewed the problem of estimating atomic multipoles and presented examples of use of the atomic multipole expansion method to the problem of molecular recognition in biology. More recently, Liang and Lipscomb (1986) considered the problem of transferabilities of atomic multipoles in atomic multipole expansions. 

Sokalski WA, Poirier RA (1983) Cumulative atomic multipole representation of the molecular charge distribution and its basis set dependence. Chem Phys Lett 98 86-92... 

Sokalski WA, Sawaryn A (1987) Correlated molecular and cumulative atomic multipole moments. J Chem Phys 87 526-534... 

For covalently bonded atoms the overlap density is effectively projected into the terms of the one-center expansion. Any attempt to refine on an overlap population leads to large correlations between p>arameters, except when the overlap population is related to the one-center terms through an LCAO expansion as discussed in the last section of this article. When the overlap population is very small, the atomic multipole description reduces to the d-orbital product formalism. The relation becomes evident when the products of the spherical harmonic d-orbital functions are written as linear combinations of spherical harmonics ( ). 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

The contribution to the predicted electrostatic potential of the anisotropic atomic multipoles (Q , / > 0), which represent the lone pair and n-electron density, rapidly become less important as the distance between the molecules increases. This not only results from the inverse power of R increasing with I, but also from the cancellation between the contributions from different multipoles and different atoms. For example, there is generally an atomic dipole component along a bond that opposes the polarity implied by the atomic charges, as shown in the results of distributed multipole analyses (DMAs) of the azabenzenes. ° Thus, the accuracy gained by using a distributed multipole model is very dependent on the relative separation and orientation of the molecules, as well as the actual distribution of charge in the molecule. 

Equations [22]-[24] illustrate why the derivation and programming of the forces and torques, and second derivatives for all the terms up to R in the atom-atom multipole expansion of the electrostatic energy is a nontrivial exercise in classical mechanics. It has been described in detail by Popelier and Stone,and, with the additional derivatives required for modeling molecular crystal structures, by Willock et al. ... 

W. A. Sokalski and A. Sawaryn,/. Chem. Phys., 87, 526 (1987). Correlated Molecular and Cumulative Atomic Multipole Moments. 

J. P. Ritchie and A. S. Copenhaver, J. Comput. Chem., 16, 777 (1995). Comparison of Potential-Derived Charge and Atomic Multipole Models in Calculating Electrostatic Potentials and Energies of Some Nucleic Acid Bases and Pairs. 

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