Multipoles, atomic, estimation

Results presented in the Table 6 indicate that the physical nature of pKa shifts is dominated at least in 99.8% by electrostatic term, with very small contribution of delocalization term not exceeding 0.2 %, whereas exchange component is completely negligible. Inexpensive atomic multipole estimates of the electrostatic term ApK may reasonably approximate costly SCF... 

It appears that in the case where minimum contacts R are quite short the electrostatic term still dominates, but this time delocalization and exchange components are not negligible and atomic multipole estimate (M)... 

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. 

W. A. Sokalski and R. A, Poirer, Chem. Phys, Lett., 98, 86 (1983). Cumulative Atomic Multipole Representation of the Molecular Charge Distribution and Its Basis Set Dependence. D. E. Williams and D. J. Craycroft, J. Phys, Chem., 89, 1461 (1985). Estimation of Dimer Coulombic Intermolecular Energy and Site Charge Polarization by the Potential-derived Method. 

Results presented in Table 8 indicate that for R2 > PTTS3 reaction step [39], the protonation of HIS 12D would have most pronounced catalytic effect. Similar conclusion may be obtained from the inspection of inexpensive atomic multipole (M) estimate of A. This may replace investi-... 

Since empirical force fields do not accurately estimate the true interatomic forces, it is difficult a priori to say how accurate the fast multipole approximation to the exact Coulomb potential and forces (exact in terms of the sum over partial charges) should be. Probably a good rule is to make sure that at each atom the approximate electrostatic force is within a few percent relative error of the true electrostatic force, obtained by explicitly summing over all atom pairs, i.e., IF — FJ < 0.05 F , for all atoms i, where F is the... 

Estimations indicate that if the wavelength of the radiation is large when compared with the size of the source (this is the case for all atoms and the overwhelming majority of ions), then the probability for radiation of a photon of multipolarity k is a rapidly decreasing function of k. Therefore, in the expansions over multipoles of certain parity it is usually sufficient to take into account the first non-zero term. However, contributions of Ek- and M(k — l)-transitions may be of the same order. 

In the case of valence band photoemission, the atomic model cannot directly be applied to the numerical estimation of the effect, but rather for a qualitative consideration only. For valence bands, the initial state is no longer described by a single spinor spherical harmonic as it was done in [32] but it can be expanded for a certain k value in a series of spherical harmonics [44] due to their completeness. This procedure will influence the values of the state multipoles and the dipole matrix elements in Eq. 5.6, but the general Eqs. 5.5 and 5.7 will remain unchanged. In particular, they should correcdy describe the dependence of MDAD on the angle of photon incidence. 

The charge distribution of the molecule can be represented either as atom-centred partial charges or as a multipole expansion. For a neutral molecule, the lowest order approximation considers only the dipole moment. This may be a quite poor approximation, and fails completely for symmetric molecules that do not have a dipole moment. For obtaining converged results, it is often necessarily to extend the expansion up to order six or more, i.e. including dipole, quadrupole, octupole, etc., moments. Furthermore, only for small and symmetric molecules can the approximation of a spherical or ellipsoidal cavity be considered realistic. The use of the Bom/ Onsager/Kirkwood models should therefore only be considered as a rough estimate of the solvent effects, and quantitative results can rarely be obtained. 

Electrostatic multipole moments of molecules, i.e., dipoles, quadrupoles, or octupoles, can also be obtained from QM wave functions. Methods like distributed multipole analysis (DMA) [84] or AIM [85] assign multipole moments to each atom or to specified sites of a molecule. The DMA method estimates multipole moments from QM wave functions and the highest obtained multipole moment depends on the basis set used. There are no limitations in this method on number or position of the multipoles anisotropic effects due to lone pairs or n electrons can also be considered. 

By symmetry arguments it can be shown that for nuclei the next multipole term possible is the hexadecapole. The magnitude of a hexadecapole term can be estimated to be of the order of 10" of the quadrupole term. Experimental evidence for the existence of hexadecapole moments for a number of nuclei has been obtained from atomic beam experiments as well as proton scattering studies. 

We simplify our considerations for the calculation of the van der Waals constant a furthermore in that we assume that the estimated interaction is valid up to a sharp boundary Rq of the molecule and oo for smaller distances. Without any doubt the higher multipoles and especially the exponentially decreasing repulsion forces of the first order manifest themselves in the medium distance, and apart fix)m this we know that the repulsion forces start gradually and that a sharply defined (i.e. temperature-independent) molecular radius does not exist. All the same our procedure reflects the same idealisations that we used previously in the discussion of this problem. For H atoms, where the force field is known, the situation can be checked in this respect and [we] find that the correct value deviates by 50% from the one calculated based on these simplifications. We are therefore not too unreasonable if we assume an uncertainty of 50% in our calculations, especially when we realise that the van der Waals equation itself allows for only a crude representation of the behaviour of real gases. But in my opinion it makes little sense to bring our presently very imprecise knowledge of molecular forces into agreement with more precise formulations of the equation of state. 

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