Charge distribution multipole moment

Overall, the QM results indicate both a large quadrupole and out-of-plane character in the charge distribution of a water molecule in the liquid phase. Only SSDQOl has multipoles consistent with both features, although somewhat too little for the former and too much for the latter. Furthermore, it was shown that multisite models require at least six points to reproduce moments consistent with the QM results [55], as has also been found for polarizable multisite models [37, 57]. On the other hand, it was also shown [55] that multipole models are able to reproduce electrostatic potentials due to the QM charge distribution with moments up to the octupole. 

Unfortunately this is still too severe a requirement in many cases. It is often possible for molecules to approach to distances at which such spheres would overlap, without encountering the repulsive part of the potential and even when they do not overlap, they may approach so closely that convergence becomes very slow. For this reason, it has become common to adopt a distributed multipole description, in which each molecule is divided into a number of regions, each described by its own multipole moments. There are many ways of determining these distributed multipole moments [6-11] many authors have used distributed charges alone, but it is now widely... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

The three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

The molecular dipole moment is perhaps the simplest experimental measure of charge density in a molecule. The accuracy of the overall distribution of electrons in a molecule is hard to quantify, since it involves all of the multipole moments. Experimental measures of accuracy are necessary to evaluate results. The values for the magnitudes of dipole moments from AMI calculations for a small sample of molecules (Table 4) indicate the accuracy you may... 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Multipole moments are useful quantities in that they collectively describe an overall charge distribution. In Chapter 0, I explained how to calculate the electrostatic field (and electrostatic potential) due to a charge distribution, at an arbitrary point in space. 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

The charge distribution of the molecule can be represented either as atom centred charges or as a multipole expansion. For a neutral molecule, the lowest-order approximation considers only the dipole moment. This may be quite a poor approximation, and fails completely for symmetric molecules which do not have a dipole moment. For obtaining converged results it is often necessarily to extend the expansion up to order 6 or more, i.e. including dipole, quadrupole, octupole, etc. moments. 

Typical properties of the charge distribution are summarized by its various electric multipole moments. The electric dipole moment p. induced in the system by the external field is obviously... 

For polarizable charge distributions, additional classical-type interactions arise from the induced dipole, quadrupole, and higher moments on each monomer, which are proportional to the fields created by the asymmetric charge distribution on the other monomer. The proportionality constants for each multipole field are the monomer polarizabilities aa and ah (a111 for dipole fields, a(Q) for quadrupole fields, etc.). The leading two induction interactions are ... 

The spherical harmonic density functions are referred to as multipoles, since the functions with 1 = 0, 1, 2, 3, 4, etc., correspond to components of the charge distribution p r) which give nonzero contributions to the monopole (/ = 0), dipole (/ = 1), quadrupole (/ = 2), octupole (/ = 3), hexadecapole (/ = 4), etc., moments of the atomic charge distribution. 

The atomic electrostatic moments of an atom are obtained by integration over its charge distribution. As the multipole formalism separates the charge distribution into pseudoatoms, the atomic moments are well defined. 

If the moments are referred to the nuclear position, only the electronic part of the charge distribution contributes to the integral. According to the multipole formalism of Eq. (3.32),... 

The Electrostatic Potential Outside a Charge Distribution in Terms of the Multipole Moments... 

The summation in Eq. (8.53) is slowly converging if a molecular charge distribution is represented by a single set of moments. However, the expression can be written as the summation over the distributed moments, centered at the nuclei j, which is precisely the information available from the multipole analysis ... 

To evaluate this expression for distributions expressed in terms of their multipolar density functions, the potential <1> and its derivatives must be expressed in terms of the multipole moments. The expression for charge distribution has been given in chapter 8 [Eq. (8.54)]. Since the potential and its derivatives are additive, a sum over the contributions of the atom-centered multipoles is again used. The resulting equation contains all pairwise interactions between the moments of the distributions A and B, and is listed in appendix J. 

The electrostatic interaction between two nonoverlapping charge distributions A and B, consisting of NA and NB atoms, respectively, and each represented by their atom-centered multipole moments, is given by (using the Einstein summation convention for the indices a, / , y) (Buckingham 1978)... 

FIM can also be used to study properties, such as the surface induced dipole moment and the effective polarizability of some surface atoms, kink site atoms and adsorbed atoms etc. The charge distribution of a surface atom is obviously completely different from that of a free atom because of its interaction with the surface and in addition surface atoms are partially shielded by itinerant charges of the surface. The charge distribution of a surface atom can be described by the magnitudes of the electric multipoles of the atom. 

The Born equation is based on the simple model of a spherical ion with a single charge at its centre. Such an ion has no dipole moment and no higher multipole moments, but real molecular ions are of course much more complex. Since the electrical charge is distributed among all the atoms of the... 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The potential outside the charge distribution and due to it is simply related to the moments, as is the interaction energy when an external field is applied.14 The multipole moments are thus very useful quantities and have been extensively applied in the theory of intermolecular forces, particularly at long range where the electrostatic contribution to the interaction may be expanded in moments. Their values are related to the symmetry of the system thus, for instance, a plane of symmetry indicates that the component of n perpendicular to it must be zero. Such multipoles are worth calculating in their own right. 

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,... 

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