Tensor spherical multipole

By using the multipole expansion, we in fact replace the exact radial expansion coefficients A (7 ) in Eq. (1-124) by the approximate coefficients A poi(-K), which are power series in R 1. Closed expressions for the latter have been given149 161 in terms of the irreducible spherical tensors of multipole moments and polarizabilities. 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

For spherical tensors, the multipole moments are defined relative to the local axis system of the rigid molecule and so the energy of interaction between two multipole moments on different atoms is a function of the atom-atom separation R and some of the scalar products between this vector R and the unit local axis vectors on each atom (xi,yi,Z ,x2,y2,Z2). For example, the interaction between the out-of-plane component of the quadrupole tensor (represented by Q20 or zz) on site 1 and the x component of dipole (Qi ic = /xx) on site 2 has the form ... 

With this notation, the electric charge qo of a monopole equals Qoo-Cartesian dipole components px, py, pz, are related to the spherical tensor components as Ql0 = pz, Qi i = +(px ipy)/y/2, with i designating the imaginary unit. Similar relationships between Cartesian and spherical tensor components can be specified for the higher multipole moments (Gray and Gubbins 1984). 

The operator V,, b is physically interpreted as representing the interaction of the instantaneous moment with respect to center A with the instantaneous 2>B moment with respect to center B and can be expressed in terms of irreducible spherical or reducible Cartesian tensor operators of multipole moments. The operator V,, can be written as... 

The spherical form of the multipole expansion is very useful if we are looking for the explicit orientational dependence of the interaction energy. However, in some applications the use the conceptually simpler Cartesian form of the operators V1a 1b may be more convenient. Moreover, unlike the spherical derivation, the Cartesian derivation is very simple, and can be followed by everybody who knows how to differentiate a function of x, y and z 149. To express the operator V,, in terms of Cartesian tensors we have to define the reducible, with respect to SO(3), tensorial components of multipole moments,... 

Gray CG, Lo BWN (1976) Spherical tensor theory of molecular multipole moments and polarizabilities. Chem Phys 14 73—87... 

State multipoles are components of a spherical tensor, which is the following tensor product (3.99), averaged over unobserved magnetic quantum numbers. 

The multipole operators are here defined as spherical tensors ... 

The electrostatic interaction energies are evaluated using the multipole expansion formulas for each intermolecular pair of sites. Explicit expressions for all terms up to R are given in Ref. 117, and for terms up to R in Ref. 118. Stone has provided a general formulation and discussion of the spherical tensor and Cartesian tensor approaches. The program ORIENT incorporates... 

Some time ago we contributed to the development of the irreducible spherical tensor multipolar theory of light scattering [13,28], According to Ref. 13, the M component of the A th-rank dipole-arbitrary order multipole linear polarizability of a pair of interacting molecules A and B reads as... 

When induction operators of high-order multipoles are taken into account intensity calculations tend to become very cumbersome [30,31]. We propose a relatively easy way of performing these calculations using the irreducible spherical tensor theory of multipole light scattering [e.g., Eqs. (6) and (7)] together with symbolic calculations of the Wigner coefficients by computer. 

The symmetry of linear molecules imposes less constraints on the multipole polarizability tensors than does tetrahedral symmetry, so the forms of the respective tensors for linear molecules are usually more highly complicated than for tetrahedral ones. This certainly complicates interpretation of light scattering results but has much less influence on complications in running our program. By means of the same methods as in the case of tetrahedral molecules we calculate the spherical irreducible components of the dipole-dipole, dipole-quadrupole, and dipole-octopole polarizability tensors as dipole-dipole... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

By having written a point-multipole as the spherical tensor gradients passing through a point, one easily derives the particle mesh Ewald method for point multipoles. The main differences occur in the calculation of the structure factor, which requires spherical tensor gradients of the Cardinal B-spline weight, and the calculation of the short-range real-space correction (see Section 1.6.3). 

The sum in (8) runs over all multipolar centers (nuclei and bond barycenters) of fragments A and B. The interaction energy tp between the multipoles on two sites with position vectors A and B is calculated by summing the product of a geometrical tensor T and each multipole moment Q. Multipole moments are expressed using spherical harmonics with rank /, k, where / = 0, 1, 2. T is used to relate the local axis systems of the multipolar sites A and B separated by vector r. ... 

The exact atomic electrostatic potential was calculated and compared to a multipole expansion using spherical tensors. The authors prove that the convergence of this expansion is faster than previously assumed, even for com-... 

Let us consider a molecule placed in a cavity surrounded by a dielectric continuum (fig. 1). The relative dielectric permittivity of the continuum is assumed to be e and in the cavity it is taken as equal to the permittivity of a vacuum. In the following we shall assume that the charge distribution of the solute is represented by a single center multipole expansion. An equivalent distributed multipole 2,3] representation may be used without further difficulty. We shall use the spherical tensors formalism [4,5] for the multipoles in which the 2/4 1 components of the multipole of rank / at the origin are defined from unnormalized spherical harmonics [6] by the equation ... 

It has recently become clear that classical electrostatics is much more useful in the description of intermolecular interactions than was previously thought. The key is the use of distributed multipoles, which provide a compact and accurate picture of the charge distribution but do not suffer from the convergence problems associated with the conventional one-centre multipole expansion. The article describes how the electrostatic interaction can be formulated efficiently and simply, by using the best features of both the Cartesian tensor and the spherical tensor formalisms, without the need for inconvenient transformations between molecular and space-fixed coordinate systems, and how related phenomena such as induction and dispersion interactions can be incorporated within the same framework. The formalism also provides a very simple route for the evaluation of electric fields and field gradients. The article shows how the forces and torques needed for molecular dynamics calculations can be evaluated efficiently. The formulae needed for these applications are tabulated. 

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