Molecular multipoles

Time-dependent correlation functions. Similar pair and triplet distributions, which describe the time evolution of a system, are also known [318]. These have found interesting uses for the theory of virial expansions of spectral line shapes, pp. 225 ff. below [297, 298], 

A molecule may be viewed as a number N of nuclear charges, Z e, of a certain arrangement given by their position vectors, Rt for i = 1... JV, surrounded by an electronic cloud of charge density p(r) of finite dimensions. The potential of the electrostatic field at the point R outside the electronic cloud is given by 

The integration is over the whole charge cloud. It is common practice to express this in terms of a multipole series (Jackson 1984), 

In general, the Q m are complex quantities.1- The Y(m are spherical harmonics an asterisk designates the conjugate complex. Equation 2.41 

For a determination of the permanent multipole moments, Eq. 2.42 must be averaged over the ground state nuclear vibrational wavefunction transition elements are also often of interest which are matrix elements between initial and final rotovibrational states. For example, for a diatomic molecule with rotovibrational states vJM), the transition matrix elements (v J M Q(m vJM) will be of interest a prime designates final states. 

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Molecular multipole moments, quantities important for collisional in-... 

Molecular multipole components are best described in the molecular (i.e., body-fixed) frame x, y, z. For example, a dipole when aligned with the z-axis is characterized by a single number, the strength p of the dipole. The other two independent components can then simply be expressed in terms of Euler angles or, in this case, of azimuthal and polar angle, q> and 3, between molecular (x, y, z) and laboratory-fixed frame (X, Y, Z). 

For some of the more common molecules, the low-order molecular multipole moments are known [166, 378] (Landolt-Bornstein 1974). Collision-induced absorption of molecular gases arises mainly from multipolar induction. Studies of collision-induced absorption in the molecular gases provides, therefore, useful information on multipole moments [38]. 

On a scale of the order of atomic size, molecular multipole fields vary strongly with orientation and separation. As a consequence, one will generally find induced dipole components arising from field gradients of first and higher order which interact with the so-called dipole-multipole polarizability tensor components, such as the A and E tensors. 

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

D. E. Stogryn and A. P. Stogryn, Molecular Multipole Moments, Molec. Phys., 11 371, 1966. 

G. Birnbaum and E. R. Cohen. Determination of molecular multipole moments and potential function parameters of non-polar molecules from far infrared spectra. Molec. Phys., 32 161, 1976. 

E. R. Cohen. Infrared spectral moment and molecular multipoles in collision-induced absorption. Can. J. Phys., 54 475, 1976. 

One of the advantages of the molecular (one center) and multicenter multipole expansions of the MEP is that, through truncating the series after some terms, one can get an analytical expression. The molecular multipole expansion, in contrast with the multicenter multipole expansion, diverges at distances of chemical interest. The various multicenter multipole expansion [87, 89, 93-96] practically equivalent to each other [97]. 

The first and second terms are the contributions of the atomic nuclei and the electrons, respectively. The calculation of the multipole integrals in the second term is simpler and faster than that of the potential integrals. After rearrangement of Eq. (12) a molecular multipole moment can be expressed as a sum of the atomic contributions (ukvlwm i ... 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

Mulder F (1978) Ab initio calculations of molecular multipoles, polarizabilties and Van der Waals interactions. PhD thesis, University of Nijmegen, Nijmegen... 

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

Electric and magnetic properties of microsystems. Definition of multipoles electrostatics of permanent multipoles interaction energies for two multipoles induced molecular multipoles interaction energies of induced multipoles. Tables of point groups tensor elements of multipoles vector elements of multipoles tensor elements of polarizabilities. 

D. W. Oxtoby. Local polarization theory for field-induced molecular multipoles. J. Chem. Phys., 72 5171-5176 (1980). 

S. Kielich. Nonlinear refractive index and light scattering due to fluctuations of molecular multipole electric fields. Optics Communic., 34 367-374 (1980). 

In order to extend the CoMMA approach to account for the lipophilicity of the molecule, the - Leo-Hansch hydrophobic fragmental constants [Abraham and Leo, 1987] have been proposed as a set of atomic lipophilic weights for the calculation of lipophilic molecular multipole moments (hydropoles) [Burden and Winkler, 1999a]. 

The first term always represents the exchange repulsion and the second term the London dispersion attraction. The Coulomb term is sometimes added in order to represent the electrostatic interactions among molecules, with fractional atomic point charges qx° and qxt> used. Alternatively, one has included these electrostatic contributions by adding the leading molecular multipole-multipole interaction term to an atom-atom potential of 12-6 or exp -6 type. 

Mulder, F., Ab Initio Calculations of Molecular Multipoles, Polarisabilities and van der Waals Interactions, Ph.D. Thesis, Nijmegen, 1978. 

An alternative coupling scheme, that proves to be advantageous in many regards, and is now the preferred theoretical method, is the multipolar framework [32, 33] of molecular QED. In this formulation, molecules couple directly to the radiation fields through their molecular multipole moments. Because the Maxwell fields obey Einstein causality, interactions between species is properly retarded, which is mediated by the emission and absorption of transverse photons. It is convenient to partition the total many-body QED matter-field multipolar Hamiltonian into a sum of parficle, radiation field, and interaction terms as follows ... 

Except in simple cases such as these, it is not generally possible to define net atomic charges from molecular multipoles alone. There usually will be more atoms in the molecule than the known molecular multipole moments, and very often no symmetry is present to reduce the number of charges needed. To allocate atomic charges in a reasonable way, it is necessary to devise rules which... 

A molecular multipole expansion is poorly converged at distances of chemical interest, for instance, at distances between atoms of molecules in a dimer or crystal. To obtain electrostatic interaction energy between heteronuclear diatomic molecules at close distance it is always better to model observed molecular point dipoles by a distributed dipole model that places net charges on the atoms. In general, a distributed multipole model of a molecule consists of a set of sites with each site having its own multipoles. Obviously distributed multipole models are not uniquely defined. For instance, at which site in a polyatomic ion is the ionic charge (monopole) located Do you spread it around evenly among the sites ... 

The electric potential resulting from any charge distribution can be represented by a convergent multicenter multipole expansion.Molecular moments of isolated molecules can be obtained experimentally, such as from the Stark effect on the microwave spectra of gas molecules. Studies of van der Waals clusters and molecular association in general require a multicenter model for short-range electrostatic interaction. Mulder and Huiszoon, for instance, found that a molecular multipole expansion was not satisfactory for the electrostatic interaction of... 

Thus any molecular multipole moment may be decomposed into additive atomic multipole moments (AAMMs)... 

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