Polarizability, multipole

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

The properties available include electrostatic charges, multipoles, polarizabilities, hyperpolarizabilities, and several population analysis schemes. Frequency correction factors can be applied automatically to computed vibrational frequencies. IR intensities may be computed along with frequency calculations. 

Studies of the internuclear distance dependence of multipole polarizabilities are... 

The role of quadmpole polarizabilities is less pronounced. Jens Oddershede, e.g., has studied the quadmpole polarizability of N2 [10]. Furthermore, there are studies which point out the need for calculations of quadmpole polarizabilities, e.g., for the interpretation of spectra obtained by surface-enhanced Raman spectroscopy [42,43]. Generally the interest in multipole polarizabilities increases due to new experimental data. We decided, therefore, to also study how different linear response theory methods perform in the calculation of quadmpole polarizabilities. 

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. 

The symbol aylyx)Lx (0) denotes the irreducible component of the multipole polarizability... 

The multicenter expansion of the dispersion energy in terms of the distributed multipole polarizabilities can be obtained is same way starting from Eq. (1-94). The final expression reads ... 

Under the assumption of the existence of ideal ionic crystals, built up from point charges and point multipoles, the NQR spectrum is completely determined by the crystal field of the electric multipoles. The experimental results of NQR can be explained within the frame of this model. Refinements of the model, such as the dependence of multipole polarizabilities upon the crystal field or the influence of overlapping of the electron clouds, are not yet understood quantitatively. 

Hitherto, theoretical calculations have been carried out for a few multipole polarizabilities only, i.e. for field-gradient-induced quadrupole polarizabilities of atoms and ions [defined by the tensor of... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

For neutral molecules, the dipole polarizabilities and hyperpolarizabilities are invariant to the choice of the moment center. Other multipole polarizabilities may be invariant in certain cases of high molecular symmetry. The changes that may occur in P , P ,. . . upon shifting an evaluation center are determined by the changes in the moments or moment operators. If a particular origin translation leads to... 

In the application of DHF to multipole polarizabilities, the first task is assembling the derivative Fock operators because in SCF the Fock operator is the effective (one-electron) Hamiltonian for the system. The parameters of differentiation are the elements of the expansion of the electrical potential, Vy, Fj, F, V y,.. ., . From Eqn. (28), one may see that the... 

One simple but very effective logic procedure developed for DHF keeps track of the derivatives. An integer list is constructed for each particular derivative where there is one integer in the list for each parameter. For first-and second-multipole polarizabilities, there will be nine parameters, three for the first moment and six for the second moment, after ignoring equivalent elements in V. One way of ordering them is V, V, V, Vj,y, Vyy, Vy, and K22, and then nine integers, ordered the same, are associated with each a derivative ... 

DHF/ELP Basis Multipole Polarizabilities and Hyperpolarizabilities of trons-NjHj... 

Spelsberg, D. (1999) Dynamic multipole polarizabilities and reduced spectra for OH, J. Chem. Phys., Ill, 9625-9633. 

M is a first-rank polytensor. The usual multipole polarizabilities can be arranged in a second-rank (two index) polytensor, by using the ordering of element labels in M for the rows and columns in For instance, pS or pPj is a dipole(A) quadrupole(xx) polarizability tensor element. Hyperpolarizabilities... 

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