Taylor series dipole moment

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

As for the change of dipole moment, the change of polarizability with vibrational displacement x can be expressed as a Taylor series... 

Taylor series 260 torque, correlation functions 28 transfer time, rotational relaxation 51 transitions dipole moment 30 forbidden 30 non-adiabatic 130 translational velocity v 6... 

This integral of the electronic dipole moment operator is a function of a nuclear coordinate Q. The integral may be expanded in a Taylor series with respect to Q (equation 4) and... 

Similarly, the interaction with a magnetic field can be written in terms of a magnetic dipole, quadrupole, and other moments. However, the first derivative in the Taylor series expansion, in the presence of magnetic field, is the permanent magnetic dipole moment the second derivative is the magnetizability. For a further study on the electrical and magnetic properties, one can refer to Dykstra [9]. 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

Hyper)polarizabilities are defined as the coefficients in the Taylor series expansion of the dipole moment - or the energy - in the presence of static and/or oscillating electric fields ... 

We can examine how induced polarization behaves as a function of an applied electric field, ( , by considering the induced electric dipole moment (cf. Section 6.1.2.2), as a Taylor series expansion in... 

We start by considering the origin of the dipole moment, which represents the lowest order nonzero term in a Taylor series expansion of the electrostatic potential arising from a neutral body (i.e., a molecule). For an assembly of n discrete charges, the electrostatic potential at a coordinate r may be written... 

By expanding the above dipole moment operators around the origins of qa and qfe, in the Taylor series gives... 

We have not failed to recognize that appropriately designed (6,0) carbon and C/B/N nanotubes may display considerably enhanced nonlinear optical activity. This term refers to the response of the dipole moment of a molecule (or the polarization of bulk material) to the oscillating electric field of electromagnetic radiation.82 85 The component of the dipole moment along an axis i in the presence of an electric field e can be represented by a Taylor series ... 

The (instantaneous) dipole moment p varies during the vibration. It can be expanded into a Taylor series at the equilibrium geometry ... 

The microscopic polarization of a molecule in an external field (or the dipole moment, i.e., the positions of the charges in the molecules averaged over the molecular volume) can be expanded in a Taylor series ... 

In section 6.8.2 we described and solved the Schrodinger equation for a harmonic oscillator, equation (6.178). The potential energy was expressed in terms of a vibrational coordinate q which was equal to R - Re, Re being the equilibrium bond length. The dependence of the electric dipole moment on the internuclear distance may be expressed as a Taylor series,... 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

In the FF procedure, the dipole moment ( ) of a molecule in its ground state, in the presence of a static external electric field (E), is expanded as a Taylor series ... 

This is because the electric dipole operators, ftp, act only on the electronic coordinates, the contribution to the transition moment from the nuclear coordinates being negligible. The resulting matrix elements, 9ge(Q), are still parametrically dependent on Q through the electronic states. This dependence is supposed to be a weak function of the internuclear coordinates and is therefore described by a rapidly converging Taylor series expanded about the equilibrium configuration, Q = 0, of the ground electronic state ... 

The nonrigidity of a molecule causes the dipole moment and other properties to differ in different vibrational and rotational states. Within the Bom-Oppenheimer approximation the property may be expanded as a Taylor series in the normal coordinates and expectation values taken for particular vibration/rotation states. For a diatomic... 

Thus far we have focused on using CVPT to follow the arrows down in Fig. 1. In other words, given a potential and dipole moment operator, expanded in a Taylor series about the equilibrium geometry of the molecule, we have calculated a variety of observables. It should be stressed that CVPT will not provide accurate results for all molecules. In particular, if the molecule is undergoing large-amplitude motion about multiple minima, our approach is not recommended. 

However, Equations (1) and (2) are approximations generally, polarizability cannot be regarded as a constant and the induced polarization is a non-linear function of field strength. This non-linearity becomes increasingly important at very intense electric fields. The non-linear dependence of a dipole moment on field can be expressed as a Taylor series as shown in Equations (3) and (4)... 

The bulk susceptibilities from Eq. [2] are to be related to molecular response properties denoted as a, p, and -y. These molecular properties can be defined from the Taylor series of the response of the molecular dipole moment as follows ... 

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