Electric dipole moment, component matrix element

The polarizability in (1) is conveniently expressed in terms of the second-order matrix element of the electric dipole moment component Dz... 

The selection rules for the Raman spectrum turn out to depend not on the matrix elements of the electric dipole moment, but on the matrix elements of the molecular polarizability, which we now define. The application of an electric field E to a molecule gives rise to an induced electric dipole moment djnd (which is in addition to the permanent dipole moment d). If E= "> 1 + yl+ >zk, then the induced dipole moment has the components... 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

Electric dipole polarizability a is a second-order tensorial (3x3 matrix) quantity whose element describes the change of dipole moment component p,- induced by a change of electric field in direction j, where ij = x,y,z denote arbitrarily chosen Cartesian directions. Formally, if p, is the field-free dipole moment, and Ap,y is the change in dipole moment induced by a static electric field AFj in the j direction, then OLy can be evaluated as the limiting ratio... 

Here 0) is the frequency of the radiation, i denotes the initial state of the molecule, and f labels the final state of the photofragments, dv = Pj dE, where is the density of final states. Usually, the wavelength of the radiation considerably exceeds the size of the molecule, and one can use the dipole approximation (see, e.g., ref. 17). Then Hf d. fi (d is the component of the dipole moment along the external electric field) and the problem reduces to the analysis of the dipole matrix element... 

The electric dipole transition moments for the Cameron system are shown in the Fig. 9. The dependences for the M and Mg transition matrix elements on the inter-nuclear distances are completely different from one another. At the short internuclear distances the values of the mo transition matrix elements are more than three time larger than for m, but with the increase of internuclear distances the difference between the components is decreasing, changes sign and then increases drastically in the opposite direction. 

Assume a perturbing electric field of frequency co in the v-direction with perturbation matrix elements (V + W) = 2DV in (16), and consider the perturbation of the magnetic dipole moment s u-component. With the help of the definitions... 

The numerical approach to the calculation of electric polarizabilities requires much higher accuracy of the total SCF energy than in standard applications. The convergence threshold on the density matrix elements was set up as 10In this work we are only interested in the longitudinal component, a, of the polarizability tensor gi, the z-axis being directed along tSe chain. The value of is obtain from the first derivative of the field-dependent dipole moment y(E)jwith respect to E in the limit of zero-field. 

Given the expressions for rotational energy levels, the subsequent step is defining the selection rules that tell us which are the rotational transitions that take place. The key point is the interaction between the molecular electric dipole components (fixed in the rotating body) and the electric components (fixed in space) of the radiation field. Without going into detail, from the nonvanishing matrix elements of transition dipole moment, the selection rules governing rotational transitions are derived to be [1, 2]... 

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