Operator dipole

Analogous considerations can be used for magnetic dipole and electric qiiadnipole selection rules. The magnetic dipole operator is a vector with tln-ee components that transfonn like R, R and R. The electric... 

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

As for the Imear response, the transitions occur tlnough the electric-dipole operator and are characterized by the matrix elements hr equation Bl.5.30, the energy denominators involve the energy differences... 

The function/( C) may have a very simple form, as is the case for the calculation of the molecular weight from the relative atomic masses. In most cases, however,/( Cj will be very complicated when it comes to describe the structure by quantum mechanical means and the property may be derived directly from the wavefunction for example, the dipole moment may be obtained by applying the dipole operator. 

Wfi cri you perform a single poin t sem i-cmpirical or ah initio ealeu-laliori, you obtain th c en ei gy and tli e first dci ivalives of the eu ei gy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a n urn ber of oth er molecti lar properties th at could be available to you. Molecularproperties arc basically an average over th e wave fun ction of certain operatorsdescribin g the property. For exam pie, the electron ic dipole operator is basical ly ju st the operator for the position of an electron and the electron ic con tribution to the dipole iTi otTi en t is... 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

The faet that the El approximation to af i eontains matrix elements of the eleetrie dipole operator between the initial and final states makes it elear why this is ealled the eleetrie dipole eontribution to afwithin the El notation, the E stands for eleetrie moment and the 1 stands for the first sueh moment (i.e., the dipole moment). 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

The n ==> n transition thus involves ground Ai) and exeited Ai) states whose direet produet (Ai x Ai) is of Ai symmetry. This transition thus requires that the eleetrie dipole operator possess a eomponent of Ai symmetry. A glanee at the C2v point group s eharaeter table shows that the moleeular z-axis is of A symmetry. Thus, if the light s eleetrie field has a non-zero eomponent along the C2 symmetry axis (the moleeule s z-axis), the n ==> 71 transition is predieted to be allowed. Light polarized along either of the moleeule s other two axes eannot induee this transition. 

In eontrast, the n ==> 71 transition has a ground-exeited state direet produet of B2 X Bi = A2 symmetry. The C2v s point group eharaeter table elearly shows that the eleetrie dipole operator (i.e., its x, y, and z eomponents in the moleeule-fixed frame) has no eomponent of A2 symmetry thus, light of no eleetrie field orientation ean induee this n ==> 71 transition. We thus say that the n ==> 71 transition is El forbidden (although it is Ml allowed). 

The third integral vanishes because the derivative of the dipole operator itself p = Zi e rj + Za Za e Ra with respect to the coordinates of atomic centers, yields an operator that contains only a sum of scalar quantities (the elementary charge e and the... 

The result of using these identities as well as the Heisenberg definition of the time-dependenee of the dipole operator... 

In this form, one says that the time dependenee has been reduee to that of an equilibrium averaged (n.b., the Si pi I)i ) time correlation function involving the eomponent of the dipole operator along the external eleetrie field att = 0(Eo p) and this eomponent at a different time t (Eo p (t)). 

Here, d is the electric dipole operator, tp (,v) are the wave functions of the intragap states with energies w/2, and C is an -independent coefficient (for small w, we can neglect the weak tw-dependence of the real pan of the dielectric constant). 

Thus coefficients with an even total order I + m + n are real and coefficients with an odd total order I m + n are pure imaginary. In the following we consider only dipole hyperpolarizabilities. In this case the four operators A, B, C and D are cartesian components of the dipole operator and the odd dispersion coefficients vanish. 

First consider the dipole operator (O = r). The matrix elements on rhs of eq. 17 are thus just the dipole transition moments, and the commutator becomes C = -ip. As the exact solution (complete basis set limit) to the RPA is under consideration, we may use eq. 10 to obtain... 

It was shown in the preceding section that PECD can be anticipated to have an enhanced sensitivity (compared to the cross-section or p anisotropy parameter) to any small variations in the photoelectron scattering phase shifts. This is because the chiral parameter is structured from electric dipole operator interference terms between adjacent -waves, each of which depends on the sine of the associated channels relative phase shifts. In contrast, the cross-section has no phase dependence, and the p parameter has only a partial dependence on the cosine of the relative phase. The distinction between the sine... 

At a fundamental level, it has been shown that PECD stems from interference between electric dipole operator matrix elements of adjacent continuum f values, and that consequently the chiral parameters depend on the sine rather than the cosine of the relative scattering phases. Generally, this provides a unique probe of the photoionization dynamics in chiral species. More than that, this sine dependence invests the hj parameter with a greatly enhanced response to small changes in scattering phase, and it is believed that this accounts for an extraordinary sensitivity to small conformational changes, or indeed to molecular substitutions, that have only a minimal impact on the other photoionization parameters. 

Here p is dipole operator and E(t) is electric field in optical radiation. Due to the application of V(t), the system becomes nonstationary and the rate of energy absorption can be calculated ... 

An electric dipole operator, of importance in electronic (visible and uv) and in vibrational spectroscopy (infrared) has the same symmetry properties as Ta. Magnetic dipoles, of importance in rotational (microwave), nmr (radio frequency) and epr (microwave) spectroscopies, have an operator with symmetry properties of Ra. Raman (visible) spectra relate to polarizability and the operator has the same symmetry properties as terms such as x2, xy, etc. In the study of optically active species, that cause helical movement of charge density, the important symmetry property of a helix to note, is that it corresponds to simultaneous translation and rotation. Optically active molecules must therefore have a symmetry such that Ta and Ra (a = x, y, z) transform as the same i.r. It only occurs for molecules with an alternating or improper rotation axis, Sn. 

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