Dipole transition moments equations

From Equation (1.35), the electric dipole transition moment Afo f may be thought of as the dipole moment of the transition density The transition density is a purely quantum mechanical quantity and cannot be inferred from classical arguments. A more pictorial representation of the electric dipole transition moment equates it to the amplitude of the oscillating dipole moment of the molecule in the transient nonstationary state that results from the mixing of the initial and the final states of the transition by the time-dependent perturbation due to the electromagnetic field, and which can be written as a linear combination Cq o + + This emphasizes the fact... 

Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

In Equation 12.11, v is the frequency of incident radiation (cm-1), v o is the frequency corresponding to the energy difference between ground and excited electronic state, the sum is over all excited states, and d0n and d o are the dipole transition moments between the ground and excited state (dno = (n d 0) = / I nd I o dx, the T s are wave functions and d is the dipole moment operator). At low frequency (v -> 0) Equation 12.11 reduces to the static field expression... 

It is seen from Equation 19 that the electronic transitions take place without changing the equilibrium positions of the nuclei, and the electronic component of the dipole transition moment is non-zero only if there is no change of the vibronic state during this transition. Dg is non-zero only if the transitions occur between the vibronic states within one electronic state, and the selection rules of Equation 16 are derived from the conditions for a non-vanishing matrix element in Dg. ... 

The transition moment, Equation 4.20, is not related to the change in dipole moment upon exitation. If the orbitals S, and Pj do not overlap in space, then Afey -=0. 

Equation (9) has a simple physical interpretation. The electric dipole transition moment for an electronic transition is defined as... 

In order to saturate a transition with a dipole transition moment /i.a j = 1 Debye somewhere in a band of 50 MHz, the pulse power must be sufficiently large to fulfill Equation 49 where Aoj/27r= 25 MHz. The waveguide formula... 

Equation 64 indicates that the gain in S/N ratio for the pulse method is essentially given by the square root of the ratio of the total sweep width and a characteristic line width in the spectrum. The actual gain in sensitivity in microwave spectroscopy depends on j, the value of the dipole transition moment, because the degree of polarization obtained from Equation 51 depends on Kj. Equation 64 assumes that the maximum of this function can be reached, but this is not always the case. With this limitation in mind, the following points may serve to summarize the advantages of the pulse... 

We note again that the magnetic dipole operator is pure imaginary, and the multiplication by i results in a real value. In the denominator of eq. (20) we have also neglected the magnetic dipole transition moments in eq. (14) compared to the usually larger electric dipole moments. This equation is sometimes written as... 

The approximation inherent in Equation [5b] means that the nuclear kinetic operator, -hyi) d ldQ ), has no influence on the electronic wavefunction, namely, the terms, -h d g/dQ)ei[S/dQi) and (-fi2/2)(a2 aQ2), which operate on the electronic part of the wavefunction, are omitted. This implies that infinitesimal changes in the nuclear configuration do not afiect the electronic wavefunction, g. This approximation is, however, not good enough in evaluations of the magnetic dipole transition moments. The disturbance to the electronic state caused by changes in the nuclear configuration has to be taken... 

Equation [8] is then substituted into [7] and [6]. Only first-order terms are kept in the wavefunction and in the derived matrix elements, and an average energy approximation is invoked to permit closure of the suras over vibrational states. For the electronic parts of the dipole transition moments, one obtains, for the /th normal mode... 

We mentioned in previous pages that (IIlB-22) was not the most efficient for use with electric-dipole-allowed transitions. The reason for this is that electric-dipole transition moments can be obtained experimentally from spectra they do not have to be estimated from wave functions pertaining to a hypothetical, completely isolated group. Therefore, it is wise to write our equations in a way which makes the best use of the experimentally determined transition moments and frequencies. 

From Equation (1.35), the electric dipole transition moment be... 

The value of Equation [12] is therefore at most a few times the ratio of a magnetic to an electric dipole transition moment. The largest observed values of A/// are of the order of 5x10, and values of 10 or less are more common. 

The theoretical foundation for all chiroptical techniques lies in the Rosenfeld equation (2) which expresses the rotatory strength of a transition, assumed to be between states 0 and n, as the imaginary part of the scalar product of the electric dipole and magnetic dipole transition moments for the transition. 

There is only one other ab initio implementation of the theory of optical activity to calculate optical rotatory strengths, that due to Hansen and Bouman, based on the random-phase approximation (RPA) and implemented in the program package, RPAC. The RPA method is intended to include those first-order correlation effects that are important both for electronic transition intensities and for excitation energies. The electric and magnetic dipole transition moments in RPA are given by equations (14), (15), and (16) (analogous to equations 7, 8, and 9, above). 

The electrostatic interaction of two chromophores with excitation energies (in wavenumbers), u, and vj, electric dipole transition moments, ixi and Jxj, situated at points, R, and Rj, is given by equation (18),... 

Insertion of equation (20) into equation (14) yields an expression in which the sum over states does not appear, namely the interesting result that the electronic contribution to the magnetic dipole transition moment is proportional to the overlap of two derivatives of the ground state electronic wavefunction ... 

In practice, the electric and magnetic dipole transition moments are usually expressed as summations of atomic properties, namely the atomic polar tensor (APT), P", and atomic axial tensor (AAT), MJ, which, in the VCT approach, can be extracted from equations (15) and (16) as. 

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