Rotatory strength equations

The computational route to the calculation of two-photon circular dichroism spectra was opened only recently [115], when it was recognized that the molecular parameters entering the TPCD rotatory strength equation 2.83 can be obtained, as for the two-photon transition amplitude in Eq. 2.78, as single residues of appropriate quadratic response functions [82] ... 

Ae(A) is related to the rotatory strength nR of the transition between the ground state 0 and the nth excited state through the equation... 

Using the formalism of response theory [24,31], the scalar rotatory strength for a transition from the ground state 0) to an excited state n) can be evaluated as the residue of the linear response function. In the velocity gauge formulation, nR is given by the equation... 

Within these three tensors final Stephens and Devlins equations for the rotatory strength Pgigo(i) of the fundamental excitation of i-th mode is the following [82] ... 

A transition will be characterized by a helical displacement of the electron if the magnetic and electric transition dipoles are aligned. This is reflected in the Rosen-feld equation for the CD intensity or rotatory strength, TZa- j, for a transition from a ground state n to an excited state j in a collection of randomly-oriented molecules ... 

These results are now combined in the Rosenfeld equation to yield the rotatory strength of both exciton states ... 

Experimental aspects of BCD as well as qualitative models such as sector rules for chromophores have been extensively reviewed. " A very brief description of experimental instrumentation and measurement is provided in Section 2. A number of successful ab initio and empirical approaches to BCD have been implemented as computer programs and permit the prediction of optical rotatory strengths by equation (2). These are described in Section 4. Examples of applications are provided in Section 5. 

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 form of eq. (11) is similar to that encountered in CD spectroscopy. Finally, it should be noted that this equation has been derived for measurements at a particular wavelength, X expressions involving integrated (over the electronic band) rotatory and absorption strengths may also be developed [2],... 

It follows from Equation (80) that given the partial rotatory dispersion curve one can, in principle, find the associated partial ellipticity curve and hence the rotational strength of the transition under consideration. However, one is never so fortunately endowed. At best one has only the superposition of the partial rotatory dispersion curve of a transition and the sum of the Drude tails coming from all the other transitions which fall off from their band centers... 

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