Rosenfeld equation

The next section (Sect. 2) is devoted to a lengthy discussion of the molecular hypothesis from the point of view of quantum field theory, and this provides the basis for the subsequent discussion of optical activity. Having used linear response theory to establish the equations for optical activity (Sect. 3), we pause to discuss the properties of the wavefunctions of optically active isomers in relation to the space inversion operator (Sect. 4), before indicating how the general optical activity equations can be related to the usual Rosenfeld equation for the optical rotation in a chiral molecule. Finally (Sect. 5), there are critical remarks about what can currently be said in the microscopic quantum-mechanical theory of optical activity based on some approximate models of the field theory. 

In order to make the connection with the usual discussion of optical rotation based on the Rosenfeld equation for a molecule49, one must express the exact states ip, of the chiral medium in terms of the states of the elementary excitations in the system, using the machinery of quantum field theory discussed in Sect. 2. Before considering this problem however it is instructive to consider first the role of the space-inversion operator P in optical activity. 

The resulting relation for rotational strength K is called Rosenfeld equation and has been derived in 1928 [78] ... 

The relatively simple Rosenfeld equation (8.6) determines the relation between the structure of a molecule and its interaction with circularly polarized radiation. Different methods are used for the computation of R, including the direct calculation by ab initio methods from first principles. However, at least within a limited range of applications, simplified approaches can be used that make a priori assumptions about a decisive mechanism by which optical activity of a molecule originates. 

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

In a chiral molecule the electron rearrangement during a transition is always helical which requires that p and m have a parallel component. The Rosenfeld equation for the CD intensity (or CD strength or rotational strength, R) of a transition in a collection of randomly oriented chiral molecules is (4) ... 

This is one form of the celebrated Rosenfeld equation, uncorrected for disparities between E and E. Secondly, the notion of partial quantities introduced above in connection with the polarizabilities may be extended to any parameter whose value depends upon the... 

As in previous chapters, the wavefunction for the initial state is written on the right in the matrix elements in Eq. (9.11), and the wavefunction for the final state on the left. For the electric dipole operator the order is immaterial because (f feljllf a) = (f a jl fe), but this is not the case for the magnetic dipole operator. Here, interchanging the orbitals changes the sign of the integral. Thus, because ( i, m fl) = —[Ta m Pb), the Rosenfeld equation (Eq. 9.11) also can be written ... 

To derive the Rosenfeld equation, we must consider the linked time dependence of the electric and magnetic fields of circularly polarized light. For light propagating along the z axis, the electric field can be written ... 

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. 

The rotational strength R is also formulated by the Rosenfeld equation ... 

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