Rotational strength

Because circular dichroism is a difference in absorption for left and right circularly polarized light, its theoretical description includes subtraction of the transition probabilities induced by left and right circularly polarized radiation. The interaction Hamiltonian that determines transition probability includes electric, , and magnetic, B, fields of electromagnetic circularly polarized radiation, and the electric, /i, and magnetic, m, dipole moments of the molecule. 

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. 

The rJ of the free acid decreases linearly with a small positive slope as the Z values rise. In contrast, the rJ of the uncomplexed anion, the species participating In complexatlon, drops sharply between Z values of 80 and 83, varying little above and below these values. Thus, the conformation of the anion tends toward one of two metastable states depending upon solvent Z value. 

The role of the solvent in determining equilibrium solution conformation can best be understood in terms of functional group stabilization. In polar protic media the equilibrium conformation of the uncomplexed anionic ionophore is determined by the solvation of the carboxylate anion and the polar llgandlng groups. 

ACS Symposium Series American Chemical Society Washington, DC, 1980. 

CD was utilized to obtain the solvent dependency of the conformation of the cation-lonophore complex as well as Kp s. Saturation Isotherms were plotted from linear computer fits of 1/[cation] versus 1/ArJ the slopes yielded Kjj s while extrapolation of Rq to infinite cation concentration provided the rJ s of the cation-saturated lonophore. It Is Important to note that the cation Itself is a significant vlnclnal moiety, which by virtue of Its charge, polarizability and location with respect to the chromophore of concern, can modify the rotational strength of the chromophore. 

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Natural circular dichroism (optical activity). Although circular dichroism spectra are most difficult to interpret in terms of electronic structure and stereochemistry, they are so very sensitive to perturbations from the environment that they have provided useful ways of detecting changes in biopolymers and in complexes particularly those remote from the first co-ordination sphere of metal complexes, that are not readily apparent in the absorption spectrum (22). It is useful to distinguish between two origins of the rotational strength of absorption bands. 

Table 9 gives some cases where the rotational strengths of absorption bands have been measured in metalloproteins. At the present time these changes are not used to diagnose the nature of the ligands of the metal but rather they have been used to follow minor changes at the metal when substrates or inhibitors interact with the metals. The sensitivity of CD and MCD measurements to very small changes in the metal environment make them very attractive for protein/metal complex studies. 

Table 9. Rotational strengths of absorption bands of metalloproteins...

The planar C2h and C2V geometries of the 1,3-butadiene moiety are achiral structures and obviously they cannot show optical activity (i.e. ORD and CD). This has, of course, a spectroscopic origin. The optical activity of a transition Pq — Pi is determined by its Rotational Strength (R)1 defined as the scalar product... 

As shown in the Appendix (in Section V), in the C2h point group, the 1Ag - 1Bll (i.e. the monoelectronic r - jr excitation) possesses only an electric dipole moment, while in the C2V structure the electric and magnetic dipole moments, both non-vanishing, are orthogonal. In both cases the product in equation 1 leads to zero rotational strength. 

The rotational strength calculated for I is as large as that of a butadiene twisted by 20°. In II, with an out-of-plane methyl, R increases by a factor of about 2. This shows that the contributions to R of dissymmetric substituents of chiral cisoid dienes may be comparable to and even outweigh the contributions arising from the intrinsic dissymmetry of the chromophore. 

It is apparent that the jr systems in 30a and 30b are distorted in opposite senses. The measured rotational strength is thus the weighted average over these two conformations, which depends on the relative populations as well as on the individual rotational strengths. 

The main interest of this molecule resides in the fact that the principal source of rotational strength of the it - it lowest energy transition has been attributed40 to the twist of one of the two double bonds (a = —136°, as in fraws-cyclooctene) rather than to the twist of the 1,3-butadiene moiety (6 = +50.2°)... 

Dienes in quasi-s-fraws conformation are found only in cyclic structures where perfect planarity is hindered. The DR also holds valid for this kind of conformation, as demonstrated by the considerations of Section II.D.l.a and also confirmed by all the reported calculations. Indeed, contrary to what is sometimes found for cisoid systems, the rotational strength evaluated by many types of calculation is invariably found to follow the diene rule for transoid systems. However, very small skew angles are usually found in real molecules and this implies that the main contribution to the observed optical activity cannot come from the weak intrinsic distortion, but is more likely to stem from the dissymmetric perturbations, notably of the allylic axial substituents. 

In the skewed form, instead, the transition is allowed both electrically and magnetically, with parallel transition moments. The product in equation 1 is hence non-vanishing, implying that this transition has finite rotational strength. This observation leads to the conclusion that skewed 1,3-butadiene is an intrinsically dissymmetric chromophore. 

We are now able to calculate the rotational strength according to equation 3 ... 

The rotational strength is then easily calculated by the equation R = fi m... 

Vibrational circular dichroism arises from the interference of the electric dipole transition moment (p joi and the magnetic dipole transition moment (m )io and is proportional to the rotational strength, / ,o, where... 

In this expression Im refers to the imaginary part of the scalar product, since (m )io is a pure imaginary quantity. The rotational strength is positive or neg-... 

The dipole and rotational strengths are related to the experimentally observed intensities by... 

Denoting the electric and magnetic dipole transition moments of oscillators a and b as p, lUg and m, lUb, respectively, the rotational strength of the coupled oscillator is given by (34)... 

The first term in eq. [21] is the contribution of the intrinsic rotational strengths if oscillators a and/or b are themselves chiral. The second term is the coupled oscillator contribution due to the intrinsic moments and the third term is the coupled oscillator contribution due to the geometric arrangement of the two electric dipole oscillators. The latter two contributions give rise to a conservative bisignate couplet in the observed spectrum, if the coupled modes are sufficiently separated in frequency such that the positive and negative contributions do not cancel. 

In the trans conformation (a), the CH2CH2 moiety is achiral, pi and fi2 colinear and no VCD can be generated, that is, the rotational strength is zero. In the two gauche conformations (b) and (c), the CH2CH2 units have opposite chirality, and generate coupled oscillator VCD intensity of opposite sign. 

The sign patterns of the calculated rotational strengths for both the OH and C=0 couplets were in agreement with experiment only for the rotamer with trans OH groups, 1 and the two possible intramolecular hydrogen-bonding patterns, shown in 2 and 3. 

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