Profile desymmetrization

Fig. 5.6. Comparison of classical and quantal profiles computed from the same dipole and potential function, for He-Ar at 295 K (thin and thick solid lines, respectively). Classical profiles desymmetrized according to the procedures P-1 through P-4 are also shown.

Zopiclone is a chiral cyclopyrrolone with hypnotic properties, possessing a pharmaceutical profile of high efficacy and low toxicity, similar to that of benzodiazepines. Zopiclone has been commercialized as a racemic mixture however, the (S)-enantiomer is more active and less toxic than the (R)-enantiomer [11]. Although enzymatic hydrolysis of esters or transesteriflcation processes of alcohols have been widely applied for enzymatic resolution or desymmetrization... 

For any given potential and dipole function, at a fixed temperature, the classical and quantum profiles (and their spectral moments) are uniquely defined. If a desymmetrization procedure applied to the classical profile is to be meaningful, it must result in a close approximation of the quantum profile over the required frequency band, or the procedure is a dangerous one to use. On the other hand, if a procedure can be identified which will approximate the quantum profile closely, one may be able to use classical line shapes (which are inexpensive to compute), even in the far wings of induced spectral lines a computation of quantum line shapes may then be unnecessary. 

We will briefly consider several desymmetrization procedures that have been mentioned in the literature. These may simply employ various factors applied to the symmetric, classical profile, G(co), or alternatively attempt to correct the classical dipole autocorrelation functions in the time domain. 

In these formulae, Gd is the desymmetrized profile, Gc is the classical (symmetric) line profile c and T are angular frequency and temperature. In all cases, the desymmetrized function Go(ft>) obeys Eq. 5.73 exactly. We note that at low frequencies, h(o < kT, the four expressions are practically equivalent. However, at high frequencies the results of these desymmetrizations differ strikingly. One needs only to compare the magnitude of the factors of Gc of the upper three defining equations, for ft) — +oo, to realize enormous differences among these. Hence, the question arises as to which one (if any) of these procedures approximates the exact quantum profile, G(co). We note that in EgelstafFs procedure the desymmetrization is accomplished in the time domain rather than the frequency domain. The classical correlation function, Cci(t), and spectral function, Gci (co), are related by Fourier transform. 

Table 5.3. Desymmetrized classical profiles of He-Ar at 295 K for comparison with the quantum profile (last column) obtained from an identical input (dipole [278] and potential [12] functions).

The desymmetrization procedures are compared in Fig. 5.6 and Table 5.3, using the classical He-Ar profile at 295 K as an example (lowermost curve, solid thin line in the figure column 2 in the Table). The quantum profile is also shown for comparison (heavy solid line last column). At positive frequencies, all four procedures mentioned enhance the wing of the classical line shape toward that of the quantum profile. Specifically,... 

Fig. 5.8. Root mean square relative errors of model line shapes fitted to a quantum profile, the quadrupole-induced (XL = 23) component [69], The abscissa gives the ratio of peak intensity and wing intensity of the fitted portion of the exact profile. The superiority of the BC model (lower set of data points) over the desymmetrized Lorentzian (upper set) is evident.

It is a straightforward matter to fit various model profiles to realistic, exact computed profiles, selecting a greater or lesser portion near the line center of the exact profile for a least mean squares fit. In this way, the parameters and the root mean square errors of the fit may be obtained as functions of the peak-to-wing intensity ratio, x = G(0)/G(comax)- As an example, Fig. 5.8 presents the root mean square deviations thus obtained, in units of relative difference in percent, for two standard models, the desymmetrized Lorentzian and the BC shape, Eqs. 3.15 and 5.105, respectively. 

---