Spectral function rotators

McCain D 0 and Markley J L 1986 Rotational spectral density functions for aqueous sucrose ... 

If the resolving capacity of the instruments is ideal then vibrational-rotational absorption and Raman spectra make it possible in principle to divide and study separately vibrational and orientational relaxation of molecules in gases and liquids. First one transforms the observed spectrum of infrared absorption FIR and that of Raman scattering FR into spectral functions... 

We demonstrate that the spectral function of valence harmonic vibrations of a diatomic group that effects rotational reorientations is broadened by w. The vector of atom C displacements relative to the atom B (see Fig. A2.1) may be represented as x(t)e(t), where x(t) is the change in the length of the valence bond oriented at the time t along the unit vector e(/). Characteristic periods of valence vibrations are much shorter than periods of changes in unit vector orientations. As a consequence, the GF of the displacements defined by Eq. (4.2.1) can be expressed approximately as ... 

Less symmetric molecules require a considerably more complicated treatment, but in the end their spectral transitions arc functions of their three moments of inertia (see Section 10.3.5). From a computational standpoint, then, prediction of rotational spectral lines depends only on the moments of inertia, and hence only on the molecular geometry. Thus, any method which provides good geometries will permit an accurate prediction of rotational spectra within the regime where tlie rigid-rotor approximation is valid. 

The translational spectral function, g(v), may be considered a (very diffuse) spectral line centered at zero frequency which arises from transitions between the states of relative motion of the interacting pair. It is the free-state analog of the familiar vibrational and rotational transitions of bound systems, with the difference that the motion is here aperiodic the period goes to zero due to the lack of a restoring force. The negative fre-... 

In other words, the spectral function is written as a sum over rotational line profiles centered at the rotational transition frequencies ojr r,. For each set of expansion parameters (c) = A1I2AL, the quantities a, satisfy the selection rules appropriate for the (c) component, and are chosen such that... 

Spectral function pertinent to librating and rotating particles... 

In the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

We have obtained the expression given in GT, p. 225 for the spectral function of free rotors moving in a homogeneous potential in the interval between strong collisions see also VIG, Eqs. (7.12) and (7.13). So, the subscript F means free. The subscript R in Eq. (74c) is used as an initial letter of restriction. Indeed, as it follows from the comparison of Eq. (77) with Eq. (74a), the second term of the last equation expresses the steric-restriction effect arising for free rotation due to a potential wall. If we set, for example, p = 7t, what corresponds to a complete rotation (without restriction) of a dipole-moment vector p, then we find from Eqs. (74a)-(74c) that LR z) = 0 and L(z) = Lj,(z). This result confirms our statement about restriction. ... 

Thus, we should discriminate three subensembles. We mark the quantities referring to the librators by the superscript -, and we mark hindered and free rotators by the common superscript °. So, we represent the total spectral function (SF) as the sum... 

We shall obtain here expression for the spectral function of the dipoles performing complete rotation by using an interpolation approximation, which allows us to ignore the distinction between hindered and free rotations. Both types of motion we mark by the same superscript °. 

If the angle (3 is much less than 1, then, in accord with Figs. 7 and 9, the most part of the rotators move freely under effect of a constant potential U0, since their trajectories do not intersect the conical cavity. A small part of the rotators moves along a trajectory of the type 1 shown in Fig. 10. However, at d > (3—that is, in the most part of such a trajectory—they are affected by the same constant potential U0- Therefore, for this second group of the particles the law of motion is also rather close to the law of free rotation. For the latter the dielectric response is described by Eq. (77). We shall represent this formula as a particular case of the general expression (51), in which the contributions to the spectral function due to longitudinal A) and transverse KL components are determined, respectively, by the first and second terms under summation sign. Free rotators present a medium isotropic in a local-order scale. Therefore, we set = K . Then the second term... 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

The corresponding spectral functions, denoted L(z) and L(z), are derived, as well as the SF L for the rotators, in Section V.E in the form of simple integrals from elementary functions over a full energy of a dipole (or over some function of this energy). The total spectral function is thus represented as... 

Finally, a simplified expression for the spectral function of the rotators is given by... 

Using this result, we may simplify calculation of the spectral function Liz) by neglecting the precessional contribution to L. We shall estimate also in this approximation the peak frequencies X ib and xrot of the absorption bands determined by the librational and the rotational subensembles. 

The multiplier 2 here accounts for existence of two potential wells with oppositely directed symmetry axes (the case of mirror symmetry is assumed). The derivation of the spectral function of the rotators, namely the last term in Eq. (170), will be given in Section V.E.6. 

Then, as shown in Appendix 2, integral (220) yields the spectral function (174). 6. Spectral Function of Rotators... 

Integration over l gives the formula (179) for the spectral function L of the rotators. 

We account for only the torque proportional to the string s expansion AL, which produces the main effect considered in this work. For calculation we employ the spectral function (SF) Lstr(Z), which is linearly connected with the spectrum of the dipolar ACF (see Section II), with Z x Y being the reduced complex frequency. Its imaginary part Y is in inverse proportion to the lifetime tstr of the dipoles exerting restricted rotation. The dimensionless absorption Astr is related to the SF Lstr as... 

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