Vibrational averaging

Measurements of Stark splittings in microwave and radiofrequency spectra allow tliese components to be detennined. The main contribution to tire dipole moment of tire complex arises from tire pennanent dipole moment vectors of tire monomers, which project along tire axes of tire complex according to simple trigonometry (cosines). Thus, measurements of tire dipole moment convey infonnation about tire orientation of tire monomers in tire complex. It is of course necessary to take account of effects due to induced dipole moments and to consider whetlier tire effects of vibrational averaging are important. 

It is also possible to measure microwave spectra of some more strongly bound Van der Waals complexes in a gas cell ratlier tlian a molecular beam. Indeed, tire first microwave studies on molecular clusters were of this type, on carboxylic acid dimers [jd]. The resolution tliat can be achieved is not as high as in a molecular beam, but bulk gas studies have tire advantage tliat vibrational satellites, due to pure rotational transitions in complexes witli intennolecular bending and stretching modes excited, can often be identified. The frequencies of tire vibrational satellites contain infonnation on how the vibrationally averaged stmcture changes in tire excited states, while their intensities allow tire vibrational frequencies to be estimated. 

For linear moleeules, the vibrationally averaged dipole moment pave lies along the moleeular axis henee its orientation in the lab-fixed eoordinate system ean be speeified in terms of the same angles (0 and ([)) that are used to deseribe the rotational funetions Yl,m (0,(1)). Therefore, the three eomponents of the <(l)ir Pave I (1)6 integral ean be written as ... 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

The equilibrium geometries produced by electronic structure theory correspond to the spectroscopic geometry R, which assumes that there is no nuclear motion. Contrast this to the Rg geometry, defined via the vibrationally-averaged nuclear positions. 

The effect of vibrational averaging is particularly significant for the carbon hyperfine coupling constant. 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

Table 4 Comparison of various ab initio results and experimental estimates for the dispersion coefficients of the electronic hyperpolarizabilities 7jj and 7 of methane. (All results in atomic units. Results for the dispersion coefficients refer to single point calculations at the equilibrium geometry. Where available, dispersion coefficients for the vibrational average are given in parentheses.)...

The similarity in the behaviour of coupling constants as a function of e in both radicals allows to discuss vibrational averaging effects simply in terms of the potential governing the out-of-plane motion. 

In general, all observed intemuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. 

Consider a pair of atoms i andj frozen at their equilibrium positions and denote the connection between them as the local z-axis. In this case r. = r,. In a vibrating molecule the nuclear positions can be averaged over the vibrational states. In that case the distances between them—the so-called vibrational average or rv-distances—are then defined in the following way6 ... 

In fact, the barrier for proton transfer in the maleate anion appears to lie below the zero-point vibrational energy level (W. M. Westler, private communication). Thus, vibrationally averaged properties of the maleate anion will correspond to a symmetrically bridged Cjv transition-state structure rather than to either of the asymmetrically bridged equilibrium structures in Fig. 5.22. For present purposes this interesting feature of the potential surface can be ignored. 

Abstract Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. 

An application of the basic ideas briefly reviewed in Sections 12.2 and 12.3 to NMR shielding has been formulated by Jameson. The isotope effect on the shielding (cr) depends on vibrationally averaged bond lengths and angles according to Equation 12.16... 

PEC, are in general in better agreement with the CAS results than the pure SOPPA results. This applies to the equilibrium geometry results as well as to the vibrational averaged results. This shows that SOPPA(CCSD) performs better in the calculation of polarizabilities for LiH than SOPPA as might have been expected [36,41]. 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

In Table 7 we compare the ZPVCs for the dipole and quadrupole polarizabilities of HF. In the same way as for LiH, we have calculated the vibrational averages for each method with two different wavefunctions - one obtained from the PEC of the same or related method as used in the calculation of the property curve and the other obtained from the loo CAS PEC. Compared with the equivalent results for LiH we observe significant differences between the calculations on the two molecules. Eirst of all the vibrational corrections are smaller than in LiH but roughly in the same ratio as the polarizabilities. The influence of the PEC is larger than in LiH. 

SOPPA(CCSD) calculations with the CCSD or MCSCF PEC are also larger. In general the differences in the ZPVC are larger between the different PEC than between the different linear response methods. The SOPPA(CCSD) results for the equilibrium geometry as well as the vibrationally averaged polarizabilities are in both molecules in better agreement with the MCSCF results than the pure SOPPA values. 

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