Temperature-Dependent Chemical Shifts in the NMR Spectra of Gases

Vibrational Corrections to Transition Properties

“…Since the calculation of αvibv , v ' f o r p o l y a t o m i c m o l e c u l e s m a y represent quite a task, its simple estimates become quite valuable. The estimation method proposed in this paper is found to give quite reasonable values of α vib v , v ' It is of importance that this estimate involves the vibrational transition moment and zero-point vibrational corrections [6,16] to the electronic dipole moment in vibrational states linked by the Raman transition, i.e., the quantities which are accessible either experimentally or by computations. The first of these quantities can be either calculated theoretically or determined from infrared band intensities.…”

“…The single-quantum vibrational transitions are known to be governed by the (first-order) transition matrix element Owing to the assumption of the Born-Oppenheimer approximation one can define the pure electronic contribution to the h-dependent first-order transition property (6). In the present case this electronic contribution is simply the expectation value of the operator V in the state 0(r; Q, λ):…”

“…The evaluation of (19) requires only some lower-order data, i.e., the first-order vibrational transition property and vibrational averages of P( 1 ) , et. All these data can be expressed in terms of spectroscopic constants and the first-order derivative of p (1),el with respect to the normal coordinate under consideration [5,6]. In the case of polyatomic molecules the use of normal coordinates would require the assumption of the small amplitude vibrations.…”

“…Bishop, in his review of the subject [5], stressed that accounting for the influence of molecular vibrations (and rotations) on the calculated molecular properties is much more than just a simple averaging with respect to the given rovibrational (v, J) state. For properties of the second-and higher-orders the usual rovibrational correction δPv,J [6], which follows from averaging the so-called electronic contribution, Pel, to the total property value P in the given electronic-vibrational-rotational state of the molecule, needs to be supplemented by additional terms, δPrv, of rovibrational origin. These terms arise from the fact that in the presence of some external perturbation both the electronic and nuclear motions are affected.…”

Hyperfine Parameters for Aluminum Hydride: An ab Initio Molecular Orbital Study

“…For a diatomic molecule, the value of a parameter at the equilibrium bond length, P(eq), is obtained from the value reported for a particular vibrational (v) and rotational (J ) state according to (66) P(eq) = P(v, J ) − v + [9] where ξ = (r − r e )/r e , ω e is the harmonic vibrational frequency, and a, the cubic force constant, is defined as −(1 + α e ω e /6B 2 e ). The first and second derivatives of P with respect to ξ may be estimated by calculating P at bond lengths that are slightly smaller than and larger than r e .…”