G factor vibrational

The rotational and vibrational g factors of net electrically neutral molecule AB" with an indicated electric polarity show a dependence [66] on masses of atomic centres of forms... 

There exists no significant comprehensive fit of spectral data of H2 with which we might here make comparison. Our discussion above demonstrates that, as for GaH above, application of an algorithm based on Dunham s algebraic approach to analysis of vibration-rotational spectral data of H2, especially through implementation of hypervirial perturbation theory [30,72] that allows the term for the vibrational g factor in the hamiltonian in formula 29 to be treated directly in that form, proves extremely powerful to derive values of fitting parameters that not only have intrinsic value in reproducing experimental data of wave numbers of transitions but also relate to other theoretical and experimental quantities. 

Like formulae for contributions to total adiabatic corrections from individual atomic centres above, the corresponding coefficients for the linear term have the same sign and comparable magnitude, hut for subsequent coefficients agreement is lacking. The maximum region of validity of the experimental functions is the same as for the vibrational g factor, specified above, whereas the region for which the calculated points define the contributions and total adiabatic correction [122] is i /10 °m=[l,3]. 

The effective hamiltonian in formula 29 incorporates approximations that we here consider. Apart from a term V"(R) that originates in nonadiabatic effects [67] beyond those taken into account through the rotational and vibrational g factors, other contributions arise that become amalgamated into that term. Replacement of nuclear masses by atomic masses within factors in terms for kinetic energy for motion both along and perpendicular to the internuclear axis yields a term of this form for the atomic reduced mass. 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

Quantum-Chemical Calculations of Radial Functions for Rotational and Vibrational g Factors, Electric Dipolar Moment and Adiabatic Corrections to the Potential Energy for Analysis of Spectra... 

Computational spectrometry, which implies an interaction between quantum chemistry and analysis of molecular spectra to derive accurate information about molecular properties, is needed for the analysis of the pure rotational and vibration-rotational spectra of HeH in four isotopic variants to obtain precise values of equilibrium intemuclear distance and force coefficient. For this purpose, we have calculated the electronic energy, rotational and vibrational g factors, the electric dipolar moment, and adiabatic corrections for both He and H atomic centres for intemuclear distances over a large range 10 °m [0.3, 10]. Based on these results we have generated radial functions for atomic contributions for g g,... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

The nuclear contribution to the rotational or vibrational g factor becomes for a diatomic molecule AB containing nucleus A of protonic number Zg along the z- axis jit Za = l a cmL nd nucleus B with protonic number at... 

Calculation of rotational and vibrational g factors by linear response methods using multiconfigurational self-consistent-field wave functions is described in detail elsewhere [18,27]. 

For a molecular ion with charge number Q a transformation between isotopic variants becomes complicated in that the g factors are related directly to the electric dipolar moment and irreducible quantities for only one particular isotopic variant taken as standard for this species these factors become partitioned into contributions for atomic centres A and B separately. For another isotopic variant the same parameters independent of mass are still applicable, but an extra term must be taken into account to obtain the g factor and electric dipolar moment of that variant [19]. The effective atomic mass of each isotopic variant other than that taken as standard includes another term [19]. In this way the relations between rotational and vibrational g factors and and its derivative, equations (9) and (10), are maintained as for neutral molecules. Apart from the qualification mentioned below, each of these formulae applies individually to each particular isotopic variant, but, because the electric dipolar moment, referred to the centre of molecular mass of each variant, varies from one cationic variant to another because the dipolar moment depends upon the origin of coordinates, the coefficients in the radial function apply rigorously to only the standard isotopic species for any isotopic variant the extra term is required to yield the correct value of either g factor from the value for that standard species [19]. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

Equations (9) and (11) indicate how the auxiliary radial function for the rotational factor becomes separable into contributions from atomic centres of types A and B. An analogous separation is practicable for both the vibrational g factor and the total adiabatic corrections for the latter quantity this separation is effected in the original quantum-chemical calculations. Accordingly we express these calculated values of rotational and vibrational g factors, presented in Table 1, and adiabatic corrections, presented in Table 3, of He H" to generate coefficients of radial functions for atomic centres of either type. He or H. The most useful variable for these functions is z, defined in terms of instantaneous R and equihbrium R internuclear distances as... 

Fig. 2. Rotational and vibrational g factors of in electronic ground state X as...

The most striking features of the radial function for the vibrational g factor, gjji), are a minimum at an internuclear distance of about 1.4X 10 m and a maximum... 

In Fig. 4 we present the energies and matrix elements for the first three excited states and in Fig. 5 we show the contributions of the five lowest excited states to the electronic contribution of the vibrational g factor, equation (3). The terms with n = 1, 2, 3 in equation (3) are displayed with hollow symbols, whereas the solid symbols and lines are the result of summation over n from 1 to 2, from 1 to 3, from 1 to 5 and all n in equation (3). According to Fig. 4 the energy of the first three excited states exhibits no atypical behaviour, but that the NACME to the first... 

Fig. 5. Contributions from the lowest five excited states to the electronic contribution to the vibrational g factor, equation (3), of as a function of internuclear distance R.

According to Fig. 5 the maximum and minimum in the NACME to the first excited state produce a minimum and maximum in the corresponding contribution to the electronic contribution to the vibrational g factor. The extrema are at the same internuclear distances and have positions near the extrema in the total electronic contribution to g R), but are not as steep. The contributions from the second, third, and up to the fifth excited states modify slightly the position and the form of the extrema but introduce no fundamental modification. We, therefore, conclude that the extrema in the vibrational g factor reflect extrema in the first-order NACME to the first excited state, and not in the energy of the excited state. The exact position of the minimum in the vibrational g factor is, however, influenced by more highly excited states. 

Eigure 2 shows directly that both rotational and vibrational g factors of He H approach a common value of 0.8 as R becomes large. This behaviour is characteristic of He H, as for both neutral diatomic molecular species H2 [18] or Li H for which we have undertaken analogous calculations the asymptotic value of both g and is zero. For He H the corresponding value of gr at m is 0.3. 

Defining a total vibrational g factor of a diatomic molecule (Herman and Asgharian, 1966 Herman and Ogilvie, 1998)... 

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