Adiabatic correction term

Figure 4.13 Adiabatic hyperspherical potentials, Eq. (101), in a.u. without the adiabatic correction term for He of symmetries 1,3SC and 1,3P° converging to the asymptotic limit e + He+(n = 2), plotted against the hyperradius R in a.u. Each potential supports an infinite number of Rydberg states of Feshbach resonance, of which the lowest level is indicated by a horizontal line. Figure from Ref. [90], Note the difference in notation.

Figure 4.15 Hyperspherical potentials without the adiabatic correction term for Hef P0) converging to the asymptotic limits e + He+(n = 4,5,6). Each potential supports an infinite Rydberg series of Feshbach resonances. Some of them exhibit typical cases of inter-series overlapping resonances illustrated in Figures 4.9 and 4.10. Figure from Ref. [69].

Figure 4.16 Hyperspherical potentials without the adiabatic correction term (full curves) for systems of unnatural parity e+He+(Pc) (left) and e+He+(D°) (right) converging to the asymptotic limits e+ + He+(n = 5-8) and He2+ + Ps(n = 2,3). The diabatic broken curves are for the He2+ + Ps configurations only see text. The vertical positions of the symbols He+(n) and Ps(n) on the right roughly indicate the asymptotic threshold energies. The calculated resonance levels are shown by horizontal bars, some of which are unexpected from the adiabatic potentials. Adapted from Ref. [66].

Le Roy et al. [OSRloy] give an alternative treatment of the frequency data (Direct-potential fit) which is based on the eigenvalues of the radial Schrddinger equation. The potential is expressed in form of a modified Morse function where the exponent is written in form of a power series expansion. The coefficients and those in the non-adiabatic correction terms are determined in a direct fit to the experimental eigenvalue differences, see [05Roy] for details and results. 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

Production of an acceptable value of reduced standard deviation of a fit required multiple parameters m, up toj = 6. Values of these parameters with j>2 likely reflect not only adiabatic corrections their association with a particular term in... 

The sign and magnitude of the linear term in formula 76 concur moderately satisfactorily with those properties of a corresponding term in formula 77, but agreement between other corresponding terms is lacking. We compare the total adiabatic corrections for Li H from a sum of corrections of separate atomic centres in formulae 74 and 76 divided by their masses,... 

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]. 

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... 

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... 

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