The Adiabatic Correction

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 )) 

The important fact that must be remembered is that in the Born-Oppenheimer approximation, Equation 2.8, the potential energy for vibrational motion is Eeiec(S) which is independent of isotopic mass of the atoms. In the adiabatic approximation, the potential energy function is Eeiec(S)+C and this potential will depend on nuclear mass if C depends on nuclear mass. 

In the context of Section 2.4, consider a chemical equilibrium in the gas phase 

In each case the uncorrected potential lies to the left, the corrected to the right 

The Keiec value for an isotopic exchange reaction resulting from a failure of the Born-Oppenheimer approximation is sometimes referred to as Kboele. With the notation employed above A AC is the value of AEeiec for the reaction (see Fig. 2.1). 

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The principal advantage of SCS in comparison with IOS is that the adiabaticity of collisions may be taken into account. The difference between actual cross-sections and their purely non-adiabatic estimation is not large but increases with rotational frequency. As shown in Fig. 5.5 the adiabatic correction improves even qualitatively the high-frequency alteration of cross-sections by minimizing the discrepancy between SCS... 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

The only difficulty with using this method is the lack of heat capacity data. With the wide spread use of the Picker et al. (129) heat capacity calorimeter one can usually find published heat capacities for most systems of interest (3) at 25°C. Since the Cp does not attribute much to the adiabatic correction, this is not a serious limitation. 

Fig. 2.1 The adiabatic correction to the Born-Oppenheimer approximation for H2 and HD schematic, not to scale AC = C(H2)-C(HD). In each case the uncorrected potential lies to the left, the corrected to the right...

In Chapter 4 we will learn to calculate the equilibrium constant for an exchange reaction like Equation2.15 using the Born-Oppenheimer approximation. If, in addition, the adiabatic correction is included, the equilibrium constant calculated in the Born-Oppenheimer approximation must be multiplied by a correction factor containing the energy difference AAC. 

To give the reader some further appreciation of the adiabatic correction, we next discuss the so-called Rydberg correction of the hydrogenic atom. The kinetic energy of the electron in the hydrogen atom is expressed by Equation 2.5, where mj is set equal to the electron mass, me. It is well known that the Schrodinger equation for the hydrogen atom separates into two equations, one of which deals with the motion of the center of mass of the system and the other with the motion of the electron... 

The BO approximation, which assumes the potential surface on which molecular systems rotate and vibrate is independent of isotopic substitution, was discussed in Chapter 2. In the adiabatic regime, this approximation is the cornerstone of most of isotope chemistry. While there are BO corrections to the values of isotopic exchange equilibria to be expected from the adiabatic correction (Section 2.4), these effects are expected to be quite small except for hydrogen isotope effects. 

The adiabatic correction to the Born-Oppenheimer potential energy for a diatomic molecule A-B is simply given by the sum of the expectation values of the nuclear... 

By calculating A.U (R) and Al/ (i ) separately, we can straightforwardly calculate the total adiabatic correction V (R) for any isotopes of A and B. The adiabatic corrections are calculated by numerical differentiation of the multi-configurational self-consistent field (MCSCF) wave functions calculated with Dalton [23]. The nurnerical differentiation was performed with the Westa program developed 1986 by Agren, Flores-Riveros and Jensen [22],... 

The adiabatic corrections for the individual atomic centres Fie and H, listed as a function of R in Table 3, are combined in the final column of that table into a total correction according to this formula... 

The adiabatic corrections to the ground state of H2, HD, and Di we shall calculate using second-order Rayleigh-Schrodinger many-body perturbation theory (RS-... 

Another approach has been proposed and employed to a number of molecules by Handy et al. (Handy et al., 1986 loannou et al., 1996 Handy and Lee, 1996). In this method the adiabatic correction is computed without separation of center of mass motion and ... 

The adiabatic corrections for a double-minimum state are characterised by their large values in the region where drastic change in the character of the wavefunction occurs i.e. in the vicinity of the potential barrier. This is the case for the EF, GK and H/f states. The largest values of the adiabatic corrections for the EF, GK and YiH states are 499, 681 and 859 cm, respectively (Wolniewicz and Dressier, 1986). A new effect, however, has been observed for the h and g Instates (Rychlewski, 1989b Kolos and Rychlewski, 1990). The adiabatic corrections for... 

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].

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