Second-order correction to the Bigeleisen–Mayer equation due to the nuclear field shift

Abstract

The nuclear field shift affects the electronic, rotational, and vibrational energies of polyatomic molecules. The theory of the shifts in molecular spectra has been studied by Schlembach U between an aqueous phase and an anion-exchange resin. The aqueous phase is principally U(IV); the resin phase is principally U(VI), hydrated or complexed uranyl ion. In each of these exchange reactions the heavy isotope concentrates in the aqueous phase, that is U(IV). The nonlinearity and the fact that the heavy isotope concentrates in the U(IV) species arise from the shift in the minimum of the electronic potential energy due to the different nuclear sizes and shapes of the isotopomers. This is known as the nuclear field shift. A quantitative calculation of this effect (9) is in excellent agreement with experiment.The calculation is based on the addition of a term due to the nuclear field shift to the Bigeleisen-Mayer equation for the logarithm of the isotopic separation factor and the reduced partition-function ratio. Analogous to the Bigeleisen-Mayer function, ᐉn(s͞sЈ)f 0 , we define ᐉn(s͞sЈ)f ns as the nuclear field shift correction and ᐉn(s͞sЈ)f T as the total reduced partitionfunction ratio corrected for the nuclear field shift.The nuclear field shift correction is simply,where (E H°Ϫ E L°) is the difference in the ground state electronic energy.The Bigeleisen-Mayer equation is based on the BornOppenheimer approximation and uses an isotope-independent potential energy; it also assumes simple harmonic vibrations. Because the vibrational force constants are the second derivatives of the electronic energy with respect to the mutual distances between atoms, a correction needs to be applied to ᐉn(sЈ͞sЈ)f 0 for the nuclear field shift. The formal theory of the nuclear field shift on the equilibrium force constant of a diatomic molecule has been given by Schlembach and Tiemann (10). From the theory and their data on the electronic and rotational spectra of diatomic Pb chalcogenides and Tl halides, Tiemann et al. (11) estimated the relative vibrational frequency shifts in these compounds resulting from the nuclear field shifts to be of the order of 10 Ϫ6. I now show that a relative frequency shift of this order of magnitude is a negligible, second-order correction to ᐉn(s͞sЈ)f 0 .In my study of the nonlinearity of the isotope separation factor, I found it convenient to use the first finite polynomial approximation to ᐉn(s͞sЈ)f 0 because this utilizes the isotopomer masses directly.For the present purpose it is more convenient to use the Bigeleisen-Mayer free energy function, G(u ), for the calculation of the correction to ᐉn(s͞sЈ)f 0 . We writewhere u boc ϭ h boc ͞kT and boc is the vibrational frequency corrected for the nuclear field shift and G(u) ϭ 1/2 Ϫ 1͞u ϩ (e u Ϫ 1) Ϫ1). To calculate (Ј boc Ϫ boc ) we set the nuclear field shift correction in the heavy isotope equal to zero. Then,where Ј BO is the Born-Oppenhemier frequency and ⌬Ј p is the perturbation due to the nuclear field shift. In Table 1 S). The correc...

Role of nuclear volume in driving equilibrium stable isotope fractionation of mercury, thallium, and other very heavy elements

“…Thus it appears that basis-set and structural errors are relatively small. The insensitivity of the nuclear-volume shift to bond length also supports earlier evidence that 2 nd order effects of isotope substitution on molecular structure are negligible (Bigeleisen, 1998). Exp.…”

“…Nuclear-volume fractionation has been thought to play a role in the separation of the isotopes of very heavy elements for several years (e.g., Bigeleisen, 1996a;Bigeleisen, 1998;Mioduski, 1999;Knyazev et al, 1999;Knyazev and Myasoedov, 2001), especially uranium. The present study is the first to predict the magnitude of these effects from first principles, finding that they are likely to be the dominant mechanism controlling equilibrium mercury and thalliumisotope fractionations.…”

Identification of sources and formation processes of atmospheric sulfate by sulfur isotope and scanning electron microscope measurements

“…Mass-independent isotopic compositions have been attributed to a variety of causes, including those that affect bonding directly and those that affect reaction rates for different isotopomers (kinetic isotope effects). Nuclear field shift effects can be classified as those that occur because of the way that the charge density of the nucleus affects the shape of the potential well that describes the chemical bonds involving different isotopes [e.g., Bigeleisen, 1998]. This class of effect exists because of a direct change in the chemical bonds for different isotopomers.…”

Nuclear field shift effects on stable isotope fractionation: a review

“…Since then, the NFSE has become an important correction term to the conventional BigeleisenMayer equation for heavy elements. In a following paper, Bigeleisen also noticed that the NFSE was just a second order correction in chemical bonds (Bigeleisen 1998), indicating that the NFSE has a minor effect on vibrational frequencies (i.e., on the conventional mass-dependent isotope fractionation). Importantly, the paper first pointed out that the NFSE could induce large isotopic fractionations through the variation of ground-state electronic energy alone.…”