Reduced partition function

We next show that (Q/Q ). depenis only on the symmetry number ratio and a mass factor lndependent of chemical composition. Write the Hamiltonian of the molecule in Cartesian coordinates. Then the classical partition function 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Thus the reduced partition function ratio, (s/s )f, Is just the chemical Isotope fractionation factor of the chemical species against the gaseous atom. With the convention prime Is the light Isotope It Is easy to prove that ln(s/s )f Is always positive. This follows from the fact that u. u.. 

We return now to Eq. (10) and examine the G(u) function. It is a monotonlc positive function. Its value Is 1/2 as u goes to Infinity and It approaches zero as u goes to zero. At small u It has the value u/12. For the case of small u, Eq. (11) becomes 

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The direct calculation of the reduced partition function ((e-AU kT))o,h=i is expected to be limited by the variance of the function e-AU/kT [ Iowever, representing the averaged quantity as a ratio opens possibilities for importance sampling, evaluating both numerator and denominator on the basis of a single sample designed to reduce the variance of e AU/kTm... 

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

L 1 mli / i Uli V 1 Finally, one defines a reduced partition function ratio for the transition state as... 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

Bigeleisen and Mayer (1947) simplified the reduced partition function by observing that vibrational frequency shifts caused by isotope substitution are relatively small (except when deuterium is substituted for normal hydrogen). When the dimensionless quantity hv/kr is of the order 5 or less (corresponding to a typical 1000 cm vibration at 288 K)—a condition applicable to most geochemical situations. 

O Neil JR (1986) Theoretical and experimental aspects of isotopic fractionation. Rev Mineral 16 1-40 Oi T (2000) Calculations of reduced partition function ratios of monomeric and dimeric boric acids and borates by the ab initio molecular orbital theory. J Nuclear Sci Tech 37 166-172 Oi T, Nomura M, Musashi M, Ossaka T, Okamoto M, Kakihana H (1989) Boron isotopic composition of some boron minerals. Geochim Cosmochim Acta 53 3189-3195 Oi T, Yanase S (2001) Calculations of reduced partition function ratios of hydrated monoborate anion by the ab initio molecular orbital theory. J Nuclear Sci Tech 38 429-432 Paneth P (2003) Chlorine kinetic isotope effects on enzymatic dehalogenations. Accounts Chem Res 36 120-126... 

Yamaji K, Makita Y, Watanabe H, Sonoda A, Kanoh H, Hirotsu T, Ooi K (2001) Theoretical estimation of lithium isotopic reduced partition function ratio for lithium ions in aqueous solution. J Phys Chem A105 602-613... 

Introducing the masses of exchanging isotopes (m and m°, respectively), the ratio of partition functions for crystalline components can be related to that of the primitive unit cell (Kieffer, 1982), thus defining reduced partition function ratio / (whose formulation is equivalent to that obtained by Bigeleisen and Mayer, 1947, and Urey, 1947, for gaseous molecules) ... 

Figure 11.30 Reduced partition function for various minerals calulated by Kieffer (1982) through equation 11.61 plotted against T. Heavy curve labeled H20(l) is reduced partition function of water according to Becker (1971). Dashed curve is a extrapolation of high-r reduced partition curve for quartz. Mineral abbreviations Qtz (quartz), Calc (calcite), Albt (albite), Muse (muscovite), Enst (clinoenstatite), Anor (anorthite). Diop (diopside), Pyrp (pyrope), Gros (grossular), Zron (zircon), Fors (forsterite), Andr (andra-dite), Rutl (rutile). Reprinted with permission from Kieffer (1982), Review of Geophysics and Space Physics, 20, 827-849, copyright 1982 by the American Geophysical Union.

Figure 11.30 shows the reduced partition function (1/r) X 1000 In / calculated by Kieffer (1982) by application of equation 11.61 and plotted against The adoption of this form of plot is dictated by the fact that the natural logarithm of the partition function ratio is asymptotically proportional to T at high tern-... 

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

Combining the results of Kieffer s model and of laboratory experiments, Clayton and Kieffer (1991) obtained a set of third-order equations relating the reduced partition functions of various minerals to the inverse of the squared absolute temperature (table 11.9) according to... 

Oxygen isotopic fractionation between two minerals A and B at any given T can easily be obtained by subtracting their respective reduced partition function equations—i.e.,... 

Reduced partition functions for oxides were calculated by Zheng (1991) on the basis of the modified increment methodr... 

The reduced partition function ratio of the mineral is derived from the equations above through... 

In equation 11.148, AE is regarded as an Arrhenius-slope energy term that has a value of 1 kJ/mol, and (1000/r) ln(gVg°) is the reduced partition function ratio for the reference phase (quartz). 

Zheng (1993b) also extended the modified increment method to the evaluation of reduced partition function ratios in hydroxyl-bearing silicate minerals through... 

The expression for can now be written in terms of a reduced partition function, Q, which contains only the translational, rotational, and vibrational contributions an ordinary molecule would have, and the special contribution Jr.c. ... 

In a system conlaining condensed and gases phases, Bigeleisen derived Ihe relationship between the reduced partition function ratio (RPFR), /, and the measured... 

The exact equation for isotopic reduced partition-function ratios s2/s )f, in the harmonic approximation with neglect of effects owing to condensation and quantum-mechanical rotation, is (7, 25)... 

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