This paper gives a mathematical approach to calculate the fractionation factor of isotopes in a general cluster (also known as super-molecule), which composes of necessary chemical effect within three bonds outside the interested atom(s). The cluster might have imaginary frequencies after being optimized in quantum softwares. The approach includes the contribution of the difference, which is resulted from the substitution of heavy and light isotopes in the cluster, of vibrations of imaginary frequencies to give precise prediction of isotope fractionation factor. We call the new mathematical approximation “reduced partition function ratio in the frequency complex plane (RPFR C )”. If there is no imaginary frequency for a cluster, RPFR C is simplified to be Urey (1947) or Bigeleisen and Mayer (1947) formula. Final results of this new algorithm are in good agreement with those in earlier studies.
KeywordsIsotope FractionationClusterReduced Partition Function RatioFrequency Complex Plane
Urey, H.C. and Rittenberg, D. (1933) Some Thermodynamic Properties of the H1H2, H2H2 Molecules and Compounds Containing the H2 Atom. Journal of Chemical Physics, 1, 137-143. http://dx.doi.org/10.1063/1.1749265
Urey, H.C. (1947) The Thermodynamic Properties of Isotopic Substances. The Journal of the Chemical Society, 562- 581. http://pubs.rsc.org/en/Content/ArticleLanding/1947/JR/jr9470000562
Bigeleisen, J. and Mayer, M.G. (1947) Calculation of Equilibrium Constants for Isotopic Exchange Reactions. Journal of Chemical Physics, 15, 261-267. http://dx.doi.org/10.1063/1.1746492 Rustad, J.R., Nelmes, S.L., Jackson, V.E. and Dixon, D.A. (2008) Quantum-Chemical Calculations of Carbon-Isotope Fractionation in CO2 (g), Aqueous Carbonate Species, and Carbonate Minerals. The Journal of Physical Chemistry A, 112, 542-555. http://dx.doi.org/10.1021/jp076103m http://www.ncbi.nlm.nih.gov/pubmed/18166027
Li, H. and Jensen, J.H (2002) Partial Hessian Vibrational Analysis: The Localization of the Molecular Vibrational Energy and Entropy. Theoretical Chemistry Accounts, 107, 211-219. http://dx.doi.org/10.1007/s00214-001-0317-7
Yuan, J. and Liu, Y. (2011) An Important Method for Calculating Isotope Fractionation in the Solid State Partial Hessian Vibrational Analysis (PHVA). Bulletin of Mineralogy, Petrology and Geochemistry, 30, 472-476.
Stern, M.J. and Wolfsber, M. (1966) Simplified Procedure for Theoretical Calculation of Isotope Effects Involving Large Molecules. The Journal of Chemical Physics, 45, 4105. http://dx.doi.org/10.1063/1.1727463
Driesner, T., Ha, T.K. and Seward, T.M (2000) Oxygen and Hydrogen Isotope Fractionation by Hydration Complexes of Li+, Na+, K+, Mg2+, F?, Cl?, and Br?: A Theoretical Study. Geochimica et Cosmochimica Acta, 64, 3007-3033. http://dx.doi.org/10.1016/S0016-7037(00)00407-5
Liu, Y. and Tossell, J.A. (2005) Ab Initio Molecular Orbital Calculations for Boron Isotope Fractionations on Boric Acids and Borates. Geochimica et Cosmochimica Acta, 69, 3995-4006. http://dx.doi.org/10.1016/j.gca.2005.04.009
Li, X.F., Zhao, H., Tang, M. and Liu, Y. (2009) Theoretical Prediction for Several Important Equilibrium Ge Isotope Fractionation Factors and Geological Implications. Earth and Planetary Science Letters, 287, 1-11. http://dx.doi.org/10.1016/j.epsl.2009.07.027
Pople, J.A., Schlegel, H.B., Krishnan, R., Defrees, D.J., Binkley, J.S., Frisch, M.J., Whiteside, R.A., Hout, R.F. and Hehre, W.J. (1981) Molecular-Orbital Studies of Vibrational Frequencies. International Journal of Quantum Chemistry, 20, 269-278. http://dx.doi.org/10.1002/qua.560200829
Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J.R., Montgomery, J.J.A., Vreven, T., Kudin, K.N., Burant, J.C., Millam, J.M., Iyengar, S.S., Tomasi, J., Barone, V., Mennucci, B., Cossi, M., Scalmani, G., Rega, N., Petersson, G.A., Nakatsuji, H., Hada, M., Ehara, M., Toyota, K., Fukuda, R., Hasegawa, J., Ishida, M., Nakajima, T., Honda, Y., Kitao, O., Nakai, H., Klene, M., Li, X., Knox, J.E., Hratchian, H.P., Cross, J.B., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R.E., Yazyev, O., Austin, A.J., Cammi, R., Pomelli, C., Ochterski, J.W., Ayala, P.Y., Morokuma, K., Voth, G.A., Salvador, P., Dannenberg, J.J., Zakrzewski, V.G., Dapprich, S., Daniels, A.D., Strain, M.C., Farkas, O., Malick, D.K., Rabuck, A.D., Raghavachari, K., Foresman, J.B., Ortiz, J.V., Cui, Q., Baboul, A.G., Clifford, S., Cioslowski, J., Stefanov, B.B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Martin, R.L., Fox, D.J., Keith, T., Al-Laham, M.A., Peng, C.Y., Nanayakkara, A., Challacombe, M., Gill, P.M.W., Johnson, B., Chen, W., Wong, M.W., Gonzalez, C. and Pople, J.A. (2009) Gaussian 09, Revision A.01. Gaussian, Inc., Wallingford.
Fong, C.F.C.M., Kee, D.D. and Kaloni, P.N. (2002) Advanced Mathematics for Engineering and Science. World Scientific Publishing Co. Pte. Ltd., Singapore.
Born, M. and Oppenheimer, R. (1927) On the Quantum Theory of Molecules. Annalen der Physik, 84, 457-484. http://dx.doi.org/10.1002/andp.19273892002
Wilson, E.B.J., Decius, J.C. and Cross, P.C. (1955) Molecular Vibrations: The Theory of Infrared and Raman Spectra. Dover Publications, New York.
Dovesi, R., Saunders, V.R., Roetti, C., Orlando, R., Zicovich-Wilson, C.M., Pascale, F., Civalleri, B., Doll, K., Harrison, N.M., Bush, I.J., D’Arco, P. and Llunell, M. (2006) CRYSTAL06 User’s Manual. University of Torino, Torino.
Schauble, E.A., Ghosh, P. and Eiler, J.M. (2006) Preferential Formation of 13C-18O Bonds in Carbonate Minerals, Estimated Using First-Principles Lattice Dynamics. Geochimica et Cosmochimica Acta, 70, 2510-2529. http://dx.doi.org/10.1016/j.gca.2006.02.011
Becke, A.D. (1993) Density-Functional Thermochemistry. III. The Role of Exact Exchange. Journal of Chemical Physics, 98, 5648-5652. http://dx.doi.org/10.1063/1.464913
Lee, C.T., Yang, W.T. and Parr, R.G. (1988) Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron-Density. Physical Review B, 37, 785-789. http://dx.doi.org/10.1103/PhysRevB.37.785
Vosko, S.H., Wilk, L. and Nusair, M. (1980) Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin-Density Calculations—A Critical Analysis. Canadian Journal of Physics, 58, 1200-1211. http://dx.doi.org/10.1139/p80-159
Stephens, P.J., Devlin, F.J., Chabalowski, C.F. and Frisch, M.J. (1994) Ab Initio Calculation of Vibrational Absorption and Circular-Dichroism Spectra Using Density-Functional Force-Fields. Journal of Physical Chemistry, 98, 11623-11627. http://dx.doi.org/10.1021/j100096a001