Reduced Partition Function Ratio in the Frequency Complex Plane: A Mathematical Approach
Author(s)
Jie Yuan
Affiliation(s)
Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.
Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.
ABSTRACT
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 (RPFRC)”. If there is no imaginary frequency for a cluster, RPFRC 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.
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 (RPFRC)”. If there is no imaginary frequency for a cluster, RPFRC 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.
Cite this paper
Yuan, J. (2014) Reduced Partition Function Ratio in the Frequency Complex Plane: A Mathematical Approach. Open Journal of Geology, 4, 654-664. doi: 10.4236/ojg.2014.412049.
Yuan, J. (2014) Reduced Partition Function Ratio in the Frequency Complex Plane: A Mathematical Approach. Open Journal of Geology, 4, 654-664. doi: 10.4236/ojg.2014.412049.
References
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http://dx.doi.org/10.1063/1.1749265 [2] 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 [3] 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 [4] 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 [5] 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. [6] 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.
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http://dx.doi.org/10.1016/j.gca.2005.04.009 [9] 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 [10] 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 [11] 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. [12] 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. [13] 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 [14] Levine, I.N. (1995) Physical Chemistry. 4th Edition, McGraw-Hill, Inc., New York. [15] 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. [16] 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. [17] 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 [18] 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 [19] 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 [20] 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 [21] 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
[1] 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 [2] 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 [3] 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 [4] 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 [5] 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. [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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. [12] 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. [13] 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 [14] Levine, I.N. (1995) Physical Chemistry. 4th Edition, McGraw-Hill, Inc., New York. [15] 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. [16] 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. [17] 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 [18] 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 [19] 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 [20] 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 [21] 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