_{1}

^{*}

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.

In 1933, Urey and Rittenberg [

To overcome this difficulty, Rustad et al. (2008) [

This study gives a new approach, i.e. reduced partition function ratio in the frequency complex plane (RPFR_{C}), to the calculation of isotope fractionations in general clusters. This new approach involves a more detailed physical figure of atom vibrations for the calculation than PHVA did; that is, the vibrations of all atoms due to the substitution of heavy and light isotopes in clusters are included to predict the isotope fractionations. This new approach is finally tested by studying isotope fractionation factors in liquid and mineral phases.

We firstly give the theoretical background on building a general cluster for isotope research. The general cluster (_{p}, where A and X represent the interested atom and all atoms in parts of both B and C respectively and the subscript p the number of interested atoms of the same element in the center of the cluster (

As discussed above, the super-molecule is sufficient to describe the chemical influence on isotopes at interested position; then one can use ab initio molecular orbital theory to get the frequencies. In Ref. [

force constants are defined by

where _{ } is the second energy derivatives for coordinates

where

in which

The two sets of harmonic frequencies for a super-molecule would, however, sometimes have imaginary frequencies [

For a super-molecule, we suggest that all frequencies, especially the imaginary ones, should be included in the calculation of isotope fractionation. The reasons come from the following facts [

In mathematics [

own modulus, i.e.

Based on the Born-Oppenheimer approximation (i.e. nearly harmonic approximation) [

The translational and rotational energies are in the form of

where

for diatomic and linear molecules

for nonlinear molecules, where

The vibrational energy is in the form of

quencies cannot be included in this expression and the partition function in the classical mechanism [

where

where

In Ref. [

where

Since

where the superscript “ ′ ” denotes the molecule with heavy isotopes.

Let us submit the frequencies with complex form. The number

After the cancellation of

Equation (14) is valid only when

Submitting Equations (15) and (16) into Equation (14), we obtain the Teller-Redlich product rule in the frequency complex plane:

for diatom and linear molecules and

for nonlinear molecules.

The differences for the isotopes in the super-molecule can be written as a typical chemical exchange reaction [

where

The equilibrium constant for this reaction is given by

Because different isotopes have negligible difference of volume, isotope exchange reactions do not involve significant pressure-volume work [

Let us substitute Equations (5)-(8) into Equation (19). For diatom and linear molecules, we have

and for nonlinear molecules,

where

Equation (20) can be reduced to a more general expression by using Equation (17):

where RPFR_{C} is short for reduced partition function ratio in the frequency complex plane.

Obviously, one can see that if the super-molecule is at a local minimal on the potential energy surface (i.e._{C} becomes Urey (1947) or Bigeleisen and Mayer (1947) formula. Due to the fact that the set of real numbers (i.e. frequencies here) is the subset of the set of the complex numbers [_{C}) (

The fractionation factor between two clusters can be written as:

To understand the new algorithm, we compute RPFR_{C} and/or _{C} are implemented in Gaussian09 [_{C} and

1) The geranium isotope fractionation factor _{4}-(H_{2}O)_{30} (_{4}-(H_{2}O)_{30}_D in Ref. [^{−1}; and 3) RPFR is the same if they neglected it. The values of Li et al.’s αs at different temperatures are taken as references. As shown in

tween Li et al.’s and present results is 8.2 × 10^{−5}‰ (273.15 K); this shows that present approach is very efficient to study isotope fractionation in liquids.

2) The carbon and ^{13}C-^{18}O clumped isotope fractionations in inner body of calcite are good examples of study of isotopes in solid phase. We cut a cluster (

fitted polynomials of

Results shown in Figures 7-9 indicate that our new algorithm have high accuracy. For

Super-molecule | Method/Basis_set/Scaling factor | Imaginary Frequency (cm^{−1})^{*} | ||
---|---|---|---|---|

n | Minimal | Maximal | ||

B3LYP/6-311+G^{**}/1.05^{2 } | 1 | −5.07 | −5.07 | |

Ge(OH)_{4}-(H_{2}O)_{30} | B3LYP/6-311+G^{**}/1.05^{2 } | 1 | −22.23 | −22.23 |

Calcite cluster | HF/3-21G/0.91 | 76 | −7119.47 | −11.93 |

Calcite cluster | B3LYP/6-31G/0.97 | 151 | −3321.80 | −69.95 |

^{1}http://cccbdb.nist.gov/. ^{2}See Ref. [^{*}The frequencies correspond to molecules with ^{70}Ge, ^{12}C^{16}O.

Volume (Å^{3}) | ||
---|---|---|

6.47 | 45.90 | 127.64 |

^{1}http://www.crystal.unito.it.

For a general cluster for isotope research (defined in Section 2.1), we have a new Equation (21) to calculate the isotope fractionation factor in the cluster. The calculation based on this equation has a clearer background of physical mechanism, which includes the contribution of vibrations of all atoms to the factor, than that based on PHVA. If there is no imaginary frequencies for the cluster, Equation (21) is simplified to be the Urey (1947) or Bigeleisen and Mayer (1947) formula. The examples show that our new algorithm is valid and efficient with high accuracy. Although the accuracy is mathematically high, we again address that present approach should be only used to calculate the isotope fractionation factor.

The author is grateful to Dr. Zhang Zhigang in IGGCAS for helpful discussions. All of the calculations are performed at the IGGCAS computer simulation lab. This work is supported by the National Natural Science Foundation of China (Grant No. 41303047, 41020134003 and 90914010).