Nuclear volume effects in equilibrium stable isotope fractionations of mercury, thallium and lead

The nuclear volume effects (NVEs) of Hg, Tl and Pb isotope systems are investigated with careful evaluation on quantum relativistic effects via the Dirac’s formalism of full-electron wave function. Equilibrium 202Hg/198Hg, 205Tl/203Tl, 207Pb/206Pb and 208Pb/206Pb isotope fractionations are found can be up to 3.61‰, 2.54‰, 1.48‰ and 3.72‰ at room temperature, respectively, larger than fractionations predicted by classical mass-dependent isotope fractionations theory. Moreover, the NVE can cause mass-independent fractionations (MIF) for odd-mass isotopes and even-mass isotopes. The plot of vs. for Hg-bearing species falls into a straight line with the slope of 1.66, which is close to previous experimental results. For the first time, Pb4+-bearing species are found can enrich heavier Pb isotopes than Pb2+-bearing species to a surprising extent, e.g., the enrichment can be up to 4.34‰ in terms of 208Pb/206Pb at room temperature, due to their NVEs are in opposite directions. In contrast, fractionations among Pb2+-bearing species are trivial. Therefore, the large Pb fractionation changes provide a potential new tracer for redox conditions in young and closed geologic systems. The magnitudes of NVE-driven even-mass MIFs of Pb isotopes (i.e., ) and odd-mass MIFs (i.e., ) are almost the same but with opposite signs.

In addition, Zheng et al. 33 and Ghosh et al. 34 did different experiments to estimate the NVE of mercury isotopes in the absence of light. They both assigned those mass-independent isotope fractionation signals as the consequence of the NVE. Wiederhold et al. 35 also did experimental and theoretical investigations on Hg mass-indpendent isotope fractionations. Schauble 4 and Wiederhold et al. 35 have explored small NVE-driven Hg isotope fractionations in organic Hg-bearing species in depth. Moynier et al. 9 reviewed the NVEs of Tl and U isotope systems in different natural environments, such as under low-or high-temperature conditions and in meteorites. The necessity of careful NVE evaluation during the exploration of new heavy elements is recognized by most people.
Right now, there are a few different computational methods used to investigate quantum relatistic effects associated with the NVE, e.g., Schauble 4,32 used the DIRAC and ABINIT software package, Abe et al. [5][6][7] used a four-component relativistic atomic program package-GRASP2K, Fujii et al. 18,20,21,30,31 used a software provided by Tokyo University (UTchem). Recently, Nemoto et al. 36 found a two-component realtivistic method (the finite-order Douglas-Kroll-Hess method with infinite-order spin-orbit interactions for the one-electron term and atomic-mean-field spin-same-orbit interaction for the two-electron term, i.e., IODKH-IOSO-MFSO) with almost equivalent accuracy but 30 times faster than the previous four-component method by DIRAC software package. They also predicted the IODKH-IOSO-MFSO method could compute larger system for future NVE calculation.
Here we calculate the NVE-driven fractionation factors of Hg-, Tl-and Pb-bearing species by using full-electron quantum chemistry calculation methods. Our method is similar to that of Schauble 4 , in which quantum relativistic effects have been carefully evaluated via four-component Dirac equation formalism 37,38 . Not only more new Hg-and Pb-bearing species (e.g., HgBr 4 2− , HgCl 3 − , HgBr 3 − and many Pb 4+ -bearing species) are calculated here, but more mass-independent fractionations are investigated in light of recent findings on even-number Hg isotope MIFs 39,40 . Large fractionations (up to ca. 4‰ at room temperature) between Pb 4+ -and Pb 2+ -bearing species are found for the first time.

Hg isotope system.
Tl isotope system. The NVE-driven fractionation of 205 Tl/ 203 Tl isotopes is up to 2.54‰ relative to Tl 0 and the CMDE is only 0.58‰ for Tl(H 2 O) 6 3+ and 0.07‰ for Tl(H 2 O) 3 + at 25 °C. Our NVE results show that Tl 3+ ion and Tl 3+ -bearing compounds enriches heavier isotope ( 205 Tl) relative to Tl 0 . However, Tl + ion and Tl + -bearing compounds enriches lighter isotope ( 203 Tl) compared to Tl 0 . Note that β -values of Tl + -bearing species are even smaller than the unity. This is because NVE tends to let heavier isotopes to be enriched in those atoms or ions with fewer s electrons or with more p, d and f electrons. Tl 0 has more p electrons than Tl + -bearing species does. Mass-independent fractionation of Hg and Pb isotopes. Table 4 shows NVE-driven mass-independent fractionations for 199 Hg, 200 Hg, and 201 Hg isotopes at room temperature. Those MIFs are relative to the MIF of Hg vapor (i.e., Hg 0 ). If the real MIF value of a specific Hg-bearing species is needed, one needs convert the number listed in Table 4 via the aid of experimental MIF data of Hg vapor.   For example, according to Ghosh et al. 34 , the NVE-driven Δ Hg NV 199 of Hg 0 is about 0.14‰, therefore, Δ Hg NV 199 of Hg 2+ should be − 0.59‰ (i.e., − 0.73‰ of Hg 2+ listed in Table 4  ) will fall on a straight line with the slope of 1.66 (Fig. 4), suggesting they will be changed in a proportional way. This result is almost identical to a previous theoretical result 35 (i.e., with the slope of 1.65). This special relationship can be used to study MIFs caused by other reason via distinguishing the NVE signals from them.
Moreover, NVE can also cause mass-independent fractionations for odd-mass isotope ( 207 Pb) and even-mass isotope ( 204 Pb) ( Table 5 ) among all the studied species relative to Pb 0 at 25 °C. The signs of even-mass isotope MIF (Δ Pb NV 204 ) and odd-mass isotope MIF (Δ Pb NV 207 ) are opposite to each other although their magnitudes are almost the same ( Table 5).
The calculation details, including optimized geometries, energies, harmonic vibrational frequencies, et al., have been documented in the Supplementary file for interested reader.

Discussion
One of the special features of NVE is that it can cause large isotope fractionations between isolated atoms and ions (e.g., Hg 2+ -Hg 0 , Tl 3+ -Tl + and Pb 4+ -Pb 2+ ), which there would be no fractionation at all if based on the classical mass-dependent isotope fractionation theory, because there is no difference in terms of kinetic energies for them. Moreover, it seems that ions with more extra charges (e.g., with fewer s orbital electrons) can have larger NVEs and isotope fractionation potential than those with lesser charges (e.g., Tl 3+ vs. Tl + , Pb 4+ vs. Pb 2+ ).
Comparing with previous studies (Fig. 1b and Table 1), our NVE-driven Hg isotope fractionation results are noticeably different from those of Schauble 4 and Wiederhold et al. 35 . The NVE is proportional to difference in mean square nuclear charge radius of different nuclei (i.e., NVE ∝ δ < r 2 > and δ < r 2 > = < r 2 > A − < r 2 > A′ ), as King 10 has pointed out based on spectrometric results. Therefore, we can explain the difference between Schauble 4 and our results very clearly. Schauble 4 used the nuclear charge radii of Angeli 42 (i.e., < r 2 > 1/2 of 202 Hg and 198 Hg are 5.4633fm and 5.4466fm) and the nuclear charge radius difference (δ < r 2 > = < r 2 > A − < r 2 > A′ ) is 0.182fm 2 . But we use the nuclear charge radii from             We use the same mean square nuclare radii as Wiederhold et al. 35 , but different methods (i.e., DHF vs. MP2, respectively), which lead to different results. In addition, there are suggestions that calculated results of Hg-bearing species used the mean square nuclear charge radii of Fricke and Heilig 41 are closer to the experiment results 35 . Note that different versions of the calculation software package (i.e., DIRAC04 and DIRAC13.1) have little impact on the calculated results (see Table S6).
Our NVE-driven Tl isotope fractionation results are in comparison with those of Schauble 4 and Fujii et al. 30 in terms of 1000·lnβ NV (Fig. 2b and Table 2). Our results agree with those of Schauble 4 perfectly because of using the similar methods and the same mean square nuclear charge radii (i.e., those radii from Angeli 42 39,40 . The mechanism leading to the even-number Hg isotope mass-independent fractionation is still unclear.
Gratz et al. 39    With the calculation data from this study, we find that the NVE-driven Δ Hg NV 200 cannot be the reason to explain those even-mass number Hg MIF results. First, the magnitudes of NVE-driven Δ Hg NV 200 are much smaller than those found by Gratz et al. 39 and Chen et al. 40 . Second, the sign of NVE-driven even-mass number Hg MIFs calculated here is opposite to those reported Δ 200 Hg results, meaning the NVE causes depletion of 200 Hg instead of enrichment of 200 Hg relative to 198 Hg. Therefore, the observed large positive Δ 200 Hg signals must have other reasons or processes to be produced.
Because the half-life times of uranium isotopes are all very long, e.g., 4.5Ga for 238 U and 0.7Ga for 235 U, people actually treat uranium isotope system as a stable one in many young geologic systems 47,48 . As the decayed products of uranium, Pb isotope system can also be treated as a regular stable isotope system for young and closed geologic systems with homogenized formation processes. For example, in some rocks formed less than 10 million years (or younger), or in some plants, or in any system which is young and homogenized before its formation. The equilibrium Pb isotope fractionations between two compounds in such systems can be meaningful and useful. In such systems, the radiogenic Pb isotope differences are no longer existing but homogenized to a background value. For instance, a system with inherited very high 208/206 Pb value has been homogenized in some processes. The compounds in such system will all have very high 208/206 Pb values. Meanwhile, the small differences of 208/206 Pb values among different compounds are caused by mass-driven and NVE-driven isotope fractionations. Our results can be used to explain such differences.
Fujii et al. 31 firstly reported calculated NVE-driven Pb isotope fractionation factors for Pb 0 and Pb 2+ -bearing species. We provide results of several new Pb-bearing species especially for Pb 4+ -bearing species. If comparing the NVE results between Pb 0 and Pb 2+ of Fujii et al. 31 and ours, our results are marginally larger than theirs ( 208 Pb/ 206 Pb: 0.60‰ vs. 0.393‰ and 207 Pb/ 206 Pb: 0.24‰ vs. 0.156‰) due to different methods and software packages used. In general, Pb isotope fractionations among Pb 2+ -species are very small even with the driving force of NVE. However, we find surprisingly large fractionations (ca. 2 to 4‰) between Pb 4+ -bearing species and Pb 2+ -bearing species at room temperature. The fractionation magnitudes are similar or even larger than those Fe isotope fractionations between ferric and ferrous Fe-bearing species (e.g., Fe 3+ (aq) vs. Fe 2+ (aq) ) at low temperature, which have been broadly used as tracer for the change of redox conditions. Therefore, Pb isotope fractionations probably can also be used as a new tracer to study redox condition changes in young and closed geologic systems.
The occurrence of such large isotope fractionations is because the β -values of Pb 4+ -bearing and Pb 2+ -bearing species are in different directions, as the consequence of unique nuclear volume effects. Pb 4+ -bearing species enrich heavy isotopes relative to Pb 0 . However, Pb 2+ -bearing species enrich light isotope compared to Pb 0 , meaning β -values of Pb 2+ -bearing species are smaller than the unity. It is similar to the case of Tl + -bearing species. Pb 0 is the one has more p electrons than Pb 2+ -bearing species. This finding cannot be explained if only based on classical isotope fractionation theory, which suggests all β values of any kind of isotope systems must be equal or larger than the unity.

Conclusions
In this study, quantum chemical calculations (Dirac-Hartree-Fock) confirm that the nuclear volume effect plays a dominant role in equilibrium isotope fractionation for mercury, thallium and lead systems compared to the contributions of conventional mass-dependent effect, and agree with those conclusions of previous studies 4, 30 ).

Methods
Conventional mass-dependent effect (CMDE). Bigeleisen  where ε is the isotope enrichment factor and is roughly equal to lnα 0 ; α 0 is the isotope fractionation factor; m and m′ are the masses of the heavy and light isotopes, respectively; Δ m is the relative mass difference of isotopes (i.e., Δ m= m-m′ ). When the temperature is constant, enrichment factor is proportional to Δ m/mm′ . According to this equation, the isotope fractionation of heavy elements (e.g., Hg, Tl or Pb) would be small. For an exchange reaction A′ Y + AX = A′ X + AY, the equilibrium CMDE fractionation factors is calculated 11,12 where RPFR is the reduced partition function ratio and it is expressed in term of the harmonic vibrational frequencies with isotope substitution Nuclear volume effect (NVE). Based on spectrometric results, King 10 proposed that the NVE was proportional to difference in mean square nuclear charge radius of different nuclei (i.e., NVE ∝ δ < r 2 > and δ < r 2 > = < r 2 > A − < r 2 > A′ where lnα 0 is the isotope fractionation factor under the B-M equation approximations; lnK anh is the anharmonic correction term; lnK BOELE is the correction to the Born-Oppenheimer approximation; lnK fs is the NVE term (also called nuclear field shift); lnK hf is the term for nuclear spin effect. In the terminology of Bigeleisen, the nuclear field shift actually includes both shape and size effects. However, the contribution from nuclear size is easy to calculate but that from nuclear shape is very difficult to evaluate and trivial. Therefore, people trend to use NVE instead of nuclear field shift for more precise description 4 . Because of extremely small anharmonic corrections for heavy elements, lnK anh can be safely neglected 2 . The correction to the Born-Oppenheimer approximation is related to Δ m/mm′ 49,50 . Therefore, lnα 0 and lnK BOELE are both proportional to Δ m/mm′ when temperature is a constant. Based on the investigations on U isotope exchange reactions, Bigeleisen 2 showed that nuclear spin effect was also very small and could be safely neglected.
Because the NVE is related to the difference in ground-state electronic energies, it can be written as 2 where E 0 is the ground-state electronic energy; AX and A'X represent different isotopologues; k is the Boltzmann's constant and T is in absolute temperature (K). We can see the magnitude of NVE is proportional to 1/T and to ground-state electronic energy differences due to isotopic substitutions.

Mass-independent fractionation (MIF).
Here we use Hg isotopes as an example to introduce the concept of mass-independent isotope fractionation (MIF). If we define δ A Hg as where m i , m j and m k are the masses of isotopes i, j and k, respectively; < > r i 2 , < > r j 2 and < > r k 2 are their mean square nuclear charge radii.
Unfortunately, Δ Hg NV A cannot be calculated theoretically because the value of δ 202 Hg for a specific Hg species is unknown. Instead, we calculate the relative MIF in comparison of Hg vapor (Hg 0 ): values for Hg 0 were 0.14 ± 0.01‰, 0.09 ± 0.01‰ and 0.01 ± 0.03‰, respectively. The results of equilibrium evaporation experiments of Estrade et al. 45 were similar to those of Ghosh et al. 34 and their Δ Hg The results show that these two methods can produce almost identical geometries but the former consumes much lesser time.
After geometry optimization, all Hg-, Tl-and Pb-bearing species are calculated for obtaining their ground-state electronic energies by using DIRAC 13.1. Different isotopologues will use their own Gaussian exponent ξ as in this form 59 : where the mean square nuclear charge radii (< r 2 > ) can be found from the Landolt-Boernstein Database 41 for Hg and from Angeli 42 for Tl and Pb. Different from closed shell species, we also use the complete open shell configuration interaction (COSCI) method to calculate the ground-state energies of opened shell species (Tl 0 and Pb 0 with the electron configuration as [Xe]4f 14 5d 10 6s 2 6p 1 and [Xe]4f 14 5d 10 6s 2 6p 2 , respectively ).
With the calculated ground-state electronic energies, the NVE can be calculated from "equation (5)". For example, the NVE on isotope fractionation of an HgX-Hg 0 isotope exchange reaction is 4 where the β (X) factor is the equilibrium fractionation factor between substance X and an ideal monoatomic gas 60 . The molecular geometries and harmonic interatomic vibrational frequency are calculated at pseudo-potential Hartree-Fock (HF) level by Guassian03 software package 58 . We treat inner-shell electrons of Hg, Tl, and Pb atom by using relativistic pseudo-potentials. However, valance and intermediate-shell electrons are treated with a double-zeta basis sets (cc-pVDZ-PP) and cc-pVDZ basis sets are used for H, C, O, Cl and Br atoms.
The usual isotope fractionation between substance A and substance B is defined as