Ab-initio structure, energy and stable Cr isotopes equilibrium fractionation of some geochemically...

$: Ab-initio structure, energy and stable Cr isotopes equilibrium fractionation of some geochemically relevant H-O-Cr-Cl complexes$
Geochimica et Cosmochimica Acta, Vol. 69, No. 4, pp. 851–874, 2005Copyright © 2005 Elsevier Ltd

Printed in the USA. All rights reserved

doi:10.1016/j.gca.2004.06.045

Ab-initio structure, energy and stable Cr isotopes equilibrium fractionation of somegeochemically relevant H-O-Cr-Cl complexes

GIULIO OTTONELLO and MARINO VETUSCHI ZUCCOLINI*Laboratorio di Geochimica at DIPTERIS - Università di Genova, Corso Europa 26, 16132 Genova Italy

(Received November 7, 2003; accepted in revised form June 17, 2004)

Abstract—Twelve Cr(III) molecules and six Cr(VI) molecules of geochemical interest in the systemCr-H-O-Cl were investigated by ab initio Density Functional Theory in the aim of determining stationary stategeometries, electronic energies and vibrational properties. The vibrational analysis conducted on the 50Cr,52Cr, 53Cr and 54Cr isotopomers indicates important separative effects which result in mass spectrometricmeasurable fractionation factors in a wide thermal range. The effect of the central cation on the computedvibrational modes is interpreted in terms of Redlich’s product rule with the introduction of an isotopicergodicity factor related to the effect of bulk molecular masses and momenta of inertia about the principalrotational axes. After the initial gas phase optimization, some relevant molecules were subjected to the effectsof reaction field through Onsager’s Model, Tomasi’s continuum Model and Conductor-like Screening Modelprocedures. Structures and vibrational properties prove in all cases not to be severely affected by solvation,although the effect of the reaction field is to reduce somewhat the isotopic fractionation in Cr(III)-Cr(VI)aqueous redox equilibria, with respect to the gaseous state. Some of the computed separative effects are

0016-7037/05 $30.00 � .00

applicable to geochemical investigations. Copyright © 2005 Elsevier Ltd

1. INTRODUCTION

Results of preliminary gas-phase Unrestricted Hartree-Fock(UHF) calculations in terms of fractionation factors � � K1/r

for the homogeneous (gas phase) reaction:

H�CrO4 gas� � ●Cr(H2O)6 gas

3� ⇔ H●CrO4 gas� � �Cr(H2O)6 gas

3� (1)

(where filled circles indicate the heavy isotopomers and opencircles the light ones), suggested that at ambient P, T (25°C, 1bar) isotopic fractionation may attain 0.0065 � 0.0002 pera.m.u. in gas phase (Ottonello and Vetuschi Zuccolini, 2001).This has important consequences for understanding Cr fraction-ation phenomena which could take place during redox pro-cesses such as those occurring on a large scale in aquifers orthose induced within the cellular membrane by reductionagents such as glutathione. We performed gas-phase calcula-tions for several H-O-Cr-Cl molecules of particular geochemi-cal interest by ab initio computations based on Density Func-tional Theory, as a preliminary step in the thoroughinvestigation of Cr fractionation in inorganic and organic mol-ecules in different media. The results of gas phase optimisa-tions provided the foundation for further characterization ofaqueous complexes through Reaction Field Models of solvationfor some key species. During the editorial process for thiscontribution, a new publication on the very same subject cameto our attention (Schauble et al., 2004). In the final draft of themanuscript we took into account part of these new, and sub-stantially concordant, results.

2. COMPUTATIONAL METHODS

Geometries, energies and harmonic vibrational frequenciesof Cr(III) and Cr(VI) gas-phase complexes were computed by

* Author to whom correspondence should be addressed ([email protected]).

851

all-electrons, non-relativistic, non-variational Density Func-tional Theory (DFT) procedures. The DFT algorithms are com-monly used in inorganic chemistry to compute static and dy-namic properties of molecules, clusters and crystals. They arepreferable with respect to multiconfiguration theories such asMøller-Plesset theory-based algorithms (MPn where n �2,3,4), QCSID or semiempirical theories (i.e., AM1, PM3) dueto better optimization of CPU time without excessive loss ofaccuracy and precision (Scott and Radom, 1996; Curtiss et al.,1997).

Density functionals are based on Kohn-Sham theory (Kohnand Sham, 1965) which states that the energy of a ground stateof a system can be expressed precisely by the electronic densitythrough functional relations. In the expression of the totalenergy, exchange and correlation energies are included as well.The former is related to the potential energy between twoelectrons of opposite spin in the same orbital. The latter con-sider the all-electron interaction, overstating the HF approxi-mation in which only one electron at time is considered withrespect to the mean electric field exherted by all the otherparticles. The molecules investigated in this study comprisehydrated-hydroxylated oligomers, protonated and chlorinatedstructures known to be stable in aqueous solutions in a widerange of pH values ([Cr(OH)n(H2O)6�n](3�n)� (n � 033);[CrCln(H2O)6�n](3�n)� (n � 133); [Cr2(OH)2(H2O)8]4�;[Cr3(OH)4(H2O)9]5�; [CrO2]�, [HnCrO4](2�n)� (n � 032);[ClCrO3]�; [HnCr2O7](2�n)� (n � 031)). Chromium, as othertransition metals, pose some problems of definition of observ-able and dynamical molecular properties due to the incomplete-ness of the inner orbitals. The ground states accepted forcalculations are, in Russell-Saunders coupling scheme, 4F forCr(III) (with configuration [Ar]3d3) and 1S for Cr(VI) (withconfiguration [Ar]3d0). Atomic orbitals of elements constitut-ing the investigated molecules (H, O, Cl, Cr) were described by
the same basis set in all computations. In particular the ex-

852 G. Ottonello and M. Vetuschi Zuccolini

tended basis set 6–31G**, also known as 6–31G(d,p), wasadopted. In this formalism, a wavefunction is described as alinear combination of 6 gaussians for core-electrons, 3 gauss-ians for inner valence electrons and 1 gaussian for outer va-lence electrons. Polarization of d orbitals for Oxygen andChlorine (or f orbitals, in the case of Chromium) and p orbitalsfor Hydrogen were considered as well. In our calculations weused a DFT approach with a modified hybrid (parameterized)B3LYP functional (Becke, 1993; Stephens et al., 1994), involv-ing a second Generalized Gradient Approximation (GGA), withthe 3 Becke parameter as exchange functional and the LYP(Lee et al., 1988) correlation functional. Symmetry constraintswere avoided, whenever possible, enabling a more reliablecomputation on numerical integration. All gas-phase computa-tions were based on a ultrafine grid with a pruning of gridpoints around atoms, using Mura-Knowles algorithms for radialquadrature and Lebedev scheme for angular quadrature.Adopted convergence criteria for energies, densities and gra-dients are 10�6, 10�5, 10�4, respectively, if not otherwisespecified.

Computed harmonic vibrational frequencies by ab initio codesare normally under or over-estimated with respect to fundamentalmodes captured by experimental techniques, but the shift (scalingfactor) is somewhat fixed, depending on the level theory and basisset used to approximate the orbital wavefunctions (Scott andRadom, 1996 and references herein). The definition of the scalingfactor computed by least-squares fitting methods on test sets ofmolecules leads some authors to further differentiate the scalefactor, depending on the kind of observable (fundamental modes,low frequencies, enthalpy and entropy). Due to the presence in ourmolecules of a heavy metal, which is a source of great uncertainty,and due to the lack of comparative estimates in the literature, wedecided to use a unique scaling factor (0.9620) for frequencies andthermochemistry (work in progress) according to the CCCBDB(Computational Chemistry Comparison and Benchmark DataBaseRelease 8—http://srdata.nist.gov/cccbdb) for DFT B3LYP/6-31G**).

The quantum mechanical code used to compute all gas-phasestructures is NWChem 4.5a (Kendall et al., 2000) generated by theHPCC Group at PNNL and implemented for parallel computingusing TCGMSG and Global Arrays communication libraries onthe Laboratorio di Geochimica beowulf-like cluster “BAXEICO”at DIPTERIS. Effects of the reaction field were investigated byutilizing both NWChem 4.5 for the Conductor-like ScreeningModel (COSMO, Klamt and Schüürmann, 1993) and GAUSS-IAN98 (Frisch et al., 1998) for Onsager and Tomasi’s IntegralEquation Formalism applied to a Polarized Continuum (IEFPCM;Cancès et al., 1997; Cossi et al., 1998).

2.1. Optimized Geometries and Vibrational Properties ofCr(III) Species

[Cr(H2O)6]3�, [Cr(H2O)18]3�

EXAFS measurements have detected up to three shells ofhydration of aqueous trivalent chromium, with a more diffuse

a High Performance Computational Chemistry Group, NWChem, AComputational Chemistry Package for Parallel Computers, Version 4.5
(2003), Pacific Northwest National Laboratory, Richland, Washington99352, USA.
character from the first to the third shell of coordination. A totalof 12–13 water molecules seems to form the second hydrationshell (12.94 according to Bleuzen et al., 1996). The secondhydration shell can be roughly described “as formed by pairs ofwater molecules hydrogen-bonded to each water molecule fromthe first hydration shell” (Sakane et al., 1998).

The distances between the central cation and the neighboringoxygens of the first shell range from 1.98 Å to 2.01 Å. Theoxygens of second shell of hydration are 4.01–4.02 Å distantfrom the central ion. These measurement appear to be relativelydependent upon concentration, up to a bulk molality 2.5M.

Due to the stability of [Cr(H2O)6]3�, ab initio calculationsand, more recently, Molecular Dynamics (MD) calculationshave been done by various researchers. Pappalardo et al.(1996), using ROHF ab initio calculation with the Kirkwood-Onsager procedure implemented in the SCRF (Self ConsistentReaction Field), determined geometry and energy properties ofaqueous hexa-aqua chromium. Table 1 summarizes experimen-tal and computational results for the geometry and energy ofthe [Cr(H2O)6]3� cluster. The Cr-O interatomic distance ob-tained in this study for the gaseous molecule is 2.004 Å, ingood agreement with EXAFS observations and with other DFTstudies (Table 1). The complete DFT B3LYP/6-31G** struc-ture has Th symmetry and the attained electronic energy showsthe best achievement in terms of variational principle (Table 1).The effect of a second solvation shell on the [Cr(H2O)6]3�

cluster (or “explicit solvation” according to Felipe et al., 2003)was simulated by adding 12 water molecules through hydrogenbonds initially conformed on the symmetry of the hexa-aquaspecies. After geometry optimization the second shell watermolecules were not embedded in the symmetry planes and thesymmetry of the cluster was reduced to T, in a similar mannerto that observed by Rudolph et al. (2000) for the [Al(H2O)18]3�

cluster (Fig. 1n). The distance between Cr and the oxygens ofthe first hydration sphere is 1.979 Å, and the distance betweenCr and the oxygens of the second hydration sphere is 4.154 Å.The H-bond distances are 0.995 and 0.967 in the first andsecond shell, respectively. The decrease in Cr-Oinner bondlength in the first hydration sphere is due to polarization effectsof the second H2O shell on the waters of the first shell, whichresults in an increased interaction of the central cation with thefirst shell.

When subjected to the full reaction field, through implicitsolvation procedures (see later) the bond between the centralcation and the first oxygens contracts more, while the oxygen tohydrogen bond in the inner shell water relaxes and the H2Obond angle is reduced by a few degrees (Table 1).

Apparently there is not much in the literature pertaining tothe vibrational properties of the molecule. A set of frequenciesin the range (300–335 cm�1) obtained by single-crystal Ramanspectrum of the Cr3� hexa-hydrate (Best et al., 1984) have beententatively assigned to the asymmetric modes of the hexa-hydrate. Åkesson et al. (1994) report a single Ag (�1) stretchingfrequency of 460 cm�1 for the high-spin Th form obtained atthe Harthree-Fock UHF-SCF level. Martinez et al. (1998) con-firmed a band assignment to Ag (�1) stretching frequency at 458cm�1 by restricted open shell Hartree-Fock (ROHF) ab initiocomputation. The Ag (�1) Cr-O stretching mode frequency inalums is around 540 cm�1 (Best et al., 1980) and lowers to 522
cm�1 in 1M H2S solution (Best et al., 1984).

853Ab-initio Cr isotopes fractionation

A molecular dynamics study of the Cr3� hydration by thehydrated ion water potential (HIW) performed by Martinez etal. (1998) seems to suggest that vibrational motions of the[Cr(H2O)6]3� cluster are severely affected by the medium. TheHIW-MD estimated frequency of the Ag (�1) stretching modefor the unperturbed aqueous hexa-hydrate (665 cm�1) attains740 cm�1 when this is surrounded by twelve second shell watermolecules and stabilizes to 750 cm�1 when this is fully im-mersed in solution (Table 2). The other significant peak of thepower spectra obtained by Martinez et al. (1998) from their MDstudy by Fourier transform of the MD velocity autocorrelationfunction (Easteal et al., 1989) has been tentatively assigned bythe authors to a hindered translational motion of the centralcation. Although the addition of the bulk solvent seems toaffect sensibly the rigidity of the hexa-hydrate, the quantifica-tion of this phenomenon is complicated by the large discrep-ancies arising from the various adopted MD potentials (seeTable 2).

The vibrational frequencies of the [Cr(H2O)18]3� clusterdetermined in this study are in agreement with the computa-tions of Schauble et al. (2004) which were made both at simpleHartree-Fock level and by DFT with different wave expan-sions. The comparison is more problematic for the results ofMolecular Dynamics calculations (Martinez et al., 1998; cf.Table 2).

As already observed by Rudolph et al. (2000) for the Al[6� 12] cluster, some of the Cr-O6 modes couple strongly withrestricted translational modes and librational modes of thesecond shell waters, although this (symmetry-allowed) cou-pling should be much reduced in real solutions. Despite this

Table 1. Geometry of the [Cr(H2O)6]3� and [Cr(H2O)18]3� cluster1M � 1 molal solution; FS � first coordination shell; SS � second c

Interatomic distances Bondangle

H-O-H Energy HartreeCr-O (Å) O-H (Å)

1.981.97 �1052.00 � 0.012.011.9562.0052.02 �1496.81342.0252.050 0.967 107.54 �1498.320532.012.022.002.004 � 0.001 0.978 107.7 �1502.0138961.940 1.007 110.9 �1495.3527654.024.02 � 0.084.054.06 � 0.034.06 � 0.024.114.214.154.173.974.15 0.995inner 110.1inner �2419.575862

0.967outer 105.0outer

fact, as we will discuss later, the vibrational behavior of

[Cr(H2O)18]3� and solvated [Cr(H2O)6]3� are quite similarin terms of reduced partition functions of the various isoto-pomers.

The mechanism of the ligand exchange reaction

Cr(H2O)63� � H2O ⇔ Cr(H2O)5(H2O)3� � H2O (2)

where the terms in bold denote the “spectator ligand” (LS), hasbeen investigated in detail by Rotzinger (1997, 2000). Substi-tution takes place via an associative interchange phenomenonleading to formation of a [cis � Cr(H2O)5 . . . (H2O)2

3� com-plex. The ab initio activation energy is �E� � 96.3 kJ/mol(Rotzinger, 2000). The postulated interchange phenomenon isin agreement with the observed experimental activation vol-ume, which is negative (�V� � �9.6 � 0.1 cm3/mol), and thecomputed activation energy is consistent with the experimen-tally observed activation enthalpy (�H� � 108.6 � 2.7 kJ/mol;Stranks and Swaddle, 1971; Xu et al., 1985).

[Cr(OH)n(H2O)6�n](3�n)� n � 1, 2, 3Chromium (III) hexa-aqua complex is stable in solution only

at very acidic conditions and progressive deprotonation takesplace with the increase of pH according to the followingequilibria:

Cr(H2O)63� ⇔ Cr(H2O)5(OH)2� � H� (3)

Cr(H2O)5(OH)2� ⇔ Cr(H2O)4(OH)2� � H� (4)

Conditional deprotonation constants have been obtained byalkalimetric titration in 1M NaClO4 at T � 298.15 K by Stünziand Marty (1983) and Beutler (1976) with consistent results

rimental observations are compared with results of ab-initio studies.tion shell.

References Notes

Tossell and Vaughan (1992) ExperimentalSakane et al. (1998) EXAFS, 1MMuñoz-Paez et al. (1995) EXAFS, 0.01M, FSMuñoz-Paez et al. (1995) EXAFS, 0.005–2.5M, FSBleuzen et al. (1996) DFT LDABleuzen et al. (1996) DFT NLDATossel and Vaughan (1992) HF-SCFRead and Sandström (1992) HF-SCFPappalardo et al. (1996) ROHF/DZV � MCYMartinez et al. (1998) RDF, FSMartinez et al. (1998) IWI/TIP4P RDF, FSMartinez et al. (1998) IWI/TIP4P RDF, FS/SSthis study (gaseous ion) DFT B3LYP/6–31G**this study (aqueous species) DFT B3LYP/3–21G � IEFPCMMuñoz-Paez et al. (1995) EXAFS, 0.005–2.5M, SSMuñoz-Paez et al. (1995) EXAFS, 0.01M, SSMartinez et al. (1998) 14 H2O RDF, FS/SSPappalardo et al. (1996) MC/210, SSPappalardo et al. (1996) MC/512, SSMartinez et al. (1998) IWI/TIP4P RDF, SSMartinez et al. (2000) HIWP � MCHO, SSMartinez et al. (2000) HIW � MCY, SSMartinez et al. (2000) HIW � TIP4P, SSSchauble et al. (2004) DFT B3LYP/6–31G*this study (gaseous ion) DFT B3LYP/6–31G**

s. Expeoordina

9

(pKaI � 4.29–4.30; pKa

II � 4.29–4.30). According to Pettine


Fig. 1. Ab-initio stationary point geometries of Cr(III) gaseous molecules. a: [Cr(H2O)6]3�; b: [Cr(H2O)5(OH)]2�;c: [Cr(H2O)4(OH)2]�; d: [Cr(H2O)3(OH)3]0; e: [Cr(H2O)5Cl]2�; f: [Cr(H2O)4Cl2]�; g: [Cr(H2O)3Cl3]0; h:

4� 5� � 5�
[(H2O)4Cr(OH)2Cr(H2O)4] ; i: [(H2O)5Cr(OH)Cr(H2O)5] ; l: [CrO2] ; m: [(H2O)4Cr(OH)2Cr(OH)2(H2O)Cr(H2O)4] ;n: [Cr(H2O)18]3�.

tudy


and Millero (1990), based on the observed pH dependence ofthe oxidation rate, reaction (3) has fast kinetics and takes placeas an intermediate step in the overall oxidation of Cr(III)aqueous solutions by the H2O2 agent. Minimum energy struc-tures of isolated [Cr(OH)n(H2O)6�n](3�n)� complexes areshown in Figure 1. Successive deprotonation (n � 1, 2, 3)perturbs the octahedral symmetry and modifies sensibly theCr-O bond lengths (Table 3).

[CrCln(H2O)6�n](3�n)� n � 1, 2, 3The structure of chromo-cloro-aquo complexes

[CrCln(H2O)6�n](3�n)� has been investigated by XANES andEXAFS in aqueous solutions of different ionic strength andacidity by Diaz-Moreno et al. (1996). The XANES spectra of

Table 2. Selected vibrational frequencies of

Cluster BandFrequency

(cm�1)

[Cr(H2O)6]3� �1 460 Åkess458 Marti665 Marti480 Marti515 Marti472 Otton456 this s

�2 235 Marti230 Marti225 Marti231 Otton206 this s

[Cr(H2O)6]3�·12 H2O �1 740 Marti555 Marti585 Marti515 Schau515 Schau507 Schau519 this s

�2 255 Marti280 Marti270 Marti225 Schau256 Schau205 Schau237 this s

[Cr(H2O)6]3�·512 H2O �1 750 Marti570 Marti

�2 305 Marti305 Marti

[Cr(H2O)6]3�-solvated �1 531 this s�2 176 this s

Table 3. Minimum energy structure of t

n

Interatomic distances

Cr-O (Å) O-H (Å) O-Cr-

0 2.004 0.978 90.02.005

1 1.794 0.969 83.22.110 0.972 103.2

2 1.873 0.963 77.42.061 0.978 101.6

3 1.877 0.964 75.2
2.167 0.994 106.32
the chloro-aquo complexes in solution are analogous to thatobserved in solid chromium chloride crystalline hydrateCrCl3·6H2O (an octahedron formed by four oxygen atoms at2.0 Å and two chlorine atoms at 2.3 Å) although the featuresafter the edge are less resolved (cf. Figs. 1 and 2 in Diaz-Moreno et al., 1996). EXAFS spectra show that the progressiveinclusion of Cl� ions in the inner coordination sphere (i.e., n �1, 2, 3) induces changes both in node positions and backscat-tering amplitude of the EXAFS functions. The results confirmthe X-ray diffraction study of Magini (1980) which foundCr-O1 and Cr-Cl1 average distances respectively of 1.97–2.32Å (n � 1), 1.98–2.32 Å (n � 2) and 1.96–2.30 Å (n � 3)(Table 4). Fourier Transform analysis of EXAFS spectra indi-

s and solvated [Cr(H2O)6]3� ion. �1 � Ag.

References Notes

l. (1994) UHF-SCFl. (1998) ROHFl. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MDl. (1998) 4–6–7-ROHF/MDVetuschi Zuccolini (2001) UHF-SCF

seous ion) DFT B3LYP/6–31G**l. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MDl. (1998) 4–6–7-ROHF/MDVetuschi Zuccolini (2001) UHF-SCF

DFT B3LYP/6–31G**l. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MDl. (1998) 4–6–7-ROHF/MDl. (2004) HF 6–31G*l. (2004) DFT LANL2DZl. (2004) DFT B3LYP/6–31G*

DFT B3LYP/6–31G**l. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MDl. (1998) 4–6–7-ROHF/MDl. (2004) HF 6–31G*l. (2004) DFT LANL2DZl. (2004) DFT B3LYP/6–31G*

DFT B3LYP/6–31G**l. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MDl. (1998) 4–6–12-ROHF/MDl. (1998) 4–6–7-MP2/MD

DFT B3LYP/3–21G � IEFPCMDFT B3LYP/3–21G � IEFPCM

ous [Cr(OH)n(H2O)6�n](3�n)� clusters.

Bond anglesEnergy

(Hartree)H-O-Cr H-O-H

126.2 107.7 �1502.013896

115.1 107.1 �1501.947453131.8 109.096.34 108.18 �1501.705975

127.66 111.1483.05 103.2 �1501.329128

gaseou

on et anez et anez et anez et anez et aello andtudy (ganez et anez et anez et aello andtudynez et anez et anez et able et able et able et a

tudynez et anez et anez et able et able et able et a

tudynez et anez et anez et anez et atudy

he gase

O

805

117.28 107.4

gaseou


cate the presence of a second coordination sphere at a distancevariable from 3.9 to 4.5 Å from the central cation. In the n �1 species this sphere is formed by �14 water molecules at 3.97Å. For the n � 2 species the 2nd coordination sphere is formedby 3.3 to 4.9 chloride ligands at 4.4 Å which replace 8.0 or 7.2water molecules respectively at 3.9 Å, depending upon theconcentration of the complex. With n � 3 in HCl � 12N, watermolecules are fully replaced by chloride ions located at adistance of 4.3 Å from the central cation.

As far as we know there are no systematic investigations onthe effect of Cl� substitutions for hydroxyl on the vibrationalproperties of [CrCln(H2O)6�n](3�n)�clusters.

Table 4. Geometry of the [CrCln(H2O)6�n](3�n)� ions and molecule.� first coordination shell; SS � second coordination shell.

n

Interatomic distances Bond angles

Cr-O (Å) Cr-Cl (Å) O-Cr-O Cl-Cr-O

0 2.004 90.0 �1 1.97 2.32

1.98 � 0.04 2.26 � 0.053.97 � 0.022.021 2.190 85.1 90.0 �2.082 90.1 94.9

2 1.98 2.321.98–1.99 2.26–2.303.88–3.89 4.36–4.432.035 2.265 90.0 90.0 �

3 1.96 2.301.98 � 0.07 2.29 � 0.03

4.29 � 0.022.053 2.276 88.9 83.2 �

Fig. 2. Ab-initio stationary point geometries of Cr(VI)[Cr2O7]2�; e: [HCr2O7]�; f: [CrO3Cl]�.

2.088 2.322 92.0 95.8

Cr(III) oligomersAs anticipated by Niels Bjerrum (1908) in his masterpiece,

Cr(III) is seldom present in only the monomeric form and a vastnumber of polynuclear species may form in partially neutral-ized and sufficiently concentrated Cr(III) solutions (see Rai etal., 1987; and Ziemniak et al., 1998; for a thorough appraisal ofCr(III) solubility in aqueous solutions).

Structure and reactive properties of dimeric Cr(III) species havebeen the object of attentive studies. Some data concerning polynu-clear species are also available in the literature, although informa-tion is less detailed. In the dinuclear complexes two octahedrallycoordinated metal ions are bound together by one, two or three

mental observations are compared with results of ab- initio studies. FS

gyee) References Notes

13896 this study (gaseous ion) DFT B3LYP/6–31G**Magini (1980) XRD, aqueous, FSDiaz-Moreno et al. (1996) EXAFS, aqueous, FSDiaz-Moreno et al. (1996) EXAFS, aqueous,SS

26955 this study (gaseous ion) DFT B3LYP/6–31G**

Magini (1980) XRD, aqueous, FSDiaz-Moreno et al. (1996) EXAFS, aqueous, FSDiaz-Moreno et al. (1996) EXAFS, aqueous,SS

01766 this study (gaseous ion) DFT B3LYP/6–31G**Magini (1980) XRD, aqueous, FSDiaz-Moreno et al. (1996) EXAFS, aqueous,FSDiaz-Moreno et al. (1996) EXAFS, aqueous,SS

21137 this study (gaseous molecule) DFT B3LYP/6–31G**

s molecules. a: [CrO4]�; b: [HCrO4]�; c: [H2CrO4]0; d:

Experi

Ener(Hartr

1502.0

1886.3

2270.5

2654.5


hydroxide ions by sharing a corner, an edge or a face. The bestcharacterized Cr(III) species are represented by the green bridgedmono-hydroxylate [(H2O)5Cr(OH)Cr(H2O)5]5� and the bluebridged dihydroxylate [(H2O)4Cr(OH)2Cr(H2O)4]4� aqueouscomplexes (decaaquo-�-hydroxo-dichromium ion and octa-aquodi-�-hydroxo-dichromium ion, designated respectively asSBD and DBD hereafter; cf. Thompson and Connick, 1981). It isof interest to note that neither of the two complexes have beenisolated as crystalline salts (Springborg, 1988). In addition to SBDand DBD oligomers, trinuclear and tetranuclear aqueous com-plexes form as well in aged Cr(III) solutions by polymerizationequilibria involving the hexa-aqua-chromium(III) ion. Thetrinuclear species [Cr3(OH)4]5� (water ligands omitted for clarity;hereafter referred to as TBT [Tetrally-Bridged-Trimer]) occurapparently as a single isomer (although two isomeric geometriesare possible; see later on) while the tetranuclear complex[Cr4(OH)6]6� occurs in two forms which interconvert rapidly atroom temperature (Stünzi et al., 1989). Equilibrium betweenmonomer and DBD was described long ago by Bjerrum (1908) interms of the homogeneous reaction.

2Cr3� � 2H2O ⇔ Cr2(OH)24� � 2H� (5)

Thompson and Connick (1981) report a constant reaction en-thalpy for equilibrium (5) �H5 � 12.7 � 1.5 kcal/mol and astandard state (T � 298.15 K, P � 1 bar) entropy of reaction�S5

0 � 20 � 5 cal/(mol K) roughly consistent with theestimates of Bjerrum (1908) (15.4 kcal/mol and 15 cal/(mol K), respectively).

Finholt et al. (1981) investigated the homogeneous equilib-rium

3Cr3� � 4H2O ⇔ Cr3(OH)45� � 4H� (6)

and provided conditional constants in concentrated nitric solu-tions. Conventional stability constants of the hydrolytic dimer,trimer and tetramer in 1 M (NaClO4) solutions were laterproposed by Stünzi et al. (1989). According to Baes and Mes-mer (1976) logK5 � �5.06 and logK6 � �8.15. More recentestimates of Rai et al. (1987), while confirming the previousestimates for equilibrium (5), assign a lower reaction constantfor the formation of trimer (�5.0, �10.75, respectively).

The minimum energy structures of gaseous SBD and DBDobtained in this study are shown in Figure 1 and some featuresare summarized in Table 5. The flexible hydroxo bridge in

Table 5. Geometry of the investigated Cr(III) oligomers SBD � [([Cr3(OH)4(H2O)9]5�; TBT-linear � [Cr3(OH)4(H2O)10]5�. O* � oxygthe 6–31G** basis set.

Cluster

Interatomic distances (Å)

Cr-O* Cr-O# H-O* H-O#

SBD 2.011 2.116 0.974 0.9752.036 0.980

DBD 2.032 1.951 0.973 0.9712.046 1.978 0.977

TBT-et 2.015 1.933 0.975 0.9732.069 2.139 0.979 0.975

TBT-linear 2.027 1.945 0.972 0.9702.057 2.007 0.977 0.971

Cr(III) complexes does not impose strain and the first coordi-

nation spheres of the two metal centers are close to beingundistorted octahedra (Springborg, 1988).

As already noted in the introductory section of this study,relativistic electron effects were excluded for computationaleconomy. Energy contributions to the minimum energy gas-eous structure arising from the coupling of the spins of theelectrons on adjacent chromium atoms were thus neglected.Spin coupling effects could be present in the dinuclear com-plexes. It should be noted however that ESR spectra on Cr(III)dimers give rise to spectroscopic splitting factors g for SBDand DBD (1.976 and 1.916, respectively; Thompson and Con-nick, 1981) which are inconsistent with either the expectedspin-only value (2.003) or the g calculated in an octahedral fieldas indicated by Jørgensen (1962) (g � 1.962; see Thompsonand Connick, 1981). Concerning the trimeric species, two dif-ferent arrangements have been proposed for TBT: an equilat-eral triangle with the chromium atoms at the vertices or a lineararray of chromium atoms (Finholt et al., 1981; see their Fig. 1for details; see also Fig. 1 in this study). While the existence ofa TBT species was clear from the analysis of the existingabsorption spectra (Thompson and Connick, 1981; Finholt etal., 1981), preliminary attempts to discriminate the two TBTisomers by existing ESR spectra (Thompson and Connick,1981) and adoption of the Kambe (1950) procedure to definethe magnetic Hamiltonian and the ensuing magnetic suscepti-bility gave ambiguous results. Nevertheless, according toStünzi and Marty (1983) each chromium in the equilateraltriangle structure (TBT-et) is bound to other two by a total ofthree hydroxide bridges, and this gives a particular stability tothe cluster.

Our gas-phase calculations confirm that a stationary state isreadily attained in both TBT-et and TBT-linear conformation atDFT/6-31G** level. As in the case of SBD and DBD struc-tures, the flexible hydroxo bridge in Cr(III) complexes does notimpose strain and the first coordination spheres of the threemetal centers are close to being undistorted octahedra in bothTBT-et and TBT-linear configurations. Details of the TBT-etand TBT-linear gaseous structures are visible in Figure 1 andTable 5.

[CrO2]�

Chromium dioxide negative ion, [CrO2]�, has a relativelysimple, although not yet precisely defined, structure. The O-Cr-O bond angle determinations in 52Cr isotopomers vary from

r(OH)Cr(H2O)5]5� DBD � [(H2O)4Cr(OH)2Cr(H2O)4]4�; TBT-et �H2O; O# � hydroxyls oxygens. All calculations are at DFT level with

Bond anglesEnergy

(Hartree)O*-Cr-O# Cr-O#-Cr H-O*-H

88.2 161.4 106.2 �2850.65832295.7 107.092.8 105.6 106.4 �2927.00973898.8 105.7 107.888.3 97.7 105.9 �4122.658869

100.1 112.1 106.689.1 105.8 106.1 �4199.21908195.4 106.3 108.1

H2O)5Cens of

105° in Serebrennikov and Mal’tsev (1975) to 112° in Almond

ueous


and Hahne (1988). A more open bond angle of 128 � 5 hasbeen deduced by Chertihin et al. (quoted in Wenthold et al.,1997) on the basis of observed stretching frequencies in 53Crand 54Cr isotopomers. For the gaseous [CrO2]� ion the opti-mized geometry at DFT B3LYP/6-31G** level indicate a bondangle of 132°, in agreement with the estimates of Chertihin etal. and an interatomic Cr-O distance of 1.646 Å. The structureof the gaseous molecule is only slightly affected by the reactionfield. The interionic Cr-O distance in solvated [CrO2]� ob-tained with the IEFPCM procedure is identical to that of thegaseous ion with a somewhat larger bond angle (135°). Withthe COSMO approach the bond angle further widens to 137°and the bond distance increases to 1.662 Å.

Most studies of vibrational properties were focused on thevalue of the asymmetric stretching frequency �3 (Table 6) andrelatively fewer values are available for the symmetric stretchand bending modes �1 and �2 (Table 6). Scaled vibrationalfrequencies obtained in this study at DFT B3LYP/6-31G**level are in good agreement with the negative ion ultravioletphotoelectron spectroscopy determinations of Wenthold et al.(1997). The overall effect of the reaction field is to somewhatreduce the frequency of all the three normal modes (Table 6).

2.2. Optimized Geometries and Vibrational Properties ofCr(VI) Species

Aqueous complexes of hexavalent chromium have been theobject of intensive studies, mainly due to their toxicity and theirrole in mutagenesis and carcinogenesis (Connett and Wetter-hahn, 1985; Brauer and Wetterhahn, 1991; Mazurek et al.,1991). Monocromate complexes generally have a tetrahedralsymmetry and bichromate complexes are formed by distortedtetrahedra linked at one apex by a bridging oxygen (Fig. 2).

In acidic solutions with sufficient dissolved Cr(VI) polychro-mate complexes form, based on the following equilibria

Table 6. Vibrational frequencies of [CrO2]�. Experimental obse0.9620

ModeCalculated

(cm�1)Observed(cm�1) R

�1 895 � 20 Wenthold et960.1 Almond and914.4 Chertihin et a

895.4 this study (ga834.2 this study (aq850.4 this study (aq

�2 220 � 20 Wenthold et224.1 this study (ga220.4 this study (aq224.1 this study (aq

�3 978 Serebrenniko971 Darling et al.971 Poliakoff et a971 Almond and969.8 Almond and965.4 Chertihin et a964 Almond et al

950.3 this study (ga877.9 this study (aq905.2 this study (aq

(Cieslak-Golonka, 1991):

3Cr2O72� � 2H� ⇔ 2Cr3O10

2� � H2O (7)

4Cr3O102� � 2H� ⇔ 3Cr4O13

2� � H2O (8)

A comparison of Cr-O interionic bond lengths in Cr(VI) aque-ous complexes lead to the following generalizations (Pressprichet al., 1988).

a) terminal bondings and internal bondings of tetrahedral com-plexes (types T and I, respectively, Pressprich et al., 1988)increase smoothly with the degree of polymerisation;

b) non-bridging bondings (type N, Pressprich et al., 1988)show the opposite trend.

The formation of Cr(VI) oligomers has been described byBrito et al. (1977) as due to equilibria of type:

pH� � qCrO4� ⇔ Hp(CrO4)q

(p�2q) (9)

with p,q integers.

[Hn(CrO4)](2�n)� n � 0, 1, 2The chromate ion (n � 0) is probably one of the most

investigated among tetraoxo-metal complexes. These investi-gations endeavor to define molecular structures, geometries andvibrational modes of ground states, both in neutral species andcomplex ion, and of excited states as well. Experimental datameasurements of interatomic Cr-O distances range from 1.65 Åto 1.66 Å (Table 7).

According to Hoffmann et al. (2001a, 2001b) the -CrO3

structural unit is nearly identical in bichromate and dichromatestructures, particularly in terms of Cr to terminal O bonddistances, and this results in similar vibrational attitudes inrelation to the asymmetric stretching of the structural unit(Tables 7 and 8). Due to the importance of this complex as aligand, a different set of computational data are available in theliterature, obtained by different ab initio procedures and with

are compared with results of ab-initio studies. Scaling factor �

ces Notes

7) UV Photoeletron Spectroscopy(1988) Matrix IR7) Matrix IRon) DFT B3LYP/6–31G** (scaled)species) DFT B3LYP/6–31G** � COSMO (scaled)species) DFT B3LYP/6–31G** � IEFPCM (scaled)7) UV Photoeletron Spectroscopyon) DFT B3LYP/6–31G** (scaled)species) DFT B3LYP/6–31G** � COSMO (scaled)species) DFT B3LYP/6–31G** � IEFPCM (scaled)

al’tsev (1975) Matrix IRMatrix IR

2) Matrix IR(1988) Matrix IR(1988) Matrix IR7) Matrix IR) Matrix IRon) DFT B3LYP/6–31G** (scaled)species) DFT B3LYP/6–31G** � COSMO (scaled)species) DFT B3LYP/6–31G** � IEFPCM (scaled)

rvations

eferen

al. (199Hahnel. (199seous iueousueousal. (199seous iueousueousv and M(1974)l. (198DownsHahnel. (199. (1985seous iueous

more or less extended basis sets. The reference experimental


data pertain to different states (single crystals, powders, aque-ous and gas-phase) posing some comparison problems.

In Brito et al. (1997) the Cr-O bond length is the shortestwith respect to all bond lengths in the datasets and is indicativeof a precise tetrahedral symmetry. This, as noted also by Dobbsand Hehre (1987a, 1987b), is probably a computational artifactdue to neglect of correlation effects.

Interelectronic effects were analyzed by Stückl at al. (1997),who also investigated the electronic structure of excited statesof [CrO4]2�. Results of different DFT methods (Table 7) are ingood agreement with experimental data, using the harmonicapproximation.

The chromate ion is one of the chromium oxo-anions com-

Table 7. Geometry of [Hn(CrO4)](2�n)� clusters. Experimental observvalence with pseudopotentials. EDZV � extended double zeta valence.comparative purposes (values in brackets).

n

Interatomic distances Bond angles

Cr-OCr-OH(Cr-Cl) O-H O-Cr-O

Cr-O-H(Cr-O-Cl

0 1.66 Td

1.6601.651.651.656 � 0.003

Td

1.591 Td

1.604 Td

1.661.681.661.661.6611.7021.6841.6621.667

Td

1.651 Td

1.635 Td

1 1.624 1.82 � 0.021.513 1.853 106.57 119.5

111.931.537 1.7941.603 1.825 0.967 106.8 106.4

1.598 111.71.599 1.787 0.981 106.2 113.9

1.597 110.8(1.593) (2.246) C3v (107.8)

(1.596) (2.202) C3v (107.8)

2 1.434 108.69 133.511.745 110.661.560 0.970 107.9 113.9

1.751 111.4

puted by Bell and Dines (2000) with a variety of basis sets

(split valence or none) and methods (HF-SCF, MP2 and DFT).Their main datum of reference for the Cr-O bond length wasthe crystallographic determination of dCr-O in calcium chro-mate of Weber and Range (quoted in Bell and Dines, 2000)(dCr-O � 1.647 Å). According to the comparative estimates ofBell and Dines (2000) it is apparent that calculations at HFlevel yield a rather short bond length, averaging 1.6 Å, whileMP2 calculations yield excessively large bond distances, typ-ically around 1.75 Å (cf. Table 1 in Bell and Dines, 2000). Thebest results, either in terms of geometry and calculated vs.observed vibrational frequencies are those achieved by SCF-DFT methods in conjunction with the LanL2DZ basis set.

Bridgeman and Cavigliasso (2001) investigated a general

re compared with results of ab-initio studies. DZV/ECP � double-zetaometry of [CrO3Cl]� gaseous and aqueous clusters is also reported for

EnergyHartree) References Notes

Dobbs and Hehre (1987a) ExperimentalKálmán (1971) ExperimentalKrebs and Hasse (1976) ExperimentalPressprich et al. (1988) ExperimentalHoffmann et al. (2001b) Experimental

329.9148 Connor et al. (1972) EDZV309.0614 Brito et al. (1997) ECP/DVZ

Dobbs and Hehre (1987a) 3–21GBridgeman and Cavigliasso (2001) LDA/FCTZBridgeman and Cavigliasso (2001) P86/FCTZBridgeman and Cavigliasso (2001) 3LYP/CEPBridgeman and Cavigliasso (2001) 3LYP/DVZStückl et al. (1997) LDAStückl et al. (1997) GGA (Ex)Stückl et al. (1997) GGA (Exc)Stückl et al. (1997) Harm. Approx.Bell and Dines (2000) DFT LanL2DZ

327.9192 Connor et al. (1972) Minimum STO345.29566 this study (gaseous ion) DFT B3LYP/6–

31G**345.68884 this study (solvated ion) DFT B3LYP/6–

31G** �COSMO

Hoffmann et al. (2001b) Experimental309.81 Brito et al. (1997) ECP/DZV

Martin-Zarza et al. (1997) RHF 3–21G*346.05260 this study (gaseous ion) DFT B3LYP/6–

31G**

346.13692 this study (solvated ion) DFT B3LYP/6–31G** �IEFPC

730.46217 this study (gaseous ion) DFT B3LYP/6–31G**

730.55486 this study (solvated ion) DFT B3LYP/6–31G** �COSMO

310.3292 Brito et al. (1997) ECP/DZV

346.58125 this study (gaseous molecule) DFT B3LYP/6–31G**

ations aThe ge

()

�1�

�1�1

�1

�

�1

�1

�1

�1

�

�1

class of oxide and oxoanions of MO4 stoichiometry, including

udy (aq


Cr(III) among metals, and computed their gas-phase electronicproperties. Their results are in good agreement with experimen-tal measurements in crystalline materials. The Cr-O interatomicdistance obtained in this study is in good agreement with theexperimental observations of Krebs and Hasse (1976) andPressprich et al. (1988) and a perfect tetrahedral symmetry(unconstrained) is attained. Successive protonation of [CrO4]2�

destroy the Td symmetry with a shortening of Cr-O distances in[HCrO4]� and more complex effects in [H2CrO4]0 (see Table7 and Fig. 2).

Raman and infrared spectroscopy seem to confirm that[CrO4]2� in aqueous solution has tetrahedral symmetry (Td)with four main vibrational modes.

Table 8 lists calculated vibrational frequencies of gaseous[CrO4]2� and experimentally observed ones (under differentcondensation states) according to various investigations. Toallow a better comparison with experimental observations com-putational frequencies obtained in this study are listed in a

Table 8. Vibrational frequencies of CrO42�. Experimental observatio

�4 � T2. Scaling factor � 0.9620.

ModeCalculated

(cm�1)Observed(cm�1)

�1 898–903–923 Mülle855 Mülle846 Kiefer844 Heyns847 Ramse846 Mülle

857 Stückl853 Bell a870 this st863 this st892 this st

�2 351 Mülle349 Kiefer348 Ramse347 Mülle

330 Stückl333 Bell a335 this st335 this st

341–344 this st�3 908 Mülle

871 Mülle890 Kiefer880 Hoffm884 Ramse891 Mülle


899–901–901 this st�4 378–380–388 Mülle

398 Mülle378 Kiefer368 Ramse368 Mülle


373–378–403 this st

scaled notation (see Scott and Radom, 1996).

The isotope fine structure B2 �(Cr-O) in K2CrO4 (D2d sym-metry) calculated by Beattie et al. (1982) who adopted a forceconstant FB2 � 4.87 mdyn Å�1 and 2� � 96°16= is incomplete agreement with infrared band assignments made bythe same authors and emphasizes the role of the mass of thecentral cation on the vibrational mode (50Cr � 901.2 cm�1;52Cr � 894.9 cm�1; 53Cr � 891.9 cm�1; 54Cr � 889.0 cm�1).Our scaled frequencies for degenerate �4 vibrations are 50Cr �910.2 cm�1; 52Cr � 903.7 cm�1; 53Cr � 900.9 cm�1; 54Cr �898.0 cm�1. In terms of the separative effect (see later), theexpected isotopic fractionation within the K2CrO4 structure,based on experimental infrared band assignments, is 0.01394per a.m.u. The one calculated in this study for gaseous[CrO4]2� is 0.01391 per a.m.u., thus virtually identical, withinuncertainties to that of its (matrix-isolation) crystalline analog.It should be noted however, that the nature of the associatedcounter-cation in crystalline chromates affects sensibly thevibrational properties of the [CrO ]2� unit (see the extensive

compared with results of ab-initio studies. �1 � Å1, �2 � E, �3 � T2

rences Notes

(1974) 53CrO42� in K2SO4

(1976) K2CrO4

ernstein (1972) solution, K� salt1999) solution, K� salt. (2001) solution(1974) solution1997) LDA/TZPs (2000) DFT LanL2DZseous ion) DFT B3LYP/6–31G** (scaled)ueous ion) DFT B3LYP/6–31G** � Onsager (scaled)ueous ion) DFT B3LYP/6–31G** � IEFPCM (scaled)(1976) K2CrO4

ernstein (1972) solution, K� salt. (2001) solution(1974) solution1997) LDA/TZPs (2000) DFT LanL2DZseous ion) DFT B3LYP/6–31G** (scaled)ueous ion) DFT B3LYP/6–31G** � Onsager (scaled)ueous ion) DFT B3LYP/6–31G** � IEFPCM (scaled)(1974) 53CrO4

2� in K2SO4

(1976) K2CrO4

ernstein (1972) solution, K� saltal. (2001b) solution. (2001) solution(1974) solution1997) LDA/TZPs (2000) DFT LanL2DZseous ion) DFT B3LYP/6–31G** (scaled)ueous ion) DFT B3LYP/6–31G** � Onsager (scaled)ueous ion) DFT B3LYP/6–31G** � IEFPCM (scaled)(1974) 53CrO4

2� in Cs2SO4

(1976) K2CrO4

ernstein (1972) solution, K� salt. (2001) solution(1974) solution1997) LDA/TZPs (2000) DFT LanL2DZseous ion) DFT B3LYP/6–31G** (scaled)ueous ion) DFT B3LYP/6–31G** � Onsager (scaled)ueous ion) DFT B3LYP/6–31G** � IEFPCM (scaled)

ns are

Refe

r et al.r et al.and Bet al. (y et al

r et al.et al. (

nd Dineudy (gaudy (aqudy (aqr et al.and By et al

r et al.et al. (

nd Dineudy (gaudy (aqudy (aqr et al.r et al.and Bann ety et al

r et al.et al. (

nd Dineudy (gaudy (aqudy (aqr et al.r et al.and By et al

r et al.et al. (

nd Dineudy (gaudy (aq

4

data tabulation in Müller et al., 1976). Comparing the gaseous

Tab

le9.

Geo

met

ryof

grou

ndst

ate

[Hn(C

r 2O

7)]

(2�

n)�

clus

ters

.E

xper

imen

tal

obse

rvat

ions

are

com

pare

dw

ithre

sults

ofab

-ini

tiost

udie

s.

Inte

rato

mic

dist

ance

s(Å

)B

ond

angl

esE

nerg

y(H

artr

ee)

Ref

eren

ces

Not

esC

r-O

TC

r-O

CC

r-O

HO

-Cr-

OC

r-O

-HC

r-O

-Cr

1.62

4�

0.00

31.

824

�0.

0212

3.5

Hof

fman

net

al.

(200

1b)

Exp

erim

enta

l;E

XA

FS1.

56–1

.64

1.76

–1.8

112

2–13

9M

artin

-Zar

zaet

al(1

995)

Exp

erim

enta

l1.

610

1.76

810

9.4–

109.

618

0�

2615

.590

61th

isst

udy

(gas

eous

)D

FTB

3LY

P/6–

31G

**1.

485

1.91

21.

795

106.

5012

3.1

162.

6B

rito

etal

.(1

997)

EC

P/D

ZV

1.62

811

2.00

1.56

91.

681

1.78

010

6.2

109.

914

0.3

�26

16.2

448

this

stud

yD

FTB

3LY

P/6–

31G

**1.

595

1.87

211

3.0


state vibrations of chromate with the solid state behavior maybe thus somewhat misleading. The effect of the reaction field isto eliminate the degeneracy of E and T2 modes (Td 3 C3v

symmetry lowering effects; cf. Cieslak-Golonka, 1991), butthis effect is obviously not detected at the Onsager level. Thereaction-field-induced degeneracy elimination at IEFPCM levelis however quite evident, and frequency shifts between nondegenerate modes are similar to what was observed experimen-tally by Müller et al. (1974) for the motions of 50CrO4

2� and53CrO4

2� units in crystalline sulphate hosts (cf. Table 8).

[CrO3Cl]�

The existence of Trioxychlorochromate(VI) was confirmedby Palmer et al. (1987) by potentiometric investigation of theaqueous homogeneous equilibrium:

CrO42� � 2H� � Cl� ⇔ CrO3Cl� � H2O (10)

after preliminary spectrophotometric observations by Cohenand Westheimer (1952) in Cl bearing acetic acid solutions(86.5%) and Haight et al. (1964) in H2O-HCl solutions. Ac-cording to Palmer et al. (1987) Trioxychlorochromate(VI) ap-parently forms preferentially in NaCl brines at high T and lowpH, where it partially replaces the bichromate ion. Our gas-phase computation indicate that the molecule is an asymmetrictop of C3v symmetry (three symmetry planes along the rota-tional axis). Cr-O distances are not much affected by substitu-tion of one Cl� for one O2� (Table 7). The reaction field doesnot modify the spatial group.

[Hn(Cr2O7)](2�n)� n � 0, 1Experimental observations on dichromate complexes are

scanty. Results of EXAFS investigations of Cr aqueous solu-tions at various T and pH conditions indicate that Cr to terminalO distances are similar to those of bichromate (1.624 � 0.003A) and the Cr to bridging oxygen distances are much larger(1.82 � 0.02 A cf. Table 9). According to Hoffmann et al.(2001b) the two -CrO3 units in the aqueous cluster are notcoaxial but entail a Cr-O-Cr angle of 123.5°. Results of ourgas-phase calculations indicate a more symmetric gaseous mol-ecule with three identical bond lengths for chromium linked toterminal oxygens (OT) and two identical (and longer) lengthsfor chromium linked to the central oxygen (OC).

Our results for [HCr2O7]� are consistent with the geometryobtained by Brito et al. (1997) with the double valence set andan effective core potential in terms of O-Cr-O bond angles, butthere is some discrepancy in the way double protonation affectsthe relative geometry of the two CrO4 monomeric units (seeTable 9 and Fig. 2). Various attempts to converge to a station-ary point for H2Cr2O7 gave inconsistent results at DFT B3LYP/6-31G** level, indicating that the gaseous molecule is eventu-ally unstable. Due to the limited geochemical interest for thiscomplex we did not explore alternative ab initio procedures.

2.3. Cr-Fractionation

The theory of isotopic fractionation is extensively treated inthe geochemical literature and need not be repeated here. Werefer the interested reader to the fundamental works of Urey(1947) and Bigeleisen and Mayer (1947). Recent appraisals
may also be found in Driesner et al. (2000) and Schauble et al. n 0 1


(2004). Here we simply recall that, because the potential energyof isotopomers is similar, one may define a separative effectbased on the ratio of the partition functions of heavy and lightisotopomers (Q●, Q� respectively) in such a way that thetranslational and rotational contributions cancel out:

�s●

s��f � �s●

s��Q●

Q��m�

m●�3⁄2

� �i

Xi●

Xi�

exp (�Xi● ⁄ 2) ⁄ �1 � exp (�Xi

�)�exp (�Xi

�) ⁄ 2) ⁄ �1 � exp (�Xi●)�

(11)

Xi● �

h�i●

kT; Xi

� �h�i

�

kTin equation (11) are undimensionalized

frequencies of heavy and light isotopomer, respectively (with h� Planck’s constant and k � Boltzmann’s constant). Thesymmetry numbers ratio s●/s� in (11) reflects the probability offorming symmetric or asymmetric molecules and m●, m� arethe masses of molecules undergoing isotopic exchange. Whenhomogeneous redox reactions of type (1) take place amongCr(VI) and Cr(III) species, symmetry factors and relativemasses cancel out and isotopic fractionation factors are readilyobtained applying:

1000 ln � � �aA � aB� � �bA � bB� � 106 ⁄ T2

� �cA � cB� � 1012 ⁄ T4 (12)

where aA, aB, bA, bB and cA, cB coefficients are obtained bypolynomial interpolation over reciprocal temperature (withinan appropriate T-range) of separative effects computed at var-ious T for heavy and light isotopomers of A, B molecules(Table 10).

In Figure 3 we see an exemplified application of equation(12) to the most emblematic redox equilibrium, i.e., that onerepresented by equation (1). The solid lines encompassing thevarious points for the 6 distinct isotopic couples in Figure 3 arethe result of the adopted interpolation. Comparing the fraction-ation factors for the 52Cr/50Cr and 54Cr/52Cr isotopomers wemay observe that the fractionation factor per a.m.u. is notconstant over the inspected isotopic mass range. This is basi-cally the reason why all the 6 possible fractionation coupleshave been listed in Table 10. Looking at the 53Cr/52Cr frac-tionation we may note that the gas-phase separative effectscomputed at 0°C for [Cr(H2O)18]3� and [CrO4]2� are 7.9 and15.9, respectively. These are in substantial agreement with theresults of Schauble et al. (2004) (8.0 and 15.5, respectively).

3. DISCUSSION

For clarity we subdivide the discussion into two main themesdealing with distinct aspects of this investigations, with somegeneral remarks.

3.1. Isotopic Fractionation in the Gaseous State

When substituting a D atom for a H atom in a diatomicmolecule the observed shift in the single normal mode isproportional to the inverse of the square root of the ratio ofmasses (i.e., 1/2). This is basically due to the fact that thepotential energy function and the configuration of the molecule
are affected in a negligible way by the isotopic substitution.
The way the vibrational frequencies of polyatomic isotopicmolecules are affected by the mass of the substituted isotopesis understood as a generalization of what one observes at thelevel of a simple diatomic molecule. The explicit account ofpotential energy and kinetic energy effects in terms of externalcoordinates (i.e., linear combinations of Cartesian displace-ments) leads to the following “product rule” (Bright Wilson etal., 1955) for a non linear molecule:

�i�1

3N�6 �i●

�i�

� �i�1

3N �m�

m●�1⁄2�M●

M��3⁄2� Ix●Iy

●Iz●

Ix�Iy

�Iz��1⁄2

(13)

where M are the bulk masses of the isotopic molecules andIx,y,z are the momenta of inertia with respect to the x,y and zprincipal axes. The second term in the product on the right ofequation (13) accounts for the effect of translational motions,and the third one accounts for the rotational contributions, i.e.:

�T●

�T�

� �M�

M●�1⁄2

(14)

�R●

�R�

� � I�

I●�1⁄2

(15)

Although translational and rotational terms are not normalvibrational modes, they nevertheless appear in the product ruleequation when considering the result of applying weak forceswhich convert the motions of translation and rotation intooscillatory motions of low frequency (of which equations (14)and (15) represent the limit of vanishing forces; see BrightWilson et al. (1955) for a detailed treatment). In the light ofequation (13) we cannot expect to observe a simple inverseproportionality between frequencies of isotopic molecules andmasses of substituted isotopes. Nevertheless, since the mo-menta of inertia are not affected by significant changes ofsymmetry and potential energy function, their bulk effect onisotopic frequency shifts may be expected to be represented bya simple functional form.

Let us consider, as an example, hexa-aqua-chromium. The[Cr(H2O)6]3� gaseous cluster, taking the hydrogens into ac-count, has Th symmetry, and the irreducible representation ofvibrational modes is �vib � 3ag � au � 3eg � eu � 5fg � 8fu.In Figure 4 the frequency ratios computed for the gaseousmolecule [Cr(H2O)6]3� are plotted against the reciprocal of thesquare root of the masses of the substituted isotopes. Thefrequency ratios are, in this diagram, grouped into four classes,corresponding to 3 asymmetric (infrared active) wagging H-O-H motions, 3 asymmetric O-Cr-O displacements, 3 asym-metric wagging H-O-H plus 3 asymmetric rocking H-O-Hmotions. Altogether, these modes constitute one half of the 8irreducible fu modes (4fu). The last class groups all ineffective(from an isotopic fractionation point of view) normal modes,most of which belongs to H-O-H and O-H high frequencies.

It is thus apparent that the way the vibrational motions of themolecule are affected by the mass of the central cation may betranslated into a single parameter that we would like to identifyas an “isotopic ergodicity factor” with the form:

�i● mi

● 1⁄2

�i�

� a � (1 � a)�mi�� (16)

4


The vibrational modes with the lowest ergodicity factor arethose which are more affected by the isotopic substitution whilethose with an ergodicity factor 1 are virtually unaffected by theisotopic substitution.

In Figure 5 we see that a molecule with lower symmetry suchas [HCrO ]� (a symmetric top with variable interionic dis-

Table 10. Coefficients of the T-dependent sepa

Species Couples a b10�6 c10�12

(1) 52Cr/50Cr 0.0128 1.2701 �0.009453Cr/50Cr 0.0193 1.8742 �0.013954Cr/50Cr 0.0249 2.4510 �0.018153Cr/52Cr 0.0065 0.6041 �0.004554Cr/52Cr 0.0123 1.1808 �0.008754Cr/53Cr 0.0055 0.5767 �0.0042









(1) � [Cr(H2O)6]3�; (2) � [Cr(H2O)5(OH)]2�; (3) � [Cr(H2O)4(OH(7) � [Cr(H2O)3Cl3]0; (8) � [(H2O)5Cr(OH)Cr(H2O)5]5�; (9) � [(H2O(11) � [CrO2]�; (12) � CrO4

2�; (13) � [HCrO4]�; (14) � [H2C[Cr(H2O)18]3�

4

tances for the protonated and unprotonated oxygens) has a

more complex isotopic ergodicity diagram, where virtuallyeach normal mode has a distinct ergodicity factor. The mainutility of this type of plot is its ability to discriminate in asimple fashion between the influence of the adopted ab initioprocedure or wave expansion in the resulting isotopic fraction-ation. We may thus see that substituting plane-polarized wave

ffects for the investigated gaseous molecules.

Species Couples a b10�6 c10�12










) � [Cr(H2O)3(OH)3]0; (5) � [Cr(H2O)5Cl]2�; (6) � [Cr(H2O)4Cl2]�;H)2Cr(H2O)4]4�; (10) � [(H2O)4Cr(OH)2 Cr(OH)2(H2O)Cr(H2O)4]5�;(15) � [Cr2O7]2�; (16) � [HCr2O7]�; (17) � [CrO3Cl]�; (18) �

rative e

)2]�; (4)4Cr(OrO ]0;

functions 6-31G** with 3-21G expansions leads, at DFT level,

Solid li


to a decrement of the isotopic ergodicity factors for the frac-tionation-determining vibrational modes (open symbols in Fig.4) and that an opposite effect is obtained by the simple UHF

Fig. 3. Fractionation factors for redox reaction 1.

Fig. 4. Isotopic ergodicity diagram for [Cr(H2O)6]3�, rel
root of substituted masses. Filled symbols: DFT/6-31G** Ultrafinpruning.
computation with plane-polarized wave functions (open sym-bols in Fig. 5). Finally, the adoption of an insufficient gridpruning results in an unwarranted complexity of the isotopic

nes are the computed interpolants of type (12).

e ratio of vibrational modes to the reciprocal of the square
ating th e grid pruning. Open symbols: DFT/3-21G Ultrafine grid

ltrafine


ergodicity diagram, due to non-linearly mass-dependent mo-menta of inertia (not shown). These results suggest that extremecaution should be used in assessing ab initio equilibrium frac-tionation factors between coexisting molecules that are basedon heterogeneous theories and different wave expansions.

A glance at the computed separative effects of the variousinvestigated molecules shows the effect of the force constant ofthe central cation on the isotopic substitution. Gaseous com-plexes of Cr(III) exhibit more or less the same separativeeffects, independently of the geometry of the investigated clus-ter, and so do the gaseous complexes of Cr(VI) (Table 10). Dueto this fact, the various investigated redox reactions lead toessentially similar isotopic fractionations. By examining theredox equilibrium (1), which is rather important in a wide rangeof Eh-pH conditions, we may observe that the Mulliken pop-ulation analysis indicates �1.495 charges localized on thecentral Cr in the [Cr(H2O)6]3� cluster and �0.821 chargeslocalized on the six oxygen first neighbors. In [HCrO4]� theresidual charges on the central cation are reduced to �1.116and those on the 4 oxygen first neighbors are in the range�0.688 to �0.579. The degree of covalence of the Cr-O bondin [HCrO4]� is thus effectively much higher with respect to[Cr(H2O)6]3�, and the ensuing increase in the force constant(also indicated by the substantial shortenings of interatomicdistances; see Tables 1 and 7) leads to an enrichment of theheavy isotope in the tetrahedral Cr(VI) complex with respect tothe Cr(III) molecule.

3.2. Effects of the Reaction Field

Although we have already seen that the energy-structure ofthe investigated gaseous molecules is not severely affected bythe reaction field in aqueous solutions, the slight differencesencountered in applying theories at different levels of sophis-

Fig. 5. Isotopic ergodicity diagram for [HCrO4]�, relatroot of substituted masses. Filled symbols: DFT/6-31G** Upruning (Ottonello and Vetuschi Zuccolini, 2001).

tication to the assessment of vibrational properties warrants

some clarification. As noted by Cramer and Truhlar (1999), themain concern in equilibrium solvation approaches is the highfrequency solute modes where the solvent motion may be “tooslow to track the solute motion.” Besides the “explicit solva-tion” technique, consisting of assigning appropriate structuralconformations to the various solvation shells, the simplesttheory one can apply in depicting solvation effect is the On-sager procedure, in which the solute is assumed to occupy afixed spherical cavity of radius a0 within the solvent field. TheOnsager procedure evaluates solely the mutual effects of thesolute dipole on the solvent structure and of the solvent dipoleon the solute. The intrinsic limit of the Onsager method is thatsymmetric molecules having a null dipole moment will notexhibit reaction field effects on the molecular structure and itsvibrational properties. The Onsager procedure is superseded bythe Polarized Continuum Models (PCM), developed primarilyby the Pisa group led by Tomasi (Tomasi and Persico, 1994),which constrain the Self Consistent Reaction Field formalismthrough boundary conditions determined by apparent surfacecharges on the solute cavity. With the Conductor-like Screen-ing Model (COSMO) of Klamt and Schüürmann (1993) asolvent accessible surface area (SASA) is first modeled. TheSASA is generated by approximating a hypersurface, generatedby superposition of atom-centered spheres, with a meshedpolyhedron. Each of the simple elements of this hypersurface isused to compute the screening charge. The radius of the spheresaround atoms generally corresponds to the van der Waalsradius (if known), plus a quantity of �0.5 Å representing halfof the first shell of water molecules surrounding the centralatom/ion. For Cr, in its different oxidation states, this radius isunknown and must be based on geometries of the gaseousclusters. In the case of hydrated and chlorinated trivalent chro-mium the solvent around the cation is modeled by mixing

ratio of vibrational modes to the reciprocal of the squaregrid pruning. Open symbols: UHF/6-31G* Ultrafine grid

ing the

cybotactic effects (those due to the first solvation shell and

l.


described by explicit water molecules) and by implicitly de-fined bulk solvent effects. In IEFPCM the PCM equations arerecast in an Integral Equation Formalism of general validity forboth isotropic and anisotropic dielectrics (Cancès et al., 1997;Cossi et al., 1998). Again the solute molecules are embedded incavities of molecular shape surrounded by a dielectric contin-uum, whose polarization is represented by point charges local-ized on discrete “tesserae” covering the cavity surface. Herehowever the atomic radii of spheres are optimised throughenergy parameterization on several test molecules.

Our gas-phase investigation delineates a strong controloperated by the oxidation state of the central cation and,subordinately, by the strength of the first-ligand (OH2, O2�,OH�, Cl�). Investigating in detail the effect of the reactionfield (2nd and higher order shells) on a few representativemolecules ([Cr(H2O)6]3�; [CrO2]�; [CrO4]2�; [HCrO4]�;[CrO3Cl]�) one achieves a satisfactory appraisal on solva-

Fig. 6. Fractionation factors of the heterogeneous reaVibrational properties computed at B3LYP/6-31G** leve

3�
Fig. 7. Isotopic ergodicity diagram for [Cr(H2O)6] , relating throot of substituted masses at B3LYP/6-31G** level. Filled symb
tion effects on all possible Cr-bearing moieties in the systemCr-O-H-Cl and may conform in this way the isotopic frac-tionation surface in the pH-Eh space of intrinsic stability ofaqueous solutions. The effect of the reaction field on thevibrational properties of the solvated molecules is best rep-resented in terms of few heterogeneous isotopic exchangereactions:

H�CrO4 aq� � H●CrO4 gas

� ⇔ H●CrO4 aq� � H�CrO4 gas

� (17)

�Cr(H2O)6 aq3� � ●Cr(H2O)6 gas

3� ⇔ ●Cr(H2O)6 aq3� � �Cr(H2O)6 gas

3�

(18)

�CrO2 aq� � ●CrO2 gas

� ⇔ ●CrO2 aq� � �CrO2 gas

� (19)

�CrO3Claq� � ●CrO3Clgas

� ⇔ ●CrO3Claq� � �CrO3Clgas

� (20)

Moreover, to better appreciate the effects of the different hy-

etween gaseous and aqueous Cr(H2O)63� isotopomers.
ction b
e ratio of vibrational modes to the reciprocal of the squareols: gaseous ion; Open symbols: solvated ion.

of thented by


dration shells on the vibrational properties of [Cr(H2O)6]3�, thefollowing homogeneous equilibrium is also relevant:

�Cr(H2O)18 gas3� � ●Cr(H2O)6 gas

3� ⇔●Cr(H2O)18 gas

3� � �Cr(H2O)6 gas3� (21)

with [Cr(H2O)18]3� representing an explicit (2nd shell) solva-tion state of hexa-aqua-chromium.

Since the B3LYP/6-31G** IEFPCM and COSMO compu-tations are plagued by convergence problems (not perceived atvariational level but solely at the vibrational analysis), weattempted a simpler computational approach avoiding diffusefunctions and performed an explicit solvation assuming theinitial Cr[6-12] structure already investigated by Martinez et al.(2000). The interaction energy between the Cr3� hexahydrateand its hydration shell, obtained from the Harthree-Fock ener-gies of the interacting molecules:

Eint � ECr(H2O)18 gas3� � ECr(H2O)6 gas

3� � 12 � E(H2O)gas0 (22)

(�329.5 kcal/mol) is analogous to that one obtained by Mar-tinez et al. (2000) with the Møller-Plesset 2nd order perturba-tion theory (�328.9 kcal/mol) and it represents 73% of theentire [Cr(H2O)6]3� solute – H2O solvent interaction computedby the IEFPCM (�456.93 kcal/mol electrostatic � 5.51 kcal/mol non-electrostatic contributions). Coupling between the firstand second solvation shell seems to account for a good part ofthe entire reaction field not only in terms of solvation energy

Fig. 8. Isotopic fractionation factors induced by solvatiothe reaction field are limited to dipolar interaction, in part ain part b. Interpolating functions are satisfactorily represe

but also in terms of vibrational motions. As a matter of fact we

may see in Figure 6 that the gaseous molecule [Cr(H2O)18]3�,exhibits with respect to [Cr(H2O)6]3� the same sort of vibra-tional behavior in terms of isotopic fractionation when com-pared to the reaction field (i.e., Eqn. 21 and 18 are analogous).The bulk effect of the reaction field on the relative stability ofthe various [Cr(H2O)6]3� isotopomers is to enhance the sepa-rative effects. In terms of fractionation factors (Eqn. 18 and 21)the maximum enhancement at the limiting T (0°C) is approx-imately � 0.5 per a.m.u.

This overall enhancement is due to a decoupling of waggingand rocking H-O-H vibrational motions, and to several centralmass-dependent O-Cr-O bending motions which contribute in amore substantial way to the overall separative effect of thesolvated molecule (Fig. 7).

The fractionation induced by the heterogeneous equilibrium(17) is of opposite sign with respect to that one induced byequilibrium (18), attaining a value of roughly �0.4 per a.m.u.at the limiting T of 0°C. The simple dipolar interaction is itselfresponsible for half of the bulk effect of the reaction field inreducing the separative effects of the various [HCrO4]� isoto-pomers (cf. Figs. 8a, 8b).

Finally, Figure 9 summarizes the effects of the reaction fieldon the 53Cr/52Cr fractionation for the various investigated mol-ecules. First, we may note that both the COSMO procedure ofKlamt and Schüürmann (1993) and Tomasi’s IEFPCM (Cancèset al., 1997; Cossi et al., 1998) give quite consistent results for[CrO4]2� at B3LYP/6-31G** level, and that explicit (B3LYP/

ts on the [HCrO4]� molecule. Non-electrostatic effects offigure, while they include cavitation end repulsion effectsequation (24).

n effec

6-31G**) and implicit (B3LYP/3-21G) solvation of the

levant


[Cr(H2O)6]3� result in nearly identical solvent effects. Onemay also observe that OH2, O2�, OH�, and Cl� ligands havequite distinct coupling attitudes with the solvent motions. In[Cr(H2O)6]3� where relatively strong hydrogen bonds areformed between terminal H atoms of the first shell and thesecond shell water molecules (Bleuzen et al., 1996), couplingof vibrational motions results in augmented fractionation ef-fects, since the second shell molecules must vibrate accordingto characteristic solvent motion. When the charge of the centralcation (distributed notation state) is higher and the first neigh-bor is a progressively weaker ligand, as in [CrO4]2� [HCrO4]�,covalent bonding with the central cation prevails and the sol-vent effect is much reduced or opposite in sign. The case of[CrO2]� is quite peculiar. Although the conformation of soluteand solvent molecules is similar (both are C2v molecules withthree non degenerate motions), their vibrational frequenciesspan completely distinct ranges. While the overall effect of thereaction field is to lower to some exent the vibrational frequen-cies of the solvated molecule with respect to its gaseous coun-terpart, the effect of the central mass is complex. Both asym-metric and symmetric stretching frequencies shifts appear toenhance the isotopic fractionation (lower isotopic ergodicityfactor of the stretching motions of the solvated molecule withrespect to its gaseous counterpart) while the mass-dependentshifts of the symmetric bending motion have an opposite effect(higher isotopic ergodicity factor of the bending motion of thesolvated molecule). Although the distributed notation state ofthe central cation is identical, the net result of the reaction fieldin terms of Cr isotopic fractionation is thus for [CrO2]� oppo-site to what observed for hexa-aqua-chromium, inducing in thisway a sensible perturbation of the fractionation surface in thefO2-pH region of predominance of Cr(III) species (see later).

Consider now equilibria (1), (17) and (18). It is quite evident

Fig. 9. 53Cr/52Cr gas/water fractionation of some reniques.

that adding equilibrium (1) and equilibrium (17) and then

subtracting equilibrium (18) we obtain the equivalent homoge-neous isotopic exchange reaction in the aqueous phase, i.e.:

H�CrO4 aq� � ●Cr(H2O)6 aq

3� ⇔ H●CrO4 aq� � �Cr(H2O)6 aq

3� (23)

Denoting with �aA, �bA, �cA and �2B, �bB, �cB, respectivelythe differences in the separative effect coefficients betweengaseous and solvated state we have, for equilibria (17) and (18):

1000 ln �17 � aA� bA

� 106 ⁄ T2 � cA� 1012 ⁄ T4 (24)

1000 ln �18 � aB� bB

� 106 ⁄ T2 � cB� 1012 ⁄ T4 (25)

and, for equilibrium (23)

1000 ln �23 � (aA � aB � aA� aB

) � (bA � bB � bA� bB

)

� 106 ⁄ T2 � (cA � cB � cA� cB

) � 1012 ⁄ T4 (26)

The delta coefficients of the solvated Cr(III) and Cr(VI) mol-ecules are listed in Table 11. These values will be used laterwhen conforming the 53CrCr(H2O)6�� surface for the aque-ous phase in the Eh, pH field.

4. APPLICATIONS

We have seen that solvation effects (i.e., effects of thereaction field in a pure water solvent) are limited in altering theseparative effects of Cr isotopomers for both Cr(III) and Cr(VI)species. However, in geological applications, where even minorisotopic fractionations may give important clues, the effect ofthe reaction field cannot be dismissed. In the following calcu-lations we will assume thus that the computed �a, �b, �c

coefficient for the hexa-aqua-ion [Cr(H O) ]3� apply as well to

Cr(III), Cr(VI) species, with different solvation tech-

2 6

all the other Cr(III) species of interest, with the exclusion of

ion


[CrO2]� (i.e., the deprotonated forms [Cr(H2O)5(OH)]2�;[Cr(H2O)4(OH)2]�; [Cr(H2O)3(OH)3]0, the SBD, DBD, TBToligomers and the chlorinated forms [Cr(H2O)5Cl]2�;[Cr(H2O)4Cl2]�; [Cr(H2O)3Cl3]0). We will adopt for[Cr2O7]2� and [HCr2O7]� the �a, �b, �c coefficient obtainedfor [CrO4]�, and for [H2CrO4]0 the coefficients of [HCrO4]�.

Although 6 distinct fractionation factors may be established,based on the four different masses of the central cation inthe various species (50, 52, 53, 54), we will focus the discus-sion on the 53/52 fractionation, expressed in terms of53CrCr(H2O)6��, with hexa-aqua-chromium ion taken as areference species. Following Ohmoto (1972), in analogy withhis investigation of sulfur isotopic systematics (similarly af-

Table 11. Reaction-field coef

Species Couples �a

[Cr(H2O)6]3� 52Cr/50Cr 0.00240.0094

53Cr/50Cr 0.00350.0136

54Cr/50Cr 0.00460.0177

53Cr/52Cr 0.00110.0042

54Cr/52Cr 0.00220.0081

54Cr/53Cr 0.00110.0042

[CrO2]� 52Cr/50Cr �0.016553Cr/50Cr �0.026454Cr/50Cr �0.032053Cr/52Cr �0.009854Cr/52Cr �0.015254Cr/53Cr �0.0055

[HCrO4]� 52Cr/50Cr �0.0093�0.0160

53Cr/50Cr �0.0136�0.0234

54Cr/50Cr �0.0177�0.0305

53Cr/52Cr �0.0043�0.0074

54Cr/52Cr �0.0084�0.0145

54Cr/53Cr �0.0041�0.0071

[CrO4]2� 52Cr/50Cr �0.01730.0038

53Cr/50Cr �0.02530.0056

54Cr/50Cr �0.03300.0074

53Cr/52Cr �0.00790.0018

54Cr/52Cr �0.01590.0035

54Cr/53Cr �0.00780.0017

[CrO3Cl]� 52Cr/50Cr 0.009053Cr/50Cr 0.013054Cr/50Cr 0.017053Cr/52Cr 0.004154Cr/52Cr 0.008054Cr/53Cr 0.0039

(1) � IEFPCM; (2) � COSMO; (3) � Onsager; (4) explicit solvat

fected by redox equilibrium), we will define as �i the relative

isotopic enrichment factor between the species i and[Cr(H2O)6]3�:

53Cri � 53CrCr(H2O)6�� i (27)

Denoting 53Cr�Cr the mean isotopic composition of chromiumin solution

53CrCr � i

53Cri �mi

mCrT

� i

53CriXi (28)

with mi � molality of the ith aqueous Cr species and mTCr �

moles of Cr in solution, we arrive at the identity (cf. Eqn. 43 in

of the isotopic fractionation.

�b10�6 �c10�12 Solvation method

0.1184 �0.0015 (1)0.1179 �0.0017 (4)0.1763 �0.0022 (1)0.1723 �0.0024 (4)0.2287 �0.0028 (1)0.2266 �0.0031 (4)0.0580 �0.0007 (1)0.0543 �0.0007 (4)0.1103 �0.0015 (1)0.1087 �0.0015 (4)0.0524 �0.0007 (1)0.0544 �0.0007 (4)

�0.1672 0.0036 (2)�0.2944 0.0060 (2)�0.3222 0.0069 (2)�0.1273 0.0024 (2)�0.0278 0.0033 (2)�0.0278 0.0009 (2)�0.0609 0.0016 (3)�0.0996 0.0027 (1)�0.0898 0.0024 (3)�0.1443 0.0039 (1)�0.1176 0.0031 (3)�0.1923 0.0052 (1)�0.0289 0.0008 (3)�0.0448 0.0013 (1)�0.0567 0.0015 (3)�0.0927 0.0025 (1)�0.0278 0.0007 (3)�0.0480 0.0012 (1)�0.0387 0.0026 (2)

0.0087 �0.0004 (1)�0.0564 0.0039 (2)

0.0132 �0.0006 (1)�0.0746 0.0050 (2)

0.0179 �0.0008 (1)�0.0178 0.0012 (2)

0.0044 �0.0002 (1)�0.0358 0.0024 (2)

0.0092 �0.0004 (1)�0.0180 0.0012 (2)

0.0047 �0.0002 (1)0.2008 �0.0027 (2)0.2955 �0.0040 (2)0.3872 �0.0052 (2)0.0948 �0.0012 (2)0.1864 �0.0024 (2)0.0917 �0.0012 (2)

ficients

Ohmoto, 1972):

redomin


53Cri � 53CrCr � i � n

nXn (29)

with the summation extended to all n species in solution.It is rather evident from the form of equation (29) that the

isotopic fractionation of each of the n species of interest de-pends linearly upon the bulk isotopic composition of the solu-tion and upon the isotopic composition of the reference species.The following discussion will thus be mainly devoted to thereference species [Cr(H2O)6]3� at 53Cr�Cr � 0.00.

Figure 10b shows the complex form of the 53CrCr(H2O)6��

surface in a wide fO2-pH range at T � 25°C and zero ionicstrength. Part a of the same Figure is included merely forcomparative purposes, since by comparing parts a and b of thesame Figure one has an immediate perception of the role ofsolventb. Aqueous speciation calculations were conducted onthe basis of selected literature values of the Gibbs free energyof formation from the elements at standard state of the variousspecies of interest (Latimer, 1952; Garrels and Christ, 1965;Baes and Mesmer, 1976; Dellien et al., 1976; Vasil’ev et al.,1977; Barner and Scheuerman, 1978; Wagman et al., 1982;Palmer et al., 1987; Brito et al., 1997; Cruywagen et al., 1998),with some assumed values for the Cr oligomer SBD and for[Cr(H2O)3Cl3]0. Speciation calculations at I�0 involved theestimate of individual ionic activity coefficients with the mod-ified Debye-Hückel equation (Helgeson and Kirkham, 1974)Different choices of standard state thermodynamic properties,such as the set of internally consistent values recently proposedby Ball and Nordstrom (1998) do not significantly alter theoverall picture.

From Figure (10) it is quite evident that the abrupt changein the isotopic composition of the reference species withfO2, is dictated by the redox equilibria of the various Cr(VI)forms, and a major inflection at high pH and low fO2 where

b It should be noted that Figures a and b are built upon an identicalspeciation state for comparative purposes, while actually the gas-phase

Fig. 10. Conformation of the 53CrCr(H2O)6 surface in thregions where Cr(VI) forms are predominant (high fO2) of predominance of Cr(III) forms. The limited step at intermpH � low fO2). The effect of the reaction field (right pafractionation and to enhance the step within the Cr(III) p

speciation could be substantially different from that in the aqueousphase.

[CrO2]� is the predominant Cr(III) form (substantially am-plified by the reaction field; cf. parts a and b of the samefigure) is also apparent. The features of the aqueous frac-tionation surface are better delineated in the two projectionsof Figures (11,12). We may thus appreciate that dramaticfractionation changes may take place both due to limited pHmodifications at constant T, fO2 conditions (Fig. 11), and tolimited Eh modifications at constant T, pH conditions(Fig. 12)c.

The first obvious application of our computational resultsconcerns the separation procedure adopted in Mass Spectro-metric determinations of the isotopic composition of Cr-bearing solutions. In fact, it is common practice to submitthe sampled specimens to a double chromatographic extrac-tion-purification step by anionic and cationic resin beds(quaternary ammonium functional groups and sulfonatedgroups, respectively, attached to a styrene divinylbenzenecopolymer lattice), after an initial oxidation of the dissolvedCr to the Cr(VI) form. These procedures alter sensibly thepH-Eh state of the solution and are not quantitative. As a netresult the isotopic composition of the purified Cr-bearingaliquots may be dramatically different from those of theinitial specimen. It should be noted that reduction of Cr(VI)during chromatographic separation through resin columns isnot limited to anionic exchangers, but applies to cationicexchangers as well. During elution through cationic ex-changers in H� form (such as Dowex 50W-X8, for example)the pH condition is controlled by the bulk amount of ex-changeable cations in solution. For a 5 ppm Cr(III) (CrCl3)� 5 ppm Cr(VI) (K2Cr2O7) solution, the initial pH is 3.415.Through speciation calculations it may be deduced that theinitial redox state of the solution is Log fO2 � �12.661.Flowing 100cc of the above solution through a 10cc colum-nar resin bed, 30% of the initial Cr(VI) is converted to

c Effects of the bulk amount of Cr in solution, within reasonable

fO2 � pH space (a � gas phase; b � aqueous phase). InH2O)6 is highly negative, while is near zero in the regionpH delimitates the predominance region of [CrO2]� (highe figure) is to reduce somewhat the bulk Cr(III)/Cr(VI)ance region

e Log53CrCr(

ediatert of th

concentration ranges, and of the ionic strength of the solution are muchsubordinated and virtually ininfluent.

ation e


Cr(III) and retained by the column (Accornero, 2004). At theend of the process the new pH is 3.327 and Log fO2 is raisedto �12.573. According to our calculations, the53CrCr(H2O)6�� of an initial solution at 53

Cr�Cris �3.490,

while the 53CrCr(H2O)6�� of the (6N HCl) eluate is�2.453 (Figs. 11 and 12). Lowering the initial bulk chro-

Fig. 11. effect of pH on 53CrCr(H2O)6�� at various re25°C, I � 0. Paths of reductive retention of Cr through a ctext).

53
Fig. 12. Effect of redox state on CrCr(H2O)6 at various pH conI � 0. Paths of reductive retention of Cr through a cation exchan
mium concentration in solution by a factor 10�3, does notalter the results much. From an initial 53CrCr(H2O)6�� of�3.532 one arrives to �2.540 (Figs. 11 and 12). In bothcases the enhancement of the initial 53CrCr(H2O)6�� isaround � 1.

Fractionations are even more dramatic whenever anionic

ditions (Log fO2 values quoted in bold in figure) at T �xchange sulphonated resin column are superimposed (see

dox con

ditions (pH values quoted in bold in figure) at T � 25°C,ge sulphonated resin column are superimposed (see text).


exchangers are adopted in the first stage of Cr purification(Götz and Heumann, 1988; Ball and Bassett, 2000). The effectof incomplete reductive elution of Cr(VI) through anionic resincolumns is satisfactorily reproduced by the following equation:

53CrCr(H2O)6�� 53CrCr � 6.925

� 7.034 · XCr(VI) � 0.065 · pH · XCr(VI) (30)

where XCr(VI) is the fractional amount of Cr(VI) retained in thecolumn. Although Götz and Heumann (1988) claim that thereductive elution through anionic resin beds is quantitative,Ball and Bassett (2000) stated that only 75% of the initialCr(VI) is reductively eluted. Based on our calculations thiswould result in an enhancement of � 1.74 of the initial53CrCr(H2O)6��.

Our experience with anionic exchangers (Accornero, 2004)seem to confirm that a quantitative reductive elution of Cr(VI)(i.e., XCr(VI) 0.01) is easily attained, whenever channelingphenomena are avoided. The fact that Ball and Bassett (2000) didnot observe any appreciable effect of incomplete reductive elutionon Cr isotope systematics may indicate that their appraisal of theactual performance of the resin was too pessimistic. Nevertheless,extreme caution should be used in attaching any geological sig-nificance to the Cr isotopic systematics, whenever chromato-graphic purification steps are involved in sample preparation.

A further caveat on the application of the computed equilib-rium isotopic fractionations concerns the kinetic rates of attain-ment of equilibrium. Although it is common practice to assumethat homogeneous aqueous phase exchange equilibria are ratherfast (especially at high T), with respect to flow rates, thisassumption would be based on quantitative argumentationswhich cannot be addressed in the present study. The treatmentof relative reaction velocities of isotopic molecules in theframework of Transition State Theory (TST) has been ad-dressed long ago by Bigeleisen (1949) to whom we refer for adetailed account (see also Felipe et al. (2003) for a practicalapplication of ab initio methods to geochemically relevantisotopic exchange kinetics).

5. CONCLUSIONS

Our gas-phase ab initio structure-energy optimization of twelveCr(III) molecules and six Cr(VI) molecules in the system Cr-H-O-Cl by DFT and plane-polarized wave functions 6-31G** leadsto stationary states in substantial agreement with existing experi-mental observations and previous computational studies. The vi-brational analysis of the normal modes and the ensuing separativeeffects for 50Cr, 52Cr, 53Cr and 54Cr isotopomers are shown to beconsistent with the Redlich “product rule” systematics (BrightWilson et al., 1955), through the adoption of an “isotopic ergo-dicity factor” which enables one to translate into a simple visualfashion the effects of bulk molecular masses and moments ofinertia about the principal axes. The computed fractionation fac-tors are dominated by the force constant of the central cation andthis leads to a certain constancy in the isotopic fractionation factorsof Cr(III)-Cr(VI) redox equilibria involving distinct molecules.The effect of the reaction field is to reduce somewhat the Cr(III)-Cr(VI) fractionation in the aqueous phase with respect to thegaseous state. The computed fractionation factors are in any case
extremely significant at ambient conditions and well within the
analytical sensitivity limits of mass spectrometry. We do not thinkthat the few literature data existing on the Cr isotopic signature ofwater samples are sufficient to warrant any comparative discussionand we limited ourselves to raising some cautionary argumenta-tion against the utilization of non-quantitative chromatographicseparation procedures in SIMS measurements.

Acknowledgments—We are indebted to associate editor Dr. David We-solowski and the three anonymous reviewers for their insightful commentsand suggestions. We wish also to thank Prof. Jacopo Tomasi and Dr.Christian Pomelli (DCCI Department, University of Pisa) for their pre-cious advice and factual help on handling a difficult convergence byIEFPCM procedures on hexa-aqua-chromium. Finally we acknowledgeCINECA supercomputing center (Bologna, Italy) for the CPU time grantallocated to us for this project on its cluster IBM SP Power4.

Associate editor: D. Wesolowski

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