10773
J. Phys. Chem. 1992, 96, 10773-10779
Theoretical ab Initio SCF Study of Binding Energies and Ligand-Field Effects for the Hexahydrated Dlvaient Ions of the First-Row Transition Metals R d f Akessoht Lars C. M. Pettersson,*JMagnus SandstrBm,*,t Per E. M. Siegbahn,* and Ulf Wahlgrent Department of Inorganic Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden, and Institute of Theoretical Physics, University of Stockholm, Vanadisviigen9, S-1 I 3 46 Stockholm, Sweden (Received: May 26, 1992)
Theoretical binding energies and M-O distances have been obtained by ab initio SCF calculations performed on the isolated clusters [M(H20)6]2+,for the divalent metal ions of the first-row transition series with M = Ca, Sc, ..., Zn, using a large water basis set. For the octahedrally coordinated hexahydrated metal ions, the binding energy variation follows a doublehumped curve of similar shape as for the experimental solution enthalpies, although the Ca2+value deviates probably due to a higher hydration number in solution. The variation of the calculated M-O bond distances is also consistent with the trend expected from ligand-field theory. The results are discussed in terms of electronic factors in order to evaluate the effects of an octahedral ligand field. The bonding is mainly ionic, although with an increasing donation of ligand electrons to the metal s and p orbitals with increasing atomic number. The metal ion 3d orbitals are found to have an essentially atomic character, which would explain the qualitatively good agreement with the energy splittings of the d orbitals predicted with an octahedral crystal-field model. The additional energy gain by Jahn-Teller effects is small compared to the ligand-field contributions.
IatrOdUCtiOn
The properties of complexes of metal ions from the transition series are clearly influenced by the variations in the d-electron configuration. In solution or in comparable solid compounds, changes in the geometrical structure, chemical reactions, and thermodynamical properties of the complexes along the series display similar trends for different ligands. Typical examples showing profound effects are found for reaction rates, stability constants (e& the Irving-Williams series), ionic radii and bond strengths, heats of formation, lattice energies, electronic spectra, etc.' These variations are often ascribed to the influence of the surrounding atoms on the d-electron shell of the metal atom. The crystal-field and ligand-field theories have been successfully used for correlations with observed trends, with the hydrated transition-metal ions as important and often cited textbook examples of octahedral ligand-field effectsa2s3 Despite the fundamental importance of the subject, relatively few theoretical studies have been performed on the hydration of transition-metal ions, which due to their open d shells may have been considered as too difficult to handle by simple theoretical models. The present paper is one of a series in which different aspects of hydration are systematically treated and modeled for transition-metal ions. The aim is to understand, qualitatively and quantitatively, transition-metal ion reactivity in solution, but also other aspects of the hydrated complexes, such as the connection between geometric and electronic structures, including ligand-field and Jahn-Teller effects, will be covered. Previously, we have studied fust-orderJahn-Teller effects for the hexahydrated clusters of Cu2+,Cr2+,and Mn3+43 and secondsrder Jahn-Teller effects for Zn2+, Cd2+, and Hg2+.6 A natural starting point for a study of the hydration of transition-metal ions is to consider hexacoordinated [M(H20),l2+ions, since 6 is known to be the predominant hydration There is also a wealth of experimental information available for these ions against which theoretical models can be compared and tested. Hydration enthalpiesIO and geometric structures9 are experimentally known or estimated for all divalent ions (except Sc2+) of the fit-row transition metals. There is a basic increase of the hydration energy as the nuclear charge is increased, but another variation is superposed, leading to a doublehumped curve. The maxima occur at vanadium-hromium and at nickel, with an intermediate minimum at manganese. It has been a primary goal of the project covered in the present paper to see how well Royal Institute of Technology. *University of Stockholm.
this trend can be reproduced by the calculations and to understand the origin of the enthalpy variations. In the crystal-field model, where the overlap of metal and ligand orbitals is neglected, the five metal d orbitals split into a low-energy (t2J and a high-energy (e,) set in the octahedral field formed by the ligands. The splitting can be expressed in terms of the spectrochemical parameter A (or 10Dq in the crystal-field model)?" the difference in one-electron energy between the t2 and egorbitals. For the fmt-row divalent transition-metal M(H28k2+ ions, the splitting is smaller than the spin-pairing energy in all cases, giving high-spin ground states. The three t2, and two eg orbitals are stabilized and destabilized by 4Dq and 6Dq, respectively, in this simple picture. Therefore, octahedral Ca2+(do), Zn2+(dl0), and Mn2+(d5: t2S3 complexes are not favored by any additional ligand-field stahhation, while V2+(ta3) and Ni2+ (t2g6 ):e complexes are those most stabilized in terms of the Dq parameter.2 However, some care should be taken in the analogies between the simplified crystal-field model and the ab initio computational description, because the largest contribution to the splitting of the d orbital energy levels is, in fact, not caused by the electrostatic ligand field but rather by the Pauli repulsion between the d electrons and the electron pairs of the ligands." Semiempirical ligand-field parameters have proven to be useful in describing the spectroscopic properties of transition-metal compounds."J2 If accurate reproduction of the ligand-field parameters should be a requirement of the computationalmodel used, simple SCF theory would hardly be sufficient for reproducing the experimental trend, at least not quantitatively, since these parameters are known to be strongly dependent on electron correlation. Clearly, a result where the SCF method is found adequate for the description of the bonding without introducing a correlation treatment would considerably simplify further theoretical modeling of hydration effects of these ions. The calculated binding energy for a hexahydrated metal ion,
et)
= E[M(H20)6(g)12' - E[M2'(g)1 - 6E[H20(g)I (1) componds to a gas-phase reaction at 0 K and would be directly comparable with experimental gas-phase hydration energies, AHm, after (relatively small) standard-state and zero-point energy corrections. However, due to the high second ionization energy of the metal ions, experimental gas-phase solvation data can only be obtained for monovalent ions (see, e.g., ref 13 for a study of successive hydration energies of alkali-metal ion clusters M(H20),+). An early attempt to obtain Lw,, values from a Born-Haber cycle involving lattice energies resulted in some incomplete data.14 The single-ion hydration enthalpy values, m b n id
0022-365419212096-10773$03.00/00 1992 American Chemical Society
10774 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Akesson et al.
TABLE I: Calculated"aod Experiwnt.lbM-O Distances for the Hexahydrated [M(H20)Bt Ions M2+ R,,(SCF)IA 2.399 Ca2+ 2.307 sc2+ 2.252 Ti2+ 2.201 V2+ 2.1548 Cr2+(eq) 2.366i Cr2+(ax) 2.233 (2.250)' Mn2+ 2.185 Fe2+ 2.143 co2+ 2.108 Ni2+ 2.0638 (2.077)8sk CU2+(eq) 2.238i (2.248Pk Cu2+(ax) 2.123 Zn2+
RMAexo)lA 2.40-2.49ee 2.147f 2.03e' 2Sh-J 2.20-2.22" 2.12'-d
[email protected] 2.04-2.07'.d 1.94-1.99d-8*' 2.3-2.4dJ*' 2.08-2.1@,d
-
With medium-sized water basis set. bSolution data. 'LAXS measurements. dReferences 7 and 8. cReference 9. /Mean value of X-ray diffraction studies on crystals, refs 21 and 22. gEquatorial M-O distance in tetragonally elongated octahedral coordination. EXAFS measurements. Reference 20. 'Axial M-O distance. Calculated with large water basis set (see text). 'Reference 4.
'
mohyd, obtained mostly from calorimetric measurements,'J5J6 form a more complete data set. However, for a comparison with the @bind values, additional (large) corrections for the heat of vaporization of water, AHvap, and the solvation of the gas-phase hydrates, AHwln,should be made to account for the effect of the infinite surrounding medium on the hydrated ion, Le., mhyd @bind + 6 m v a p + m w l n (2) according to the Born-Haber cycle below:
Calculations to estimate the solvation energy, AHwIn,are presently being performed," but the discussion in the present paper will focus on the ligand-field effects displayed by the variations of the experimental and theoretical curves. The determination of an absolute scale of single-ion hydration enthalpies can only be made indirectly by deriving a reference value for one particular hydrated ion, usually H+.I5 The recommended reference value, obtained by the method of Halliwell and Nyburg,I8 is whyd[H+(aq)] = -1091 kJ m ~ l - ' . ' ~ + The ' ~ compilation by Smithloof w h y d values for the transition-metal ions seems to be the most complete and reliable, with most data comparable to those given earlier by Burgess.Is Quite large discrepancies, even when differences in reference states are taken into account, are found in some of the older literature. The ligand-field stabilization will also affect the effective ionic radii of the hwramrdinated transition-metalions, and comparisons of calculated and experimental metal-oxygen distances can give information on this influence. Metal-oxygen distances in solution have been obtained by LAXS (large-angle X-ray scattering) or EXAFS (extended X-ray absorption fine structure) techniques for most divalent ions in the series,7-9,20except for Sc2+,Ti2+,and V2+, which are rapidly oxidized in solution. Numerous studies of hexaaquametal(I1) ions in the solid state have been made, and in the first transition series, only [sc(H20)6]2+and [Ti(H20),12+ are unknown. Typical examples are the isomorphous Tutton salts (M')2[M(H20)6](S04)2(where M' = Cs+,NH4+,Rb+, etc. and M is a divalent metal ion),21for which a mean V-O distance of 2.147 A has been obtained for the [V(H20)6]2tion.22923 The precision with which the M-O distances have been determined is as a rule higher in the solid state than in solution, but systematic deviations due to packing effects and thermal corrections can be rather large.24a.b For the same coordination number, the solution distances are, in principle, more directly comparable with the calculated values; see Table I.
P
W
u
Figure 1. Hexahydrated metal ion in Thsymmetry.
Theoretical binding energies for a few hexahydrated divalent transition-metal ions have previously been obtained by ab initio methods,25g26although with much smaller basis sets than in the present study. Smaller basis sets lead to unrealistic values of the dipole moment and polarizability of the water molecules and generally to overestimated binding energies.
Method of Calculation Optimized binding energies and bonding distances have been computed for all the hexahydrated dipositive transition-metal ions in the series from Ca2+ to Zn2+. The calculations have been performed at the SCF level of approximation or, in some cases CASSCF, using the MOLECULE-SWEDEN27 system of programs. The metal ions were described using the primitive basis sets optimized by Wachters,28 extended with two diffuse p functions28and one d function. The basis sets were contracted to [5s4p3d] using the Raffenetti contraction schemeZ9based on the neutral atom SCF orbitals, leaving the outermost (one s, two p, and two d) functions uncontracted. This description is quite satisfactory for the dipositive ions; the metal basii set superposition errors (BSSE) as measured by computing the energy differences of the ion with and without the surrounding water basis sets were found to be smaller than 100 cm-' in all cases. Two differently sized basis sets were used for the description of the water molecules. In the calculations with the medium-sized basis set, the oxygen atoms were described using the Huzinaga [9sSp] basis,3oextended with a diffuse p function and contracted to [3s4p] using the atomic SCF orbitals.29 The hydrogen atoms were described by the Huzinaga [4s] basis,3O contracted to two functions. A substantially larger water basis set was constructed by adding two uncontracted d functions to the oxygen basis and a p function to the hydrogen. This basis set gives results for the dipole moment and dipole polarizability of H 2 0very close to the Hartree-Fock limit, which is essential for a reliable estimate of the SCF-level solvation energy. The internal water geometry was = 0.957 A, HOH as previously kept at the gas-phase values (kH = 104.5°).31 For all the hexahydrated clusters, except Cr2+and Cu2+,the M-O distances were optimized in the highest symmetry possible (Th) with trigonal orientation of the coordinated water molecules, Figure 1. Octahedral hexahydrates of Cr2+ and Cu2+ are Jahn-Teller unstable, and for these systems, DZhsymmetry was used in the optimizations in order to allow for tetragonal distortions of the M - 0 6 coordination octahedra to remove the degeneracy of the eg orbitals. For a detailed discussion of geometries and energies obtained theoretically for Jahn-Teller active hexahydrated ions, see ref 5. In addition, Jahn-Teller distortionscan be expected for the dl, d2, high-spin d6 and d7 electron configurations due to degeneracies in the t2eorbitals, although in these cases the energy gain should be insignificant compared to the overall binding en-
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10775
Hexahydrated Divalent Ions of Transition Metals TABLE II: Totd EM!* E Of the [M(HzO)$+md M*+IOIM,' Tbeoretierl Binding Energies AEW Experlmeatal Single-Ion Hydration Enthrldes in G r s - P h A H - a d in Solution AH', M2+ -
E[M(H%%612+
Ca2+ -1 132.830450 sc2+ -1215.801 025 Ti2+ -1304.452 932f V2+ -1398.934024 Cr2+ -1499.321 147 Mn2+ -1605.835 928 Fez+ -1718.404 371 co2+ -1837.343 2 6 v Ni2+ -1962.799 521 cu2+ -2094.843 206 Zn2+ -2233.702 790
EM^+ -676.114327 -759.081 508 -847.717 182 -942.162 767 -1042.556 498 -1 149.082 197 -1261.625 761 -1380.559821 -1505.980 910 -1638.026639 -1 776.890 003
Ubindb"
-1002 -1089 -1 148% -1225 -1208 -1 179 -1245 -12879 -1350 -1344 -1334
-1289 -1527 -1560 -1565 -1502
-1862 -1918 -1904 -1841 -1946 -1996 -2105 -2100 -2046
(15) (15) (6) (6) (6) (6) (6) (6) (6)
Large water basis set (see text) and atomic units (1 hartree = 4.359 75 lo-'* J) have been used. b hkJ mol-'. ' h E b i n d E p 4 ( ~ ~ 0 ) ~-1 2EM^+ + 6EHp with EHto = -76.050761 hartree. dReference 14. 'Reference 10. fSCF single-determinant energy (see text) with large water basis. #Corrected for the mixing of atomic states for Ti2+and CoZcin the hexahydrates by subtracting 16 and 30 kJ mol-', respectively (see text), from the single-determinantvalue. X
2.1,
Ca
I
I
I
I
I
I
r
SC
Ti
V
Cr
Mn
Fe
Co
NI
tu
Zn
Figure 2. Calculated M a distances of regular [M(H20)6]2+complexes as a function of increasing atomic number of the metal atom M, obtained with the medium-sized water basis set (for Cr2+ and Cu2+,the mean values, 2.225 and 2.121 A, respectively, are used).
ergies. Moreover, geometric distortions with this origin have not compound [V(H20)6][H5OZ](CF3S03)4,and an explanation based been v d i e d experimentally; see,e.g., crystal structures containing on a Jahn-Teller distortion of the degenerate (t2J2 configuration hexahydrated [Fe(Hz0)6]2+(d6) and [CO(H~O),]~+ (d7) ions.2' has been ~uggested.'~ Finally, a point of technical importance in connection with D3dsymmetry of an [M(H20)6]complex can also be achieved calculations on open d-shell ions should be noted, namely, the fact with trigonally coordinated water molecules in an 'all-vertical" that arbitrary occupationsof the d orbitals in a singledeterminant geometry (see Figure 1, ref 34). This type of structure has been picture do not in general lead to angular momentum eigenfunctions proposed to be energetically favorable for d' ions, because the d or, equivalently, to pure atomic states. However, for the atom, electron a n enter an algorbital oriented to give minimal repulsion a single-determinant description of some component of the from the ligand^.'^ Preliminary calculations on [ S C ( H ~ O ) ~in] ~ + high-spin ground state can always be found. In all cases, except the all-vertical D3dsymmetry indicate slightly lower energies than for Tiz+(3F)and C O ~ + ( ~ the F ) , occupation required for this dein Thsymmetry, but the difference (a few kJ mol-') is not sigscription coincides with that required by the ligand field. For Tiz+, nificant.l7 the ligand field requires two t, orbitals, e.g., (d,,, d,,), to be Jahn-Teller distorsions were recently studied theoretically for occupied, which in a single-determinant description would corthe isoelectronic [Ti(HZO),I3+ion at the MR-SDCI level with respond to a mixture of the ion 3F (80%) and 3P (20%) states. a metal atom basis (43321/43/31) + polarizing functions and In the limit of an infinitely strong ligand field, this is also the an STO-3G basis for the water molecules.35 The D3dstructure correct occupation. In the absence of a field, the correct dewas found to have slightly (-3 kJ mol-') lower energy than a scription of the 3F state using these orbitals also requires a concompressed octahedron in D2 symmetry, with the equatorial T i 4 figuration with the d+; and d, orbitals occupied, i.e., a twodistance slightly (0.042 A) longer than the axial. However, an configurational description (with a fixed ratio 2:l between the unreasonably large energy deviation (Jahn-Teller energy) was coefficients in the expansion). The ligand field raises the energy obtained from the regular Th symmetry (76.6 kJ mol-'), which of the dX+2 e, orbital over that of the t, orbitals and affects this probably is a result of the insufficient water basis set used. ratio or equivalently induces a mixing of the ionic 3Fand )P states. Results and Discussion The extent of this mixing thus depends on the strength of the I. Metal-Oxygen Distances. The experimental ionic radii of ligand field, and to allow for this general mixing or coupling of Ca2+,Mn2+, and ZnZ+in 6-coordination, 99, 78, and 72 pm, the pure 3d orbitals, we have utilized a two-configuration derespecti~ely,~~ decrease smoothly as expected from the increasing scription of the solvated Ti2+and Coz+ions (for which a similar nuclear ~ h a r g e s , ~with J ~ the radii for the intermediate divalent argument applies). It should be emphasized that this improved ions falling below the connecting curve. The calculated M - O description only affects the coupling of the d orbitals in the metal distances for the hexahydrated complexes (Figure 2) follow the ion and does not imply that electron correlation has been included same trend as found for comparable experimental M-O values for these systems. (Table I), consistent with an additional stabilization of high-spin The medium-sized water basis set was generally used in the ions in an octahedral ligand field. Thus, minima in the M-O geometry optimizations, and the energy in the optimized geometry distances occur for V2+ and Ni2+,which have the highest ligwas then recomputed using the larger water basis set. In the latter and-field stabilization energiesaZThe Ni-O distance is, in fact, case, a modified version of the direct SCF program DISC032,33 the shortest in the whole series. was used in a parallellized version running either on an Alliant Generally, the calculated M-O distances appear to be somewhat FX/2800 with 24 CPU's or on a Cray X-MP/416. To test the longer than those obtained experimentally, in particular for the influence on the geometry:" reoptimizationswere performed with the larger basis sets for [Mn(H20)6]2+(Th)and [ C U ( H ~ O ) ~ ] ~ +large water basis, giving values closer to the HartreeFock limit.* This can partly be related to the omission of a hydrogen-bonded (DZh)and resulted in small (
