CASSCF study of the relation between the Fe charge and the Mössbauer isomer shift

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Chemical Physics Letters 454 (2008) 196–200 www.elsevier.com/locate/cplett

CASSCF study of the relation between the Fe charge and the Mo¨ssbauer isomer shift Aymeric Sadoc a, Ria Broer a, Coen de Graaf b,* a

Theoretical Chemistry, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands b ICREA Research Professor at the Department of Physical and Inorganic Chemistry, Universitat Rovira i Virgili, Marcellı´ Domingo s/n, 43007 Tarragona, Spain Received 20 November 2007; in final form 11 February 2008 Available online 15 February 2008

Abstract Multiconfigurational wave functions are calculated for a series of Fe complexes. We find a linear correlation between the experimental Fe Mo¨ssbauer isomer shift and the calculated electron density at the Fe nucleus. However, the analysis of the wave function in valence bond terms shows that there is no straightforward relation between the density at the nucleus and the Fe charge. The analysis of the CASSCF wave function expressed in localized orbitals shows that the isomer shift is very sensitive to the weight of charge transfer configurations and hence to the covalency, rather than to the absolute charge. Ó 2008 Elsevier B.V. All rights reserved.

57

1. Introduction Mo¨ssbauer spectroscopy is nowadays an important tool to characterize transition metal (TM) ions either in organic or inorganic chemistry [1]. Especially, 57Fe Mo¨ssbauer spectroscopy is widely applied and useful insights have been obtained for the electronic structure of, for example, spin-crossover compounds [2]. Usually the Mo¨ssbauer spectra are interpreted in terms of two parameters: the isomer shift (IS) and the quadrupole splitting. However, the spectral features remain difficult to interpret in many cases and computational approaches may play a role to elucidate the relation between the spectra, the mentioned parameters and the electronic structure of the TM complexes. The direct determination of Mo¨ssbauer parameters with quantum chemical methods is still a difficult task. A promising approach has recently been published by Kurian and Filatov [3,4]. It is, however known for a long time that there exists a linear correlation between calculated electron *

Corresponding author. E-mail address: [email protected] (C. de Graaf).

0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.02.034

densities at the Fe nucleus, the electron contact density, and the experimentally measured IS. Such correlation was established by Nieuwpoort and collaborators [5] for restricted open-shell Hartree–Fock (ROHF) wave functions based on earlier work of Blomquist [6] and Duff [7]. More recently, this relationship was also found for density functional theory (DFT) calculations [8–11]. This indirect determination of the IS opens a way to relate the calculated electronic structure to experimental observations. In this study we establish a linear correlation between the measured IS and the electron contact density at an Fe point nucleus qð0Þ calculated from accurate multiconfigurational wave functions obtained by complete active space self-consistent field (CASSCF) calculations and subsequent complete active space second-order perturbation theory (CASPT2) treatment of the dynamical electron correlation effects. We analyze the CASSCF wave functions in terms of localized orbitals [12,13] and determine the effective number of electrons in the Fe-3d orbitals for a series of Fe complexes. It is shown that there exists no simple relation between the IS and the effective number of d-electrons. Hence, the interpretation of IS in terms of Fe oxidation states should be taken with caution.

A. Sadoc et al. / Chemical Physics Letters 454 (2008) 196–200

2. Computational information The geometry of the Fe complexes is taken from Ref. [8] and is shortly reviewed in Table 1, which also lists the symmetry of the electronic ground state for which qð0Þ is calculated. In most cases, the point group of the complex is nonAbelian. In these cases the highest Abelian symmetry subgroup is exploited in the calculations. The quantum chemistry code MOLCAS 6.4 [14] is used. The one-electron basis sets employed are of the atomic natural orbital (ANO) type. The (21s, 15p, 10d, 6f) primitive basis set for iron is contracted to a (6s, 5p, 4d, 2f) basis. For carbon, nitrogen and oxygen, the (14s, 9p, 4d) primitive functions are contracted to a (4s, 3 p, 1d) basis. Finally for Cl and H, we apply the (17s, 12p, 5d/ 5s, 4p, 1d) and (8s, 4p/ 3s, 1p) basis sets, respectively [15,16]. The N-electron wave function is computed within the CASSCF approach. The configuration expansion of the wave function is constructed by distributing a limited number of active electrons over a set of valence orbitals, the socalled active space. All other electrons are in doubly occupied, inactive orbitals. The active space contains five orbitals of mainly Fe-3d character and a set of virtual orbitals of the same symmetry character as the Fe-3d orbitals, which become the so-called Fe-3d0 orbitals in the orbital optimization and account for part of the radial 3d electron correlation [17,18]. In the case of (quasi-)octahedral complexes, the active space is extended with two occupied orbitals of eg -like symmetry with mainly L-2p character to describe the r interaction between Fe and the ligand. The p interaction through the t2g orbitals is much weaker. For the tetrahedral complexes, we add a set of e and t2 occupied orbitals that have mainly ligand character [19]. Due to the 2 strong covalent bonds in ½FeO4  [20], we opt here for a smaller active space with twelve electrons and ten orbitals, namely the bonding and anti-bonding e and t2 orbitals. This choice of active space ensures a balanced and unbiased treatment of the most important electronic configurations [13,19,21–24]; the non charge transfer (NCT) Fe-3dn , the charge transfer (CT) Fe-3dnþ1 L1 , and the double CT (DCT) Fe-3dnþ2 L2 , n being the number of 3d electrons according to the ionic model.

197

To determine the effective number of Fe-3d electrons, we perform a unitary transformation of the active orbitals to express the wave function in localized orbitals. This transformation is based on pair-wise orbital rotations as described in Ref. [13]. We compare this way of calculating the effective Fe charge to the more traditional Mulliken and Bader charges [25] and the recently developed LoProp charges [26]. The latter scheme is based on a series of transformations that lead to localized orthogonal orbitals from which atomic charges can directly be assigned. 3. Results 3.1. Calibration of the CASSCF results Fig. 1 relates the CASSCF electron densities at the Fe point nucleus (qð0Þ) with the experimental Mo¨ssbauer IS. qð0Þ is obtained as the sum of the natural orbital densities at r ¼ 0 multiplied by the natural occupation numbers. This contact density is decomposed in Fig. 2 into contributions of the Fe-1s, 2s, 3s, and ligand orbitals. Before all, we mention that the calculated value of qð0Þ is strongly dependent on the choice of the basis set. The addition of s-type functions with large exponents leads to a drastic increase of qð0Þ. The main contribution arises, as expected, from the Fe-1s orbital. It contributes about 90% to the total contact density. However, Fig. 2 shows that the differential effect of the q1s ð0Þ is rather small; Dq1s ð0Þ ¼ 0:19 between 2 2þ ½FeO4  (IS = 0.69) and ½FeðH2 OÞ6  (IS = 1.39), which is approximately 2% of the total Dqð0Þ. Dq2s ð0Þ and Dq3s ð0Þ contribute with 2% and 36% to Dqð0Þ. These differential effects are hardly affected by the addition of the tight functions to the basis set. Hence, it can be concluded that the standard ANO basis set is adequate to establish an accurate relation between qð0Þ and IS. Moreover, because the Fe-1s and 2s contributions to the variation in qð0Þ are minor, we expect relativistic corrections to be small [8]. Fig. 2 also indicates that the smallest contribution to qð0Þ 2.00

1.50

Complex

Symmetry

½FeðH2 OÞ6 2þ ½FeF6 4 ½FeCl4 2 ½FeðH2 OÞ6 3þ ½FeF6 3 ½FeCl4  ½FeðCNÞ6 4 ½FeðCNÞ6 3 ½FeO4 2

D2h Oh ðD2h Þ T d ðC 2v Þ D2h Oh ðD2h Þ T d ðC 2v Þ Oh ðD2h Þ Oh ðD2h Þ T d ðC 2v Þ

State

Bond lengths

IS

5

2.087, 2.136, 2.156 2.060 2.265 1.900 1.920 2.165 1.900 1.900 1.650

1.39 1.34 0.90 0.50 0.48 0.19 0.02 0.13 0.69

B1g T2g 5 A1 6 Ag 6 A1g 6 A1 1 A1g 6 A1g 3 T2 5

The experimental IS (in mm/s) is also given.

1.00 IS [mm/s]

Table 1 Local (Abelian) symmetry (see text), Russell Saunders states and bond ˚ ) of the electronic wave functions of a series of Fe complexes lengths (in A

0.50

0.00 IS =-0.2761 ρ(0) +3264.1342 -0.50

R2=0.984

-1.00 11818 11819

11820

11821

11822

11823

11824

11825

11826

11827

ρ(0) [au-3]

Fig. 1. Calibration of CASSCF results for the prediction of the 57Fe IS. qð0Þ is plotted versus the observed IS for the complexes listed in Table 1.

198

A. Sadoc et al. / Chemical Physics Letters 454 (2008) 196–200 Table 2 Calculated q(0) (in au3) and extrapolated IS (in mm/s) of several Fe systems with different charge and spin coupling

10700 10699

985

984 983

ρ(0) [au-3]

137

136

System

qð0Þ

IS

Ionic charge

State

Fe2þ Fe3þ ½FeBr4  ½FeðNH3 Þ6 3 ½FeðCOÞ6 2þ Fe(1) in CaFeO3 Fe(2)in CaFeO3

11817.715 11820.324 11822.837 11819.653 11823.595 11824.194 11825.009

1.76 1.05 0.36 1.23 0.15 0.01 0.23

+2 +3 +3 +2 +2 +4 +4

5

D S 6 A1 5 T2g 1 A1g 5 Eg 5 Eg 6

135

also included in the forthcoming analysis of the relation between IS and electronic structure.

7

6

3.2. Relation between isomer shift and Fe charge 5 4

3 2 1 -0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

IS [mm/s]

Fig. 2. Contribution to qð0Þ of the Fe-1s (diamonds), Fe-2s (squares), Fe3s (triangles), and ligand (circles) orbitals. Solid lines are a guide to the eyes.

– arising from Fe-4s and the tails of the totally symmetric ligand orbitals – causes the main differential effect, in agreement with the findings reported in Ref. [8]. It is, however, not possible to separate these two contributions due to the large overlap between Fe-4s orbital and L-2p orbitals. In line with findings for ROHF [5] and DFT [8], Fig. 1 shows a linear correlation between the calculated qð0Þ and the experimental IS. The slope is equal to 0.276 mm/s. A linear relationship was also found for the first order CASPT2 wave function, be it with a different slope. The slope is also dependent on the basis set. This means that the calibration of qð0Þ against the observed IS only has a meaning within a method and cannot be compared quantitatively with other methods. The validity of the calibration can be assessed by calculating the IS from the relationship derived for a complex that was not included in the calibration set. The Mo¨ssbauer spectrum of the CaFeO3 low temperature phase shows two signals, interpreted as originating from two distinct iron sites. The reported IS are 0.31 and 0.03 mm/s [27]. Embedded cluster calculations give qð0Þ ¼ 11824:194 and 11825.009 for the two iron sites (for details see Ref. [12]). With the relation given in Fig. 1, we obtain IS = 0.01 and 0.23 mm/s, in reasonable agreement with the experimental values. Especially the difference between the IS reported for the two Fe sites in the crystal is nicely reproduced. Table 2 shows the extrapolated IS for some Fe-containing systems with different oxidation state and spin coupling. These complexes are

The atomic charge is an essential ingredient of many successful qualitative models. However, the assignment of a charge to atoms in molecules or crystals is not unique. The quantity is not measurable and also theoretically, there seems to be no consensus about the best way to extract these charges from electronic structure calculations given the numerous schemes to calculate charges. It is common practice to relate the Mo¨ssbauer IS with the oxidation state of Fe, and hence, also with its charge. Here, we check this assumption comparing the IS with Fe charges deduced from our calculations according to three standard schemes. Table 3 lists the formal ionic charges as well as Mulliken, Bader and LoProp charges of the Fe complexes mentioned in the previous section. The effect of dynamical correlation effects on the charges has been checked by comparing Mulliken and LoProp charges for CASSCF and CASPT2, which turn out to be virtually the same. Mulliken charges are strongly basis set dependent. They are generally considered to yield at best an indication of trends. Here, we observe that the gross Mulliken charges indeed follow more or less the trends given by the Bader and LoProp charges, but also some marked (unpredictable) deviations are observed, e.g. the differences of about 1 electron in the water complexes. For this reason, we will focus Table 3 Relation between experimental IS (in mm/s) and formal or calculated Fe charges 2þ

½FeðH2 OÞ6  ½FeF6 4 ½FeðNH3 Þ6 2þ ½FeCl4 2 ½FeðH2 OÞ6 3þ ½FeF6 3 ½FeBr4  ½FeCl4  ½FeðCOÞ6 2þ ½FeðCNÞ6 4 ½FeðCNÞ6 3 ½FeO4 2

IS

Formal

Mulliken

Bader

LoProp

1.39 1.34 1.23 0.9 0.5 0.48 0.25 0.19 0.15 0.02 0.13 0.69

2 2 2 2 3 3 3 3 2 2 3 6

2.53 1.77 1.67 0.98 3.41 1.87 0.72 0.83 0.67 0.75 1.44 1.40

1.67 1.74 1.52 1.34 2.28 2.39 1.90 1.79 0.76 0.87 1.60 1.98

1.56 1.58 1.45 1.26 2.24 2.29 1.56 1.71 0.67 0.68 1.79 1.82

A. Sadoc et al. / Chemical Physics Letters 454 (2008) 196–200

our attention on the Bader and LoProp charges, which give rather similar values and are (almost) independent of the basis set size. Comparison of the calculated charges of the complexes with the smallest and largest IS clearly illustrates the problem of relating this parameter directly to a charge of the Fe ion in the complex. The IS of +1.39 mm/s in ½FeðH2 OÞ6 2þ is connected to a Fe charge of approximately +1.6, while the Fe charge in ½FeO4 2 with an IS of 0.69 mm/s is around +1.9, a difference of only 0.3 electrons. Table 3 also shows that the IS cannot be related to the formal ionic charges. Instead, the comparison of the IS with the difference between formal ionic charge and calculated charge shows a regular pattern as demonstrated in Fig. 3. Small deviations from the ionic model as in ½FeðH2 OÞ6 2þ and ½FeF6 4 lead to large positive IS, whereas the highly covalent ½FeO4 2 complex has a large negative IS. Combined with the observation that the largest differential effect to qð0Þ arises from the orbitals centered at the ligands, it might be interesting to see whether the IS measures to some extent the contribution of the ligand to metal CT excitations. To

formal charge - calculated charge

5.0

4.0

3.0 0.5

2.0 0.0 -0.3

0.0

0.3

0.6

0.9

1.2

1.0

-0.5

answer this question, the multiconfigurational wave function is re-expressed in localized orbitals and a valence bond type analysis is made of the electronic structure. Table 4 decomposes the CASSCF wave function in terms of the aforementioned NCT, CT and DCT configurations, extended to the triple CT (TCT) and quadruple CT (QCT) configurations. The analysis of the wave function in terms of localized orbitals confirms that there is no correlation between the ionic charge of the TM and the measured IS. There is, however, a clear tendency of increasing importance of the charge transfer effects along the series with decreasing IS. Whereas the rather ionic 4 ½FeF6  complex has a 94% contribution of the NCT configuration and only 6% of the wave function can be ascribed to CT configurations, the NCT contribution has decreased to less than 50% for the more covalent ½FeCl4  complex. The only exceptions to the regular pat4 2þ tern are the low-spin ½FeðCNÞ6  and ½FeðCOÞ6  complexes. Here, the formal electronic configuration of Fe is ½1s2    3d6 ðt62g e0g Þ. The empty Fe-3d(eg) shell favours the transfer of electrons from ligand-centered eg orbitals into these Fe orbitals, leading to a strong CT contribution and also an exceptionally strong DCT contribution. 4. Conclusion

1.0

0.0 -1.0

199

0.0

0.5

1.0

1.5

IS [mm/s]

Fig. 3. Difference between calculated and formal ionic Fe charge as function of the IS. Squares are Bader charges and diamonds represent LoProp charges.

Although the direct calculation of Mo¨ssbauer isomer shifts is still a hard task, relating qð0Þ to this spectroscopic parameter opens a way to interpret the (sometimes rather complicated) Mo¨ssbauer spectra. The calibration of the calculated density at the nucleus against well-established experimental isomer shifts shows a linear relation between these two quantities as reported in previous studies [5–11]. It should be noted that this relation is both dependent on the computational scheme (CASSCF, CASPT2, DFT with different functionals, etc.) and the basis set used. However, the relation can be derived for relatively small complexes and is not difficult to obtain.

Table 4 Decomposition (in %) of the CASSCF wave function in terms of NCT and CT configurations and configurations in which two (DCT), three (TCT) or four (QCT) electrons are transferred to the metal Cluster

NCT 2þ

½FeðH2 OÞ6  ½FeF6 4 ½FeðNH3 Þ6 2þ ½FeCl4 2 ½FeðH2 OÞ6 3þ ½FeF6 3 ½FeBr4  ½FeCl4  ½FeðCOÞ6 2þ ½FeðCNÞ6 4 ½FeðCNÞ6 3 ½FeO4 2

93.2 94.1 89.5 78.6 77.1 80.0 38.1 45.5 16.6 14.6 43.8 0.0

CT

6.5 5.6 9.9 19.7 21.5 19.0 51.8 45.9 43.6 42.0 47.3 0.2

DCT

0.1 0.1 0.2 1.02 1.0 0.7 8.7 7.3 31.2 33.2 8.6 8.2

TCT

0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.3 7.3 8.7 0.0 44.5

QCT

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 37.0

d-Count Formal

Calculated

6 6 6 6 5 5 5 5 6 6 5 2

6.05 6.05 6.08 6.18 5.22 5.19 5.62 5.56 7.19 7.32 5.63 5.30

The number of Fe-3d electrons (d-count) derived from this analysis and the formal ionic d-count is also given. The Fe complexes are ordered by decreasing IS values.

200

A. Sadoc et al. / Chemical Physics Letters 454 (2008) 196–200

The largest contribution to qð0Þ arises from the electrons in the Fe-1s orbitals, but the largest differential effect originates from the contribution of the tails of the liganddominated orbitals. The analysis of the CASSCF wave function in terms of atomic-like orbitals confirms this observation. The complexes with the smallest contribution of CT configurations have the largest positive isomer shift. This shift reduces with increasing importance of the CT determinants to reach the extreme value of IS = 0.69 2 for the ½FeO4  complex, whose wave function is completely dominated by CT configurations. Finally, we observed that the isomer shift can be related to the covalency in the Fe-ligand bond via the difference between the formal ionic charge of the Fe ion and its computed charge. In other words, the isomer shift is a measure of the deviation from the ionic model. Large positive shifts are expected for highly ionic complexes where the computed charge is close to the formal charge, while small and negative shifts indicate strong deviations from the ionic model. Acknowledgements Financial support has been provided by the Spanish Ministry of Education and Science (CTQU2005-08459C02-02/BQU), and the Generalitat de Catalunya (2005SGR-00104). References [1] J.F. Derry, E. Bill, E. Bothe, S. DeBeer George, B. Mienert, F. Neese, K. Wieghardt, Science 312 (2006) 1937. [2] P. Gu¨tlich, H.A. Goodwin, in: Spin Crossover in Transition Metal Compounds I, Springer Verlag, Berlin, 2004, p. 1.

[3] M. Filatov, J. Chem. Phys. 127 (2007) 084101. [4] R. Kurian, M. Filatov, J. Chem. Theory Comput. 4 (2008) 278. [5] W.C. Nieuwpoort, D. Post, P.Th. van Duijnen, Phys. Rev. B 17 (1978) 91. [6] J. Blomquist, B.O. Roos, M. Sundbom, J. Chem. Phys. 55 (1970) 141. [7] K.J. Duff, Phys. Rev. B 9 (1973) 66. [8] F. Neese, Inorg. Chim. Acta 337 (2002) 181. [9] S. Sinnecker, L.D. Slep, E. Bill, F. Neese, Inorg. Chem. 44 (2004) 2245. [10] T. Liu, T. Lowell, W.-G. Han, L. Noodleman, Inorg. Chem. 42 (2003) 5244. [11] Y. Zhang, J. Mao, E. Oldfield, J. Am. Chem. Soc. 124 (2002) 7829. [12] A. Sadoc, C. de Graaf, R. Broer, Phys. Rev. B 75 (2007) 165116. [13] A. Sadoc, R. Broer, C. de Graaf, J. Chem. Phys. 126 (2007) 134709. [14] G. Karlstro¨m et al., Comput. Mater. Sci. 28 (2003) 222. [15] R. Pou-Ame´rigo, M. Mercha´n, I. Nebot-Gil, P.-O. Widmark, B.O. Roos, Theor. Chim. Acta 92 (1995) 149. ˚ . Malmqvist, B.O. Roos, Theor. Chim. Acta 77 [16] P.-O. Widmark, P.-A (1990) 291. [17] B.H. Botch, T.H. Dunning Jr., J.F. Harrison, J. Chem. Phys. 75 (1981) 3466. [18] K. Andersson, B.O. Roos, Chem. Phys. Lett. 191 (1992) 507. [19] M.A. Buijse, E.J. Baerends, J. Chem. Phys. 93 (1990) 4192. [20] A. Al-Abdalla, L. Seijo, Z. Barandiara´n, J. Chem. Phys. 109 (1998) 6396. [21] G.J.M. Janssen, W.C. Nieuwpoort, Int. J. Quantum Chem. 22 (1988) 679. [22] C. de Graaf, R. Broer, W.C. Nieuwpoort, Chem. Phys. 208 (1996) 35. [23] K. Pierloot, E. van Praet, L.G. Vanquickenborne, B.O. Roos, J. Phys. Chem 97 (1993) 12220. [24] M.-C. Heitz, C. Daniel, Chem. Phys. Lett. 246 (1995) 488. [25] R. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, Oxford, 1994. [26] L. Gagliardi, R. Lindh, G. Karlstro¨m, J. Chem. Phys. 121 (2004) 4494. [27] S. Nasu, T. Abe, K. Yamamoto, S. Endo, M. Takano, Y. Takeda, Hyperfine Interact. 144/145 (2002) 119.