Ab initio density matrix renormalization group study of magnetic coupling in dinuclear iron and chromium complexes Travis V. Harris, Yuki Kurashige, Takeshi Yanai, and Keiji Morokuma Citation: The Journal of Chemical Physics 140, 054303 (2014); doi: 10.1063/1.4863345 View online: http://dx.doi.org/10.1063/1.4863345 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Geometries and electronic structures of the ground and low-lying excited states of FeCO: An ab initio study J. Chem. Phys. 137, 244303 (2012); 10.1063/1.4769283 Density functional estimations of Heisenberg exchange constants in oligonuclear magnetic compounds: Assessment of density functional theory versus ab initio J. Chem. Phys. 131, 224316 (2009); 10.1063/1.3264570 High-performance ab initio density matrix renormalization group method: Applicability to large-scale multireference problems for metal compounds J. Chem. Phys. 130, 234114 (2009); 10.1063/1.3152576 Quantum chemistry using the density matrix renormalization group II J. Chem. Phys. 119, 4148 (2003); 10.1063/1.1593627 Density-functional study of intramolecular ferromagnetic interaction through m-phenylene coupling unit (II): Examination of functional dependence J. Chem. Phys. 113, 10486 (2000); 10.1063/1.1290008
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THE JOURNAL OF CHEMICAL PHYSICS 140, 054303 (2014)
Ab initio density matrix renormalization group study of magnetic coupling in dinuclear iron and chromium complexes Travis V. Harris,1 Yuki Kurashige,2 Takeshi Yanai,2 and Keiji Morokuma1,a) 1 2
Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan
(Received 29 October 2013; accepted 13 January 2014; published online 3 February 2014) The applicability of ab initio multireference wavefunction-based methods to the study of magnetic complexes has been restricted by the quickly rising active-space requirements of oligonuclear systems and dinuclear complexes with S > 1 spin centers. Ab initio density matrix renormalization group (DMRG) methods built upon an efficient parameterization of the correlation network enable the use of much larger active spaces, and therefore may offer a way forward. Here, we apply DMRG-CASSCF to the dinuclear complexes [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ . After developing the methodology through systematic basis set and DMRG M testing, we explore the effects of extended active spaces that are beyond the limit of conventional methods. We find that DMRGCASSCF with active spaces including the metal d orbitals, occupied bridging-ligand orbitals, and their virtual double shells already capture a major portion of the dynamic correlation effects, accurately reproducing the experimental magnetic coupling constant (J) of [Fe2 OCl6 ]2− with (16e,26o), and considerably improving the smaller active space results for [Cr2 O(NH3 )10 ]4+ with (12e,32o). For comparison, we perform conventional MRCI+Q calculations and find the J values to be consistent with those from DMRG-CASSCF. In contrast to previous studies, the higher spin states of the two systems show similar deviations from the Heisenberg spectrum, regardless of the computational method. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863345] term is added as follows:
I. INTRODUCTION
The development and application of theoretical methods for describing magnetic interactions in molecules and materials is a highly active field of research.1 The magnetic behavior is typically modeled by the phenomenological HeisenbergDirac-van Vleck (HDVV) Hamiltonian, defined for a system with two magnetic centers as Hˆ = −2J Sˆa Sˆb ,
(1)
where Sˆa and Sˆb are the spin operators for centers a and b. J is a coupling constant that is positive for ferromagnetic (F) and negative for antiferromagnetic (AF) spin alignment, and its magnitude indicates the strength of the interaction. Qualitatively correct J values are routinely calculated using broken-symmetry density functional theory (BS-DFT).2, 3 The strong dependence of the results on the adopted DFT functionals in some cases leads to quantitative accuracy, but it is not always possible to a priori choose the functional that gives the “correct” answer for a given system. Since only the high spin (HS) state can be represented by a single determinant, DFT J values are derived using BS and HS states. The J values can then be used to parameterize the HDVV Hamiltonian, and the spectrum of low-energy spin states will follow the Landé interval rule: E(S) − E(S − 1) = −2SJ.
(2)
However, some magnetic systems deviate from this pattern dictated by the bilinear equation (1), and thus a biquadratic a) Email: [email protected]
0021-9606/2014/140(5)/054303/10/$30.00
Hˆ = −2J Sˆa Sˆb + j (Sˆa Sˆb )2 ,
(3)
where j is the biquadratic coupling constant.4, 5 The origin of this deviation has been attributed to excited atomic states (i.e., non-Hund states) of transition metal complexes.6 Complete active space self consistent field (CASSCF)7, 8 calculations offer rigorous descriptions of the spin eigenfunctions, allowing the direct determination of J from the ground and first-excited spin state, which is ideal for numerous many-electron systems where the higher spin states are not significantly populated in the commonly used temperaturedependent magnetic susceptibility experiments. For weakly coupled systems, and those for which a more complete description of the magnetic interaction is desired, CASSCF enables the assessment of possible deviation from the Heisenberg spectrum by examining the higher spin states. However, CASSCF calculations using minimal active spaces consisting of only the singly occupied (often called magnetic) orbitals predict very weak J values, typically capturing only 20% of the coupling strength.9 A minimal active space includes the neutral and ionic (charge transfer) determinants that couple to produce the AF interaction according to Anderson’s mechanism of superexchange,10 but ligand to metal charge transfer (LMCT) states are also critical for mediating the coupling.11 The LMCT states can be considered by adding occupied bridging-ligand orbitals to the active space, but doing this alone has little effect because the LMCT and ionic configurations are too unstable; dynamic correlation is required for orbital relaxation.12, 13
140, 054303-1
© 2014 AIP Publishing LLC
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054303-2
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Multireference configuration interaction (MRCI) techniques can predict J values with high accuracy, but they are computationally demanding. Among these techniques, difference dedicated CI (DDCI)14 methods have been particularly useful for many magnetic systems,15–19 as they exclude at least the two-electron excitations from doubly occupied to virtual orbitals on the grounds that, according to quasidegenerate second order perturbation theory, those excitations do not contribute to energy differences between electronic states. DDCI-based analyses of the contributions to the magnetic interaction have shown the importance of LMCT and MLCT configurations for stabilizing the ionic determinants.20, 21 Further excitations on top of the LMCT configurations provide relaxation effects, manifesting in delocalization of the magnetic orbitals onto the bridging ligands, and stronger magnetic coupling.21, 22 Relaxing the MLCT states in AF systems has a dampening effect that also should be included for quantitative accuracy. With these in mind, more efficient methods have been devised involving truncation of the CI expansion to singles only, while adding ligand orbitals to the minimal, magnetic-orbital-only active space that is sufficient for DDCI.9, 23 However, despite these efficient CI methods, dinuclear Cr(III) complexes with 6 magnetic orbitals already approach the computational limit.24 A relatively efficient alternative to CI methods is the perturbation-theory-based CASPT2,25, 26 which qualitatively reproduces experimental J values when using a minimal active space.27 The accuracy can be improved by extending the active space with bridging ligand orbitals and virtual (doubleshell) orbitals corresponding to each occupied orbital in the active space,27 although there is still a notable failure of erroneously predicting deviations from the Heisenberg spectrum for weakly coupled (∼20 cm−1 ) systems.24 As a consequence of using extended active spaces, the practical limit of ∼16 electrons and orbitals (16,16) is quickly reached for oligonuclear systems or those containing metals with many unpaired electrons. An emerging method to overcome the active space limitation is the density matrix renormalization group (DMRG).28, 29 DMRG is commonly used in the field of condensed matter physics to study one-dimensional lattices and it has also been applied to two-dimensional magnetic systems such as the Kagome lattice.30 The development of ab initio DMRG with orbital optimization (DMRG-SCF or DMRGCASSCF)31–34 has enabled quantum chemical calculations with very large active spaces; for example, two of the authors studied a magnetic polycarbene using a (46,46) active space.35 Recently, algorithms for CASPT236 and MRCI37 based on active-space DMRG reference wavefunctions have also been developed, providing an opportunity for unprecedented accuracy in property predictions for large molecules with multireference character. In this study, we use the dinuclear, oxo-bridged molecules [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ (Figure 1) as model systems to test the capability of DMRG for predicting J values of transition-metal complexes. Although these complexes are only dinuclear, Fe(III) and Cr(III) centers already present a challenge for multireference methods. To the best of our knowledge, DDCI calculations have not been ac-
J. Chem. Phys. 140, 054303 (2014)
FIG. 1. Representations of the studied complexes. Hydrogen atoms have been omitted for clarity.
complished for any di-iron system, and extended-active-space CASPT2 is certainly out of reach, considering that the magnetic orbitals with double shell already put the active space at 20 orbitals without accounting for the ligands. Using DMRG, we can systematically examine the effect of including doubleshell orbitals in active spaces that exceed the conventional CASSCF limit. The magnetic coupling in [Fe2 OCl6 ]2− has been the subject of previous UHF,38 DFT,39, 40 and internally contracted MRCI (IC-MRCI)41 studies, the last of which found the unusual behavior of deviation from the Heisenberg spectrum in opposing directions with CASSCF and IC-MRCI. [Cr2 O(NH3 )10 ]4+ has also been studied using DFT42 and IC-MRCI43, 44 methods. IC-MRCI does not relax the coefficients of the reference configuration state functions (CSFs) in response to coupling with the doubly external CSFs, as it is the doubly external space that is internally contracted.45 Thus, some relaxation effects that contribute to the magnetic coupling are missing, unlike in the case of fully variational DDCI methods.46 Further errors are introduced by not including many internal orbitals in the correlation, especially the 3s,3p of first-row transition metals.47 Since the previous IC-MRCI studies of [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ , a more efficient algorithm has been devised that enables the correlation of many more internal orbitals than was previously possible.48 Here, we use this method, which we will call MRCI, to update the previous findings and to provide points of comparison for the DMRG calculations. II. COMPUTATIONAL DETAILS
Atomic coordinates for [Fe2 OCl6 ]2− were taken from the crystal structure of [PhCH2 N(CH3 )3 ]2 [Fe2 OCl6 ], which showed two conformations of the anion, linear and bent, referring to the Fe-O-Fe angle.49 The computational model was based on the bent structure with a Fe-O-Fe angle of 144.6◦ and symmetric Fe-O bond lengths of 1.761 Å. The model of [Cr2 O(NH3 )10 ]4+ was based on the crystal structure of the chloride salt with a linear Cr-O-Cr angle and Cr-O bond lengths of 1.821 Å. Ammonia ligands were placed in octahedral positions, each at the crystallographic average distance of 2.12 Å from Cr. Hydrogen positions were optimized using density functional theory at the B3LYP50, 51 /TZVP52 level with GAUSSIAN 0953 software. The ab initio calculations were carried out with relativistic atomic natural orbital (ANO-RCC) basis sets, using a series of contractions defined in Table I. Scalar relativistic effects were included via the second-order Douglas-KrollHess Hamiltonian (DKH2).54–57 In this study, all the DMRG
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J. Chem. Phys. 140, 054303 (2014)
TABLE I. Definition of ANO-RCC basis set contractions.a Name BS0 BS1 BS2 BS3 BS4 BS5 a
Fe,Cr
O
Cl
[4s,3p,2d] [5s,4p,3d] [6s,5p,3d,2f,1g] [6s,5p,3d,2f,1g] [7s,6p,4d,3f,2g,1h] [8s,7p,5d,4f,3g,2h]
[4s,3p,1d] [4s,3p,1d] [4s,3p,2d,1f] [4s,3p,2d,1f] [5s,4p,3d,2f,1g] [6s,5p,4d,3f,2g]
[4s,3p] [4s,3p] [5s,4p,2d] [5s,4p,2d,1f] [6s,5p,3d,2f,1g]
N
H
[2s,1p] [3s,2p] [3s,2p] [4s,3p,2d]
[1s] [2s] [2s] [2s]
Additional basis sets used only in the CAS(16,13)SCF study of basis set saturation (Figure 1) are given in Table S1.
calculations were carried out with the DMRG-CASSCF procedure31–34 to describe active space (or multireference) correlation. Hereafter, we refer to DMRG-CASSCF as simply DMRG, unless otherwise noted. Active orbitals were localized for DMRG calculations, because localized orbitals are convenient for mapping onto sites in a one-dimensional DMRG lattice and they are also advantageous in terms of computational efficiency. DMRG calculations were performed with a parallel code developed by Kurashige and Yanai,58 with spin-adapted wavefunctions.59 The initial localized orbitals were derived by Pipek-Mezey populationlocalization60 of natural orbitals from B3LYP calculations of the high-spin states (i.e., S = 5 for [Fe2 OCl6 ]2− , S = 3 for [Cr2 O(NH3 )10 ]4+ ). Several active spaces consisting of metal and bridging-ligand orbitals were employed and are described in Sec. III. The orbital order within the active space is known to affect the energy convergence with respect to the number of renormalized basis states (M). In general, for optimal efficiency and accuracy the orbitals should be ordered atom-wise and by alternating between valence and virtual correlating orbitals (i.e., double shell) of the same symmetry. The localized orbitals for the largest active space used for [Fe2 OCl6 ]2− are shown, in order, in Figure S1 of the supplementary information.66 Also note that generally the DMRG energy variationally converges to the exact value with increasing M. Conventional CASSCF and MRCI calculations were performed with MOLPRO.61, 62 The recently developed internally contracted MRCI algorithm48 was used, which allows correlation of more than the previous limit of 32 closed orbitals; thus, all valence and semi-core (metal 3s,3p) closed orbitals were correlated. The MRCI energies were corrected for sizeconsistency errors via the relaxed-reference Davidson method (+Q). State-specific orbitals were used in all calculations.
basis set dependence until BS5, where the DMRG result is in error due to an insufficient M value for such a large basis set. M = 1000 was the highest possible value, due to memory limitations. The CASSCF J is nearly converged with BS5, which is a quintuple-ζ basis set for Fe and O, quadruple-ζ for Cl (see Table I; additional basis sets used only for CASSCF are given in Table S1). Although on the basis of DFT calculations,39 the peripheral Cl ligands have very little influence on the magnetic coupling, the J values differ by 4-5 cm−1 between BS2 and BS3, in which only the Cl basis is increased from doubleζ to triple-ζ . Overall, BS3, with 310 basis functions and a J value
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THE JOURNAL OF CHEMICAL PHYSICS 140, 054303 (2014)
Ab initio density matrix renormalization group study of magnetic coupling in dinuclear iron and chromium complexes Travis V. Harris,1 Yuki Kurashige,2 Takeshi Yanai,2 and Keiji Morokuma1,a) 1 2
Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan
(Received 29 October 2013; accepted 13 January 2014; published online 3 February 2014) The applicability of ab initio multireference wavefunction-based methods to the study of magnetic complexes has been restricted by the quickly rising active-space requirements of oligonuclear systems and dinuclear complexes with S > 1 spin centers. Ab initio density matrix renormalization group (DMRG) methods built upon an efficient parameterization of the correlation network enable the use of much larger active spaces, and therefore may offer a way forward. Here, we apply DMRG-CASSCF to the dinuclear complexes [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ . After developing the methodology through systematic basis set and DMRG M testing, we explore the effects of extended active spaces that are beyond the limit of conventional methods. We find that DMRGCASSCF with active spaces including the metal d orbitals, occupied bridging-ligand orbitals, and their virtual double shells already capture a major portion of the dynamic correlation effects, accurately reproducing the experimental magnetic coupling constant (J) of [Fe2 OCl6 ]2− with (16e,26o), and considerably improving the smaller active space results for [Cr2 O(NH3 )10 ]4+ with (12e,32o). For comparison, we perform conventional MRCI+Q calculations and find the J values to be consistent with those from DMRG-CASSCF. In contrast to previous studies, the higher spin states of the two systems show similar deviations from the Heisenberg spectrum, regardless of the computational method. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863345] term is added as follows:
I. INTRODUCTION
The development and application of theoretical methods for describing magnetic interactions in molecules and materials is a highly active field of research.1 The magnetic behavior is typically modeled by the phenomenological HeisenbergDirac-van Vleck (HDVV) Hamiltonian, defined for a system with two magnetic centers as Hˆ = −2J Sˆa Sˆb ,
(1)
where Sˆa and Sˆb are the spin operators for centers a and b. J is a coupling constant that is positive for ferromagnetic (F) and negative for antiferromagnetic (AF) spin alignment, and its magnitude indicates the strength of the interaction. Qualitatively correct J values are routinely calculated using broken-symmetry density functional theory (BS-DFT).2, 3 The strong dependence of the results on the adopted DFT functionals in some cases leads to quantitative accuracy, but it is not always possible to a priori choose the functional that gives the “correct” answer for a given system. Since only the high spin (HS) state can be represented by a single determinant, DFT J values are derived using BS and HS states. The J values can then be used to parameterize the HDVV Hamiltonian, and the spectrum of low-energy spin states will follow the Landé interval rule: E(S) − E(S − 1) = −2SJ.
(2)
However, some magnetic systems deviate from this pattern dictated by the bilinear equation (1), and thus a biquadratic a) Email: [email protected]
0021-9606/2014/140(5)/054303/10/$30.00
Hˆ = −2J Sˆa Sˆb + j (Sˆa Sˆb )2 ,
(3)
where j is the biquadratic coupling constant.4, 5 The origin of this deviation has been attributed to excited atomic states (i.e., non-Hund states) of transition metal complexes.6 Complete active space self consistent field (CASSCF)7, 8 calculations offer rigorous descriptions of the spin eigenfunctions, allowing the direct determination of J from the ground and first-excited spin state, which is ideal for numerous many-electron systems where the higher spin states are not significantly populated in the commonly used temperaturedependent magnetic susceptibility experiments. For weakly coupled systems, and those for which a more complete description of the magnetic interaction is desired, CASSCF enables the assessment of possible deviation from the Heisenberg spectrum by examining the higher spin states. However, CASSCF calculations using minimal active spaces consisting of only the singly occupied (often called magnetic) orbitals predict very weak J values, typically capturing only 20% of the coupling strength.9 A minimal active space includes the neutral and ionic (charge transfer) determinants that couple to produce the AF interaction according to Anderson’s mechanism of superexchange,10 but ligand to metal charge transfer (LMCT) states are also critical for mediating the coupling.11 The LMCT states can be considered by adding occupied bridging-ligand orbitals to the active space, but doing this alone has little effect because the LMCT and ionic configurations are too unstable; dynamic correlation is required for orbital relaxation.12, 13
140, 054303-1
© 2014 AIP Publishing LLC
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054303-2
Harris et al.
Multireference configuration interaction (MRCI) techniques can predict J values with high accuracy, but they are computationally demanding. Among these techniques, difference dedicated CI (DDCI)14 methods have been particularly useful for many magnetic systems,15–19 as they exclude at least the two-electron excitations from doubly occupied to virtual orbitals on the grounds that, according to quasidegenerate second order perturbation theory, those excitations do not contribute to energy differences between electronic states. DDCI-based analyses of the contributions to the magnetic interaction have shown the importance of LMCT and MLCT configurations for stabilizing the ionic determinants.20, 21 Further excitations on top of the LMCT configurations provide relaxation effects, manifesting in delocalization of the magnetic orbitals onto the bridging ligands, and stronger magnetic coupling.21, 22 Relaxing the MLCT states in AF systems has a dampening effect that also should be included for quantitative accuracy. With these in mind, more efficient methods have been devised involving truncation of the CI expansion to singles only, while adding ligand orbitals to the minimal, magnetic-orbital-only active space that is sufficient for DDCI.9, 23 However, despite these efficient CI methods, dinuclear Cr(III) complexes with 6 magnetic orbitals already approach the computational limit.24 A relatively efficient alternative to CI methods is the perturbation-theory-based CASPT2,25, 26 which qualitatively reproduces experimental J values when using a minimal active space.27 The accuracy can be improved by extending the active space with bridging ligand orbitals and virtual (doubleshell) orbitals corresponding to each occupied orbital in the active space,27 although there is still a notable failure of erroneously predicting deviations from the Heisenberg spectrum for weakly coupled (∼20 cm−1 ) systems.24 As a consequence of using extended active spaces, the practical limit of ∼16 electrons and orbitals (16,16) is quickly reached for oligonuclear systems or those containing metals with many unpaired electrons. An emerging method to overcome the active space limitation is the density matrix renormalization group (DMRG).28, 29 DMRG is commonly used in the field of condensed matter physics to study one-dimensional lattices and it has also been applied to two-dimensional magnetic systems such as the Kagome lattice.30 The development of ab initio DMRG with orbital optimization (DMRG-SCF or DMRGCASSCF)31–34 has enabled quantum chemical calculations with very large active spaces; for example, two of the authors studied a magnetic polycarbene using a (46,46) active space.35 Recently, algorithms for CASPT236 and MRCI37 based on active-space DMRG reference wavefunctions have also been developed, providing an opportunity for unprecedented accuracy in property predictions for large molecules with multireference character. In this study, we use the dinuclear, oxo-bridged molecules [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ (Figure 1) as model systems to test the capability of DMRG for predicting J values of transition-metal complexes. Although these complexes are only dinuclear, Fe(III) and Cr(III) centers already present a challenge for multireference methods. To the best of our knowledge, DDCI calculations have not been ac-
J. Chem. Phys. 140, 054303 (2014)
FIG. 1. Representations of the studied complexes. Hydrogen atoms have been omitted for clarity.
complished for any di-iron system, and extended-active-space CASPT2 is certainly out of reach, considering that the magnetic orbitals with double shell already put the active space at 20 orbitals without accounting for the ligands. Using DMRG, we can systematically examine the effect of including doubleshell orbitals in active spaces that exceed the conventional CASSCF limit. The magnetic coupling in [Fe2 OCl6 ]2− has been the subject of previous UHF,38 DFT,39, 40 and internally contracted MRCI (IC-MRCI)41 studies, the last of which found the unusual behavior of deviation from the Heisenberg spectrum in opposing directions with CASSCF and IC-MRCI. [Cr2 O(NH3 )10 ]4+ has also been studied using DFT42 and IC-MRCI43, 44 methods. IC-MRCI does not relax the coefficients of the reference configuration state functions (CSFs) in response to coupling with the doubly external CSFs, as it is the doubly external space that is internally contracted.45 Thus, some relaxation effects that contribute to the magnetic coupling are missing, unlike in the case of fully variational DDCI methods.46 Further errors are introduced by not including many internal orbitals in the correlation, especially the 3s,3p of first-row transition metals.47 Since the previous IC-MRCI studies of [Fe2 OCl6 ]2− and [Cr2 O(NH3 )10 ]4+ , a more efficient algorithm has been devised that enables the correlation of many more internal orbitals than was previously possible.48 Here, we use this method, which we will call MRCI, to update the previous findings and to provide points of comparison for the DMRG calculations. II. COMPUTATIONAL DETAILS
Atomic coordinates for [Fe2 OCl6 ]2− were taken from the crystal structure of [PhCH2 N(CH3 )3 ]2 [Fe2 OCl6 ], which showed two conformations of the anion, linear and bent, referring to the Fe-O-Fe angle.49 The computational model was based on the bent structure with a Fe-O-Fe angle of 144.6◦ and symmetric Fe-O bond lengths of 1.761 Å. The model of [Cr2 O(NH3 )10 ]4+ was based on the crystal structure of the chloride salt with a linear Cr-O-Cr angle and Cr-O bond lengths of 1.821 Å. Ammonia ligands were placed in octahedral positions, each at the crystallographic average distance of 2.12 Å from Cr. Hydrogen positions were optimized using density functional theory at the B3LYP50, 51 /TZVP52 level with GAUSSIAN 0953 software. The ab initio calculations were carried out with relativistic atomic natural orbital (ANO-RCC) basis sets, using a series of contractions defined in Table I. Scalar relativistic effects were included via the second-order Douglas-KrollHess Hamiltonian (DKH2).54–57 In this study, all the DMRG
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.174.255.116 On: Fri, 16 Jan 2015 20:16:43
054303-3
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J. Chem. Phys. 140, 054303 (2014)
TABLE I. Definition of ANO-RCC basis set contractions.a Name BS0 BS1 BS2 BS3 BS4 BS5 a
Fe,Cr
O
Cl
[4s,3p,2d] [5s,4p,3d] [6s,5p,3d,2f,1g] [6s,5p,3d,2f,1g] [7s,6p,4d,3f,2g,1h] [8s,7p,5d,4f,3g,2h]
[4s,3p,1d] [4s,3p,1d] [4s,3p,2d,1f] [4s,3p,2d,1f] [5s,4p,3d,2f,1g] [6s,5p,4d,3f,2g]
[4s,3p] [4s,3p] [5s,4p,2d] [5s,4p,2d,1f] [6s,5p,3d,2f,1g]
N
H
[2s,1p] [3s,2p] [3s,2p] [4s,3p,2d]
[1s] [2s] [2s] [2s]
Additional basis sets used only in the CAS(16,13)SCF study of basis set saturation (Figure 1) are given in Table S1.
calculations were carried out with the DMRG-CASSCF procedure31–34 to describe active space (or multireference) correlation. Hereafter, we refer to DMRG-CASSCF as simply DMRG, unless otherwise noted. Active orbitals were localized for DMRG calculations, because localized orbitals are convenient for mapping onto sites in a one-dimensional DMRG lattice and they are also advantageous in terms of computational efficiency. DMRG calculations were performed with a parallel code developed by Kurashige and Yanai,58 with spin-adapted wavefunctions.59 The initial localized orbitals were derived by Pipek-Mezey populationlocalization60 of natural orbitals from B3LYP calculations of the high-spin states (i.e., S = 5 for [Fe2 OCl6 ]2− , S = 3 for [Cr2 O(NH3 )10 ]4+ ). Several active spaces consisting of metal and bridging-ligand orbitals were employed and are described in Sec. III. The orbital order within the active space is known to affect the energy convergence with respect to the number of renormalized basis states (M). In general, for optimal efficiency and accuracy the orbitals should be ordered atom-wise and by alternating between valence and virtual correlating orbitals (i.e., double shell) of the same symmetry. The localized orbitals for the largest active space used for [Fe2 OCl6 ]2− are shown, in order, in Figure S1 of the supplementary information.66 Also note that generally the DMRG energy variationally converges to the exact value with increasing M. Conventional CASSCF and MRCI calculations were performed with MOLPRO.61, 62 The recently developed internally contracted MRCI algorithm48 was used, which allows correlation of more than the previous limit of 32 closed orbitals; thus, all valence and semi-core (metal 3s,3p) closed orbitals were correlated. The MRCI energies were corrected for sizeconsistency errors via the relaxed-reference Davidson method (+Q). State-specific orbitals were used in all calculations.
basis set dependence until BS5, where the DMRG result is in error due to an insufficient M value for such a large basis set. M = 1000 was the highest possible value, due to memory limitations. The CASSCF J is nearly converged with BS5, which is a quintuple-ζ basis set for Fe and O, quadruple-ζ for Cl (see Table I; additional basis sets used only for CASSCF are given in Table S1). Although on the basis of DFT calculations,39 the peripheral Cl ligands have very little influence on the magnetic coupling, the J values differ by 4-5 cm−1 between BS2 and BS3, in which only the Cl basis is increased from doubleζ to triple-ζ . Overall, BS3, with 310 basis functions and a J value
