Assessing the density functional theory-based multireference configuration interaction (DFT/MRCI) method for transition metal complexes Daniel Escudero and Walter Thiel Citation: The Journal of Chemical Physics 140, 194105 (2014); doi: 10.1063/1.4875810 View online: http://dx.doi.org/10.1063/1.4875810 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A combined DFT and restricted open-shell configuration interaction method including spin-orbit coupling: Application to transition metal L-edge X-ray absorption spectroscopy J. Chem. Phys. 138, 204101 (2013); 10.1063/1.4804607 The electronic spectrum and photodissociation of dinitrogen tetroxide ( N 2 O 4 ) : Multireference configuration interaction studies J. Chem. Phys. 133, 144311 (2010); 10.1063/1.3498899 Benchmarks for electronically excited states: Time-dependent density functional theory and density functional theory based multireference configuration interaction J. Chem. Phys. 129, 104103 (2008); 10.1063/1.2973541 Configuration interaction based on constrained density functional theory: A multireference method J. Chem. Phys. 127, 164119 (2007); 10.1063/1.2800022 Density functional theory and multireference configuration interaction studies on low-lying excited states of Ti O 2 J. Chem. Phys. 126, 034313 (2007); 10.1063/1.2429062
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THE JOURNAL OF CHEMICAL PHYSICS 140, 194105 (2014)
Assessing the density functional theory-based multireference configuration interaction (DFT/MRCI) method for transition metal complexes Daniel Escuderoa) and Walter Thiela) Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mülheim an der Ruhr, Germany
(Received 10 March 2014; accepted 29 April 2014; published online 20 May 2014) We report an assessment of the performance of density functional theory-based multireference configuration interaction (DFT/MRCI) calculations for a set of 3d- and 4d-transition metal (TM) complexes. The DFT/MRCI results are compared to published reference data from reliable highlevel multi-configurational ab initio studies. The assessment covers the relative energies of different ground-state minima of the highly correlated CrF6 complex, the singlet and triplet electronically excited states of seven typical TM complexes (MnO4 − , Cr(CO)6 , [Fe(CN)6 ]4− , four larger Fe and Ru complexes), and the corresponding electronic spectra (vertical excitation energies and oscillator strengths). It includes comparisons with results from different flavors of timedependent DFT (TD-DFT) calculations using pure, hybrid, and long-range corrected functionals. The DFT/MRCI method is found to be superior to the tested TD-DFT approaches and is thus recommended for exploring the excited-state properties of TM complexes. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4875810] I. INTRODUCTION
The theoretical study of electronically excited states is significantly more challenging for transition metal (TM) complexes than for organic and main-group compounds. This is due to the more complicated electronic structure of TM complexes, which involves the d shell of the metal surrounded by ligands with different acceptor and donor properties. Consequently, there are electronic transitions of very different nature, i.e., (i) metal-centered (MC), (ii) metal-to-ligand charge transfer (MLCT), (iii) ligand-to-metal CT (LMCT), and (iv) intra-ligand (IL) transitions, which may all contribute to the UV/Vis spectra of TM complexes. A computational method that aims at an accurate and balanced description of these excited states needs to deal with dynamic and nondynamic correlation effects equally well, since these states often have an inherently multi-configurational character. Therefore, theoretical calculations of electronic spectra of TM complexes have not yet reached a “black box” status.1 Multiconfigurational methods, such as the complete-active-space second-order perturbation theory (CASPT2) or the restrictedactive-space PT2 (RASPT2) method, are in principle well suited for these complicated electronic structure situations. However, these methods are not yet the “gold standard” for the spectroscopy of TM complexes, since their success is critically dependent upon their judicious application, particularly with regard to the selection of the active space,2 and also since they are still restricted to small and medium-size systems. Excited-state methods based on density functional theory (DFT), such as the SCF method or time-dependent DFT (TD-DFT) are less costly, but also less reliable, esa) Authors to whom correspondence should be addressed. Electronic ad-
dresses: [email protected] and [email protected]
0021-9606/2014/140(19)/194105/8/$30.00
pecially when compared to the performance of DFT for ground-state properties of TM complexes. An alternative DFT-based method for electronically excited states makes use of Kohn-Sham (KS) orbitals in a multi-reference configuration interaction (MRCI) framework: In this DFT/MRCI method,3 dynamic correlation effects are captured by the KS-DFT treatment while non-dynamic correlation effects are treated at the MRCI level. To alleviate problems arising from the possible double counting of dynamical correlation effects, some off-diagonal elements are scaled by incorporating five global empirical parameters, which have been determined by fitting to experimental data of ten reference molecules.3, 4 The DFT/MRCI method has the potential of being used in a “black-box” manner, and it has already been proven successful for a number of organic chromophores.5 A parallel code has been recently developed6 to facilitate the calculation of electronic spectra of larger molecules. However, there are yet only few DFT/MRCI applications to TM complexes, for example, tests for ferrocene,3 heptacyanovanadate(III),7 and the uranyl(VI) ion.8 In this contribution, we present a DFT/MRCI study of the electronically excited states of several TM complexes, for which reliable highly accurate ab initio data have recently become available in the literature. To ensure that the DFT/MRCI results are directly comparable to these reference data, we use the same geometries as in the original papers. For most of the studied complexes, we also provide an assessment of TD-DFT methods using different functionals (i.e., pure, hybrid, and long-range corrected functionals). Non-dynamic correlation effects are not only crucial for TM spectroscopy, but also for the bonding in ground-state TM complexes. Therefore, we also evaluate the performance of the DFT/MRCI method to obtain the relative energies of different structural minima of CrF6 , which has been recently revisited using the RASPT2 method.9
140, 194105-1
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II. COMPUTATIONAL DETAILS
The geometries of all compounds included in this study were taken either from available X-ray data or from groundstate optimizations reported previously. Single-point DFT/ MRCI and TD-DFT calculations were performed at these geometries. In the DFT/MRCI case, the initial KS-BHLYP calculations were carried out with the TURBOMOLE program,10 and the subsequent MRCI calculations with the DFT/MRCI code3 using the def2-TZVP basis set (unless noted otherwise). Initial reference configurations were generated by promoting up to two electrons in an active space of ten electrons in ten orbitals, and an energy-based selection criterion (with a threshold value of 1.0 hartree) was used to disregard configurations. Standard DFT/MRCI parameters for singlet and triplet states were employed. The DFT/MRCI code supports only Abelian groups, and hence most of the computations were done in C2v symmetry. The TD-DFT calculations were performed using the Gaussian0911 program, the 6-311G* basis set and the following functionals: (i) the pure PBE functional, (ii) the hybrid PBE0 functional (25% of exact exchange), and (iii) the long-range corrected CAMB3LYP functional.
III. RESULTS AND DISCUSSION
This section is organized as follows: in Subsection III A, the DFT/MRCI relative energies of different structural minima of CrF6 (1) are presented. Subsections III B–III D are dedicated to TM spectroscopy. Thus, the DFT/MRCI vertical excitation energies and oscillator strengths of a benchmark set of molecules are compared to the best theoretical estimates from the literature. Subsection III B addresses the tetrahedral MnO4 − complex (2). In Subsection III C the spectroscopic properties of the octahedral Cr(CO)6 (3) and [Fe(CN)6 ]4− (4) complexes are investigated. Subsection III D deals with the spectroscopy of larger complexes, i.e., [Fe(CN)5 (py)]3− (5, py:pyridine), [Fe(bpy)3 ]2+ (6, bpy:bipyridine), trans(Cl)Ru(bpy)Cl2 (CO)2 (7, Ru-complex-1), and a second Rucomplex-2 (8). Figure 1 shows the benchmark set of molecules. The selection of this set is motivated by the availability of reliable theoretical data in the literature and by the objective to cover different 3d and 4d metals, different oxidation states, and excited states of different character, in order to allow for a general assessment.
A. TM bonding: Octahedral and trigonal prismatic structures of CrF6
The CrF6 molecule is a textbook example of how nondynamical correlation effects may affect TM complexes. The high oxidation state of chromium (+VI) leads to highly covalent metal-ligand bonds and consequently to strong static correlation effects already in the ground-state configuration. Two local minimum structures, i.e., the octahedral (Oh ) and trigonal prismatic (D3h ) complexes, are found to be adiabatically close-lying (see Figure 1). Their relative energies at different computational levels are given in Table I. Pier-
FIG. 1. Benchmark set of TM complexes used in this study.
loot and Roos studied this challenging system already in the 1990s.12 Their early CASPT2(10,10) calculations with a minimum active space (10 electrons, 10 orbitals) overestimated the stability of the Oh structure, as compared to CCSD(T) results (coupled-cluster singles and doubles with perturbative triples.) According to recent RASPT2 calculations,9 this deficiency of the CASPT2(10,10) results was caused by the small size of the original (10,10) active space. To get accurate relative energies for CrF6 , it is crucial to include the whole set of F 2p orbitals, which are involved in strong static correlation effects. Such an extension of the active space is prohibitive for CASPT2, but still affordable (though very expensive) for RASPT2. The RASPT2(42,26) values of 17.8 and 16.5 kcal/mol9 in Table I are considered as our best theoretical estimates. The relative energy (E) at the DFT/MRCI level was computed using the D3d and Oh geometries from previous studies.9, 12 The DFT/MRCI value of 22.0 kcal/mol (Table I) is in fair agreement with the best RASPT2 estimate, which supports the use of DFT/MRCI for describing TM bonding in electronically complicated ground-state complexes. Concerning computational effort, we note that the DFT/MRCI wavefunction for CrF6 involves 165 815 configuration state functions (CSFs) compared with several million CSFs in the RASPT2(42,26) wavefunction (with the exact TABLE I. Relative energy (kcal/mol) of the D3h minimum structure with respect to the Oh minimum structure of CrF6 .
CASPT2(10,10)a E (kcal/mol) a b
49.0
RASPT2(42,4,0;16,10,0)b RASPT2(42,6,6;21,0,5)b DFT/MRCI 17.8 16.5
22.0
From Ref. 12. From Ref. 9. Value in italics taken from RASPT2 calculation specified in italics.
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TABLE II. Selected electronic transition energies (in eV) and oscillator strengths (values in parentheses) of MnO4 − at different levels.
State (main configuration)
RASPT2(24,17)a CASPT2 in 17ob
1 1 T1 (1t1 → 2e) 11 T2 (1t1 → 2e) 2 1 T1 (2t2 → 2e) 2 1 T2 (1t1 → 3t2 , 2t2 → 2e) 1 1 E (1t1 → 3t2 ) 1 1 A2 (1t1 → 3t2 ) 2 1 A1 (2t2 → 3t2 ) 3 1 T1 (1t1 → 3t2 ) 3 1 T2 (2t1 → 2e, 2t2 → 3t2 ) a b c
1.93 (0.000) 2.33 (0.004) 2.28 3.39 (0.000) 3.53 (0.002) 3.46 3.90 (0.000) 3.89 (0.000) 5.10 (0.000) 3.93 (0.000) 4.20 (0.006) 3.90
DFT/MRCI
SAC-CIc
TD-DFT (SAOP)a
2.38/2.38/2.49 (0.000) 2.74/2.80/2.80 (0.013)
2.18 (0.000) 2.57 (0.020)
2.66 (0.000) 3.08 (0.007)
4.05/4.05/4.10 (0.000) 4.21/4.21/4.23 (0.004)
3.33 (0.000) 3.58 (0.045)
3.91 (0.000) 4.12 (0.002)
4.38/4.39 (0.000) 4.41 (0.000) 4.81 (0.000) 4.37/4.37/4.37 (0.000) 4.81/4.81/4.84 (0.022)
3.41 (0.000) 4.46 (0.000) 5.41 (0.000) 4.12 (0.000) 3.72 (0.136)
4.50 (0.000) 4.31 (0.000) 5.72 (0.000) 4.44 (0.00) 5.01 (0.009)
From Ref. 20. Values in italics taken from Ref. 19. From Ref. 13(a).
number depending on the chosen RASPT2 options). Hence, the use of DFT/MRCI leads to very large computational savings.
B. Spectroscopic properties of a tetrahedral complex: MnO4 −
The geometry of MnO4 − was taken from experiment, as in other benchmark studies.13 The electronic spectrum of permanganate is among the most studied in TM spectroscopy. It has recurrently been used as a benchmark system for assessing the performance of quantum-chemical methods for excited states. Early Hartree-Fock (HF) and smallscale CI studies14 were followed by DFT studies, both at the SCF15,13(b) and TD-DFT16 levels. MnO4 − has also been investigated with coupled cluster (CC) theory, particularly with the equation-of-motion CC (EOM-CC), similarity transformed EOM-CC (STEOM-CC), and extended-STEOM-CC methods,17 as well as with the symmetry-adapted cluster CI (SAC-CI) approach.13(a) It is the paradigm of how important non-dynamic correlation effects may become in TM spectroscopy. As for CrF6 , problems arise from the high covalency of the metal-ligand bonds in permanganate (due to the high oxidation state of Mn, i.e., +VII), which gives rise to strong correlation effects already in the ground state, involving excitations from occupied O 2p orbitals to virtual Mn 3d orbitals. Furthermore, these correlation effects vary notably in the excited states,18 some of which have important contributions from doubly excitations.19 For these reasons, singlereference methods face considerable difficulties for MnO4 − , as demonstrated for the otherwise very accurate extendedSTEOM-CC method (with errors of up to 1 eV).17 Due to its inherent complexity the assignment of the permanganate spectrum has actually undergone several changes throughout the years, and a general consensus has been reached only very recently. Reliable theoretical results for the electronic spectrum of MnO4 − were obtained in the last few years using the
CASPT219 and RASPT220 methods in combination with extended active spaces. As in CrF6 , the inclusion of many doubly occupied O 2p orbitals was crucial to get converged results. The best theoretical estimates for permanganate from the RASPT2(24,17) level of theory20 show perfect agreement with the experimental data. The active space in these RASPT2 calculations included the 12 highest doubly occupied O 2p orbitals and the 5 virtual antibonding orbitals of mainly Mn 3d character. Table II lists the best published RASPT2 and CASPT2 results for the excitation energies and oscillator strengths of selected excited states of MnO4 − , as well as the presently computed DFT/MRCI values and both SAC-CI and TD-DFT values from the literature. The spectrum of MnO4 − is dominated by dipole-allowed LMCT excitations, which in Td symmetry correspond to T2 states. Results for the lowest T1 , T2 , A1 , A2 , and E states are collected in Table II. We note that in C2v symmetry (used in the DFT/MRCI calculations) the T1 , T2 , and E states are decomposed into a1 + b1 + b2 , a2 + b1 + b2 , and a1 + a2 states, respectively. DFT/MRCI consistently overestimates the excitation energies compared with RASPT2; the overall agreement is reasonable, with maximum absolute deviations of ca. 0.65 eV (see, e.g., 2 1 T1 and 2 1 T2 ). Compared with DFT/MRCI, the SAC-CI results are generally somewhat closer to RASPT2, with maximum absolute errors up to 0.6 eV (see, e.g., 1 1 A2 ), but the deviations are less uniform. The TD-DFT results show the worst agreement with RASPT2, with maximum absolute deviations of ca. 0.8 eV (i.e., the 3 1 T2 state). Concerning the oscillator strengths of the dipole-allowed transitions, only the DFT/MRCI and TD-DFT values reproduce the RASPT2 reference data well, while the SAC-CI values are much too high. In conclusion, only high-level multi-configurational methods appear capable of reproducing the spectrum of permanganate within an error bar of 0.1 eV. A likely explanation of the problems of the DFT/MRCI method in the highly correlated permanganate is the small CI expansion, with a (10,10) active space, which is not able to recover all the non-dynamic correlation effects.
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TABLE III. Selected singlet and triplet electronic transition energies (in eV) and oscillator strengths (in parentheses) of Cr(CO)6 at different levels of theory.
State/assignment 1 1 T1g 1 1 T1u 2 1 T1u 1 3 T1g 1 3 T2g
(1 MC) (1 MLCT) (1 MLCT) (3 MC) (3 MC)
RASPT2(10,0,4;0,10,14)a CASPT2(10,10)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
4.98 (0.000)a 4.50 (0.20)a 5.42 (2.57)a 4.28b 4.64b
4.90/4.76/4.75 (0.000) 4.71/4.70/4.70 (0.074) 5.71 (0.768) 4.53/4.54/4.62 4.54/4.71/4.72
4.99 (0.000) 4.83 (0.029) 6.42 (0.702) 4.48 4.76
5.08 (0.000) 4.71 (0.021) 6.31 (0.065) 4.50 4.79
5.37 (0.000) 4.14 (0.009) 5.78 (0.513) 4.89 5.01
Expt.c
4.44 (0.25) 5.48 (2.30)
a From Ref. 9. RAS(n,l,m;I,j,k) notation: n is the number of active electrons, l is the maximum number of holes in RAS1; m is the maximum of electrons in RAS3; and i, j, and k are the number of orbitals in RAS1, RAS2, and RAS3, respectively. b From Ref. 21. c From Ref. 26.
C. Spectroscopic properties of octahedral complexes: Cr(CO)6 and [Fe(CN)6 ]4−
The geometries of Cr(CO)6 and [Fe(CN)6 ]4− were again taken from experiment, as in other previous benchmark studies, see Refs. 21 and 22, respectively. The electronic spectrum of Cr(CO)6 has been much studied, both experimentally and in particular theoretically, using SC-DFT,23 TD-DFT,24 CASPT2,21 and coupled-cluster methods.25 The assignment of the spectra has been less controversial (apart from the position of the MC states) than in the case of MnO4 − , consistent with the less pronounced multi-configurational character of the ground and excited states of Cr(CO)6. A recent RASPT2 study with an extended active space, i.e., (10e, 24o), has provided converged excitation energies close to the experimental data (within 0.05 eV).9 These RASPT2 values are adopted as our best theoretical estimates. The experimental spectrum is dominated by the symmetry-allowed 1 T1u excitations of MLCT character, but there are also symmetry-forbidden MC excitations.26 Though being optically forbidden, the MC states are very important in Cr(CO)6 photochemistry, since they can be populated in the course of photochemical deactivation, similarly to other metal-carbonyl complexes. They are located higher in energy than the T1u states. Table III lists spectroscopic data for the main singlet and triplet electronic states of Cr(CO)6 obtained at the RASPT2, DFT/MRCI, and TD-DFT levels (employing different functional in the latter case). Due to the C2v symmetry constraints in the DFT/MRCI calculations, the original Oh states are unfolded into states of lower symmetry. At the DFT/MRCI level, the position of the lowest 1 MC state (1 1 T1g ) is well reproduced (within 0.1–0.2 eV), and the 1 MLCT transitions are also predicted with acceptable accuracy (within 0.25 eV of the RASPT2 values). We note that an even closer agreement for the MLCT states has been reported in previous DFT/MRCI calculations using newly optimized parameters.27 On the other hand, there are much larger discrepancies in the case of the TD-DFT values, which are moreover very functional-dependent. Among the tested functionals, the long-range TD-CAM-B3LYP functional outperforms the other functionals for the MC excitation energies. To the contrary, the hybrid TD-PBE0 and the pure TD-PBE functionals best reproduce the MLCT transitions. Hence, in spite of being designed for accurately describing CT states in organic compounds, TD-CAM-B3LYP is apparently not suited for dealing with MLCT states in TM
spectroscopy, as has been pointed out previously for other TM complexes.28, 29 TD-PBE0 (TD-PBE) overestimate (underestimate) the excitation energy of the 11 T1u state by ca. 0.2 and 0.35 eV, respectively. Even larger are the discrepancies for the high-lying MLCT state (predicted too high by ca. 0.35, 0.8, and 1.0 eV with PBE, PBE0, and CAM-B3LYP, respectively). Concerning the triplet excited states, DFT/MRCI provides reliable energies for the 3 MC states (see Table III). The performance of the TD-DFT methods for the 3 MC states follows the same trends as observed for the 1 MC states. Finally, oscillator strengths are generally underestimated at both the DFT/MRCI and TD-DFT levels (compared with RASPT2). Among all the DFT-based methods, DFT/MRCI gives the most realistic ratio between the intensities of the symmetry-allowed transitions. Gathering all data on electronic excitation energies and oscillator strengths, DFT/MRCI emerges as the best DFTbased approach to reproduce the spectrum of Cr(CO)6 with acceptable accuracy. Given their rather poor performance for Cr(CO)6 , it seems advisable to use TD-DFT methods only with much caution for the assignment of electronic spectra of metal-carbonyl complexes. The electronic spectrum of [Fe(CN)6 ]4− has been studied theoretically (CASPT2)22 and experimentally.30 It could be rather well assigned already in the 1990s on the basis of CASPT2(10,10) calculations with a relatively small active space. Compared with MnO4 − , [Fe(CN)6 ]4− has less covalent metal-ligand bonds, and consequently the effect of enlarging the active space to improve the description of non-dynamic correlation effects is less pronounced. The interpretation of its electronic spectrum is also facilitated by the fact that the 1 MC and 1 MLCT states are energetically well separated, contrary to the situation in Cr(CO)6 . Table IV lists the computed excitation energies and oscillator strengths for the main electronic states of [Fe(CN)6 ]4− , either of singlet and triplet character, obtained at different levels of theory. The experimental values are also included for comparison since the symmetry-allowed MLCT excitations were not computed in the CASPT2 study. The position of the low-lying 1 MC state (1 1 T1g ) is underestimated by ca. 0.4 eV at the DFT/MRCI level, whereas TDCAM-B3LYP and TD-PBE0 (but not TD-PBE) better match the CASPT2 value. This is also the case for the higher lying 1 MC state (1 1 T2g ), where the DFT/MRCI value is too high by ca. 0.65 eV compared with CASPT2 (but within 0.35–0.2 eV of the experimental data). One should thus explore whether
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TABLE IV. Selected singlet and triplet electronic transition energies (in eV) and oscillator strengths (in parentheses) of [Fe(CN)6 ]4− at different levels of theory. State/assignment 1 1 T1g 3 1 T1u 4 1 T1u 1 1 T2g 1 3 T1g a b
(1 MC) (1 MLCT) (1 MLCT) (1 MC) (3 MC)
CASPT2(10,10)a
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
Expt.b
3.60 ... ... 4.33 2.67
3.23/3.20/3.20 (0.000) 5.46/5.46/5.47 (0.060) 6.05/6.04/6.04 (0.072) 4.98/4.98/4.99(0.000) 2.58/2.58/2.61
3.65 (0.000) 6.97 (0.072) 7.66 (0.125) 4.40 (0.000) 2.92
3.64 (0.000) 6.53 (0.055) 7.16 (0.121) 4.33 (0.065) 2.83
4.15 (0.000) 5.27 (0.024) 5.85 (0.100) 4.23 (0.000) 3.52
3.80–3.94 5.69–5.89 6.20 4.43–4.77 2.94
From Ref. 22. From Ref. 30.
the (10,10) active space is large enough to describe this state properly, especially in view of the missing 3d double-shell correlation effects, which are crucial for the proper description of the MC states.19 In the case of the 1 MLCT states, only DFT/MRCI is able to predict them correctly within an accuracy of 0.2 eV; as in the case of Cr(CO)6 , the TD-DFT values are much too high at the CAM-B3LYP and PBE0 levels (e.g., by up to 1.4 eV for CAM-B3LYP) and moderately too low for PBE. Consequently, only DFT/MRCI is able to describe all the different electronic excitations of [Fe(CN)6 ]4− in a balanced manner. Concerning the low-lying triplet excited state, i.e., 1 3 T1g , the DFT/MRCI value is slightly too low compared with CASPT2 and experiment. D. Spectroscopic properties of larger complexes: [Fe(CN)5 (py)]3− , [Fe(bpy)3 ]2+ , Ru-complexes 1 and 2
The progress in the field of multi-configurational methods within the last years has made it feasible to calculate the excited states of larger TM complexes, even though these rather time-demanding and intricate computations have not yet achieved a routine status. Hence, only few large TM complexes, some of which are included in this subsection, have been studied with CASPT2/RASPT2 methods using extended active spaces that are capable of accounting for all desired correlation effects. Despite their limited accuracy, TD-DFT methods are instead often used for routine calculations on the excited states of such TM complexes, which may contain different ligands giving rise to different kinds of excitations (i.e., LMCT, MLCT, MC, and LC) in the UV/Vis region. There-
fore, it is crucial to describe all these excitations in a balanced manner in order to correctly assign the experimental spectrum and to identify the relevant photodeactivation channels. In this subsection, we will assess the performance of the DFT/MRCI method and compare it to different flavors of TD-DFT. Polypyridyl complexes of Fe(II) and Ru(II) have attracted much interest due to their unique photophysical and magnetic properties with potential use as photosensitizers, as magnetic materials, or as materials with interesting nonlinear optical properties. Hence, there is an urgent need to compute reliable spectroscopic and photochemical data for these complexes. The geometries of [Fe(CN)5 (py)]3− ,31 [Fe(bpy)3 ]2+ ,32 Rucomplex-1,28 and Ru-complex-233 were taken from the original benchmark papers. Table V summarizes the main results for [Fe(CN)5 (py)]3- at different levels of theory. Many lowlying MLCT states of different symmetry dominate the lowenergy region of the UV/Vis spectrum, whereas the MC states are found at higher energies. For the MLCT and MC states, the agreement between the DFT/MRCI and CASPT2(14,14) results is remarkable. Indeed, DFT/MRCI is the only DFTbased method capable of reproducing the CASPT2 ordering of states. For the lowest lying MLCT states, there are some rather minor differences (redshifts of some of the states by ca. 0.2 eV), as previously observed for [Fe(CN)6 ]4− . DFT/MRCI places the other states (including all the MC states) within 0.1 eV of the CASPT2 values. By contrast, in the TDDFT calculations, none of the tested functionals is able to accurately describe the excited states of [Fe(CN)5 (py)]3− , especially the MLCT states, or to get the correct ordering of states. TD-CAM-B3LYP consistently overestimates
TABLE V. Selected electronic transition energies (in eV) and oscillator strengths (in parentheses) of [Fe(CN)5 (py)]3− at different levels of theory.
11 A1 (1 MLCT) 1 1 A2 (1 MLCT) 1 1 B1 (1 MLCT) 2 1 B1 (1 MLCT) 2 1 A2 (1 MLCT) 1 1 B2 (1 MLCT) 2 1 B2 (1 MC) 3 1 B1 (1 MC) 2 1 A1 (1 MC)
CASPT2(14,14)a /(10,10)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
0.96 (0.01)a,c 0.72 (0.00)a,c 0.77 (0.00)a,c 1.34 (0.00)a,c 1.36 (0.00)a,c 1.49 (0.00)a,c 2.89 (0.00)b,c 3.03 (0.00)b,c 3.49 (0.00)b,c
1.16 (0.27) 0.53 (0.00) 0.52 (0.00) 1.38 (0.00) 1.33 (0.00) 1.39 (0.00) 2.82 (0.00) 2.93 (0.00) 3.34 (0.00)
2.19 (0.17) 1.74 (0.00) 1.61 (0.00) 2.76 (0.00) 2.62 (0.00) 2.49 (0.00) 2.92 (0.00) 2.96 (0.00) 3.77 (0.00)
1.99 (0.18) 1.27 (0.00) 2.22 (0.00) 2.47 (0.00) 2.09 (0.00) 2.01 (0.00) 3.07 (0.00) 3.11 (0.00) 3.53 (0.01)
1.98 (0.17) 0.84 (0.00) 0.79 (0.00) 1.11 (0.00) 1.07 (0.00) 1.36 (0.00) 3.33 (0.00) 3.63 (0.01) 3.58 (0.00)
a
From Ref. 31. From Ref. 31. c Oscillator strengths obtained at the PCM-CASPT2(14,14) level of theory from Ref. 31. b
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TABLE VI. Selected electronic transition energies (in eV) and oscillator strengths (in parenthesis) of [Fe(bpy)3 ]2+ at different levels of theory.
2 1 A (1 MLCT) 3 1 A (1 MC) 4 1 A (1 MLCT) 5 1 A (1 MLCT) 1 1 B (1 MC) 2 1 B (1 MC) 3 1 B1 (1 MC) 4 1 B (1 MLCT) 19 1 B (1 LC) a b
CASPT2(10,15)a /(12,12)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
2.44a 2.51a 2.59a 2.72a 2.47a 2.48a 2.59a 2.65a ∼4.65b
1.69 (0.00) 2.41 (0.00) 2.59 (0.03) 2.74 (0.00) 1.69 (0.00) 1.88 (0.00) 2.41 (0.00) 2.58 (0.00) 4.40 (0.26)
3.23 (0.00) 2.73 (0.00) 3.38 (0.03) 3.63 (0.05) 2.73 (0.00) 2.84 (0.00) 3.83 (0.01) 3.23 (0.00) 4.52 (0.07)
2.80 (0.00) 2.60 (0.00) 2.92 (0.02) 3.08 (0.05) 2.60 (0.00) 2.76 (0.00) 3.65 (0.00) 2.82 (0.00) 4.58 (0.21)
1.78 (0.00) 3.15 (0.01) 1.98 (0.01) 2.18 (0.04) 3.15 (0.01) 3.29 (0.00) 3.81 (0.00) 1.68 (0.00) 3.51 (0.00)
From Ref. 32. From Ref. 32.
the excitation energies of all MLCT states by at least ca. 1 eV. TD-PBE0 behaves analogously, but in a less systematic manner, with errors ranging from ca. 0.55 eV (see, e.g., 1 1 A2 ) to ca. 1.5 eV (see, e.g., 1 1 B1 ). Both TD-CAMB3LYP and TD-PBE0 accurately locate the MC states. TDPBE shows a somewhat better overall performance (i.e., being closer to the CASPT2 results), in line with a previous TD-DFT evaluation for Fe(II) complexes.34 Still, the errors for the MLCT states are inconsistent, since TD-PBE overestimates the excitation energy of the bright MLCT state of A1 symmetry (by 1.0 eV) while predicting the other 1 MLCT transitions with acceptable accuracy. The MC states are also overestimated at this level (by 0.1–0.6 eV) so that TD-PBE predicts an incorrect ordering of the states. The TD-DFT and DFT/MRCI methods give similar oscillator strengths, which cannot be directly compared with the CASPT2 values because the latter were computed in the presence of solvent. We conclude that it may be misleading to rely too much on TD-DFT results in the interpretation of electronic spectra of complexes such as [Fe(CN)5 (py)]3− . Recently, the excited states of the larger [Fe(bpy)3 ]2+ complex have been studied using the CASPT2 method.32 Table VI collects the results for the electronically excited states of [Fe(bpy)3 ]2+ at the CASPT2, DFT/MRCI, and TDDFT levels (note that the DFT/MRCI calculations employed the def2-SV(P) basis set and C2 symmetry constraints). Since the LC state contributes to the near-UV region of the spectrum (see Table VI), this complex is well suited to evaluate the performance of the DFT/MRCI and TD-DFT methods (not only for LC, but also for MC and MLCT states). DFT/MRCI again underestimates the excitation energies of the lowest excited states (i.e., 2 1 A, 1 1 B, and 2 1 B) uniformly by ca. 0.7 eV relative to the CASPT2 values, which thus appears to be a common trend for all the Fe(II) complexes studied herein. The remaining states, i.e., 3 1 A, 4 1 A, 5 1 A, 3 1 B, 4 1 B, and 19 1 B (i.e., the lowest bright LC state) are accurately located at the DFT/MRCI level, regardless whether they are reached by MC, LC, or MLCT excitations. Notably, the ordering of states is the same for DFT/MRCI and CASPT2. On the other hand, the TD-DFT results are again very functional-dependent. The TD-CAM-B3LYP excitation energies are overestimated, especially for the MLCT states, and the lowest excited state is thus a MC state, in disagree-
ment with CASPT2 and DFT/MRCI; however, the LC state is predicted rather well (albeit with lower oscillator strength). Overall, TD-PBE0 performs best among the tested functionals, with a reasonably balanced description of MC, MLCT, and LC states (and an especially good description of the LC state, in terms of energy and oscillator strength). Still, TDPBE0 predicts some of the MC states too high by more than 1 eV (i.e., 3 1 B1 ) and gives a state ordering different from CASPT2 and DFT/MRCI. TD-PBE shows the opposite behavior: the computed excitation energies are underestimated by 0.6–1.0 eV for the MLCT states and overestimated by ca. 0.7 eV for the MC states; TD-PBE also fails in correctly locating the LC state. In conclusion, the TD-DFT results for the different Fe(II) complexes studied herein reveal a dependence of the computed spectroscopic properties on the choice of the functional. TD-PBE appears to perform best for the negatively charged compounds [Fe(CN)5 (py)]3− and [Fe(CN)6 ]4− , while TD-PBE0 appears superior for the positively charged compound [Fe(bpy)3 ]2+ . The reasons for these discrepancies are unknown. Such erratic behavior of the tested TD-DFT methods makes it difficult to generally recommend any functional as being best for describing the spectroscopic properties of Fe(II) complexes. On the contrary, the DFT/MRCI method usually predicts the correct order of states and treats all kind of excited states in a balanced manner, while showing a systematic error (redshift) for the lowest lying excited states. Finally, we discuss the spectroscopic properties of some 4d-TM complexes, namely, Ru-complex-1 and Ru-complex2 (see Figure 1), which have previously been studied by some of us28 and others33 with the RASPT2 method. Table VII contains the results for selected electronically excited states obtained at different levels of theory. As in the case of [Fe(bpy)3 ]2+ , electronic transitions of different character contribute to the UV/Vis spectrum of Ru-complex1 (i.e., MLCT, MC, and LC transitions). The lowest lying 1 1 B1 and 2 1 A1 states are of MC and MLCT character, respectively. These states are almost degenerate at the RASPT2 level. They are also very close in energy at the DFT/MRCI level, but too low by ca. 0.4–0.5 eV. This is a common trend for all 3d- and 4d-TM complexes studied herein. The higherlying MLCT state (4 1 B2 ) is well predicted by DFT/MRCI, while the LC state is computed too high by ca. 0.4 eV. Concerning the TD-DFT results originally reported in Ref. 28,
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TABLE VII. Selected electronic transition energies (in eV) and oscillator strengths (in parentheses) of trans(Cl)-Ru(bpy)Cl2 (CO)2 (Ru-complex-1) and Rucomplex-2 at different levels of theory. Ru-complex-1 1 1 B1 (1 MC) 2 1 A1 (1 MLCT) 5 1 B2 (1 LC) 4 1 B2 (1 MLCT) Ru-complex-2 FDAES- 21 A (1 MLCT)
MS-RASPT2(16,3,3;4,7,4)a
DFT/MRCI
TD-CAM-B3LYP a
TD-PBE0 a
TD-PBEa
3.36 (0.01) 3.34 (0.06) 3.70 (0.31)
2.92 (0.01) 2.81 (0.03) 4.09 (0.33)
3.39 (0.00) 3.38 (0.00) 4.67 (0.26)
3.31 (0.00) 2.60 (0.01) 4.44 (0.20)
3.09 (0.00) 1.56 (0.01) 4.02 (0.07)
3.96 (0.01)
3.99 (0.02)
4.51 (0.02)
3.49 (0.00)
2.19 (0.00)
RASPT2(18,18)b 2.84 (0.22)
DFT/MRCI 2.71 (0.36)
TD-CAM-B3LYP 3.50 (0.40)
TD-PBE0 2.78 (0.34)
TD-PBE 2.12 (0.36)
a
From Ref. 28. RAS(n,l,m;I,j,k) notation: n is the number of active electrons; l is the maximum number of holes in RAS1; m is the maximum number of electrons in RAS3; and i, j, and k are the number of orbitals in RAS1, RAS2, and RAS3, respectively. b From Ref. 33.
the excitation energies computed for the MC state do not vary much for the tested functionals, whereas those for the MLCT and LC states are rather more functional-dependent. The TD-PBE excitation energy is quite accurate for the LC state, but much too low for the MLCT transitions (e.g., by up to 1.8 eV for the 4 1 B2 state). TD-CAM-B3LYP and TDPBE0 also have problems with the MLCT states, which are calculated notably too high and too low, respectively. Concerning the oscillator strengths, DFT/MRCI offers the best agreement with respect to the RASPT2 values. On the other hand, none of the tested functionals is able to reproduce the trends in the relative intensities of the states. Overall, the PBE0 functional (and generally hybrid functionals with intermediate amounts of exact exchange) provide the best compromise in terms of excitation energies and intensities for the excited states of the Ru-complex-1. Table VII also includes the results for the first dipole-allowed excited state (FDAES) of Ru-complex-2. Both DFT/MRCI and TD-PBE0 closely reproduce the RASPT2 excitation energy value of this MLCT state, which is notably overestimated and underestimated by TD-CAM-B3LYP and TD-PBE, respectively. The oscillator strengths from DFT/MRCI and TD-DFT are somewhat higher than our best RASPT2 reference value.
IV. CONCLUSIONS
TM complexes are prototypes of systems where nondynamic correlation effects in the ground and excited states may become extremely important. This makes it especially complicated to handle them computationally, and it is thus often difficult to obtain correct spin-state energies and a balanced description of the excited states. This is especially true when non-innocent ligands are involved or in the case of complexes possessing highly covalent metal-ligand bonds. In principle, multi-configurational methods are preferred for such systems, since they can deal with any kind of electronic structure problem. However, their application is still restricted to small or medium-size systems and requires a judicious choice of suitable computational options. Hence, there is a demand for less expensive methods that can be used in a “black-box” manner to compute the excited-state properties of TM complexes. In this paper, we have assessed the performance of the DFT/MRCI method and several TD-DFT approaches. For highly multi-reference situations, such as the
permanganate anion, none of the tested methods reproduces the UV/Vis spectrum with acceptable accuracy, which in the case of DFT/MRCI can be attributed to the small size of the chosen CI expansion that it is not able to recover all important static correlation effects. For the other complexes studied herein, DFT/MRCI performs reasonably well for describing the excited-state properties of TM complexes, both for singlet and triplet excited states. In general, DFT/MRCI systematically underestimates the energies of the lowest excited states, but provides the correct order of states and a balanced description of excited states of different character. By contrast, the present TD-DFT results are less accurate and generally depend strongly on the chosen functional. In the series of Fe(II) complexes, the TD-DFT errors are not systematic so that it is difficult to recommend any of the tested functionals—in general, none of them gives the correct order of states or a balanced description of all excited states of different character. In the majority of cases, the PBE0 hybrid functional with an intermediate amount of exact exchange provides TDDFT results of higher quality than the pure PBE functional and the long-range corrected CAM-B3LYP hybrid functional. We conclude that the assignment of UV/Vis spectra of TM complexes exclusively based on TD-DFT data may be dangerous without an initial careful validation. Interpretations based on a simple matching of experimental and theoretical bands may be misleading, because there may be accidental matches due to the large number of low-lying transitions and possibly large errors in the TD-DFT predictions. The limited accuracy of TD-DFT is also a critical issue in its potential use for computing excited-state potential energy surfaces during non-adiabatic excited-state dynamics simulations of such systems. Finally, we have also evaluated the performance of the DFT/MRCI method for TM ground-state bonding in the demanding case of CrF6 . The DFT/MRCI results are reasonably close to those from extensive RASPT2 calculations. Further evaluation of the DFT/MRCI method for other ground-state problems involving TM complexes, such as spin-state energetics, are currently under evaluation in our group. 1 K.
Pierloot, Int. J. Quantum Chem. 111, 3291 (2011); L. González, D. Escudero, and L. Serrano-Andrés, ChemPhysChem 13, 28 (2012). 2 K. Pierloot, Mol. Phys. 101, 2083 (2003). 3 S. Grimme and M. Waletzke, J. Chem. Phys. 111, 5645 (1999). 4 S. Grimme, Chem. Phys. Lett. 259, 128 (1996).
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D. Escudero and W. Thiel
R. Silva-Junior, M. Schreiber, S. P. A. Sauer, and W. Thiel, J. Chem. Phys. 129, 104103 (2008); C. M. Marian and N. Gilka, J. Chem. Theory Comput. 4, 1501 (2008). 6 M. Kleinschmidt, C. M. Marian, M. Waletzke, and S. Grimme, J. Chem. Phys. 130, 044708 (2009). 7 V. Schmid, R. Linder, and C. M. Marian, Eur. J. Inorg. Chem. 2006, 1588 (2006). 8 F. Réal, V. Vallet, C. Marian, and U. Wahlgren, J. Chem. Phys. 127, 214302 (2007). 9 S. Vancoillie, H. Zhao, V. T. Tran, M. F. A. Hendrickx, and K. Pierloot, J. Chem. Theory Comput. 7, 3961 (2011). 10 R. Ahlrichs, M. Bär, M. Häser, H. Horn, and C. Kölmel, Chem. Phys. Lett. 162, 165 (1989). 11 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., Gaussian 09, Revision A.1, Gaussian, Inc., Wallingford, CT, 2009. 12 K. Pierloot and B. O. Roos, Inorg. Chem. 31, 5353 (1992). 13 (a) H. Nakai, Y. Ohmori, and H. Nakatsuji, J. Chem. Phys. 95, 8287 (1991); (b) A. C. Stückl, C. A. Daul, and H. U. Güdel, Int. J. Quantum. Chem. 61, 579 (1997). 14 I. H. Hillier and V. R. Saunders, Chem. Phys. Lett. 9, 219 (1971); H. Hsu, C. Peterson, and R. M. Pitzer, J. Chem. Phys. 64, 791 (1976); H. Johanson, Mol. Phys. 49, 1209 (1983). 15 R. M. Dickson and T. Ziegler, Int. J. Quantum Chem. 58, 681 (1996). 16 S. J. A. Van Gisbergen, J. A. Groenveld, A. Rosa, J. G. Snijders, and E. J. Baerends, J. Phys. Chem. A 103, 6835 (1999). 17 M. Nooijen and V. Lotrich, J. Chem. Phys. 113, 494 (2000). 18 M. A. Buijse and E. J. Baerends, J. Chem. Phys. 93, 4129 (1990). 19 K. Pierloot, “Calculation of electronic spectra of transition metal complexes,” in Computational Photochemistry (Elsevier, Amsterdam, 2005), pp. 279–315.
J. Chem. Phys. 140, 194105 (2014) 20 J.
Su, W.-H. Xu, C.-F. Xu, W. H. E. Schwarz, and J. Li, Inorg. Chem. 52, 9867 (2013). 21 K. Pierloot, E. Tsokos, and L. G. Vanquickenborne, J. Phys. Chem. 100, 16545 (1996). 22 K. Pierloot, E. Van Praet, L. G. Vanquickenborne, and B. O. Roos, J. Phys. Chem. 97, 12220 (1993). 23 C. Pollak, A. Rosa, and J. E. Baerends, J. Am. Chem. Soc. 119, 7324 (1997). 24 A. Rosa, E. Baerends, S. J. A. van Gisbergen, A. van Lenthe, J. A. Groeneveld, and J. G. Snijders, J. Am. Chem. Soc. 121, 10356 (1999); P. Hummel, J. Oxgaard, W. A. Goddard III, and H. B. Gray, Inorg. Chem. 44, 2454 (2005). 25 N. Ben Amor, S. Villaume, D. Maynau, and C. Daniel, Chem. Phys. Lett. 421, 378 (2006); S. Villaume, A. Strich, C. Daniel, S. A. Perera, and R. J. Bartlett, Phys. Chem. Chem. Phys. 9, 6115 (2007). 26 N. A. Beach and H. B. Gray, J. Am. Chem. Soc. 90, 5713 (1968). 27 R. Crespo-Otero and M. Barbatti, J. Chem. Phys. 134, 164305 (2011). 28 D. Escudero and L. González, J. Chem. Theory Comput. 8, 203 (2012). 29 T. Le Bahers, E. Bremond, I. Ciofini, and C. Adamo, “The nature of vertical excited states of dyes containing metals for DSSC applications: Insights from TD-DFT and density based indexes,” Phys. Chem. Chem. Phys. (published online). 30 J. B. Ayers and W. H. Wagoner, J. Inorg. Nucl. Chem. 33, 721 (1971). 31 A. L. B. Formiga, S. Vancoillie, and K. Pierloot, Inorg. Chem. 52, 10653 (2013). 32 C. de Graaf and C. Sousa, Chem. Eur. J. 16, 4550 (2010). 33 B. J. Coe, A. Avramopoulos, M. G. Papadopoulos, K. Pierloot, S. Vancoillie, and H. Reis, Chem. Eur. J. 19, 15955 (2013). 34 I. Georgieva, A. J. A. Aquino, N. Trendafilova, P. S. Santos, and H. Lischka, Inorg. Chem. 49, 1634 (2010).
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THE JOURNAL OF CHEMICAL PHYSICS 140, 194105 (2014)
Assessing the density functional theory-based multireference configuration interaction (DFT/MRCI) method for transition metal complexes Daniel Escuderoa) and Walter Thiela) Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mülheim an der Ruhr, Germany
(Received 10 March 2014; accepted 29 April 2014; published online 20 May 2014) We report an assessment of the performance of density functional theory-based multireference configuration interaction (DFT/MRCI) calculations for a set of 3d- and 4d-transition metal (TM) complexes. The DFT/MRCI results are compared to published reference data from reliable highlevel multi-configurational ab initio studies. The assessment covers the relative energies of different ground-state minima of the highly correlated CrF6 complex, the singlet and triplet electronically excited states of seven typical TM complexes (MnO4 − , Cr(CO)6 , [Fe(CN)6 ]4− , four larger Fe and Ru complexes), and the corresponding electronic spectra (vertical excitation energies and oscillator strengths). It includes comparisons with results from different flavors of timedependent DFT (TD-DFT) calculations using pure, hybrid, and long-range corrected functionals. The DFT/MRCI method is found to be superior to the tested TD-DFT approaches and is thus recommended for exploring the excited-state properties of TM complexes. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4875810] I. INTRODUCTION
The theoretical study of electronically excited states is significantly more challenging for transition metal (TM) complexes than for organic and main-group compounds. This is due to the more complicated electronic structure of TM complexes, which involves the d shell of the metal surrounded by ligands with different acceptor and donor properties. Consequently, there are electronic transitions of very different nature, i.e., (i) metal-centered (MC), (ii) metal-to-ligand charge transfer (MLCT), (iii) ligand-to-metal CT (LMCT), and (iv) intra-ligand (IL) transitions, which may all contribute to the UV/Vis spectra of TM complexes. A computational method that aims at an accurate and balanced description of these excited states needs to deal with dynamic and nondynamic correlation effects equally well, since these states often have an inherently multi-configurational character. Therefore, theoretical calculations of electronic spectra of TM complexes have not yet reached a “black box” status.1 Multiconfigurational methods, such as the complete-active-space second-order perturbation theory (CASPT2) or the restrictedactive-space PT2 (RASPT2) method, are in principle well suited for these complicated electronic structure situations. However, these methods are not yet the “gold standard” for the spectroscopy of TM complexes, since their success is critically dependent upon their judicious application, particularly with regard to the selection of the active space,2 and also since they are still restricted to small and medium-size systems. Excited-state methods based on density functional theory (DFT), such as the SCF method or time-dependent DFT (TD-DFT) are less costly, but also less reliable, esa) Authors to whom correspondence should be addressed. Electronic ad-
dresses: [email protected] and [email protected]
0021-9606/2014/140(19)/194105/8/$30.00
pecially when compared to the performance of DFT for ground-state properties of TM complexes. An alternative DFT-based method for electronically excited states makes use of Kohn-Sham (KS) orbitals in a multi-reference configuration interaction (MRCI) framework: In this DFT/MRCI method,3 dynamic correlation effects are captured by the KS-DFT treatment while non-dynamic correlation effects are treated at the MRCI level. To alleviate problems arising from the possible double counting of dynamical correlation effects, some off-diagonal elements are scaled by incorporating five global empirical parameters, which have been determined by fitting to experimental data of ten reference molecules.3, 4 The DFT/MRCI method has the potential of being used in a “black-box” manner, and it has already been proven successful for a number of organic chromophores.5 A parallel code has been recently developed6 to facilitate the calculation of electronic spectra of larger molecules. However, there are yet only few DFT/MRCI applications to TM complexes, for example, tests for ferrocene,3 heptacyanovanadate(III),7 and the uranyl(VI) ion.8 In this contribution, we present a DFT/MRCI study of the electronically excited states of several TM complexes, for which reliable highly accurate ab initio data have recently become available in the literature. To ensure that the DFT/MRCI results are directly comparable to these reference data, we use the same geometries as in the original papers. For most of the studied complexes, we also provide an assessment of TD-DFT methods using different functionals (i.e., pure, hybrid, and long-range corrected functionals). Non-dynamic correlation effects are not only crucial for TM spectroscopy, but also for the bonding in ground-state TM complexes. Therefore, we also evaluate the performance of the DFT/MRCI method to obtain the relative energies of different structural minima of CrF6 , which has been recently revisited using the RASPT2 method.9
140, 194105-1
© 2014 AIP Publishing LLC
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II. COMPUTATIONAL DETAILS
The geometries of all compounds included in this study were taken either from available X-ray data or from groundstate optimizations reported previously. Single-point DFT/ MRCI and TD-DFT calculations were performed at these geometries. In the DFT/MRCI case, the initial KS-BHLYP calculations were carried out with the TURBOMOLE program,10 and the subsequent MRCI calculations with the DFT/MRCI code3 using the def2-TZVP basis set (unless noted otherwise). Initial reference configurations were generated by promoting up to two electrons in an active space of ten electrons in ten orbitals, and an energy-based selection criterion (with a threshold value of 1.0 hartree) was used to disregard configurations. Standard DFT/MRCI parameters for singlet and triplet states were employed. The DFT/MRCI code supports only Abelian groups, and hence most of the computations were done in C2v symmetry. The TD-DFT calculations were performed using the Gaussian0911 program, the 6-311G* basis set and the following functionals: (i) the pure PBE functional, (ii) the hybrid PBE0 functional (25% of exact exchange), and (iii) the long-range corrected CAMB3LYP functional.
III. RESULTS AND DISCUSSION
This section is organized as follows: in Subsection III A, the DFT/MRCI relative energies of different structural minima of CrF6 (1) are presented. Subsections III B–III D are dedicated to TM spectroscopy. Thus, the DFT/MRCI vertical excitation energies and oscillator strengths of a benchmark set of molecules are compared to the best theoretical estimates from the literature. Subsection III B addresses the tetrahedral MnO4 − complex (2). In Subsection III C the spectroscopic properties of the octahedral Cr(CO)6 (3) and [Fe(CN)6 ]4− (4) complexes are investigated. Subsection III D deals with the spectroscopy of larger complexes, i.e., [Fe(CN)5 (py)]3− (5, py:pyridine), [Fe(bpy)3 ]2+ (6, bpy:bipyridine), trans(Cl)Ru(bpy)Cl2 (CO)2 (7, Ru-complex-1), and a second Rucomplex-2 (8). Figure 1 shows the benchmark set of molecules. The selection of this set is motivated by the availability of reliable theoretical data in the literature and by the objective to cover different 3d and 4d metals, different oxidation states, and excited states of different character, in order to allow for a general assessment.
A. TM bonding: Octahedral and trigonal prismatic structures of CrF6
The CrF6 molecule is a textbook example of how nondynamical correlation effects may affect TM complexes. The high oxidation state of chromium (+VI) leads to highly covalent metal-ligand bonds and consequently to strong static correlation effects already in the ground-state configuration. Two local minimum structures, i.e., the octahedral (Oh ) and trigonal prismatic (D3h ) complexes, are found to be adiabatically close-lying (see Figure 1). Their relative energies at different computational levels are given in Table I. Pier-
FIG. 1. Benchmark set of TM complexes used in this study.
loot and Roos studied this challenging system already in the 1990s.12 Their early CASPT2(10,10) calculations with a minimum active space (10 electrons, 10 orbitals) overestimated the stability of the Oh structure, as compared to CCSD(T) results (coupled-cluster singles and doubles with perturbative triples.) According to recent RASPT2 calculations,9 this deficiency of the CASPT2(10,10) results was caused by the small size of the original (10,10) active space. To get accurate relative energies for CrF6 , it is crucial to include the whole set of F 2p orbitals, which are involved in strong static correlation effects. Such an extension of the active space is prohibitive for CASPT2, but still affordable (though very expensive) for RASPT2. The RASPT2(42,26) values of 17.8 and 16.5 kcal/mol9 in Table I are considered as our best theoretical estimates. The relative energy (E) at the DFT/MRCI level was computed using the D3d and Oh geometries from previous studies.9, 12 The DFT/MRCI value of 22.0 kcal/mol (Table I) is in fair agreement with the best RASPT2 estimate, which supports the use of DFT/MRCI for describing TM bonding in electronically complicated ground-state complexes. Concerning computational effort, we note that the DFT/MRCI wavefunction for CrF6 involves 165 815 configuration state functions (CSFs) compared with several million CSFs in the RASPT2(42,26) wavefunction (with the exact TABLE I. Relative energy (kcal/mol) of the D3h minimum structure with respect to the Oh minimum structure of CrF6 .
CASPT2(10,10)a E (kcal/mol) a b
49.0
RASPT2(42,4,0;16,10,0)b RASPT2(42,6,6;21,0,5)b DFT/MRCI 17.8 16.5
22.0
From Ref. 12. From Ref. 9. Value in italics taken from RASPT2 calculation specified in italics.
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TABLE II. Selected electronic transition energies (in eV) and oscillator strengths (values in parentheses) of MnO4 − at different levels.
State (main configuration)
RASPT2(24,17)a CASPT2 in 17ob
1 1 T1 (1t1 → 2e) 11 T2 (1t1 → 2e) 2 1 T1 (2t2 → 2e) 2 1 T2 (1t1 → 3t2 , 2t2 → 2e) 1 1 E (1t1 → 3t2 ) 1 1 A2 (1t1 → 3t2 ) 2 1 A1 (2t2 → 3t2 ) 3 1 T1 (1t1 → 3t2 ) 3 1 T2 (2t1 → 2e, 2t2 → 3t2 ) a b c
1.93 (0.000) 2.33 (0.004) 2.28 3.39 (0.000) 3.53 (0.002) 3.46 3.90 (0.000) 3.89 (0.000) 5.10 (0.000) 3.93 (0.000) 4.20 (0.006) 3.90
DFT/MRCI
SAC-CIc
TD-DFT (SAOP)a
2.38/2.38/2.49 (0.000) 2.74/2.80/2.80 (0.013)
2.18 (0.000) 2.57 (0.020)
2.66 (0.000) 3.08 (0.007)
4.05/4.05/4.10 (0.000) 4.21/4.21/4.23 (0.004)
3.33 (0.000) 3.58 (0.045)
3.91 (0.000) 4.12 (0.002)
4.38/4.39 (0.000) 4.41 (0.000) 4.81 (0.000) 4.37/4.37/4.37 (0.000) 4.81/4.81/4.84 (0.022)
3.41 (0.000) 4.46 (0.000) 5.41 (0.000) 4.12 (0.000) 3.72 (0.136)
4.50 (0.000) 4.31 (0.000) 5.72 (0.000) 4.44 (0.00) 5.01 (0.009)
From Ref. 20. Values in italics taken from Ref. 19. From Ref. 13(a).
number depending on the chosen RASPT2 options). Hence, the use of DFT/MRCI leads to very large computational savings.
B. Spectroscopic properties of a tetrahedral complex: MnO4 −
The geometry of MnO4 − was taken from experiment, as in other benchmark studies.13 The electronic spectrum of permanganate is among the most studied in TM spectroscopy. It has recurrently been used as a benchmark system for assessing the performance of quantum-chemical methods for excited states. Early Hartree-Fock (HF) and smallscale CI studies14 were followed by DFT studies, both at the SCF15,13(b) and TD-DFT16 levels. MnO4 − has also been investigated with coupled cluster (CC) theory, particularly with the equation-of-motion CC (EOM-CC), similarity transformed EOM-CC (STEOM-CC), and extended-STEOM-CC methods,17 as well as with the symmetry-adapted cluster CI (SAC-CI) approach.13(a) It is the paradigm of how important non-dynamic correlation effects may become in TM spectroscopy. As for CrF6 , problems arise from the high covalency of the metal-ligand bonds in permanganate (due to the high oxidation state of Mn, i.e., +VII), which gives rise to strong correlation effects already in the ground state, involving excitations from occupied O 2p orbitals to virtual Mn 3d orbitals. Furthermore, these correlation effects vary notably in the excited states,18 some of which have important contributions from doubly excitations.19 For these reasons, singlereference methods face considerable difficulties for MnO4 − , as demonstrated for the otherwise very accurate extendedSTEOM-CC method (with errors of up to 1 eV).17 Due to its inherent complexity the assignment of the permanganate spectrum has actually undergone several changes throughout the years, and a general consensus has been reached only very recently. Reliable theoretical results for the electronic spectrum of MnO4 − were obtained in the last few years using the
CASPT219 and RASPT220 methods in combination with extended active spaces. As in CrF6 , the inclusion of many doubly occupied O 2p orbitals was crucial to get converged results. The best theoretical estimates for permanganate from the RASPT2(24,17) level of theory20 show perfect agreement with the experimental data. The active space in these RASPT2 calculations included the 12 highest doubly occupied O 2p orbitals and the 5 virtual antibonding orbitals of mainly Mn 3d character. Table II lists the best published RASPT2 and CASPT2 results for the excitation energies and oscillator strengths of selected excited states of MnO4 − , as well as the presently computed DFT/MRCI values and both SAC-CI and TD-DFT values from the literature. The spectrum of MnO4 − is dominated by dipole-allowed LMCT excitations, which in Td symmetry correspond to T2 states. Results for the lowest T1 , T2 , A1 , A2 , and E states are collected in Table II. We note that in C2v symmetry (used in the DFT/MRCI calculations) the T1 , T2 , and E states are decomposed into a1 + b1 + b2 , a2 + b1 + b2 , and a1 + a2 states, respectively. DFT/MRCI consistently overestimates the excitation energies compared with RASPT2; the overall agreement is reasonable, with maximum absolute deviations of ca. 0.65 eV (see, e.g., 2 1 T1 and 2 1 T2 ). Compared with DFT/MRCI, the SAC-CI results are generally somewhat closer to RASPT2, with maximum absolute errors up to 0.6 eV (see, e.g., 1 1 A2 ), but the deviations are less uniform. The TD-DFT results show the worst agreement with RASPT2, with maximum absolute deviations of ca. 0.8 eV (i.e., the 3 1 T2 state). Concerning the oscillator strengths of the dipole-allowed transitions, only the DFT/MRCI and TD-DFT values reproduce the RASPT2 reference data well, while the SAC-CI values are much too high. In conclusion, only high-level multi-configurational methods appear capable of reproducing the spectrum of permanganate within an error bar of 0.1 eV. A likely explanation of the problems of the DFT/MRCI method in the highly correlated permanganate is the small CI expansion, with a (10,10) active space, which is not able to recover all the non-dynamic correlation effects.
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194105-4
D. Escudero and W. Thiel
J. Chem. Phys. 140, 194105 (2014)
TABLE III. Selected singlet and triplet electronic transition energies (in eV) and oscillator strengths (in parentheses) of Cr(CO)6 at different levels of theory.
State/assignment 1 1 T1g 1 1 T1u 2 1 T1u 1 3 T1g 1 3 T2g
(1 MC) (1 MLCT) (1 MLCT) (3 MC) (3 MC)
RASPT2(10,0,4;0,10,14)a CASPT2(10,10)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
4.98 (0.000)a 4.50 (0.20)a 5.42 (2.57)a 4.28b 4.64b
4.90/4.76/4.75 (0.000) 4.71/4.70/4.70 (0.074) 5.71 (0.768) 4.53/4.54/4.62 4.54/4.71/4.72
4.99 (0.000) 4.83 (0.029) 6.42 (0.702) 4.48 4.76
5.08 (0.000) 4.71 (0.021) 6.31 (0.065) 4.50 4.79
5.37 (0.000) 4.14 (0.009) 5.78 (0.513) 4.89 5.01
Expt.c
4.44 (0.25) 5.48 (2.30)
a From Ref. 9. RAS(n,l,m;I,j,k) notation: n is the number of active electrons, l is the maximum number of holes in RAS1; m is the maximum of electrons in RAS3; and i, j, and k are the number of orbitals in RAS1, RAS2, and RAS3, respectively. b From Ref. 21. c From Ref. 26.
C. Spectroscopic properties of octahedral complexes: Cr(CO)6 and [Fe(CN)6 ]4−
The geometries of Cr(CO)6 and [Fe(CN)6 ]4− were again taken from experiment, as in other previous benchmark studies, see Refs. 21 and 22, respectively. The electronic spectrum of Cr(CO)6 has been much studied, both experimentally and in particular theoretically, using SC-DFT,23 TD-DFT,24 CASPT2,21 and coupled-cluster methods.25 The assignment of the spectra has been less controversial (apart from the position of the MC states) than in the case of MnO4 − , consistent with the less pronounced multi-configurational character of the ground and excited states of Cr(CO)6. A recent RASPT2 study with an extended active space, i.e., (10e, 24o), has provided converged excitation energies close to the experimental data (within 0.05 eV).9 These RASPT2 values are adopted as our best theoretical estimates. The experimental spectrum is dominated by the symmetry-allowed 1 T1u excitations of MLCT character, but there are also symmetry-forbidden MC excitations.26 Though being optically forbidden, the MC states are very important in Cr(CO)6 photochemistry, since they can be populated in the course of photochemical deactivation, similarly to other metal-carbonyl complexes. They are located higher in energy than the T1u states. Table III lists spectroscopic data for the main singlet and triplet electronic states of Cr(CO)6 obtained at the RASPT2, DFT/MRCI, and TD-DFT levels (employing different functional in the latter case). Due to the C2v symmetry constraints in the DFT/MRCI calculations, the original Oh states are unfolded into states of lower symmetry. At the DFT/MRCI level, the position of the lowest 1 MC state (1 1 T1g ) is well reproduced (within 0.1–0.2 eV), and the 1 MLCT transitions are also predicted with acceptable accuracy (within 0.25 eV of the RASPT2 values). We note that an even closer agreement for the MLCT states has been reported in previous DFT/MRCI calculations using newly optimized parameters.27 On the other hand, there are much larger discrepancies in the case of the TD-DFT values, which are moreover very functional-dependent. Among the tested functionals, the long-range TD-CAM-B3LYP functional outperforms the other functionals for the MC excitation energies. To the contrary, the hybrid TD-PBE0 and the pure TD-PBE functionals best reproduce the MLCT transitions. Hence, in spite of being designed for accurately describing CT states in organic compounds, TD-CAM-B3LYP is apparently not suited for dealing with MLCT states in TM
spectroscopy, as has been pointed out previously for other TM complexes.28, 29 TD-PBE0 (TD-PBE) overestimate (underestimate) the excitation energy of the 11 T1u state by ca. 0.2 and 0.35 eV, respectively. Even larger are the discrepancies for the high-lying MLCT state (predicted too high by ca. 0.35, 0.8, and 1.0 eV with PBE, PBE0, and CAM-B3LYP, respectively). Concerning the triplet excited states, DFT/MRCI provides reliable energies for the 3 MC states (see Table III). The performance of the TD-DFT methods for the 3 MC states follows the same trends as observed for the 1 MC states. Finally, oscillator strengths are generally underestimated at both the DFT/MRCI and TD-DFT levels (compared with RASPT2). Among all the DFT-based methods, DFT/MRCI gives the most realistic ratio between the intensities of the symmetry-allowed transitions. Gathering all data on electronic excitation energies and oscillator strengths, DFT/MRCI emerges as the best DFTbased approach to reproduce the spectrum of Cr(CO)6 with acceptable accuracy. Given their rather poor performance for Cr(CO)6 , it seems advisable to use TD-DFT methods only with much caution for the assignment of electronic spectra of metal-carbonyl complexes. The electronic spectrum of [Fe(CN)6 ]4− has been studied theoretically (CASPT2)22 and experimentally.30 It could be rather well assigned already in the 1990s on the basis of CASPT2(10,10) calculations with a relatively small active space. Compared with MnO4 − , [Fe(CN)6 ]4− has less covalent metal-ligand bonds, and consequently the effect of enlarging the active space to improve the description of non-dynamic correlation effects is less pronounced. The interpretation of its electronic spectrum is also facilitated by the fact that the 1 MC and 1 MLCT states are energetically well separated, contrary to the situation in Cr(CO)6 . Table IV lists the computed excitation energies and oscillator strengths for the main electronic states of [Fe(CN)6 ]4− , either of singlet and triplet character, obtained at different levels of theory. The experimental values are also included for comparison since the symmetry-allowed MLCT excitations were not computed in the CASPT2 study. The position of the low-lying 1 MC state (1 1 T1g ) is underestimated by ca. 0.4 eV at the DFT/MRCI level, whereas TDCAM-B3LYP and TD-PBE0 (but not TD-PBE) better match the CASPT2 value. This is also the case for the higher lying 1 MC state (1 1 T2g ), where the DFT/MRCI value is too high by ca. 0.65 eV compared with CASPT2 (but within 0.35–0.2 eV of the experimental data). One should thus explore whether
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194105-5
D. Escudero and W. Thiel
J. Chem. Phys. 140, 194105 (2014)
TABLE IV. Selected singlet and triplet electronic transition energies (in eV) and oscillator strengths (in parentheses) of [Fe(CN)6 ]4− at different levels of theory. State/assignment 1 1 T1g 3 1 T1u 4 1 T1u 1 1 T2g 1 3 T1g a b
(1 MC) (1 MLCT) (1 MLCT) (1 MC) (3 MC)
CASPT2(10,10)a
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
Expt.b
3.60 ... ... 4.33 2.67
3.23/3.20/3.20 (0.000) 5.46/5.46/5.47 (0.060) 6.05/6.04/6.04 (0.072) 4.98/4.98/4.99(0.000) 2.58/2.58/2.61
3.65 (0.000) 6.97 (0.072) 7.66 (0.125) 4.40 (0.000) 2.92
3.64 (0.000) 6.53 (0.055) 7.16 (0.121) 4.33 (0.065) 2.83
4.15 (0.000) 5.27 (0.024) 5.85 (0.100) 4.23 (0.000) 3.52
3.80–3.94 5.69–5.89 6.20 4.43–4.77 2.94
From Ref. 22. From Ref. 30.
the (10,10) active space is large enough to describe this state properly, especially in view of the missing 3d double-shell correlation effects, which are crucial for the proper description of the MC states.19 In the case of the 1 MLCT states, only DFT/MRCI is able to predict them correctly within an accuracy of 0.2 eV; as in the case of Cr(CO)6 , the TD-DFT values are much too high at the CAM-B3LYP and PBE0 levels (e.g., by up to 1.4 eV for CAM-B3LYP) and moderately too low for PBE. Consequently, only DFT/MRCI is able to describe all the different electronic excitations of [Fe(CN)6 ]4− in a balanced manner. Concerning the low-lying triplet excited state, i.e., 1 3 T1g , the DFT/MRCI value is slightly too low compared with CASPT2 and experiment. D. Spectroscopic properties of larger complexes: [Fe(CN)5 (py)]3− , [Fe(bpy)3 ]2+ , Ru-complexes 1 and 2
The progress in the field of multi-configurational methods within the last years has made it feasible to calculate the excited states of larger TM complexes, even though these rather time-demanding and intricate computations have not yet achieved a routine status. Hence, only few large TM complexes, some of which are included in this subsection, have been studied with CASPT2/RASPT2 methods using extended active spaces that are capable of accounting for all desired correlation effects. Despite their limited accuracy, TD-DFT methods are instead often used for routine calculations on the excited states of such TM complexes, which may contain different ligands giving rise to different kinds of excitations (i.e., LMCT, MLCT, MC, and LC) in the UV/Vis region. There-
fore, it is crucial to describe all these excitations in a balanced manner in order to correctly assign the experimental spectrum and to identify the relevant photodeactivation channels. In this subsection, we will assess the performance of the DFT/MRCI method and compare it to different flavors of TD-DFT. Polypyridyl complexes of Fe(II) and Ru(II) have attracted much interest due to their unique photophysical and magnetic properties with potential use as photosensitizers, as magnetic materials, or as materials with interesting nonlinear optical properties. Hence, there is an urgent need to compute reliable spectroscopic and photochemical data for these complexes. The geometries of [Fe(CN)5 (py)]3− ,31 [Fe(bpy)3 ]2+ ,32 Rucomplex-1,28 and Ru-complex-233 were taken from the original benchmark papers. Table V summarizes the main results for [Fe(CN)5 (py)]3- at different levels of theory. Many lowlying MLCT states of different symmetry dominate the lowenergy region of the UV/Vis spectrum, whereas the MC states are found at higher energies. For the MLCT and MC states, the agreement between the DFT/MRCI and CASPT2(14,14) results is remarkable. Indeed, DFT/MRCI is the only DFTbased method capable of reproducing the CASPT2 ordering of states. For the lowest lying MLCT states, there are some rather minor differences (redshifts of some of the states by ca. 0.2 eV), as previously observed for [Fe(CN)6 ]4− . DFT/MRCI places the other states (including all the MC states) within 0.1 eV of the CASPT2 values. By contrast, in the TDDFT calculations, none of the tested functionals is able to accurately describe the excited states of [Fe(CN)5 (py)]3− , especially the MLCT states, or to get the correct ordering of states. TD-CAM-B3LYP consistently overestimates
TABLE V. Selected electronic transition energies (in eV) and oscillator strengths (in parentheses) of [Fe(CN)5 (py)]3− at different levels of theory.
11 A1 (1 MLCT) 1 1 A2 (1 MLCT) 1 1 B1 (1 MLCT) 2 1 B1 (1 MLCT) 2 1 A2 (1 MLCT) 1 1 B2 (1 MLCT) 2 1 B2 (1 MC) 3 1 B1 (1 MC) 2 1 A1 (1 MC)
CASPT2(14,14)a /(10,10)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
0.96 (0.01)a,c 0.72 (0.00)a,c 0.77 (0.00)a,c 1.34 (0.00)a,c 1.36 (0.00)a,c 1.49 (0.00)a,c 2.89 (0.00)b,c 3.03 (0.00)b,c 3.49 (0.00)b,c
1.16 (0.27) 0.53 (0.00) 0.52 (0.00) 1.38 (0.00) 1.33 (0.00) 1.39 (0.00) 2.82 (0.00) 2.93 (0.00) 3.34 (0.00)
2.19 (0.17) 1.74 (0.00) 1.61 (0.00) 2.76 (0.00) 2.62 (0.00) 2.49 (0.00) 2.92 (0.00) 2.96 (0.00) 3.77 (0.00)
1.99 (0.18) 1.27 (0.00) 2.22 (0.00) 2.47 (0.00) 2.09 (0.00) 2.01 (0.00) 3.07 (0.00) 3.11 (0.00) 3.53 (0.01)
1.98 (0.17) 0.84 (0.00) 0.79 (0.00) 1.11 (0.00) 1.07 (0.00) 1.36 (0.00) 3.33 (0.00) 3.63 (0.01) 3.58 (0.00)
a
From Ref. 31. From Ref. 31. c Oscillator strengths obtained at the PCM-CASPT2(14,14) level of theory from Ref. 31. b
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194105-6
D. Escudero and W. Thiel
J. Chem. Phys. 140, 194105 (2014)
TABLE VI. Selected electronic transition energies (in eV) and oscillator strengths (in parenthesis) of [Fe(bpy)3 ]2+ at different levels of theory.
2 1 A (1 MLCT) 3 1 A (1 MC) 4 1 A (1 MLCT) 5 1 A (1 MLCT) 1 1 B (1 MC) 2 1 B (1 MC) 3 1 B1 (1 MC) 4 1 B (1 MLCT) 19 1 B (1 LC) a b
CASPT2(10,15)a /(12,12)b
DFT/MRCI
TD-CAM-B3LYP
TD-PBE0
TD-PBE
2.44a 2.51a 2.59a 2.72a 2.47a 2.48a 2.59a 2.65a ∼4.65b
1.69 (0.00) 2.41 (0.00) 2.59 (0.03) 2.74 (0.00) 1.69 (0.00) 1.88 (0.00) 2.41 (0.00) 2.58 (0.00) 4.40 (0.26)
3.23 (0.00) 2.73 (0.00) 3.38 (0.03) 3.63 (0.05) 2.73 (0.00) 2.84 (0.00) 3.83 (0.01) 3.23 (0.00) 4.52 (0.07)
2.80 (0.00) 2.60 (0.00) 2.92 (0.02) 3.08 (0.05) 2.60 (0.00) 2.76 (0.00) 3.65 (0.00) 2.82 (0.00) 4.58 (0.21)
1.78 (0.00) 3.15 (0.01) 1.98 (0.01) 2.18 (0.04) 3.15 (0.01) 3.29 (0.00) 3.81 (0.00) 1.68 (0.00) 3.51 (0.00)
From Ref. 32. From Ref. 32.
the excitation energies of all MLCT states by at least ca. 1 eV. TD-PBE0 behaves analogously, but in a less systematic manner, with errors ranging from ca. 0.55 eV (see, e.g., 1 1 A2 ) to ca. 1.5 eV (see, e.g., 1 1 B1 ). Both TD-CAMB3LYP and TD-PBE0 accurately locate the MC states. TDPBE shows a somewhat better overall performance (i.e., being closer to the CASPT2 results), in line with a previous TD-DFT evaluation for Fe(II) complexes.34 Still, the errors for the MLCT states are inconsistent, since TD-PBE overestimates the excitation energy of the bright MLCT state of A1 symmetry (by 1.0 eV) while predicting the other 1 MLCT transitions with acceptable accuracy. The MC states are also overestimated at this level (by 0.1–0.6 eV) so that TD-PBE predicts an incorrect ordering of the states. The TD-DFT and DFT/MRCI methods give similar oscillator strengths, which cannot be directly compared with the CASPT2 values because the latter were computed in the presence of solvent. We conclude that it may be misleading to rely too much on TD-DFT results in the interpretation of electronic spectra of complexes such as [Fe(CN)5 (py)]3− . Recently, the excited states of the larger [Fe(bpy)3 ]2+ complex have been studied using the CASPT2 method.32 Table VI collects the results for the electronically excited states of [Fe(bpy)3 ]2+ at the CASPT2, DFT/MRCI, and TDDFT levels (note that the DFT/MRCI calculations employed the def2-SV(P) basis set and C2 symmetry constraints). Since the LC state contributes to the near-UV region of the spectrum (see Table VI), this complex is well suited to evaluate the performance of the DFT/MRCI and TD-DFT methods (not only for LC, but also for MC and MLCT states). DFT/MRCI again underestimates the excitation energies of the lowest excited states (i.e., 2 1 A, 1 1 B, and 2 1 B) uniformly by ca. 0.7 eV relative to the CASPT2 values, which thus appears to be a common trend for all the Fe(II) complexes studied herein. The remaining states, i.e., 3 1 A, 4 1 A, 5 1 A, 3 1 B, 4 1 B, and 19 1 B (i.e., the lowest bright LC state) are accurately located at the DFT/MRCI level, regardless whether they are reached by MC, LC, or MLCT excitations. Notably, the ordering of states is the same for DFT/MRCI and CASPT2. On the other hand, the TD-DFT results are again very functional-dependent. The TD-CAM-B3LYP excitation energies are overestimated, especially for the MLCT states, and the lowest excited state is thus a MC state, in disagree-
ment with CASPT2 and DFT/MRCI; however, the LC state is predicted rather well (albeit with lower oscillator strength). Overall, TD-PBE0 performs best among the tested functionals, with a reasonably balanced description of MC, MLCT, and LC states (and an especially good description of the LC state, in terms of energy and oscillator strength). Still, TDPBE0 predicts some of the MC states too high by more than 1 eV (i.e., 3 1 B1 ) and gives a state ordering different from CASPT2 and DFT/MRCI. TD-PBE shows the opposite behavior: the computed excitation energies are underestimated by 0.6–1.0 eV for the MLCT states and overestimated by ca. 0.7 eV for the MC states; TD-PBE also fails in correctly locating the LC state. In conclusion, the TD-DFT results for the different Fe(II) complexes studied herein reveal a dependence of the computed spectroscopic properties on the choice of the functional. TD-PBE appears to perform best for the negatively charged compounds [Fe(CN)5 (py)]3− and [Fe(CN)6 ]4− , while TD-PBE0 appears superior for the positively charged compound [Fe(bpy)3 ]2+ . The reasons for these discrepancies are unknown. Such erratic behavior of the tested TD-DFT methods makes it difficult to generally recommend any functional as being best for describing the spectroscopic properties of Fe(II) complexes. On the contrary, the DFT/MRCI method usually predicts the correct order of states and treats all kind of excited states in a balanced manner, while showing a systematic error (redshift) for the lowest lying excited states. Finally, we discuss the spectroscopic properties of some 4d-TM complexes, namely, Ru-complex-1 and Ru-complex2 (see Figure 1), which have previously been studied by some of us28 and others33 with the RASPT2 method. Table VII contains the results for selected electronically excited states obtained at different levels of theory. As in the case of [Fe(bpy)3 ]2+ , electronic transitions of different character contribute to the UV/Vis spectrum of Ru-complex1 (i.e., MLCT, MC, and LC transitions). The lowest lying 1 1 B1 and 2 1 A1 states are of MC and MLCT character, respectively. These states are almost degenerate at the RASPT2 level. They are also very close in energy at the DFT/MRCI level, but too low by ca. 0.4–0.5 eV. This is a common trend for all 3d- and 4d-TM complexes studied herein. The higherlying MLCT state (4 1 B2 ) is well predicted by DFT/MRCI, while the LC state is computed too high by ca. 0.4 eV. Concerning the TD-DFT results originally reported in Ref. 28,
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194105-7
D. Escudero and W. Thiel
J. Chem. Phys. 140, 194105 (2014)
TABLE VII. Selected electronic transition energies (in eV) and oscillator strengths (in parentheses) of trans(Cl)-Ru(bpy)Cl2 (CO)2 (Ru-complex-1) and Rucomplex-2 at different levels of theory. Ru-complex-1 1 1 B1 (1 MC) 2 1 A1 (1 MLCT) 5 1 B2 (1 LC) 4 1 B2 (1 MLCT) Ru-complex-2 FDAES- 21 A (1 MLCT)
MS-RASPT2(16,3,3;4,7,4)a
DFT/MRCI
TD-CAM-B3LYP a
TD-PBE0 a
TD-PBEa
3.36 (0.01) 3.34 (0.06) 3.70 (0.31)
2.92 (0.01) 2.81 (0.03) 4.09 (0.33)
3.39 (0.00) 3.38 (0.00) 4.67 (0.26)
3.31 (0.00) 2.60 (0.01) 4.44 (0.20)
3.09 (0.00) 1.56 (0.01) 4.02 (0.07)
3.96 (0.01)
3.99 (0.02)
4.51 (0.02)
3.49 (0.00)
2.19 (0.00)
RASPT2(18,18)b 2.84 (0.22)
DFT/MRCI 2.71 (0.36)
TD-CAM-B3LYP 3.50 (0.40)
TD-PBE0 2.78 (0.34)
TD-PBE 2.12 (0.36)
a
From Ref. 28. RAS(n,l,m;I,j,k) notation: n is the number of active electrons; l is the maximum number of holes in RAS1; m is the maximum number of electrons in RAS3; and i, j, and k are the number of orbitals in RAS1, RAS2, and RAS3, respectively. b From Ref. 33.
the excitation energies computed for the MC state do not vary much for the tested functionals, whereas those for the MLCT and LC states are rather more functional-dependent. The TD-PBE excitation energy is quite accurate for the LC state, but much too low for the MLCT transitions (e.g., by up to 1.8 eV for the 4 1 B2 state). TD-CAM-B3LYP and TDPBE0 also have problems with the MLCT states, which are calculated notably too high and too low, respectively. Concerning the oscillator strengths, DFT/MRCI offers the best agreement with respect to the RASPT2 values. On the other hand, none of the tested functionals is able to reproduce the trends in the relative intensities of the states. Overall, the PBE0 functional (and generally hybrid functionals with intermediate amounts of exact exchange) provide the best compromise in terms of excitation energies and intensities for the excited states of the Ru-complex-1. Table VII also includes the results for the first dipole-allowed excited state (FDAES) of Ru-complex-2. Both DFT/MRCI and TD-PBE0 closely reproduce the RASPT2 excitation energy value of this MLCT state, which is notably overestimated and underestimated by TD-CAM-B3LYP and TD-PBE, respectively. The oscillator strengths from DFT/MRCI and TD-DFT are somewhat higher than our best RASPT2 reference value.
IV. CONCLUSIONS
TM complexes are prototypes of systems where nondynamic correlation effects in the ground and excited states may become extremely important. This makes it especially complicated to handle them computationally, and it is thus often difficult to obtain correct spin-state energies and a balanced description of the excited states. This is especially true when non-innocent ligands are involved or in the case of complexes possessing highly covalent metal-ligand bonds. In principle, multi-configurational methods are preferred for such systems, since they can deal with any kind of electronic structure problem. However, their application is still restricted to small or medium-size systems and requires a judicious choice of suitable computational options. Hence, there is a demand for less expensive methods that can be used in a “black-box” manner to compute the excited-state properties of TM complexes. In this paper, we have assessed the performance of the DFT/MRCI method and several TD-DFT approaches. For highly multi-reference situations, such as the
permanganate anion, none of the tested methods reproduces the UV/Vis spectrum with acceptable accuracy, which in the case of DFT/MRCI can be attributed to the small size of the chosen CI expansion that it is not able to recover all important static correlation effects. For the other complexes studied herein, DFT/MRCI performs reasonably well for describing the excited-state properties of TM complexes, both for singlet and triplet excited states. In general, DFT/MRCI systematically underestimates the energies of the lowest excited states, but provides the correct order of states and a balanced description of excited states of different character. By contrast, the present TD-DFT results are less accurate and generally depend strongly on the chosen functional. In the series of Fe(II) complexes, the TD-DFT errors are not systematic so that it is difficult to recommend any of the tested functionals—in general, none of them gives the correct order of states or a balanced description of all excited states of different character. In the majority of cases, the PBE0 hybrid functional with an intermediate amount of exact exchange provides TDDFT results of higher quality than the pure PBE functional and the long-range corrected CAM-B3LYP hybrid functional. We conclude that the assignment of UV/Vis spectra of TM complexes exclusively based on TD-DFT data may be dangerous without an initial careful validation. Interpretations based on a simple matching of experimental and theoretical bands may be misleading, because there may be accidental matches due to the large number of low-lying transitions and possibly large errors in the TD-DFT predictions. The limited accuracy of TD-DFT is also a critical issue in its potential use for computing excited-state potential energy surfaces during non-adiabatic excited-state dynamics simulations of such systems. Finally, we have also evaluated the performance of the DFT/MRCI method for TM ground-state bonding in the demanding case of CrF6 . The DFT/MRCI results are reasonably close to those from extensive RASPT2 calculations. Further evaluation of the DFT/MRCI method for other ground-state problems involving TM complexes, such as spin-state energetics, are currently under evaluation in our group. 1 K.
Pierloot, Int. J. Quantum Chem. 111, 3291 (2011); L. González, D. Escudero, and L. Serrano-Andrés, ChemPhysChem 13, 28 (2012). 2 K. Pierloot, Mol. Phys. 101, 2083 (2003). 3 S. Grimme and M. Waletzke, J. Chem. Phys. 111, 5645 (1999). 4 S. Grimme, Chem. Phys. Lett. 259, 128 (1996).
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: 128.123.35.41 On: Thu, 21 Aug 2014 03:12:21
194105-8 5 M.
D. Escudero and W. Thiel
R. Silva-Junior, M. Schreiber, S. P. A. Sauer, and W. Thiel, J. Chem. Phys. 129, 104103 (2008); C. M. Marian and N. Gilka, J. Chem. Theory Comput. 4, 1501 (2008). 6 M. Kleinschmidt, C. M. Marian, M. Waletzke, and S. Grimme, J. Chem. Phys. 130, 044708 (2009). 7 V. Schmid, R. Linder, and C. M. Marian, Eur. J. Inorg. Chem. 2006, 1588 (2006). 8 F. Réal, V. Vallet, C. Marian, and U. Wahlgren, J. Chem. Phys. 127, 214302 (2007). 9 S. Vancoillie, H. Zhao, V. T. Tran, M. F. A. Hendrickx, and K. Pierloot, J. Chem. Theory Comput. 7, 3961 (2011). 10 R. Ahlrichs, M. Bär, M. Häser, H. Horn, and C. Kölmel, Chem. Phys. Lett. 162, 165 (1989). 11 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., Gaussian 09, Revision A.1, Gaussian, Inc., Wallingford, CT, 2009. 12 K. Pierloot and B. O. Roos, Inorg. Chem. 31, 5353 (1992). 13 (a) H. Nakai, Y. Ohmori, and H. Nakatsuji, J. Chem. Phys. 95, 8287 (1991); (b) A. C. Stückl, C. A. Daul, and H. U. Güdel, Int. J. Quantum. Chem. 61, 579 (1997). 14 I. H. Hillier and V. R. Saunders, Chem. Phys. Lett. 9, 219 (1971); H. Hsu, C. Peterson, and R. M. Pitzer, J. Chem. Phys. 64, 791 (1976); H. Johanson, Mol. Phys. 49, 1209 (1983). 15 R. M. Dickson and T. Ziegler, Int. J. Quantum Chem. 58, 681 (1996). 16 S. J. A. Van Gisbergen, J. A. Groenveld, A. Rosa, J. G. Snijders, and E. J. Baerends, J. Phys. Chem. A 103, 6835 (1999). 17 M. Nooijen and V. Lotrich, J. Chem. Phys. 113, 494 (2000). 18 M. A. Buijse and E. J. Baerends, J. Chem. Phys. 93, 4129 (1990). 19 K. Pierloot, “Calculation of electronic spectra of transition metal complexes,” in Computational Photochemistry (Elsevier, Amsterdam, 2005), pp. 279–315.
J. Chem. Phys. 140, 194105 (2014) 20 J.
Su, W.-H. Xu, C.-F. Xu, W. H. E. Schwarz, and J. Li, Inorg. Chem. 52, 9867 (2013). 21 K. Pierloot, E. Tsokos, and L. G. Vanquickenborne, J. Phys. Chem. 100, 16545 (1996). 22 K. Pierloot, E. Van Praet, L. G. Vanquickenborne, and B. O. Roos, J. Phys. Chem. 97, 12220 (1993). 23 C. Pollak, A. Rosa, and J. E. Baerends, J. Am. Chem. Soc. 119, 7324 (1997). 24 A. Rosa, E. Baerends, S. J. A. van Gisbergen, A. van Lenthe, J. A. Groeneveld, and J. G. Snijders, J. Am. Chem. Soc. 121, 10356 (1999); P. Hummel, J. Oxgaard, W. A. Goddard III, and H. B. Gray, Inorg. Chem. 44, 2454 (2005). 25 N. Ben Amor, S. Villaume, D. Maynau, and C. Daniel, Chem. Phys. Lett. 421, 378 (2006); S. Villaume, A. Strich, C. Daniel, S. A. Perera, and R. J. Bartlett, Phys. Chem. Chem. Phys. 9, 6115 (2007). 26 N. A. Beach and H. B. Gray, J. Am. Chem. Soc. 90, 5713 (1968). 27 R. Crespo-Otero and M. Barbatti, J. Chem. Phys. 134, 164305 (2011). 28 D. Escudero and L. González, J. Chem. Theory Comput. 8, 203 (2012). 29 T. Le Bahers, E. Bremond, I. Ciofini, and C. Adamo, “The nature of vertical excited states of dyes containing metals for DSSC applications: Insights from TD-DFT and density based indexes,” Phys. Chem. Chem. Phys. (published online). 30 J. B. Ayers and W. H. Wagoner, J. Inorg. Nucl. Chem. 33, 721 (1971). 31 A. L. B. Formiga, S. Vancoillie, and K. Pierloot, Inorg. Chem. 52, 10653 (2013). 32 C. de Graaf and C. Sousa, Chem. Eur. J. 16, 4550 (2010). 33 B. J. Coe, A. Avramopoulos, M. G. Papadopoulos, K. Pierloot, S. Vancoillie, and H. Reis, Chem. Eur. J. 19, 15955 (2013). 34 I. Georgieva, A. J. A. Aquino, N. Trendafilova, P. S. Santos, and H. Lischka, Inorg. Chem. 49, 1634 (2010).
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: 128.123.35.41 On: Thu, 21 Aug 2014 03:12:21
