Performance of Conventional and Range-Separated Hybrid Density

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Performance of Conventional and Range-Separated Hybrid Density Functionals in Calculations of Electronic Circular Dichroism Spectra of Transition Metal Complexes Mark Rudolph and Jochen Autschbach* Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260, United States

bS Supporting Information ABSTRACT: A number of density functionals, including ‘pure’ (nonhybrid) functionals, global hybrids, and range-separated hybrids, were used to calculate the electronic circular dichroism (CD) spectra of 10 tris-bidentate transition metal complexes. The results are compared to one another and to experimental CD spectra, in an effort to illustrate the shortcomings of particular approximations in time-dependent density functional theory (TDDFT). The use of an origin invariant formalism to calculate magnetic transition dipole moments with the help of gauge-including atomic orbitals (GIAOs) is also investigated. With valence basis sets of moderate flexibility, good agreement between GIAO results and rotatory strengths calculated from the dipolevelocity representation is obtained for selected test cases. Empirically broadened vertical CD spectra calculated with the global hybrid functionals B3LYP and PBE0 are found to agree overall the best with experimental CD spectra.

1. INTRODUCTION Time-dependent density functional theory (TDDFT) is a computationally efficient tool to obtain excitation spectra and transition moments for molecules from first principles. However, research carried out during the past decade has also demonstrated that the approximations made in most exchange-correlation (XC) potentials and the associated response kernels limit the accuracy of linear-response TDDFT excitation spectra in specific ways. There is, for instance, the lack of capability to describe electronic transitions with substantial double excitation character with a frequency-independent (adiabatic) XC response kernel derived from standard functionals.1 More important from a spectral simulation point of view are problems with traditional nonhybrid and global hybrid functionals to describe long-range charge-transfer (CT) excitations, the energies of which are notoriously underestimated. This problem is now well understood.28 For clear-cut long-range CT excitations, reliable energies can be obtained with hybrid functionals where the exchange is range-separated and switches from DFT to HartreeFock (HF) as the interelectronic distances increase.10 A frequently used separation of the operator 1/r12 in the HF exchange integrals into a long- and short-range part10 is 1 α þ βerf ðγr12 Þ 1 ½α þ βerf ðγr12 Þ ¼ þ r12 r12 r12

ð1Þ

Here, γ is the range-separation parameter (often chosen to be on the order of 0.3 bohr1), α is the fraction of HF exchange that is r 2011 American Chemical Society

present at short-range, and α + β is the fraction of HF exchange at very large r12. Conventional hybrids correspond to β = 0, with α being the fraction of global HF exchange. Fully long-range corrected (LC) functionals have α + β = 1 and switch to 100% HF asymptotically. Many excitations afford some CT character, but a full LC may not be desirable for best performance. ‘Coulomb-attenuated’ methods (CAM) have been proposed where the exchange range separation does not fully switch to HF. A popular example of this class of functionals is CAMB3LYP10 which is parametrized to give α + β = 65% HF exchange at large interelectronic distances. Another problematic issue in TDDFT calculations is the incorrect asymptotic behavior of most XC potentials which may deteriorate the accuracy of excitations involving diffuse valence states and Rydberg states. Shape corrections to XC potentials have been proposed to obtain the correct 1/r asymptotic behavior of the potential.11,12 Likewise, LC range-separated functionals afford the desired 1/r dependence and are therefore expected to perform well for excitations involving diffuse states. Range-separated density functionals have been shown to be a valuable tool for calculating a variety of molecular properties, especially long-range charge-transfer excitations but mainly in main group (organic) chemistry.7,9,1323 As benchmark studies have revealed, the performance of such functionals is strongly Received: September 28, 2011 Revised: November 12, 2011 Published: November 14, 2011 14677

dx.doi.org/10.1021/jp2093725 | J. Phys. Chem. A 2011, 115, 14677–14686

The Journal of Physical Chemistry A dependent on the parametrization, both regarding the α and β parameters, controlling the DFT and HF contributions globally, and regarding the range-separation parameter γ determining the balance of DFT to HF exchange at intermediate r12.2426 For instance, Peach et al.24 have emphasized the importance of the parameter choice in order to qualitatively and quantitatively account for different classes of vertical electronic excitation energies. The γ parameter appears to be of critical influence here. A demonstrated way to improve calculations with range-separated functionals in a more systematic manner is to treat γ as a systemdependent functional of the density that can be tuned based on fundamental DFT-based considerations.27 We have recently highlighted the challenge of calculating reliable excitation spectra of metal complexes from TDDFT28 in the context of simulating electronic circular dichroism (CD). The cited article has focused on the calculated ligand-field transitions in the complexes [Co(en)3]3+ and [Rh(en)3]3+, showing that these energies are very sensitive to the balance of exact versus DFT exchange. Moreover, calculations for the low-energy part of the CD spectrum of [Fe(phen)3]2+ demonstrated an interplay of approximations for the excitation energies and for the geometry that makes it difficult to decide which functional combination performs best. Herein, we consider a larger set of complexes and the full UVvis spectral range that has been explored experimentally. CD spectra of metal complexes in the UVvis range tend to exhibit much detail due to the changing sign patterns of the rotatory strengths. Thus, in addition to CD of metal complexes being an important topic of research in itself,29,30 CD is also a sensitive property that is useful for TDDFT benchmarks. A successful simulation of a CD spectrum requires accurate excitation energies. Further, the magnitudes of the electric and magnetic transition dipole moments as well as their relative orientations have to be correct. For a typical metal complex with unsaturated ligands, for example [Ru(bipy)3]2+, the experimentally accessible UVvis spectral range includes ligand-field (LF) transitions, metal-to-ligand and ligand-to-metal CT (MLCT, LMCT), and ligand-centered transitions that may exhibit strong exciton coupling CD. Exciton couplets in metal complexes resulting from ligand centered π-to-π* transitions were shown to be described quite well already with nonhybrid functionals,3133 while for other types of excitations in metal complexes, the results appear to be very sensitive to the quality of the XC functional.28 One should expect MLCT and LMCT transitions to be significantly improved by the use of range-separated hybrid functionals, hopefully without doing much harm to the description of ligand-centered exciton CD. Given that LF transitions in pseudooctahedral complexes have weak intensity, range-separated functionals may therefore yield significantly improved simulated CD spectra of metal complexes, in particular in frequency regimes that are not dominated by ligand centered π-to-π* transitions. The main purpose of this work is to assess the performance of range-separated XC functionals in simulations of electronic CD spectra of metal complexes over a wide spectral range, and to compare the spectra to those obtained with nonhybrid functionals and with global (fixed HF exchange) hybrids. A set of 10 tris-bidentate [M(L)3]n+ complexes with saturated and unsaturated ligands L and different metals selected from the 3d transition metal series (Co, Fe) and the heavier 4d counterparts (Rh, Ru) is investigated. The protocol for the CD spectra simulation followed here is ‘standard’ in the sense that vertical excitations are empirically broadened with Gaussian line shapes. The

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calculation of vibronic band shapes is in principle possible34,35 but computationally demanding and therefore presently not carried out routinely. We show that, for the prototype [M(L)3]n+ systems considered here, the tested range separated hybrids do not clearly outperform more commonly used global hybrids for the purpose of spectral assignments and in comparison with experimental data. This is likely due to the absence of truly long-range CT excitations in the [M(L)3]n+ complexes (where the CT improvements offered by CAM and LR functionals would be particularly advantageous), and the absence of excitations into diffuse states within the experimentally accessible frequency range (where the improvements in the asymptotic region offered by LC XC potentials would be an asset). The results therefore suggest that computationally more demanding simulations of vibronic transition, or a dynamic treatment of metal complexes, including explicit solvation, might be more promising for obtaining significantly better agreement with experiment rather than tinkering with parameters in already quite flexible functionals. Details about the computations are provided in section 2. Calculated CD spectra are presented in section 3, and the performance of the various functionals is discussed. A brief summary and an outlook can be found in section 4.

2. COMPUTATIONAL DETAILS Calculations of excitation energies and rotatory strengths were performed with two computational packages using the following Gaussian-type basis sets: H: 31G. C,N: 6-31G*. Fe,Co: Wachters +f. Ru,Rh: TZVP with a corresponding effective core potential. The UVvis spectral range of the [M(L)3]n+ complexes investigated here is dominated by valence transitions31,36 which therefore allows the use of basis sets that do not include diffuse functions. The Turbomole37 package was used for calculations employing the BeckePerdew (BP)3842 and PerdewBurke Ernzerhof (PBE)43 nonhybrid functionals, and for most calculations with the PBE0 (25% HF exchange)44 and B3LYP45 (20% HF exchange) global hybrids (the latter using the VWNIII parametrization46 in the local density approximation). A locally modified developer version of the NWChem47 package was used for computations with the range-separated hybrid functional CAM-B3LYP10 (Coulomb attenuated, α = 0.19, β = 0.46, γ = 0.33) and a fully long-range corrected variant25 of PBE0 dubbed here LC-PBE0 (α = 0.25, β = 0.75, γ = 0.30). Calculations of rotatory strengths with gauge-including atomic orbitals (GIAOs) were performed with a new implementation in NWChem described in ref 48. Spectra not calculated with GIAOs were generated from an origin invariant form of the rotatory strengths based on electric transition dipole moments in the velocity representation. A continuum solvent model, the conductor-like screening model (COSMO), was used in all calculations with default parameters and a dielectric constant for water (ε = 78). A series of NWChem calculations with the BP functional was carried out for [Co(en)3]3+ to test the influence of the ‘solvent radius’ parameter in the COSMO calculations (varied from 0 to 1 Å). The CD bands were affected in a minor way, with energy shifts of 0.1 eV or less and slight intensity changes. Therefore, the calculated spectra are not expected to vary in a significant way if different COSMO parameters chosen within reasonable limits are adopted. Because of the varying energy ranges of the available experimental data, as well as differing computational methods, a varying number of excitations were calculated per complex and per functional. Unless explicitly noted otherwise, the lowest number of 14678

dx.doi.org/10.1021/jp2093725 |J. Phys. Chem. A 2011, 115, 14677–14686

The Journal of Physical Chemistry A

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Figure 1. Calculated CD spectra of [Co(en)3]3+ (left) and [Fe(phen)3]2+ (right) with and without GIAOs. GIAO calculations were performed with NWChem. Rotatory strengths in the velocity formulation were taken from results of Turbomole calculations.

Table 1. Summary of the Performance of TDDFT Functionals for Calculated Circular Dichroisma complex/Functional BP PBE PBE0 B3LYP LC-PBE0 CAM-B3LYP [Co(en)3]3+

L

L

L

L

L

L

[Co(bipy)3]3+

X

X

(G)

G

(G)

G

[Co(phen)3]3+ [Rh(en)3]3+ b,c

X X (G) (G)

H G

G G

H G

H G

[Rh(bipy)3]3+ c

G

G

G

(G)

(G)

[Rh(phen)3]3+ c

(G) (G)

(G)

(G)

(G)

(G)

[Fe(bipy)3]2+

H

H

H

(G)

H

(L)

[Fe(phen)3]2+

G

G

G

H

H

H

[Ru(bipy)3]2+

(H) (H)

G

G

X

X

[Ru(phen)3]2+

H

G

G

H

H

G

H

a

G: Reasonably good agreement with respect to sign pattern in the lowand high-energy region of the spectra (with up to 0.5 eV band shifts deemed acceptable). L: Good performance mainly only in the lowenergy regime. H: Good performance mainly only in the high-energy regime. X: Relatively poor performance over the entire computed range. Parentheses: borderline acceptable performance. b Only LF transitions occurring in the experimentally recorded spectral range. c Spectra for this complex were not separated into a low- and high-energy regime.

singlet excitations calculated or shown extends at least 1 eV beyond the upper frequency cutoff of the experimental spectra to avoid truncation errors in the calculated broadened CD intensity. All spectra shown, computed and experimental, are for the Λ configurations of the complexes. Computed spectra were Gaussian broadened with a global σ parameter of 0.13 eV, which corresponds to 0.31 eV peak widths at half peak heights. The corresponding nonbroadened line spectra are provided in the Supporting Information. For general aspects of CD spectra simulations and further details, see ref 49.

3. RESULTS AND DISCUSSION There are typically strong CD intensity differences for the [M(L)3]n+ system between a low-energy regime with LF, MLCT, and LMCT transitions (often below 3.5 eV/above 350 nm) and a higher energy regime where MLCT and LMCT transitions may be overpowered by intense ligand centered π-to-π* exciton CD.

Therefore, for most of the complexes, the spectra have been split into two plots, separating low- and high-energy regimes where applicable. We note again that all spectra are representative of the Λ absolute configurations of the complex. In our experience, compared to nonhybrid functionals, hybrids typically generate fewer excitations per energy unit in the spectra of [M(L)3]n+ complexes such as studied herein. Further, a larger fraction of HartreeFock included in a global or rangeseparated hybrid tends to lead to more blue-shifted spectra overall. A similar trend has also been found for computed circular dichroism spectra in organic molecules such as methyloxirane or 1,10 -bi(2-naphthol).50 An exception appears to be the LF transitions in 3d complex ions such as [Co(en)3]3+ where nonhybrid functionals strongly overestimate the energies, HF strongly underestimates, and global hybrids such as PBE0 and B3LYP happen to yield excitation energies that agree well with experiment.28 For a spectral region dominated by a particular type of excitation, the sensitivity to the XC functional may allow one to artificially choose a ‘best’ type of functional. However, the occurrence of different types of excitations in [M(L)3]n+ type systems over large spectral regions28,31,33,36 complicates the situation. Before beginning a discussion of the performance of the various functionals in comparison with experimental CD spectra, it is important to establish that the choice of the velocity representation of the electric transition moments in calculations of the rotatory strengths (spectral intensities) is unproblematic with the basis sets used here. For this purpose, computations of CD spectra of [Co(en)3]3+ and [Fe(phen)3]2+ using a GIAO formalism were compared to the corresponding velocity gauge spectra. The spectra, shown in Figure 1, demonstrate that there is excellent agreement between the two methods. Thus, the GIAO formalism does not offer particular advantages over the velocity gauge for calculating the CD spectra of the metal complexes considered herein with the basis sets listed in section 2. The discussion in the remainder of this section is therefore based on CD spectra obtained from the velocity form of the rotatory strengths, which allowed us to save some computational resources in particular when obtaining results with the NWChem code. For readers less interested in details of the spectra, a summary of the performance of the various functionals investigated in this 14679



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Figure 2. Calculated CD spectra of [Co(en)3]3+ using TDDFT with the functionals BP, PBE, B3LYP, CAM-B3LYP, PBE0, and LC-PBE0. Spectra have been Gaussian broadened based on rotatory strengths calculated with the electric dipole velocity representation and are magnified as labeled. Exptl data from ref 52.

work is given in Table 1. The assessment is based mainly on how well a given functional reproduces the experimental CD sign patterns in different spectral regions. In order for a computation to be considered as reproducing the experimental spectral features, energetic shifts of CD bands of up to about 0.5 eV were considered borderline acceptable (although, certainly, not desirable). Because of possible nonuniform shifts in the positions of calculated versus experimental CD bands, we have not attempted to quantify the spectral agreement but rather used a visual assessment instead. Good (G) performance for a majority of the complexes is obtained with the two global hybrid functionals, although difficulties remain with the low-energy regions of some of the spectra. For [Co(en)3]3+, all of the functionals predicts a weak intensity pair of LMCT excitations in the high-energy region of the spectrum, followed by a more intense and energetically more separated pair of CD bands. It is unclear which set of these excitations correspond to the broad downup pattern seen between 5 and 7 eV in the experimental spectrum, and therefore none of the functionals is considered to perform well in reproducing experiment. For [Rh(en)3]3+, the overall acceptable performance of all functionals is simply due to the higher energy of LMCT excitations outside of the experimentally recorded frequency window. In comparison with the two global hybrids, the nonhybrid functionals and the range-separated hybrids yield deficient spectra more often. For [Ru(bipy)3]2+, the exciton coupling region of the spectrum is not well reproduced by the two range-separated functionals. Apart from this outlier, and a poor performance of CAM-B3LYP for the high-energy portion of the [Fe(bipy)3]2+ spectrum, the ligand-centered exciton CD of the phen and bipy complexes is reproduced reasonably well with all functionals. For the Co complexes with bipy and phen ligands, the two nonhybrid functionals give CD intensity of opposite sign compared with experiment between 3 and 3.5 eV and are therefore considered poorly performing in the low-energy spectral region. The [M(L)3]n+ complexes studied herein are relatively compact, and therefore they do not afford truly long-range CT excitations. As pointed out by Rosa et al.,51 many transitions in metal complexes that are formally labeled as charge-transfer involve occupied and unoccupied orbitals that are spatially not well separated. A likely consequence is that the good performance for CT excitations offered

by range-separated functionals does not play out in the [M(L)3]n+ CD spectra. Likewise, the asymptotic region is not of particular importance for the UVvis spectral range of these systems. The slightly less good agreement with experiment of the simulated CD spectra obtained with the range-separated hybrids is therefore potentially caused by an overall too large component of HF in the exchange (possibly leading to a trade-off of electron correlation). Alternatively, it is possible that the spectra with the range-separated functionals are the most accurate among our calculations, theoretically, but that the approximation of simulating the spectra from vertical excitations calculated for static structures would have to be lifted to bring such advantages to light. For the metals and types of chromophores encountered in our samples, simulations based on empirically broadened vertical excitations obtained with global hybrid functionals such as PBE0 or B3LYP are seen to deliver acceptable accuracy that is not surpassed by the range-separated versions. Individual ‘tuning’ of the range-separation parameter γ of eq 1 as suggested by Baer et al.27 might improve the results. We will investigate such tuning for metal complexes in a follow-up study. A discussion of the spectra of individual complexes follows. The experimental CD spectrum of [Co(en)3]3+ (Figure 2) exhibits a weak CD band around 2.5 eV with a maximum Δε below 2 L 3 mol1 3 cm1, followed by an even weaker band 1 eV higher. The excitations below 4 eV are assigned as LF transitions involving occupied and unoccupied metal d orbitals. The CD bands at the high-frequency end of the spectrum are assigned as LMCT.53 This assignment has already been obtained from nonhybrid TDDFT calculations in ref 36. We therefore focus here on how well the different functionals reproduce the experimental CD spectrum in the LF and in the LMCT regions. All of the functionals employed here reproduce well the two CD bands below 4 eV in magnitude and sign. Energetically, the CD band at 2.5 eV is best reproduced with the PBE0 functional, with B3LYP being shifted slightly higher in energy. Both range-separated functionals give red-shifted LF transitions compared to their counterparts, but not by much, resulting in a good peak placement with CAM-B3LYP as well. BP and PBE yield the most blueshifted results, both giving visually indistinguishable broadened spectra. The agreement between BP and PBE has been found for all of the complexes investigated here in both the low- and highenergy spectral regimes. For [Co(en)3]3+, in the higher energy 14680



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Figure 3. Calculated CD spectra of the lel3 and ob3 conformers of [Co(en)3]3+ using the PBE0 functional. See caption of Figure 2 and text for further details.

Figure 4. Calculated CD spectra of [Co(bipy)3]3+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

range of 4 to 7 eV none of the functional reproduces the experimental spectrum really well, as already mentioned in the discussion of Table 1. The calculated intensities are too high (although this could also be related to the sensitivity of the experimental setup at high energies), and the experiment gives no indication of a double pair of CD bands. The range-separated functionals yield blue-shifted excitations compared to the other functionals, which is reminiscent of the charge-transfer character of these excitations. There is conformational flexibility in the chelate rings of the [Co(en)3]3+ complex ion.53 One may label the chelate ring conformations based on whether the CC bond of the en ligand is oriented approximately parallel (lel) or oblique (ob) with respect to the 3-fold symmetry axis of the complex. Computationally, the lel3- and ob3-conformers of [Co(en)3]3+ exhibit quite similar spectra (Figure 3), with the exception of a more developed negative CD band in the lower energy regime for the lel3 conformer, which is due to a slightly larger splitting of the lowest-energy A2/E pair of transitions. Apart from this difference, no dramatic energy shifts or intensity changes are observed, which also renders it unlikely that spectra for the mixed conformers lel2ob1 and lel1ob2 will be much different. Regarding solvent effects, the COSMO continuum model applied in this work has been shown to lower the energies of the LMCT transitions.36 Solvent effects for this system have been reinvestigated

using a discrete nonquantum solvent model and force-field-based molecular dynamics (MD) simulations,54 yielding CD spectra that were very similar to those obtained with COSMO. It remains to be seen whether spectral simulations based on ab initio MD with explicit solvation at the quantum mechanical level are able to deliver significantly improved agreement with the experimental CD spectrum. In the case of [Co(bipy)3]3+ (Figure 4), the low-energy part of the experimental spectrum exhibits an updownup CD pattern of increasing magnitude up to 3.5 eV. None of the functional reproduces the first, very weak, positive peak. However, the global hybrids produce the following downup pattern with CD bands at roughly the correct energies. For this complex, B3LYP performs the best, energetically, with PBE0 being slightly redshifted low-energy. The CD bands obtained with the rangeseparated functionals are somewhat red-shifted with respect to the global hybrids. The pure functionals reproduce the negative part of the spectral pattern; however, they incorrectly produce a second negative band below 3.5 eV which overpowers the development of positive CD intensity between 3 and 3.5 eV. The pure density functionals also underperform in the higher energy regime, producing an ambiguous sign pattern. For the hybrid functionals, a clean exciton couplet is produced above 3.5 eV with the correct sign pattern. CAM-B3LYP and LC-PBE0 yield fewer spectral features compared with the other functionals 14681



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Figure 5. Calculated CD spectra of [Co(phen)3]3+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

Figure 6. Calculated CD spectra of [Fe(bipy)3]2+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

beyond the intense exciton peaks as seen in many [M(L)3]n+ complexes, not just for [Co(bipy)3]3+ but also for the other complexes investigated here. As found for [Co(en)3]3+, the higher energy part of the spectrum is blue-shifted with the range-separated functionals compared to B3LYP/PBE0. Overall, a very similar description of the functional behavior for [Co(bipy)3]3+ applies to [Co(phen)3]3+ as well. The spectra are shown in Figure 5. However, the range-separated functionals yield ambiguous results for the spectrum below 3.5 eV, and therefore in Table 1 acceptable performance is only listed for the high-energy regime. It is noted that even there, the intense first exciton couplet appears at significantly too high energy with the range-separated hybrid DFT calculations. B3LYP and PBE0 produce this couplet reasonably well, while PBE and BP exhibit unsatisfactory performance. CD spectra of the complexes of iron are shown in Figure 6 for [Fe(bipy)3]2+ and Figure 7 for [Fe(phen)3]2+. In both cases, the spectra from the range-separated functionals are slightly red-shifted at lower energies and blue-shifted at higher energies compared to their global hybrid counterparts. In a qualitative assessment, the results of the range-separated functionals turn out to be worse. Both PBE0 and B3LYP reproduce a clear exciton

couplet for [Fe(bipy)3]2+ whereas the range-separated versions produce an ambiguous sign pattern. It is interesting that this behavior is not seen for [Fe(phen)3]2+, nor for most other [M(L)3]3+ complexes with unsaturated ligands studied here. It is, however, seen for [Ru(bipy)3]2+, the only other +2 bipy complex investigated herein. Both variants of hybrid and nonhybrid functionals reproduce well the exciton bands of both [Fe(L)3]2+ complexes, with more or less pronounced blueand red-shifts with respect to experiment, respectively. BP and PBE perform significantly better at higher energies for the iron complexes compared with the cobalt systems discussed above. Between 2 and 3.5 eV, both [Fe(L)3]2+ systems exhibit in their CD a downupdown sign pattern of decreasing magnitude. No functional reproduces this pattern to a fully satisfactory degree. The experimental CD spectrum of ref 56 for [Rh(en)3]3+ (Figure 8) only covers the regime of LF excitations. Because of the larger ligand-field splitting of Rh, these transitions occur at energy significantly higher than those for the Co analogue (Figure 2). The trends for the Rh LF excitations obtained for the different functionals are comparable to those for the analogous Co complex, albeit with a somewhat less pronounced 14682



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Figure 7. Calculated CD spectra of [Fe(phen)3]2+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

Figure 8. Calculated CD spectra of [Rh(en)3]3+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 56.

Figure 9. Calculated CD spectra of [Rh(bipy)3]3+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

overestimation of the energies with the nonhybrid functionals BP and PBE. These functionals show the onset of an intense negative

Figure 10. Calculated CD spectra of [Rh(phen)3]3+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

CD band near 5.5 eV, unlike any of the hybrid functionals. The experimental spectrum does not indicate such an early onset of a negative CD band. Therefore, in Table 1 the performance of the nonhybrid functionals has been assessed as satisfactory, but only marginally so. The spectrum of [Rh(bipy)3]3+ (Figure 9) is for the most part nicely reproduced by all functionals; especially by PBE0 and B3LYP. Every calculation correctly reproduces the correct sign pattern for the exciton couplet; however, each of them overestimates the intensity of the exciton CD (assuming that the experimental spectrum has been recorded with a sample of 100% ee). The spectrum beyond the first exciton couplet is generally not well reproduced for these types of complexes. TDDFT calculations tend to produce a dense spectrum at higher energies (less so for the range-separated hybrids). The higher lying excitations in the LR-PBE0 spectrum appear far too blue-shifted. The PBE0 and B3LYP spectra resemble the experiment best. Compared to the bipy complex, the intensities of the calculated spectra for [Rh(phen)3]3+ (Figure 10) agree better with experiment. Again, all functionals perform well for the first exciton couplet, with varying success at energies beyond. The low-energy portions of the CD spectra of [Ru(bipy)3]2+ and [Ru(phen)3]2+ (Figures 11 and 12) experimentally exhibit a 14683



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Figure 11. Calculated CD spectra of [Ru(bipy)3]2+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

Figure 12. Calculated CD spectra of [Ru(phen)3]2+ using various functionals. See caption of Figure 2 for further details. Exptl data from ref 55.

broad updown pattern ranging from 2 to 3.4 eV with bands of similar magnitude. This pattern is reproduced by both global hybrids, with B3LYP showing the best peak placement. LC-PBE0 does not predict any excitations in this energy range while CAM-B3LYP starts to develop some excitations above 3 eV. As evidenced by the other functionals, the reason for the shift does not seem to be related to the adoption of an effective core potential for the metal. The pure functionals produce an updownupdown pattern for [Ru(bipy)3]2+ and an up downflat pattern for [Ru(phen)3]2+. Blue-shifting these by approximately 0.4 eV would place some of the peaks in-line with experiment, but the strong underestimation of the lowest excitation energies has prompted us to designate the performance of the pure functionals as unsatisfactory in the low-energy regime. As mentioned in the discussion of [Fe(bipy)3]2+, the high-energy portion of [Ru(bipy)3]2+ is poorly reproduced by the rangeseparated functionals, while the pure functionals and the global hybrids both give acceptable results.

4. CONCLUSIONS When comparing calculated broadened vertical CD spectra of 10 chiral tris-bidentate metal complexes with their experimental

counterparts, for the majority of the cases presented here the hybrid functionals PBE0 and B3LYP performed consistently well in the low- and high-energy ranges. The nonhybrid functionals BP and PBE performed well overall in the π-to-π* energy ranges but tended to underperform at lower spectral energies. Some of the spectra calculated with nonhybrid functionals were severely deficient. Using the different GGA functionals BP and PBE resulted in negligible differences. The spectra calculated with the CAM-B3LYP and LC-PBE0 range-separated hybrid functionals generally did not give better agreement with experiment compared with the spectra calculated with global hybrids. Overall, the two range-separated functionals performed very similarly for the complexes studied herein. For applications where empirically broadened calculated vertical CD spectra of transition metal complexes are needed, it appears that the added computational cost required for the range separation can be avoided. In several cases the range-separated functionals led to less good agreement with experiment than with global hybrids. Reasons why the range-separated hybrids did not outperform their global hybrid counterparts are likely the lack of truly longrange charge-transfer excitations in our samples, and the fact that the asymptotic region is not relevant for the UVvis spectral range of these complexes. Future studies regarding the 14684


The Journal of Physical Chemistry A dependence of the spectra on the value of the range separation parameter γ might reveal improved performance if γ is adjusted as a system-specific parameter.27,57 Structurally related systems such as the phen complexes with Co, Fe, and Rh have a tendency to yield similar CD spectra when calculated with the same functionals. However, a comparison of the BP and PBE spectra for [Co(phen)3]3+ and [Rh(phen)3]3+, for instance, demonstrates that this is not always the case, i.e., the different metals matter. The experimental spectra for these two complexes also show differences in the higher energy regime. Similar findings are obtained for [Co(bipy)3]3+ versus [Rh(bipy)3]3+, for instance. A major issue remains the correct position of the excitations, in particular in the low-energy regime where ligand-field transitions may dominate the spectrum. One may speculate at this point that an improved performance might eventually become apparent when simulating the spectra in a more rigorous fashion, by computing vibronic fine structure instead of broadening vertical transitions or by performing molecular dynamics simulations in the presence of solvent molecules. Work along those lines is currently pursued in our group.

’ ASSOCIATED CONTENT

bS

Supporting Information. Nonbroadened CD spectra (‘line spectra’, ‘stick spectra’) for each complex and each functional. This material is available free of charge via the Internet at http://pubs.acs.org

’ AUTHOR INFORMATION Corresponding Author

*E-mail: jochena@buffalo.edu.

’ ACKNOWLEDGMENT The authors acknowledge support from the Center for Computational Research at the University at Buffalo. M.R. is grateful for a 2010 Speyer summer fellowship from the University at Buffalo Chemistry Department. This work has received financial support from the National Science Foundation (CHE-0952253). The authors thank Dr. Monika Srebro for helpful comments. ’ REFERENCES (1) Casida, M. E. J. Chem. Phys. 2005, 122, 054111–9. (2) Dreuw, A.; Weisman, J. L.; Head-Gordon, M. J. Chem. Phys. 2003, 119, 2943–2946. (3) Marques, M. A. L.; Ullrich, C. A.; Nogueira, F.; Rubio, A.; Burke, K.; Gross, E. K. U., Eds. Time-Dependent Density Functional Theory; Vol. 706 of Lecture Notes in Physics; Springer: Berlin, 2006. (4) Maitra, N. T.; Tempel, D. G. J. Chem. Phys. 2006, 125, 184111–16. (5) Elliott, P.; Burke, K.; Furche, F. Excited states from timedependent density functional theory. In Reviews of Computational Chemistry; Lipkowitz, K. B.; Cundari, T. R., Eds.; Wiley: Hoboken, NJ, 2009. (6) Peach, M. J. G.; Benfield, P.; Helgaker, T.; Tozer, D. J. J. Chem. Phys. 2008, 128, 044118. (7) Stein, T.; Kronik, L.; Baer, R. J. Am. Chem. Soc. 2009, 131, 2818– 2820. (8) Autschbach, J. ChemPhysChem 2009, 10, 1–5. (9) Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. J. Chem. Phys. 2004, 120, 8425–8433.

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(10) Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51–57. (11) Tozer, D. J.; Handy, N. C. J. Chem. Phys. 1998, 109, 10180– 10189. (12) Schipper, P. R. T.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. J. Chem. Phys. 2000, 112, 1344–1352. (13) Peach, M. J. G.; Helgaker, T.; Salek, P.; Keal, T. W.; Lutnæs, O. B.; Tozer, D. J.; Handy, N. C. Phys. Chem. Chem. Phys. 2006, 8, 558–562. (14) Livshits, E.; Baer, R. Phys. Chem. Chem. Phys. 2007, 9, 2932– 2941. (15) Jacquemin, D.; Perpete, E. A.; Scuseria, G. E.; Ciofini, I.; Adamo, C. J. Chem. Theory Comput. 2008, 4, 123–135. (16) Wong, B. M.; Cordaro, J. G. J. Chem. Phys. 2008, 129, 214703(1)– 214703(8). (17) Lange, A. W.; Rohrdanz, M. A.; Herbert, J. M. J. Phys. Chem. B 2008, 112, 6304–6308. (18) Andzelm, J.; Rinderspacher, B. C.; Rawlett, A.; Dougherty, J.; Baer, R.; Govind, N. J. Chem. Theory Comput. 2009, 5, 2835–2846. (19) Govind, N.; Valiev, M.; Jensen, L.; Kowalski, K. J. Phys. Chem. A 2009, 113, 6041–6043. (20) Jensen, L.; Govind, N. J. Phys. Chem. A 2009, 113, 9761–9765. (21) Minami, T.; Nakano, M.; Castet, F. J. Phys. Chem. Lett. 2011, 2, 1725–1730. (22) Arago, J.; Sancho-García, J. C.; Ortí, E.; Beljonne, D. J. Chem. Theory Comput. 2011, 7, 2068–2077. (23) Kuritz, N.; Stein, T.; Baer, R.; Kronik, L. J. Chem. Theory Comput. 2011, 7, 2408–2415. (24) Peach, M. J. G.; Cohen, A. J.; Tozer, D. J. Phys. Chem. Chem. Phys. 2006, 8, 4543–4549. (25) Rohrdanz, M. A.; Herbert, J. M. J. Chem. Phys. 2008, 129, 034107(1)–034107(9). (26) Shcherbin, D.; Ruud, K. Chem. Phys. 2008, 349, 234–243. (27) Baer, R.; Livshits, E.; Salzner, U. Annu. Rev. Phys. Chem. 2010, 61, 85–109. (28) Rudolph, M.; Ziegler, T.; Autschbach, J. Chem. Phys. 2011, 391, 92–100. (29) Autschbach, J. Coord. Chem. Rev. 2007, 251, 1796–1821. (30) Autschbach, J. Calculating electronic optical activity of coordination compounds. In Comprehensive Inorganic Chemistry II; Poeppelmeier, K.; Reedijk, J., Eds.; Elsevier: Oxford, UK, Vol. 9; in press. (31) Le Guennic, B.; Hieringer, W.; G€orling, A.; Autschbach, J. J. Phys. Chem. A 2005, 109, 4836–4846. (32) Rudolph, M.; Autschbach, J. J. Phys. Chem. A 2011, 115, 2635– 2649. (33) Fan, J.; Autschbach, J.; Ziegler, T. Inorg. Chem. 2010, 49, 1355–1362. (34) Grimme, S. Calculation of the Electronic Spectra of Large Molecules. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Larter, R.; Cundari, T. R., Eds.; John Wiley & Sons, Inc.: Hoboken, 2004; Vol. 20. (35) Pritchard, B.; Autschbach, J. ChemPhysChem 2010, 11, 2409– 2415. (36) Autschbach, J.; Jorge, F. E.; Ziegler, T. Inorg. Chem. 2003, 42, 2867–2877. (37) Quantum Chemistry Group, Universitaet Karlsruhe, “Turbomole, Ver. 5.7.1”. (38) Dirac, P. A. M. Proc. R. Soc. (London) 1929, 123, 714–733. (39) Slater, J. C. Phys. Rev. 1951, 81, 385–390. (40) Vosko, S.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200–1211. (41) Becke, A. D. Phys. Rev. A 1988, 38, 3098–3100. (42) Perdew, J. P. Phys. Rev. B 1986, 77, 3865–3868. (43) Perdew, J.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3686. (44) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158–6170. (45) Becke, A. D. J. Chem. Phys. 1993, 98, 1372–1380. (46) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200–1211. 14685



ARTICLE

(47) Bylaska, E. J.; de Jong, W. A.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Valiev, M.; van Dam, J. J.; Wang, D.; Apra, E.; Windus, T. L.; et al. NWChem, A Computational Chemistry Package for Parallel Computers, Version 6.0 (2011 developer’s version), Pacific Northwest National Laboratory, Richland, WA, 2011. (48) Autschbach, J. ChemPhysChem, 2011, in press. DOI: 10.1002/ cphc.201100225. (49) Autschbach, J. Chirality 2009, 21, E116–E152. (50) Rizzo, A.; Vahtras, O. J. Chem. Phys. 2011, 134, 244109. (51) Rosa, A.; Ricciardi, G.; Gritsenko, O.; Baerends, E. J. Excitation energies of metal complexes with timedependent density functional theory. In Principles and Applications of Density Functional Theory in Inorganic Chemistry I, Kaltsoyannis, N.; McGrady, J. E., Eds.; Springer: Heidelberg, 2004; Vol. 112. (52) Mason, S. F.; Seal, R. H. Mol. Phys. 1976, 31, 755–775. (53) Kuroda, R.; Saito, Y. Circular Dichroism of Inorganic Complexes: Interpretation and Applications. In Circular Dichroism: Principles and Applications, 2nd ed.; Berova, N.; Nakanishi, K.; Woody, R. W., Eds.; VCH: New York, 2000. (54) Jensen, L.; Swart, M.; van Duijnen, P. T.; Autschbach, J. Int. J. Quantum Chem. 2006, 106, 2479–2488. (55) Mason, S. F.; Peart, B. J. J. Chem. Soc., Dalton Trans. 1973, 9, 949–954. (56) McCaffery, A. J.; Mason, S. F.; Ballard, R. E. J. Chem. Soc. 1965, 2883–2892. (57) Srebro, M.; Govind, N.; de Jong, W.; Autschbach, J. J. Phys. Chem. A 2011, 115, 10930–10949.

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Performance of Conventional and Range-Separated Hybrid Density

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