Calculation of Optical Rotatory Dispersion and Electronic Circular

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Calculation of Optical Rotatory Dispersion and Electronic Circular Dichroism for Tris-bidentate Groups 8 and 9 Metal Complexes, With Emphasis on Exciton Coupling Mark Rudolph and Jochen Autschbach* Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000

bS Supporting Information ABSTRACT: The optical rotatory dispersion (ORD, both nonresonant and resonant) and the electronic circular dichroism (CD) of tris-bidentate transition metal complexes of the form [M(L)3]nþ (M = Fe, Ru, Os, Co, Rh, Ir; n = 2, 3; L = 1,10phenanthroline, 2,20 -bipyridine) are calculated using time-dependent density functional theory (TDDFT). The exciton CD band resulting from the coupling of ligand π-to-π* transitions is investigated in detail and analyzed in terms of exciton coupling of long-axis transitions using a dipole coupling model that takes TDDFT data for a single ligand as input. Results of the coupling model agree well with the full TDDFT CD spectra. The usefulness and reliability of this model is discussed. The resonant ORDs calculated directly from analytical damped linear TDDFT response compare well with Kramers-Kronig transformations of the calculated CD spectra. For comparisons of resonant ORD with experiment, one needs to consider wavelength shifts.

1. INTRODUCTION The optical activity of tris phenanthroline and bipyridyl transition metal complexes has been the target of much research from the 1960s1 through the year 2010.2 There is continuing interest in such complexes,3-7 in particular with respect to their photophysical and photochemical properties. From a computational point, the antetype [M(L)3]nþ complexes where M is a group 8 or 9 metal and L is 1,10-phenanthroline or 2,20 -bipyridine are attractive systems for study because of their structural rigidity and the availability of crystal-structure data8-16 and experimental electronic circular dichroism (CD) spectra and optical rotatory dispersion (ORD).1,17-21 Whereas the features of the CD spectra of [M(L)3]nþ complexes are reasonably well understood, experimental single-wavelength OR, and in some cases ORD curves, have been reported for several of such complexes but have not been discussed in much detail. First-principles ORD computations have, to our knowledge, not yet been reported. A complication in the computational prediction of the UVvis frequency ORD of many metal complexes is that it requires one to treat the anomalous dispersion around electronic resonances. It was only a few years ago when first-principles computations of dampened ORD, which allows to cover the resonant region, were developed within the time-dependent density functional theory (TDDFT) framework22,23 and suggested as a potentially useful tool for determining the absolute configuration of molecules and metal complexes.23 Regarding circular dichroism, development has taken until 2003 when the first TDDFTbased CD calculations were successfully performed for chiral metal complexes.24-27 For systems as large as the [M(L)3]nþ r 2011 American Chemical Society

complexes investigated in this work, TDDFT is at present the most practical route to performing spectra and optical property calculations from first principles with a reasonable accuracy, an affordable computational cost, and including the effects from electron correlation. Following the first pioneering applications of TDDFT to chiroptical properties of metal complexes, detailed studies have been undertaken in the meantime on various [M(L)3]nþ complexes,24,28-30 some open-shell trigonal dihedral Cr(III) complexes,31 and other metal-containing systems.32-35 For additional citations and reviews of related works, see refs 36-38. Before the use of ab initio calculations to aid in the assignment of absolute configuration and interpretation of spectral data of metal complexes, various theoretical models were applied quite successfully,1,17,39-50 such as ligand field and crystal field theories for metal d-d transitions and exciton coupling models for interchromophore transitions.39-50 Le Guennic et al.26 have previously investigated group 8 metal [M(phen)3]nþ complexes (phen = 1,10-phenanthroline) and demonstrated that the intense CD band, in [Os(phen)3]2þ for example, is predominantly due to exciton coupling as has long been assumed, with a contribution on the order of 10% or less related directly to the metal orbitals. Fan et al. have recently28,29 studied the origin of the CD in trigonal dihedral cobalt complexes involving various saturated and unsaturated ligands. Fan et al. have also investigated the electronic structure and CD of Fe, Ru, and Os [M(bipy)3]nþ Received: December 2, 2010 Revised: January 21, 2011 Published: March 04, 2011 2635

dx.doi.org/10.1021/jp111484z | J. Phys. Chem. A 2011, 115, 2635–2649

The Journal of Physical Chemistry A complexes (bipy = 2,20 -bipyridine).30 In these works, a qualitative framework and understanding of the origin of optical activity using TDDFT and qualitative Molecular Orbital (MO) theory was built. Expressions for the overlaps of symmetry-adapted combinations of MOs from the free ligand and the metal in terms of their placement in the complex were derived, which yielded important information about bonding and MO contributions to the rotatory strengths. This was achieved by investigating symmetry-adapted microstates related to the free ligands excitations and to the excitations in the complexes, and by decomposing the rotatory strengths in terms of individual ligand MO overlap contributions. This work follows a somewhat similar approach in the sense that TDDFT is used to investigate both the free ligands and the full [M(L)3]nþ complexes. The overall scope and the tools used to analyze the exciton CD are different. TDDFT spectral data for the free ligands are used along with a simple exciton coupling model to determine coupled excitation spectra in ligand trimers, which are shown to agree well with first principles calculations on the full complexes and with experiment. Further, by performing a concerted study of these [M(L)3]nþ complexes, including both CD and ORD of Fe, Ru, and Os as well as Co, Rh, and Ir [M(L)3]nþ systems, we aim to develop a better fundamental understanding of the electronic structure and optical activity of these complexes. Kramers-Kronig (KK) transformations are applied to generate resonant ORD from the calculated CD spectra, which compare well with direct damped (complex) linear response computations of the optical rotation on a grid of optical frequencies. Application of the exciton coupling model shows why simple theoretical models used in the past explain their spectra so well. For example, the exciton CD couplets in the [M(L)3]nþ systems obtained from the coupling model agree quite well with full TDDFT computations on the complexes, indicating that the simple point-dipole coupling mechanism dominates the spectral features of the complexes in the 4-5 eV range even though the ligands are close enough such that a breakdown of the point-dipole approximation is indicated. The computational protocols developed here are intended to serve as a foundation for applications to much larger systems where exciton coupling models can be expected to perform well and full TDDFT computations are not feasible. The remainder of this Article is organized as follows: Section 2 outlines the computational details used for the TDDFT calculations, as well the dipole excitation coupling model. Section 3 contains a relatively detailed analysis of the free 1,10-phenanthroline (phen) and 2,20 -bipyridine (bipy) ligands (L) as well as an analysis of the results from the exciton coupling model for D3 symmetric L3 arrangements. Experimental, structural, CD, and OR(D) data of the [M(L)3]nþ complexes are collected in section 4 and compared to selected results from the computations. A discussion of the full [M(L)3]nþ CD and ORD is provided in section 5. A brief summary and some concluding remarks are provided in section 6.

2. THEORETICAL AND COMPUTATIONAL DETAILS 2.1. DFT and TDDFT Computations. The computational methods have been benchmarked previously26,30 and are known to yield good agreement with experimental CD spectra for (phen)3 and (bipy)3 complexes of group 8 metals. Geometry optimizations for phen, bipy, and the [M(L)3]nþ systems were performed using the BP functional51-55 and the triple-ζ Gaussian-type TZVP[56] basis set as implemented in the Turbomole

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v5.7.1 package.57 ORD and CD calculations were performed using the Amsterdam Density Functional package58 (ADF) with the BP functional and a Slater-type basis set combination of TZP/DZP/DZ on M/C,N/H. The 3p, 3d, and 4f cores were frozen for Fe, Ru, Os as well as the 2p, 3d, and 4f cores on Co, Rh, and Ir. The zeroth-order regular approximation59-61 (ZORA) was used to include scalar relativistic effects in the response calculations. Solvation effects were included using the conductorlike screening model62 (COSMO) using the dielectric constant for water (ε = 78) and default atomic radii. The lowest 200 symmetry-allowed excitations were calculated for phen, bipy, and the [M(L)3]nþ complexes. A large number of excitations is beneficial for use with Kramers-Kronig (KK) transformations. The KK transformation63,64 is a symmetry-adapted Hilbert transformation, which converts the real and imaginary parts of a complex response function into one another, assuming one is known in entirety. KK transformations are used in this work to convert TDDFT computed CD into ORD in addition to performing resonant ORD calculations directly with the code reported in refs 23,65,66. For more information on the applications of KK transformations, we direct the reader to refs 67-69 and our previous work on KK transformations for chiroptical response in refs 70,71. All computed absorbance and CD spectra were Gaussian broadened using a σ of 0.13 eV unless otherwise noted. Linear response ORD calculations used a corresponding half-width at halfheight for the Lorentzian dampening parameter.23 2.2. The “Matrix Method” Excitation Coupling Model. This section gives an account of the dipole coupling model used to analyze the spectra, the “Matrix Method” (MM). A full discussion of this matrix formulation has been given by Schellman et al. in ref 72 The original MM affords a blocked Hamiltonian matrix for the interacting excitations, specifically considering the transitions of peptide groups, where the blocks consist of interactions for different mechanisms, which give rise to optical activity (such as the one-electron theory of Condon, Altar, and Eyring,73 the electric-electric Kuhn and Kirkwood mechanism,74 and electric-magnetic multipole interactions). In this work, a general Hamiltonian is set up for the coupling of an arbitrary number of excitations, but only the electric dipole coupling mechanism is considered. For the purpose of detailed analysis of the [M(L)3]nþ systems, we consider specifically the coupled excitations involving three degenerate single π-to-π* long-axis polarized excitations, one for each ligand, which is the exciton modification of the Kuhn-Kirkwood mechanism discussed by Moffitt.75 Within the MM framework implemented here, higher multipole moment terms and the coupling of an essentially arbitrary number of degenerate and nondegenerate transitions can be easily included. The following discussion, for illustrative purposes, will revolve around the coupling of a single degenerate transition per subgroup. In some places in the text, however, MM calculated spectra will be shown where multiple transitions have been coupled. For a [M(L)3]nþ system, there are three degenerate subgroups (a, b, and c) corresponding to the three ligands. The Hamiltonian representing the coupling of a single excitation per subgroup for the system is written as 0 1 0 0 0 0 B B 0 Ea Vab Vac C C C ð1Þ H¼B B0 V C E V ba b bc A @ 0 Vca Vcb Ec 2636

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The Journal of Physical Chemistry A

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where Ea is the excitation energy of subgroup a, and Vab is the potential interaction between subgroups a and b. In our approach, the energies are taken from TDDFT computations on the free ligand. The zeros in the first row and column represent the ground state under the assumption that the subgroup ground states do not interact with each other or with any of the excited states. Hartree atomic units are adopted throughout this section. To the lowest order in the multipole expansion, the electrostatic coupling of two transitions, for instance Vab, reads Vab ¼

da 3 db 3ðda 3 rba Þðdb 3 rba Þ 3 5 rba rba

ð2Þ

In eq 2, da/b is the electric transition dipole vector for the excitation in subgroup a and b, respectively. Further, rba = rb - ra is the vector pointing from the a subgroup center ra to the b subgroup center rb, and rba is the scalar length of rba. For N excitations per subgroup, and M subgroups, the Hamiltonian matrix is a square symmetric matrix of linear dimension MN þ 1. The Hamiltonian is diagonalized by a unitary matrix C. This eigenvector matrix C is then used to transform the original uncoupled electric d0 and magnetic m0 dipole moment vector matrices into those for the coupled excitations, d0 and m0 . In this work, these matrices are constructed on the basis of TDDFT calculations on a free ligand and subsequent rotation of the dipole moment vectors into the local coordinate frame that the ligand adopts in the complex. The coupled electric transition dipole moment for the Nth transition from the ground state reads X X C†0I d0IJ CJN ¼ d00J CJN ð3Þ d00N ¼ I, J J where subgroup labels a, b, and c are replaced by excitation labels I, J,... for the compound system. Similarly, the magnetic part reads X X C†NI m0IJ CJ0 ¼ m0I0 CIN ð4Þ m0N0 ¼ I, J I Using the coupled moments, the rotatory strength R0N for the Nth transition of the compound system from the ground state can be calculated by X CJN CIN ðd00J 3 m0I0 Þ ð5Þ R0N ¼ IJ

When coupling only electric transition moments (excitations with no intrinsic magnetic transition dipole), the induced magnetic moment has to be considered, otherwise R0N is zero. For a subgroup excitation 0I, the magnetic dipole operator reads 1X m ^ ¼ ri p î ð6Þ 2c i where c is the speed of light, ri is the position vector from the subgroup coordinate origin to electron i, p̂i is the linear momentum operator for electron i in the subgroup, and the sum runs over all electrons in the subgroup. The induced magnetic transition dipole moment for an uncoupled excitation I of the subgroup in the aggregate is m00I ¼

1 rI p00I 2c

ð7Þ

where rI is the position vector from the coordinate origin chosen for the compound system to the centerPof the subgroup where î|Iæ evaluated within the excitation takes place, and p00I = Æ0| i p the subgroup. Equation 7 assumes that there are no intrinsic

Figure 1. Depiction of the s, h, and φ values typically used to describe the distortion of D3 complexes from octahedral symmetry. Note that φ is not the — N-M-N bite angle in the complex, but rather the angle in the projection onto the plane perpendicular to the 3-fold symmetry axis. See ref 78 for more details.

magnetic transition dipole moments for the subgroup excitation, which otherwise would have to be added to m00I. Using the relation between the linear momentum and the electric transition dipole moment for a transition from subgroup state 0 to state I: p00I ¼ iE00I d00I

ð8Þ

the expression for the induced magnetic dipole in the compound but uncoupled system used in eq 4 is m0I0 ¼

iE00I rI d00I 2c

ð9Þ

where E00I is the energy of one of the subgroup excitations. It is assumed that the wave functions are real. The coupling of degenerate transitions in the [M(L)3]nþ systems is not origin dependent, but in general the rotatory strengths calculated with the MM may be origin dependent. The MM can be written in an origin-independent formalism as outlined by Goux and Hooker76 based on suggestions by the authors of ref 72. Goux and Hooker did not calculate R0I = Im[d0I 3 mI0], but rather a “chiral strength” c0I defined as 1/(3c)Re[p0N 3 LN0], where p and L are the linear and angular momentums, respectively. The rotatory strength R0I is then obtained from c0I in atomic units as R0I = 3/(2E0I)c0I, where E0N is the excitation energy. This is easily recognized as a dipole-velocity variant of the rotatory strength, which has found application also in modern electronic structure methods to avoid an origin dependence of calculated magnetic properties of molecules.36,77 2.3. Coupling Three Degenerate Excitations in a [M(L)3]nþ System. D3 symmetric complexes such as [M(L)3]nþ have traditionally been analyzed by an “s/h” polar ratio and an azimuthal distortion φ, which gives insight on the compression/elongation and the twist on the ligands as compared to that of an ideal octahedral parent geometry78 (see Figure 1). Fan and Ziegler28 have instead used a single variable ω. For a bidentate ligand, ω is the angle between the 3-fold axis for ideal octahedral parent symmetry and the axis connecting the two chelating atoms. If we define ω as 0 when the plane of each ligand is parallel to the C3 axis of the complex and a positive ω as a clockwise rotation looking down the short axis of the ligand toward the metal, then a ( ω indicates a Δ/Λ configuration. By varying ω from zero (no chirality), the symmetry is distorted from D3h to D3, and the complex becomes chiral. This is illustrated in Figure 2. For octahedral coordination of the 2637



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Figure 2. Left: When the axes connecting the two chelating atoms of each ligand are parallel to the C3 axis, the system has D3h symmetry. Center: When the ligands are rotated about their short axes, the system becomes chiral (D3 symmetry). Right: The angle of rotation, ω.

metal, ω ≈ 35.3 (for an ideal octahedron, sin2 ω = 1/3, cos2 ω = 2/3). The main feature of the exciton CD of [M(L)3]nþ complexes can be reproduced from one intense long-axis polarized transition if L is bipy or phen (section 3 and Supporting Information). Because of the symmetry of the system, the three coupling electronic transitions can be represented by one transition via coefficients from a rotation matrix (as well as the induced magnetic transitions stemming from them). For a degenerate 3 by 3 case (coupling only one excitation per subgroup), the Cmatrix coefficients are always the same irrespective of the value of Vab (see Appendix). As in the H€uckel theory, the eigenvectors are completely determined by the topology of the Hamiltonian matrix. For the coupling of the long-axis polarized π-to-π* transitions in the ligands (polarization in the z direction in Figure 3), eq 5 leads to the expression: RE ¼

-3El rDl 0 sin ω cos ω 4c

ð10Þ

3El rDl 0 sin ω cos ω 2c

ð11Þ

RA 2 ¼

for the E and A2 rotatory strengths of the coupled set of transitions in a D3 symmetric configuration. Here, El is the energy of the long-axis transition in the free ligand, D0l is the long-axis transition dipole strength in the free ligand (the square of the electric transition dipole moment), and r is the distance from the metal center to the ligand center, where for the latter we adopt its center of nuclear charge. In eqs 10 and 11, the magnitude for RE is one-half that of RA2 because of the double degeneracy of the E transition; the components are counted here as two individual excitations. The total rotatory strength for the CD couplet sums to zero. The potential interaction V (dipole-dipole interaction of long-axis transitions; eq 2) is given as V ¼

Dl 0 pffiffiffi ð4 cos2 ω - 5 sin2 ωÞ 12 3r 3

ð12Þ

in atomic units. Mason and Norman1 previously used the following equation for the potential interaction: Dl 0 Vl ¼ ð13Þ 4rab 3 which is what eq 12 reduces to for the ω value of octahedral coordination (sin ω cos ω = (2)1/2/3). In eq 13, rab is the

Figure 3. Structure of 1,10-phenanthroline (left) and 2,20 -bipyridine (right).

distance from one ligand center (a) to another ligand center (b), which is not the same r as in eq 12. The relation is r = rab/(3)1/2. Mason and Norman’s expression for the rotatory strength of long-axis coupled transitions in [M(L)3]nþ systems also follows from eqs 10 and 11 given here if the value of ω for an ideal octahedron is adopted.

3. ANALYSIS OF THE LIGAND SYSTEM A planar phen ligand (Figure 3, left) has C2v symmetry, which, for absorption spectra, affords the following symmetry-allowed transitions: A1, B1, and B2. Aligning the ligand C2 axis with the x axis and the long axis of the ligand along z, the electric transition dipoles are oriented either along the short axis (x, A1), along the long axis (z, B1), or perpendicular to the plane of the molecule (y, B2). A planar bipy ligand (Figure 3, right) affords the same symmetry classification for its allowed transitions. The computed UV-vis spectrum for 1,10-phenanthroline compares favorably with experiment (0.1 M HCl in ethanol), as shown in Figure 4 (left). The simulated extinction coefficients depend somewhat on the chosen broadening, but overestimations of TDDFT computed spectral intensities by a factor of 2 with respect to experiment are not uncommon. The computed UV-vis spectrum for the free 2,20 -bipyridine also compares reasonably well with that of experiment (0.1 M HCl in ethanol), as shown in Figure 4 (right), although with several tenths eV of a blue shift. There is a second peak in the lowest intense band of the simulated bipy spectrum, nearly 0.5 eV lower than the second intense transition in phen. Unlike phen, bipy has a C-C σ bond for the two pyridine groups to rotate about, and the ligand is significantly more flexible than phen, which might account for the less satisfactory agreement of the broadened vertical excitation spectrum with experiment. In both ligand spectra, B1 long-axis polarized transitions dominate the spectrum below 6 eV (see the Supporting Information for additional details). Along with understanding the nature of the free ligand excitations, we are also concerned with the coupling “in-complex”. From considering three phen and three bipy ligands in the D3 orientation that they adopt in the [Os(phen)3]2þ and [Os(bipy)3]2þ complexes (see Figure 5), the r and ω variables were varied from their optimized values (r = 3.195, 2.796 and ω = -36.62, -36.56 degrees, respectively) to gauge the effect on the exciton CD. Most of this material is presented in the Supporting Information; however, some of the central features of the TDDFT þ MM model are summarized here. The main effect from increasing r with ω held fixed is a reduction in the intensity of the broadened fully coupled MM CD spectra from competing distance behaviors for V and the rotatory strengths. It is noted that for V being identical to zero there is no coupling and therefore the rotatory strengths vanish, 2638



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Figure 4. Experimental and calculated UV-vis spectrum for 1,10-phenanthroline (left) and 2,20 -bipyridine (right). Experimental data from ref 79.

Table 1. MM Excitation Energies of Coupled Transitions (from a Single Excitation for phen) and the Peak Position of the Broadened Bands dist. (Å)

Figure 5. Three phenanthrolines adopting a Λ geometry as in the Λ-[Os(phen)3]2þ complex. View along the 3-fold symmetry axis. The arrows represent the C2 axes, which the phen ligands are moved along and rotated about.

but for finite V (eqs 10 and 11) the degenerate set of transitions produces large rotatory strengths for large r because of the large induced magnetic moments. In the simulated spectra, a large r goes along with a small V (proportional to r-3) and the large rotatory strengths cancel in the broadened spectra and yield a physically meaningful distance behavior for the couplet intensity. The lowest intense band in the UV-vis spectrum, for both phen and bipy, is dominated by a relatively pure π-to-π* transition. A coupled state in a [M(L)3]nþ system with significant rotatory strength should therefore be well described by using just this single excitation per ligand. The lowest-energy intense B1 transition for phen is calculated by TDDFT at El = 4.687 eV with d0l = (-2.110,0,0) au. The corresponding values for bipy are El = 4.297 eV and d0l = (-1.693,0,0), respectively. Structural data (r and ω) for the various [M(L)3]nþ complexes can be found in Tables 4 and 5. Considering the coupling of just a single phen transition and varying the distance r, the CD band peak positions in Figure 6 appear, to the eye, to be maintained within a 10 Å range outward from the optimized geometry. The maximum/ minimum peak position of each of these couplets for this case was determined numerically. Table 1 lists the distance outward, the EE/A2 band maximum peak position and the EE/A2 energies. The

peakE

EE

peakA2

EA2

r = 3.195

4.566

4.830

4.661

4.739

rþ1

4.561

4.821

4.676

4.710

rþ2

4.558

4.818

4.681

4.699

rþ3

4.557

4.817

4.684

4.694

rþ5

4.556

4.816

4.686

4.690

r þ 10

4.556

4.816

4.687

4.688

Table 2. Average M-N Bond Lengths (Å) and Bidentate — N-M-N (deg) Taken from Reported X-ray Crystallography Data for Various Tris-bidentate 1,10-Phenanthroline Complexesa complex Λ-(-)[Fe(phen)3]2þc Λ-(þ)[Ru(phen)3]2þd Λ-(þ)[Os(phen)3]2þe Λ-(þ)[Co(phen)3]3þg Λ-(þ)[Rh(phen)3]3þ

M-N — N-M-N

r0

ω

Δbpeak

[R]b

[j]b

1.978

82.6

1.49 -37.5 0.21 -1400 -8350

2.066

84.8

1.53 -36.4 0.18

1340

2.070

79.3

1.59 -35.6 0.22

3650f 26 676f

2.127

78.1

1.65 -35.5 0.20

800

4796

0.22

650

4183

8599

Also shown is the distance r0 from the metal center to the center of the N-N vector (Å) and ω angles (ω from X-ray data for Δ structures has been multiplied by -1). Experimental A2 and E band peak splittings (Δpeak in eV) for the Λ-phen complexes are also given, along with specific and molar ([R] and [j]) rotations measured at the sodium D line unless indicated otherwise. b Spectral data taken from ref 18. c Structural data taken from ref 11. d Structural data taken from reference 16. e Structural data taken from ref 13. f Specific rotation at 546 nm. g Structural data taken from ref 9. a

difference in both the E and the A2 band peak position from the optimized r and r þ 10 is on the order of 0.03-0.05 eV, while for the energies the difference is on the order of 0.01 eV. For all cases shown, the band-peak is at least 0.1 eV lower in energy than the energy for the E transition, and 0.1 eV higher than the A2 2639



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Table 3. Experimental, Spectroscopic, and Structural Data for Various 2,20 -Bipyridine Complexesa complex Λ-(-)[Fe(bipy)3]2þc Λ-(þ)[Ru(bipy)3]2þ Λ-(þ)[Os(bipy)3]2þ Λ-(þ)[Co(bipy)3]3þe Λ-(þ)[Rh(bipy)3]3þf Λ-[Ir(bipy)3]3þg

M-N — N-M-N

r0

ω

Δbpeak

[R]b

Table 5. Structural and TDDFT Data for Optimized [M(bipy)3]nþ Complexesa metal M-N — N-M-N

[j]b

0.22 -4600 -24 122

1.981

0.21

800

4557

0.21

4000d

26 351d

1.93

83.2

1.44 -37.5 0.25

260

1371

2.015

79.7

1.55 -34.2 0.39

120

686

2.021

78.7

1.56 -33.8

Fe Ru Os Co Rh Ir

1.974 2.075 2.092 1.970 2.070 2.083

81.58 79.40 77.56 82.54 79.45 78.79

r

r0

ω

2.653 2.771 2.796 2.628 2.744 2.761

1.49 1.61 1.63 1.48 1.59 1.61

-36.5 -36.3 -36.6 -36.8 -36.6 -36.9

Δpeak-MM Δpeak-TDDFT VMM 0.26 0.26 0.26 0.26 0.26 0.26

0.26 0.32 0.29 -b 0.41 0.27

0.0241 0.0221 0.0205 0.0234 0.0216 0.0199

a See also caption for Table 4. b Data not given due to assignment of the calculated data. See discussion in section 5.

a

See also the caption of Table 2. b Spectral data taken from ref 18. c Structural data taken from ref 10. d Specific rotation at 546 nm. e Structural data taken from ref 8. f Structural data taken from ref 14. g Structural data taken from ref 12.

Table 4. Structural Data for the Optimized [M(phen)3]nþ Complexes: M-N Distances (Å), — N-M-N (deg), and ω (deg)a metal M-N — N-M-N Fe Ru Os Co Rh Ir

1.977 2.081 2.106 1.969 2.085 2.096

82.75 79.49 78.23 83.80 80.19 79.54

r

r0

ω

3.033 3.156 3.195 3.006 3.141 3.159

1.48 1.60 1.63 1.47 1.59 1.61

-36.0 -35.8 -35.6 -36.6 -37.0 -36.4

Δpeak-MM Δpeak-TDDFT VMM 0.25 0.26 0.26 0.26 0.26 0.26

0.27 0.27 0.27 —b 0.39 0.30

0.0275 0.0251 0.0252 0.0253 0.0205 0.0226

Also provided are the Δ peak maxima for the Gaussian broadened (σ = 0.13 eV) exciton couplet via MM and TDDFT, along with V calculated from the MM eq 12. b Data not given due to the complexity of this system. See discussion in section 5.

Figure 6. The coupling of a single excitation for the Λ-(phen)3 complex with varying distances between the complex center and the ligands.

transition. For a CD couplet with a Lorentzian broadening, the leading order of the peak intensity for the degenerate threetransition coupling model is r-2, resulting from the splitting energy being proportional to r-3 and the rotatory strengths increasing linearly with r. For a Gaussian broadening, the peak intensities are also fitted accurately by a r-2 dependence. Considering variations of ω at a fixed value for r, as the magnitude of |ω| decreases from the optimized value, the intensity of the broadened spectrum increases, and vice versa. Eventually, as |ω| increases, the splitting of the states reverses to A2 and then E, causing a sign change. These stark differences occur in the (10 range from the optimized ω values. Changing ω by a few degrees in one direction or the other can result in a substantial increase or decrease in the broadened CD intensity for the exciton couplet. Similar trends can be seen for TDDFT computations (also shown in the Supporting Information). CD calculations were performed on the same structures used in the various MM calculations. The trend of a weaker CD spectrum with increasing r is obtained. A decrease in |ω| from the optimized value gives a more intense CD couplet as well, while an increase in |ω| affords a weaker spectrum. In the TDDFT computations, the ordering of the states does not invert in the (10 interval. The dependence of the TDDFT spectra on r is an interesting question. For a two-level system, that is, a transition involving one occupied MO i and one unoccupied MO a, the linear response TDDFT singlet excitation energy can be written as80-82

where the ε are the MO energies and f sing XC is the exchangecorrelation (XC) response kernel. For a triplet excitation, the second term on the right-hand side reads [ai|f trip XC |ia]. The last term in eq 14 vanishes for triplet excitations unless the computation includes spin-orbit coupling variationally. A Coulomb integral between the charge densities of orbitals a and i would be obtained from an exact exchange term in the XC kernel, that is, [ai|fXC|ia] = -[aa|r-1 12 |ii] for the Hartree-Fock response kernel. With standard functionals that typically have between 0% and 20% Hartree-Fock exchange, a large fraction of the XC kernel cannot fully describe such nonlocal effects. This deficiency of the local part of the XC kernel is responsible for the well-known breakdown of TDDFT for charge-transfer excitations involving large distances when using standard non-hybrid and hybrid functionals. One might perhaps expect that there are related issues with the ability of TDDFT to describe the exciton coupling at large ligand separations. However, the Coulomb interactions involved in the exciton coupling need to be attributed to the last term of eq 14, which stems from the Coulomb operator in the TDDFT response kernel. In the simple Coulomb-only dipole coupling model, the transition dipole moment for a triplet excitation vanishes, and no long-range coupling is obtained. The last term in eq 14 also vanishes for triplet excitations. An equation resembling the dipole coupling interactions of eq 2 would be obtained as the leading term of a multipole expansion of [ai|r-1 12 |ia]. The orbitals involved here are the symmetryadapted linear combinations of the ligand π and π* orbitals in the complex for which explicit expressions have been given in refs 29,30. There is no need here to repeat the detailed analysis of

a

-1 ΔE ¼ ðεa - εi Þ þ ½aijfXC jia þ 2½aijr12 jia sing

ð14Þ

2640


The Journal of Physical Chemistry A ref 30 for the coupled ligand excitations in terms of microstates or orbital contributions. Qualitatively, it is clear that a term [ai|r-1 12 |ia] in the TDDFT excitation energy must contain inter-ligand coupling terms from a part of the orbital product (ai) centered on one ligand and another part of (ai) on another ligand, which are at long-range dominated by the local dipole moments of (ai). Numerical data have been obtained by fits of the A2/E TDDFT excitation energy splitting for the most intense transitions in the exciton CD band of a (phen)3 system as a function of r. From r between 1 and 6 Å larger than the optimized value, it is somewhat difficult to ascertain wether the TDDFT excitation energy splitting decays exponentially with r or with the expected r-3 because the MM results, which exactly follow r-3, can be fitted to an exponential function with a correlation coefficient of 0.990. An exponential fit of the TDDFT data yielded the same correlation coefficient. However, a r-3 fit yielded a better correlation coefficient of 0.999 and, when plotted, an overall better visual comparison with the numerical data, which can be found in the Supporting Information. We had to forego attempts to perform fits over a larger range of r due to convergence problems in the TDDFT computations. It is noted that for larger r, spurious CT transitions (typically with low intensities) start to contaminate the TDDFT spectra and interfere with the analysis.83,84 The use of a range-separated hybrid functional

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would be beneficial to avoid such problems. At r values below about 5 Å, the r-3 fits clearly indicated that at the ligand distances adopted in the complexes, the point-dipole approximation for the exciton coupling starts to break down. Better agreement of the MM results with the TDDFT results should therefore be expected upon inclusion of higher order multipole terms in the formalism. The MM spectra from coupling all calculated phen and bipy excitations, respectively, are shown in Figure 7. One feature that can be noticed in the coupled CD is that the trough of the first CD couplet (Λ configuration) is much less intense than the peak. This is due to the proximity of the next couplet around 5.5 eV for phen (4.6 eV for bipy), causing the negative band to largely cancel with a higher energy positive CD band. This is especially true for (bipy)3 where, as mentioned above, the energetic spacing between the lowest two intense long-axis ligand transitions is nearly 0.5 eV lower than for phen. It is plausible that upon binding in the complex, the excitation energies of these two transitions in the ligand grow more separated, allowing the lowerenergy couplet to develop more completely. Additional computations confirm this hypothesis because such a trend is seen computationally for a protonated phen ligand, as well as for an Os(phen)(NH3)4 system (see Supporting Information). As already mentioned, from an analysis of the TDDFT calculations

Figure 7. Broadened Matrix Method CD spectrum with excitation energies and rotatory strength (sticks) shown of the Λ configurations of (phen)3 (left) and (bipy)3 (right).

Figure 8. TDDFT calculations on the (phen)3 systems along with the calculated Matrix Method spectrum at the ω and r values optimized for [Os(phen)3]2þ (left) as well as ω and r þ 5 Å (right). 2641



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Figure 9. Calculated CD and ORD of Λ-[Fe(phen)3]2þ and Λ-[Fe(bipy)3]2þ along with experimental data taken from ref 18. r and ω values used for the Matrix Method exciton calculations are listed in Tables 4 and 5. Calculated CD spectra are Gaussian-broadened using σ = 0.13 eV. For the KK transformation, the CD was Lorentzian broadened with the corresponding line width to match the damping parameter used for linear response calculations. No shift has been applied to calculated responses.

(see Supporting Information), it is found that the first UV-vis band in the ligand spectra is dominated by intense long-axis polarized B1 transitions. The coupling of different combinations of the A1, B1, and B2 symmetry-allowed transitions in the Matrix Method also shows that the coupling between B1 transitions from different ligands plays the dominant role in the exciton CD, whereas the involvement of transitions with different symmetry has only a small effect on the resulting coupled MM spectra (Supporting Information). In Figure 8, it is shown that for (phen)3 at the same geometry as adopted in [Os(phen)3]2þ, TDDFT calculations also predict a weakened trough in the couplet above 4.5 eV due to the next intense couplet around 5.5 eV (left panel). The TDDFT and MM spectra match quite well over the plotted energy range regarding the sign pattern and the relative intensities of the CD bands. The peak positions for the first couplet are slightly higher in energy for TDDFT than the Matrix Method. The right panel of Figure 8 shows that at r þ 5 Å, despite the differences in the rotatory strengths, the band peaks of both intense couplets are still nearly identical for both the Matrix Method and the TDDFT. In summary, the coupling of the single transition responsible for the intensity of the [M(L)3]nþ exciton CD band yields quite reliable results when compared to TDDFT calculations on the coupled ligand systems. Although the broadened CD intensities do not quantitatively match those of the TDDFT calculated

spectra, the sign pattern and trends as a function of r and ω do. As shown in the Supporting Information, it is seen that a decrease in |ω|, from the optimized value, causes an increase in CD intensity for the dipole coupling model and the TDDFT calculation. The trend of decreasing CD with an increase in |ω| from the optimized value is reproduced with both methods as well. Results from the dipole coupling model provide an unambiguous assignment of the absolute configuration of these systems. There are two ligand orientations that produce nearly identical bands at larger deviations from the optimized ω value, due to a reordering and sign change of the A2 and E transitions. However, one of these is unphysical due to the steric interactions that would be involved.

4. EXPERIMENTAL TRENDS FOR [M(L)3]Nþ SYSTEMS AND COMPARISON WITH CALCULATED DATA Table 2 collects experimental peak positions for the A2 and E exciton CD bands for the phen complexes, as well as the specific and molar rotations along with structural data reported from X-ray crystallography data. The analogous information for the bipy-complexes is collected in Table 3. The structural data show that the [M(phen)3]nþ and [M(bipy)3]nþ complexes have very comparable M-N bond lengths and bidentate — N-M-N angles. The distance from the metal to the center-point of the 2642



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Figure 10. Calculated CD and ORD of Λ-[Ru(phen)3]2þ and Λ-[Ru(bipy)3]2þ along with experimental data. See also the caption of Figure 9.

two ligating N atoms is shown as r0 , which is not the same as the r used to calculated the rotatory strengths and splitting energies (which was chosen as the center of nuclear charge of the phen and bipy ligands). For comparison, Tables 4 and 5 collect some of the calculated structural information from the optimized structures, specifically r, r0 , and ω, values for V calculated from eq 12, as well as Δpeak from simulated TDDFT spectra and from the MM data for the case of coupling just one transition for the three ligands. From Tables 2 and 3 of the experimental data, it is gathered that the differing metal centers give structurally similar [M(phen)3]nþ and [M(bipy)3]nþ complexes, which also have similar electronic spectra in the range where the ligandbased excitations dominate. Tables 4 and 5 demonstrate that DFT yields optimized structures that are in good agreement with the crystal structure data. The exciton peak splittings obtained from the calculations are also in good agreement with experiment. The underlying splitting energies V according to eq 12 are on the order of 0.02 eV. Mason et al. have given estimates for V of the phen and bipy complex in ref 17 as 0.052 and 0.011 eV, respectively. The magnitudes agree reasonably well with our results. We point out again that the splitting energies obtained from the point dipole approximation give broadened spectra whose Δpeaks are in good agreement with those calculated from TDDFT (Tables 4 and 5) and with those from experiment (Tables 2 and 3).

5. COMPUTED CD AND ORD OF [M(PHEN)3]Nþ AND [M(BIPY)3]Nþ COMPLEXES We now turn the discussion to the full simulated CD spectra of the complete set of complexes based on TDDFT calculations, and a comparison of the ORD calculated directly and from KK transformations. CD spectra and ORD curves for [Fe(phen)3]2þ and [Fe(bipy)3]2þ are shown in Figure 9 computed with TDDFT (full spectrum, and MM using a single excitation) along with experimental data. Regarding the other metal complexes, see Ru, Figure 10; Os, Figure 11; Co, Figure 12; Rh, Figure 13; and Ir, Figure 14. All experimental and computed data are for Λ configurations. For all complexes, the KK transformation of the computed CD (lowest 200 excitations) agrees well with the linear response ORD over the entire range plotted. Any discrepancies can be attributed to not having the CD spectra available in their entirety for the exact transformation, and to convergence of the optical rotation calculations near resonant frequencies. The sign patterns and magnitudes of the ORD oscillations are reproduced well enough by the KK transformations that more involved methods of eliminating errors from truncating the CD spectra that were developed previously70,71 are not deemed to be necessary. Because the KK transformation of the (truncated) computed CD matches excellently with the linear response calculated ORD, we refer to both methods as “computed ORD”. The group 8 metal complexes studied have resonances around 589.3 nm, which makes the ORD more difficult to interpret. 2643



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Figure 11. CD and ORD of Λ-[Os(phen)3]2þ and Λ-[Os(bipy)3]2þ along with experimental data. See also the caption in Figure 9.

Difficulties also arise due to any applied wavelength (λ) or frequency (ω) corrections that may be required to line up bands in the simulated and experimental CD spectra for a variety of reasons such as systematic TDDFT errors and solvent effects that are not described well by a continuum model. As a consequence, single wavelength OR calculations are unlikely to match experiment in the resonant frequency regime unless the computational method is right for all the right reasons, which is difficult to achieve for systems as large as the [M(L)3]nþ complexes. Good agreement between calculation and experiment for a single wavelength optical rotation would therefore be fortuitous. This issue clearly highlights the advantage of concerted studies of CD and ORD. If a significant part of the ORD is accessible experimentally, it is easier to recognize if a frequency shift might be required in the calculations. Frequency or wavelength corrections for calculated optical rotations have been suggested occasionally, for instance, in refs 85,86, and can be of great utility in concerted studies of CD and ORD. Care should be exercised when applying a frequency correction to the ORD in the low energy regime as exemplified by two cases: Λ-[Rh(bipy)3]3þ has a positive low-frequency tail of the ORD that is computed to be larger in magnitude than the experimental value at 589.3 nm. The lowest calculated excitation is at 3.3 eV, more than a full eV above the sodium D line. The excitation energy of the broadened TDDFT computed CD band agrees well with experiment; however, it is overestimated in magnitude. This overestimation of the CD intensity, not a frequency shift, is the prime cause of the

overestimation of the OR at 589.3 nm. On the other hand, the computed OR at the sodium D line for Λ-[Rh(phen)3]3þ agrees much better with experiment. Looking at the exciton peak, however, the TDDFT computed exciton band is red-shifted with respect to experiment as well as slightly underestimated with regards to intensity. Application of a blue-shift to the computed spectrum and subsequent KK transformation would therefore result in an underestimation of the computed low energy ORD. For the cases of Fe, Ru, and Os, the low frequency ORD would be a useful aid for an absolute configuration (AC) assignment despite already being in the resonant regime. For both the phen and the bipy complexes, there is a pronounced trough between 2 and 3 eV (calculated) in the ORD for the Λ configuration. An AC assignment would be best made after comparing the ORD as well as the theoretical and experimental CD spectrum, to gauge if a frequency correction is needed to obtain an accurate location of the trough. A concerted approach using both CD and ORD is clearly beneficial. As discussed in section 3, the intense CD couplet around 4.5 eV in the spectra of all complexes is mainly due to the coupling of ligand long-axis transitions, which can unambiguously reproduce the sign pattern corresponding to a Δ/Λ structure. As an example, consider the experimental and calculated CD spectra and ORD for [Os(phen)3]2þ and [Os(bipy)3]2þ shown in Figure 11. A detailed analysis of the CD spectrum of the phen complex has previously been given in ref 26. Both the ligand π-to-π* exciton couplet and the low energy metal-to-ligand charge transfer (MLCT) and ligand-to-metal 2644



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Figure 12. Calculated CD and ORD of Λ-[Co(phen)3]3þ and Λ-[Co(bipy)3]3þ along with experimental data. See also the caption of Figure 9.

charge transfer (LMCT) transitions match relatively well with experiment with regards to both energies and intensities. The low energy transitions are more red-shifted with respect to experiment than the exciton couplet. This has implications for the low frequency ORD (bottom left panel of Figure 11). Comparing the computed OR at 2.27 eV (546 nm) to the experimental value, the difference is large. The lowest calculated excitation for [Os(phen)3]2þ is at 1.8 eV, meaning that the OR at 546 nm is already within the resonant region. The presence of a pair of weak excitations seen in the experimental CD confirms this. Comparison of the calculated CD with experiment indicates that the calculated ORD curve should be blue-shifted. Focusing on higher energies, the MM exciton peak, calculated from a single phen transition, matches quite well with experiment in terms of both peak placement and magnitude. This again resounds the findings from ref 26 that the metal orbitals have little to no contribution to this set of excitations as there is by definition no metal contribution in the MM data. As already mentioned, the sign pattern produced from only the calculations on the ligand system is specific to the Λ configuration. Similar agreement is found for the bipy complex as well in terms of computed CD, ORD, and exciton coupling. Another trend seen experimentally, as well as computationally, is that the exciton CD couplet is weaker in intensity for bipy than for phen. The Matrix Method also reproduces this trend, mainly due to a smaller calculated transition dipole moment

for the bipy transition and the smaller induced magnetic dipole moment because of the somewhat lesser spatial extension of the (bipy)3 ligand system as compared to (phen)3. In group 9, the cobalt phen and bipy complexes have no calculated excitations lower in energy than the sodium D line (∼2.2 eV). The lowest excitations start to appear at approximately 2.4 and 2.7 eV for [Co(phen)3]3þ and [Co(bipy)3]3þ, respectively. The rhodium and iridium complexes do not have any calculated excitations lower than 3 eV. The optical rotation in the regular dispersive nonresonant frequency regime is dominated by the sign pattern of the exciton CD couplet, with a positive OR being specific for a Λ configuration. For the iron and ruthenium complexes, the Matrix Method method produces a CD couplet at the appropriate energy with respect to experiment. No experimental CD or ORD of [Ir(phen)3]3þ or [Ir(bipy)3]3þ appear to be available in the scientific literature. The calculated CD spectra and ORD curves are reported here to complete the group 9 triad. Trends in the calculated results appear to be similar to the other metal complexes. An analysis of the cobalt spectra uncovered some interesting details (Figure 12). The experimental [Co(phen)3]3þ CD couplet around 4.4 eV for the Λ configuration has a positivenegative pattern followed by another intense positive-negative couplet around 5.6 eV. This second couplet is energetically located where the second CD couplet would be expected from both TDDFT and Matrix Method CD calculations of the phenonly system (Figure 8). However, as compared to the higher 2645



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Figure 13. Calculated CD and ORD of Λ-[Rh(phen)3]3þ and Λ-[Rh(bipy)3]3þ along with experimental data. See also the caption of Figure 9.

energy couplet and the low-energy exciton CD couplets of the other complexes as well as MM data for all systems, TDDFT predicts the low energy couplet in [Co(phen)3]3þ as having a reversed splitting of states, A2 at lower energy followed by E. The energetic splitting of the states as calculated by TDDFT is very small (V = -0.00953 eV), and the rotatory strengths in the exciton couplet largely cancel due to this. The resulting band shape is therefore largely dependent on other excitations calculated in close proximity. Structurally, Co, along with Fe, has a smaller r value than do the other metal complexes (section 4). As pointed out in section 3, the point dipole coupling starts to break down at the distances seen in these complexes. Because of the oscillations in sign for this computed spectrum in this energy range, the assignment of the spectrum is not as clear-cut as for the other systems. The Ru, Os, and Co phen complexes and the Co and Rh bipy complexes exhibit one particularly intense pair of E and A2 transitions of similar intensity for the exciton couplet in the energy range between 4 and 5 eV. For the other metal systems, however, the spectrum is determined as an envelope of several intense E and A2 transitions. In the experimental absorbance spectra18,20 at the energy of the corresponding π-to-π* transitions, slightly split peaks can be noted near the band maxima in some cases, while in other cases the band maxima are not split. We find a correspondence between the presence of only one particularly intense exciton couplet in the calculated spectra and the absence of any structure in the experimental absorption peaks

for the Ru, Os, and Co phen complexes and the Co and Rh bipy complexes. For the remaining systems, the experimental absorption spectra provide some, albeit not strong, evidence for the presence of a group of several intense transitions contributing to the exciton band.

6. CONCLUDING REMARKS Structural, spectral, and chiroptical properties of a series of [M(L)3]nþ complexes have been investigated by first-principles computations and model calculations. Full CD spectra simulated using vertical transitions obtained from TDDFT agree reasonably well with available experimental spectra. The exciton CD band resulting from the coupling of ligand π-to-π* transitions has been investigated in detail. A dipole coupling model (“Matrix Method”, MM) that takes TDDFT data for a single ligand as input was shown to yield results that agree well with TDDFT for the exciton CD, even if only a single transition per ligand is used as input. Combined, the theoretical data show that the assignment of the experimental spectra in the 4-5 eV range as resulting from a simple dipole coupling of the long-axis polarized ligand π-to-π* transitions is fully justified. Calculations performed for phen3 ligand model systems at separations up to about 10 Å have demonstrated that TDDFT yields the expected r-3 dependence for the exciton splitting according to Coulomb interaction between the transition dipoles centered at the ligands, but at separations 2646



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Figure 14. Calculated CD and ORD of Λ-[Ir(phen)3]3þ and Λ-[Ir(bipy)3]3þ . See also the caption of Figure 9.

as adopted in the complexes the point dipole approximation starts to break down. The resonant ORDs calculated directly from analytical damped linear TDDFT response compared well with Kramers-Kronig transformations of the calculated CD spectra. The anomalous dispersion of the OR in the frequency region of the exciton couplet is a clear signature of the absolute configuration of the [M(L)3]nþ complexes, just like the CD itself. At frequencies of visible light, most of the metal complexes investigated here are already in the resonant regime. In this case, single wavelength OR calculations are unlikely to match experiment. However, concerted calculations of CD and ORD can be meaningfully compared to experiment. For the rhodium and iridium complexes, the lowest excitation is well above the sodium D line, therefore making the low frequency OR much more straightforward to interpret.

If coupling only a single excitation from each ligand L in the D3 arrangement of an ML3 complex, one encounters a simple 3-by-3 eigenvalue problem. The Hamiltonian matrix H reads 0

Ea B B V H ¼ @ ba Vca

matrix method as discussed in section 2.2. However, it can be omitted from the calculations because the ground state is taken to not couple with the excited states (i.e. the first row/ column are all zeroes), and the last term in eqs 3 and 5, respectively, only involves the excited-state elements of the matrix C. In the following, we only discuss the non-zero part of the Hamiltonian.

1 Vac C Vbc C A Ec

ð15Þ

which one can separate into components A and B: H ¼ AþB 0 Ea 0 B ¼B @ 0 Eb 0 0

’ APPENDIX Exciton Rotatory Strengths in a Degenerate Three-Subsystem Complex from a Long-Axis Transition Coupling Model. Conceptually, the ground state is included in the

Vab Eb Vcb

1 0 0 0 C B BV 0C þ A @ ba Vca Ec

Vab 0 Vcb

1 Vac C Vbc C A 0

ð16Þ

Because the energies are degenerate in A, the matrix that diagonalizes B also diagonalizes A þ B. For the triply degenerate case, the potential interaction terms are all the same, thus 0

0 B B B ¼ V 3@1 1 2647

1 0 1

1 1 C 1C A 0

ð17Þ



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When coupling a single transition for the D3 symmetry trimer, the eigenvector matrix is independent of V and the excitation energy and can be written, for instance, as 1 0 -1 -1 1 pffiffiffi pffiffiffi pffiffiffi B 3C C B 2 r6ffiffiffi B 2 1 C C B pffiffiffi C ð18Þ C¼B 0 B 3 3C C B -1 1 A @ 1 pffiffiffi pffiffiffi pffiffiffi 3 2 6 Consider a scenario where the axis connecting two chelating atoms of a phen ligand is parallel to the C3 axis of the D3 complex (here taken as z) and the phen C2 axis parallel to x, then 0 1 0 B C C d1 ¼ d 3 B ð19Þ @0A 1 This structural configuration is exemplified by the left panel of Figure 2. Rotation by ω of the phen ligand about x (the ligand C2 axis) gives 0 1 0 B C B C sin ω C d1 ¼ d 3 B ð20Þ @ A cos ω Rotating d1 about z (the C3 symmetry axis of the system) by 120 and 240 will yield the other two oriented transitions of the D3 complex. The d0 matrix for the three oriented transitions is then 0

pffiffiffi 3 sin ω 2 1 - sin ω 2 cos ω

0 B B B d ¼ d3B B sin ω @ cos ω 0

1 pffiffiffi 3 sin ω C 2 C C 1 C - sin ω C A 2 cos ω

ð21Þ

from which the associated magnetic moment matrix m0 according to eq 9 is given as 0 1 pffiffiffi pffiffiffi 3 3 cos ω cos ω C B 0 2 2 C idEl r B B C 0 1 1 ð22Þ m ¼ B -cos ω 3 cos ω cos ω C C 2c B 2 2 @ A sin ω sin ω sin ω using the subgroup centers 0 B1 B r ¼ r3B B0 @ 0

1 1 1 - C p2ffiffiffi p2ffiffiffi C 3 3C C 2 2 A 0 0

ð23Þ

Using eqs 21 and 22 in conjunction with eq 18 in eqs 3-5 then yields the rotatory strengths of the coupled long-axis polarized A2 and E excitations in the aggregate as 3 ð24Þ RE ¼ - El rD0l sin ω cos ω 4c RA 2 ¼

3 El rD0l sin ω cos ω 2c

ð25Þ

in atomic units, where r is the distance between the complex center to the center of the ligand, El is the energy of the long-axis transition in the free ligand, c is the speed of light, and D0l = d2 is the dipole strength of the free ligand long-axis excitation.

’ ASSOCIATED CONTENT

bS

Supporting Information. Further discussion of the coupling of transitions. Geometry dependence (r,ω) of the MM coupled CD spectra. 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 We would like to acknowledge support from the center for computational research at the University at Buffalo. M.R. is grateful for a Speyer summer fellowship from the University at Buffalo chemistry department. This work has received financial support from the National Science Foundation (CHE 447321 and 952253). ’ REFERENCES (1) Mason, S. F.; Norman, B. J. J. Chem. Soc. A 1969, 1442–1447. (2) Beller, G.; Lente, G.; Fabian, I. Inorg. Chem. 2010, 49, 3968– 3970. (3) Bressler, C.; Milne, C.; Pham, V.-T.; ElNahhas, A.; van der Veen, R. M.; Gawelda, W.; Johnson, S.; Beaud, P.; Grolimund, D.; Kaiser, M.; Borca, C. N.; Ingold, G.; Abela, R.; Chergui, M. Science 2009, 323, 489– 492. (4) Gawelda, W.; Cannizzo, A.; Pham, V.-T.; van Mourik, F.; Bressler, C.; Chergui, M. J. Am. Chem. Soc. 2007, 129, 8199–8206. (5) Muldoon, J.; Ashcroft, A. E.; Wilson, A. J. Chem.-Eur. J. 2010, 16, 100–103. (6) Randazzo, R.; Mammana, A.; D’Urso, A.; Lauceri, R.; Purrello, R. Angew. Chem., Int. Ed. 2008, 47, 9879–9882. (7) Zhang, D.; Wu, Z.; Xu, J.; Liang, J.; Li, J.; Yang, W. Langmuir 2010, 26, 6657–6662. (8) Yanagi, K.; Ohashi, Y.; Sasada, Y.; Kaizu, Y.; Kobayashi, H. Bull. Chem. Soc. Jpn. 1981, 54, 118–126. (9) Boys, D.; Escobar, C.; Wittke, O. Acta Crystallogr. 1984, 40, 1359–1362. (10) Heilmann, J.; Lerner, H.-W.; Bolte, M. Acta Crystallogr. 2006, E62, m1477–1478. (11) Song, B.; Wu, F.; Hsieh, A. Acta Crystallogr. 1994, C50, 884– 886. (12) Hazell, A.; Hazell, R. Acta Crystallogr. 1984, C40, 806–811. (13) Demadis, K.; Dattelbaum, D.; Kober, E.; Concepcion, J.; Paul, J.; Meyer, T.; White, P. Inorg. Chim. Acta 2007, 360, 1143–1153. (14) Hauser, A.; Maeder, M.; Robinson, W.; Murugesan, R.; Ferguson, J. Inorg. Chem. 1987, 26, 1331–1338. (15) Rillema, D. P.; Jones, D. S. J. Chem. Soc., Chem. Commun. 1979, 849–851. (16) Maloney, D. J.; MacDonnell, F. M. Acta Crystallogr. 1997, C53, 705–707. (17) McCaffery, A. J.; Mason, S. F.; Norman, B. J. J. Chem. Soc. A 1969, 1428–1441. (18) Mason, S. F.; Peart, B. J. J. Chem. Soc., Dalton Trans. 1973, 9, 949–954. (19) Mason, S. F.; Peart, B. J. J. Chem. Soc., Dalton Trans. 1973, 9, 944–949. 2648


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Calculation of Optical Rotatory Dispersion and Electronic Circular

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