Advanced Review Circular : electronic Ingolf Warnke and Filipp Furche∗

First-principles calculations of electronic (ECD) are widely used to determine absolute configurations of chiral . In addition, ECD is a sensitive probe for the three-dimensional molecular structure, making ECD cal- culations a useful tool to study conformational changes. In this review, we explain the origin of ECD and optical activity using response theory. While the quantum- mechanical underpinnings of ECD have been known for a long time, efficient electronic structure methods for ECD calculations on molecules with more than 10–20 atoms have become widely available only in the past decade. We review the most popular methods for ECD calculation, focusing on time-dependent density functional theory. Although single-point vertical ECD calculations yield useful ac- curacy for conformationally rigid systems, inclusion of finite- effects is necessary for flexible molecules. The scope and limitations of modern ECD cal- culations are illustrated by applications to helicenes, fullerenes, iso-schizozygane alkaloids, paracyclophanes, β-lactams, and complexes. C 2011 John Wiley & Sons Ltd.

How to cite this article: WIREs Comput Mol Sci 2012, 2: 150–166 doi: 10.1002/wcms.55

INTRODUCTION ircular dichroism (CD) is the difference between C the absorption coefficients for left and right cir- 1 cularly polarized , εL and εR (see Figure 1),

ε = εL − εR. (1) ε as a function of the incident light frequency ω is called CD spectrum. In this review, we focus on frequencies in the visible and (UV), where electronic excitations dominate. is a necessary requirement for molecules and crystals to exhibit CD. The CD of pure enantiomers differs in sign, but not in mag- nitude. For example, Figure 2 shows the electronic circular dichroism (ECD) spectra of trans-(2S,3S)- FIGURE 1| Left-circularly polarized light propagating in dimethyloxirane and its enantiomer.2,3 There is no z-direction. The plotted electric field vectors rotate in the x–y plane, simple relation between the absolute configuration describing a left-handed helix. The magnetic field vector (not shown) is (AC) of an enantiomer and the sign of its ECD spec- perpendicular to the electric field vector. trum: CD depends on details of the electronic and geometric molecular structure. However, ab initio computed ECD spectra. In the example of Figure 2, electronic structure calculations are nowadays able the AC of the experimental spectrum is assigned to predict ECD accurately and thus allow an assign- (2S,3S) by comparison to the computed ECD spectra. ment of the AC by comparison of experimental and The assignment of ACs by computed ECD spectra is becoming increasingly popular because it is less ex- ∗Correspondence to: fi[email protected] pensive and faster than anomalous-dispersion X-ray 4,5 Department of Chemistry, University of California, Irvine, CA, diffraction and it does not require crystals. ECD- USA based determination of the AC can be complemented DOI: 10.1002/wcms.55 by calculations of the (OR). As ECD

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molecular μ is  iω  μ(1) (ω) = α (ω) E (ω) + G (ω) B (ω) j jk k c jk k k k    ∂ E (ω) 1 + P (ω) k + O , (2) jkl ∂l λ2 kl where j, k, l ∈ (x, y, z). E and B denote the elec- tric and magnetic field vectors at the position of the . Atomic units (a.u.) are used throughout, unless otherwise stated. α jk(ω) is the well-known lin- ear electric dipole–electric dipole polarizability. The linear electric dipole–magnetic dipole polarizability Gjk(ω) describes the linear response of the electric FIGURE 2| Experimental ECD spectrum of dipole moment induced by an oscillating magnetic trans-(2S,3S)-dimethyloxirane (solid) together with calculated ECD field and is the central quantity in ECD . 3 spectra of the (2S,3S) and the (2R,3R) enantiomers. (Theoretical data Pjkl(ω) is the electric dipole–quadrupole polarizability reprinted with permission from Ref 3. Copyright 2002 American tensor. The magnetic dipole and electric quadrupole Institute of Physics. Experimental data reprinted with permission from terms in Eq. (2) are of the same order in ω/c = 2π/λ, Ref 2. Copyright 1994 Elsevier.) where c is the velocity of light and λ is the of the incident radiation, but only the magnetic dipole and OR are closely related, we will briefly comment term gives rise to ECD. In typical ECD measurements, 6–10 on OR, but refer the reader to specialized reviews the wavelength λ is large compared to the molecular for details. dimensions, which justifies the neglect of higher-order ECD spectra are much more sensitive to the multipoles in Eq. (2). Gas-phase and solution experi- three-dimensional molecular structure than ordinary ments capture only the isotropic average absorption spectra because ECD intensities capture 1  angular correlations between electric and magnetic G(ω) = G (ω)· (3) 3 jj transition moments. In biology, ECD spectra are j frequently used to characterize the secondary struc- ture of .1,11–13 With accurate electronic struc- (See Ref 20 for recent work on the full electric dipole– ture ECD calculations becoming available for larger magnetic dipole tensor.) First-order response theory systems, it is possible to quantify the ECD-structure yields the sum-over-state (SOS) expression for G at relation and infer details of the three-dimensional complex frequency z, molecular structure such as conformation from  , c 1 ECD spectra. Many excellent books1 14–17 and G(z) =− 9,10,18,19 3 reviews on ECD exist. The present paper fo- n 0n   cuses on recent developments in electronic structure R0n R0n methods for ECD. In the section Basic Definitions, × − . (4) z − + iη z + + iη we set the stage by reviewing basic quantities and 0n 0n relations underlying ECD. Section Electronic Struc- Here, 0n = En − E0 denotes the excitation energy ture Methods for ECD summarizes some of the most from the ground state 0 to the nth excited state n, important methods presently available for ECD cal- and R0n is the rotatory strength of the corresponding culations. In the section Applications, we review in- transition.21 η is a small contour distortion making structive examples of ECD calculations. G(z) analytical in the upper complex plane. This en- sures causal behavior of the polarizability in the time 22 domain. R0n may be expressed in terms of the vec- BASIC DEFINITIONS tors of the electric transition dipole moment

Electric Dipole–Magnetic Dipole μ0n =− n|rˆ| 0 (5) Polarizability Consider a molecule interacting with a monochro- and the magnetic transition dipole moment matic electromagnetic field oscillating at frequency ω. 1 m =−  |Lˆ | , (6) In the frequency domain, the first-order change in the 0n 2c n 0

Volume 2, January/February 2012 c 2011 John Wiley & Sons, Ltd. 151 Advanced Review wires.wiley.com/wcms where rˆ and Lˆ are the position and the angular mo- Optical Rotation mentum operator, respectively. For real wavefunc- The real (dispersive) part of G(z), , tions 0 and n the electric transition dipole moment −  μ = μ = μ∗ β ω = ω = 2c R0n , 0n n0 0n is purely real, whereas the mag- ( ) Re[G( )] (12) ∗ 3 ω2 − 2 =− =− n 0n netic transition dipole moment m0n mn0 m0n is purely imaginary. The rotatory strength is the imag- μ is the optical rotatory (ORD). As first inary part of the scalar of the scalar product of 0n pointed out by Rosenfeld21 in 1929, the chiral resp- and m0n, onse parameter β(ω) determines the specific rotation, = μ · · R0n Im[ 0n m0n] (7) ω2 [α]ω = C β(ω), (13) Within the Born–Oppenheimer approximation, M , ◦ the molecular eigenstates 0 n are approximated with M being the molar mass. If [α]ω is in /(dm(g/ by products of electronic and nuclear wavefunctions. cm3)), M in g/mol, and ω and β are in a.u., the value To leading order in the Herzberg–Teller expansion, of the constant C is 6.469 × 103 (cf. Ref 24). the electric and magnetic transition moments of an Because the real-time response of μ is causal, allowed transition separate into a purely electronic G(z) is analytic in the upper complex plane.22 part and a Franck–Condon factor. Although the Thus, Cauchy’s theorem gives rise to Kramers–Kronig 18 electronic part governs integrated intensities, the relations18,25–27 between Re[G(ω)] and Im[G(ω)]: Franck–Condon factors determine the vibronic fine  ∞ 2 ωIm[G(ω )] structure of ECD spectra and can affect band shapes Re[G(ω)] = P dω , (14) π ω 2 − ω2 significantly, as discussed in the section Temperature 0 Effects. We will focus on the electronic part here and ,  ∞ assume in the following that the states 0 n are 2 Re[G(ω )] Im[G(ω)] =− P dω , (15) solutions of the electronic Schrodinger¨ equation. π ω 2 − ω2 0 where P denotes the Cauchy principal value. Circular Dichroism Equations (13) and (14) show that knowledge of G(z) may be decomposed into a real and an imaginary the entire ECD spectrum is equivalent to knowl- part. The imaginary part of G (at real frequency ω), edge of the entire ORD curve. Figure 3 illustrates the ω ω cπ  1 qualitative behavior of Re[G( )] and Im[G( )] as a Im[G(ω)] = (R0nδ(ω − 0n) function of ω. 3 n 0n −R δ(ω + )), (8) 0n 0n Selection Rules determines the shape of the linear ECD spectrum,18 A given electric transition has finite intensity in the ECD spectrum if the rotatory strength R [Eq. (7)] is N 0n ε(ω) = A 16πω2Im[G(ω)], (9) nonvanishing. This is possible only if the transition is ln10 spin-allowed and both the electric and magnetic tran- = where NA is Avogadro’s constant and ln10 sition dipole moments μ0n and m0n are nonzero. In 2.302585.... Equations (8) and (9) imply that the addition, a necessary condition for a nonzero rotatory rotatory strength determines the integrated intensity strength is that the scalar product μ0n· m0n does not of a single ECD band of width 2ε, vanish on symmetry grounds. The electric dipole moment operator transforms 0n+ε 3ln10 ε(ω) as a polar vector and the magnetic dipole moment R = dω · (10) 0n 2 operator transforms as an axial vector. Thus, for μ 16π cNA ω 0n −ε 0n and m0n to be nonzero, polar and axial vectors must transform according to the same irreducible represen- The rotatory strength is usually expressed − tations of the molecular point group. This is possible in 10 40 cgs units, where 1 a.u. corresponds to − if the molecular point group does not contain im- 64604.8 × 10 40 erg cm3. Rotatory strengths satisfy proper rotations, i.e., reflections, inversions, various sum rules.18 Most importantly, their sum is and higher S axes, because polar and axial vectors zero,23 n  differ by a sign in their transformation properties R0n = 0. (11) under improper rotations. For example, polar vec- n tors change their sign under an inversion, whereas

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strength:   (l) = μ(l) · , R0n Im 0n m0n (18)   (v) = μ(v) · . R0n Im 0n m0n (19)

(l) (v) Finite basis set approximations to R0n and R0n generally differ; the relative difference can be used as an indicator of basis set quality. Rotatory strengths computed in the dipole–length gauge tend to converge faster with basis set size than those computed in the (l) (v) dipole–velocity gauge. R0n and R0n will differ even in the basis-set limit if approximate response theories such as the Tamm–Dancoff 29 approximation are used that do not preserve gauge invariance. The magnetic transition dipole moment is not invariant under a change of the gauge origin R, 1 m (R) =−  |(rˆ − R) × pˆ |  0n 2c n 0 1 =− ( |rˆ × pˆ | −R × |pˆ | ) 2c n 0 n 0 i v = m (0) − 0n R × μ( ). (20) 0n 2c 0n μ(v) · × μ(v) = , Because 0n (R 0n) 0 the velocity form of the rotatory strength, Eq. (19) is independent of R, even in a finite basis. However, the length form is not |μ(v) − μ(l) | = . gauge-origin invariant, as long as 0n 0n 0 FIGURE 3| Qualitative behavior of real and imaginary part of This spurious gauge-origin dependence can be largely eliminated by using methods and basis sets where G(ω) close to an excitation energy 0n. |μ(v) − μ(l) | 0n 0n is sufficiently small, or entirely by us- axial vectors are invariant. The presence of an im- ing gauge-including atomic orbitals (GIAOs, Refs 30– proper rotation is thus equivalent to the molecule be- 34). As pointed out above, with reasonably polarized ing achiral.28 basis sets, the length and velocity forms of the elec- As enantiomers behave as mirror images, their tric transition dipole moment should not differ much, rotatory strengths must differ by a sign due to the and gauge-origin dependence is much less of a con- different transformation properties of μ0n and m0n cern in ECD and ORD calculations than it is, e.g., for under a reflection. It follows that the ECD and ORD calculations of NMR chemical shieldings. spectra of enantiomers differ in their sign. Thus, ECD and ORD spectra may be used to determine the AC. ELECTRONIC STRUCTURE METHODS FOR ECD Gauge Invariance The electric transition dipole moment may be evalu- A straightforward approach to the prediction of the ated in different gauges.18 Common choices are the electronic CD spectrum, Eq. (9), would be to com- dipole–length gauge, pute the ground and excited state wavefunctions | 0, |  (l) n , and their energies to evaluate the excitation en- μ =− n|rˆ| 0, (16) 0n ergies and rotatory strengths R0n. In practice, how- and the dipole–velocity gauge, ever, almost all modern ab inito methods for comput- ing ECD spectra rely on response theory. As opposed (v) 1 μ =−  | ˆ | , to state-based methods such as state-specific complete 0n n p 0 (17) i 0n active space self-consistent field or configuration in- where pˆ denotes the linear momentum operator. This teraction treatments, which focus on individual ex- leads to the length and velocity forms of the rotatory cited states, response theory aims at computing all

Volume 2, January/February 2012 c 2011 John Wiley & Sons, Ltd. 153 Advanced Review wires.wiley.com/wcms properties from the response of a reference state to generating the TDKS system is a unique functional of a time-dependent external perturbation. For exam- ρ(t, x). Many one-particle response properties can be ple, the electric dipole–magnetic dipole polarizability extracted from the time-dependent density; for exam- tensor Gjk(ω) is computed from linear response ple, the time-dependent dipole moment is theory as   ∂μ ω  μ(t) =− dxrρ(t, x), (22) ω = c j (B ( )) , Gjk ( ) ω ∂ ω  (21) i Bk ( ) B(ω)=0 and thus a perturbation expansion of ρ(t,x) in powers consistent with Eq. (2). This approach has a number of the external fields yields formally exact polarizabil- of important advantages: (1) The resulting frequency- ities. dependent properties reduce to the correct static The exchange–correlation (XC) piece of the limit (ω → 0); (2) important constraints, such as the TDKS potential is unknown as an explicit density Kramers–Kronig relations linking computed ECD and functional. In ECD calculations, the following ap- ORD spectra or sum rules, are automatically satisfied, proximations are used: (1) The XC potential is re- which is not true for state-specific methods; (3) ana- placed by its time-independent counterpart, evaluated lytical derivative methods35 may be used to evaluate at the instantaneous density (adiabatic approxima- response properties efficiently; (4) excited-state wave- tion, Ref 40). This corresponds to a neglect of double functions do not need to be explicitly dealt with. and higher excitations in the excitation spectrum.41 Different computational strategies are used to (2) The static XC potential is usually approximated evaluate the real and imaginary parts of G(ω): Ex- using semilocal density functionals, such as general- citation energies are obtained from the poles of the ized gradient approximation42–45 (GGA) or hybrid frequency-dependent linear response; for example, functionals.46,47 according to Eq. (4), the excitation energies 0n of As ECD is a mixed electric–magnetic response the states visible in the ECD spectrum are the poles property, the proper density functional framework of G(ω) on the real frequency axis. The rotatory to compute ECD intensities is time-dependent cur- 48–50 strengths R0n are extracted from the corresponding rent density functional theory. So far, the cur- residues of G(ω). This is accomplished in practice by rent dependence of the XC potential has been almost computing the 0n as eigenvalues of the inverse re- universally neglected. Within time-dependent density sponse function, and construct the residues from the functional linear response theory, the electric dipole– corresponding eigenvectors, as explained below. The magnetic dipole polarizability tensor at complex fre- real part of G(ω) at a nonresonant frequency may be quency is obtained as computed directly by solving a single linear system of 24 c ( j) (k) (k) first-order equations ;itisnot necessary to compute Gjk(z) =− μ |X (z), Y (z). (23) all the excitation energies and rotatory strengths in z the SOS expression [Eq. (4)] first. In the present notation, a supervector   X |X, Y= (24) Time-Dependent Density Functional Y Theory Time-dependent density functional theory (TDDFT) consists of two components X and Y, which × is presently the most widely used electronic structure are elements of the vector spaces Locc Lvirt and × method for ECD calculations. We briefly summarize Lvirt Locc spanned by products of occupied (occ) some key aspects here and refer the reader to more and virtual (virt) ground-state Kohn–Sham (KS) specialized reviews36–38 for details. molecular orbitals.51 |μ( j)=|μ( j),μ( j) contains the According to the fundamental Runge–Gross matrix elements of the electric dipole moment op- theorem,39 the time-dependent one-particle density erator. |X(k)(z), Y(k)(z) is a solution of the coupled ρ(t,x) of a many system is uniquely deter- perturbed Kohn–Sham (CPKS) equations at complex mined by a local multiplicative one-particle potential frequency, and vice versa for a fixed initial state; x = (r,σ ) de-  −  | (k) , (k) =−| (k). notes a space-spin coordinate. Runge and Gross in- ( z ) X (z) Y (z) m (25) troduced a noninteracting system, the so-called time- The 2 × 2 superoperators dependent Kohn–Sham (TDKS) system, which has the     same one-particle density as the physical interacting AB 10  = ,= (26) system at all times. The effective one-particle potential BA 0 −1

154 c 2011 John Wiley & Sons, Ltd. Volume 2, January/February 2012 WIREs Computational Molecular Science Electronic Circular dichroism contain the electric and magnetic orbital rotation charge-transfer intruder states57 and unbound Ryd- Hessians in the adiabatic approximation, berg excitations.58,59 Although intensities of individ- ual states are strongly affected by these shortcom- (A + B) = (ε − ε )δ δ + 2(ia| jb) + 2 f XC iajb a i ij ab iajb ings and can be wrong, the overall shape of com- − cx[(ib| ja) + (ij|ab)], (27) puted ECD spectra is fairly robust. Range-separated hybrid functionals (CAM-B3LYP),60 LCω-PBE61–64) 65 (A − B) = (ε − ε )δ δ − c [(ib| ja) − (ij|ab)], improve long-range CT excitations, but introduce iajb a i ij ab x an additional adjustable parameter. The B2PLYP (28) double-hybrid functional mixes TDDFT and con- figuration interaction singles with perturbative dou- where indices i,j...denote occupied and a,b...virtual bles excitation energies.66–68 Featuring an N5 scal- canonical KS spin-MOs with eigenvalues ε and ε , i a ing of computational cost with the system size N, respectively. (ia| jb) is an electron repulsion integral in B2PLYP is significantly more expensive than conven- Mullikan notation, and f XC denotes a matrix element iajb tional DFT which scales approximately as N2, but of the static XC kernel, B2PLYP does seem to improve excitation energies 2 XC δ E significantly (see the section Inherently Chiral Chro- f XC (x, x ) = . (29) iajb δρ(x)δρ(x ) mophores). A drawback for ECD spectroscopy is that B2PLYP rotatory strengths do not contain any dou- EXC is the time-independent exchange-correlation en- bles corrections.67 ergy, e.g., from a semilocal approximation. Becke’s hybrid mixing parameter cx is used to interpolate Other Methods between nonhybrid TDDFT (c = 0) and time- x Coupled-cluster response theory is occasionally used dependent Hartree–Fock (TDHF) theory (c = 1, , , ,  x to compute ECD69 70 and ORD.6 71 72 Coupled- EXC = 0). The right-hand side m(k) contains the kth cluster singles–doubles (CCSD) calculations exhibit component of the magnetic dipole moment vector. an N6 scaling of computational cost and are mostly If G(ω) is evaluated at real frequencies ω,the used for benchmarking. The approximate CCSD ORD can be computed by solving a single CPKS equa- method CC2 is more economical, particularly in con- tion [Refs 20 and 24; Eq. (25)]. For ECD calculations, junction with resolution-of-identity methods.73–75 A the imaginary part of G is obtained from the poles limitation of coupled cluster response theory for ECD and residues of G(ω) by solving the TDKS eigenvalue calculations is its lack of gauge invariance.76 problem,52,53 TDHF calculations of ECD were popular in 77 ( − 0n)|Xn, Yn=0 (30) the 1980s, but have been superseded by TDDFT. TDDFT using, for example, GGA or hybrid function- using the symplectic normalization constraint als, is considerably more accurate and less suscepti- 51,53 X , Y ||X , Y =1. (31) ble to reference state instabilities than TDHF. n n n n Semiempirical methods such as CNDO (Complete 78 The eigenvalues 0n are the excitation energies, Neglect of Differential Overlap) and related meth- whereas the electric and magnetic transition dipole ods can produce accurate valence excitation energies moments are obtained from the eigenvectors accord- on restricted data sets.79 ing to Basis Sets μ0n =Xn, Yn|μ, (32) In most ECD calculations, the molecular orbitals (MOs) ϕ are expanded in a basis of atom-centered = , | , p m0n Xn Yn m (33) functions χμ (linear combination of atomic orbitals, and Eq. (7) yields the rotatory strengths R . LCAO), 0n  The most common XC functionals used ϕp(r,σ) = Cμpσ χμ(r). (34) for ECD calculations include the Becke-Perdew86 μ (BP86)43,54 and Perdew–Burke–Ernerzhof (PBE)45,55 generalized gradient approximation (GGA) function- The most common type of basis functions for molecu- als along with hybrid functionals such as Becke’s lar ECD calculations are Gauss-type orbitals (GTOs), three-parameter hybrid (B3LYP)43,47,56 and PBE allowing analytical integral evaluation and efficient hybrid.44–46 The GGA functionals are susceptible to prescreening in polyatomic systems. Basis sets op- self-interaction error, which shows up in spurious timized for ground-state calculations are usually

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TABLE 1 Excitation Energies E (nm) Oscillator Strengths f0n chiroptical response of the energetically lowest con- −40 (Length Representation) and Rotatory Strengths R0n (10 cgs, former. In this case, the rotational strengths of only 1 Length Representation) for the 2 B State of (M)-hexahelicene. one structure have to be evaluated in order to reason- 87 The Structure was Optimized at the HF/SV(P) Level of Theory. ably reproduce the experimental ECD spectrum or The R0n were Computed Using the PBE Hybrid Functional. SVP, OR. However, the chiroptical response is sensitive to TZP, and QZVP Stand for Polarized Split Valence, Triple–ζ ,and structural changes,95 and conformer equilibria lead to Quadruple–ζ Valence Basis Sets.88–90 The Suffix D Denotes Diffuse Augmentation86 important and sometimes qualitative deviations from experiment.91,96 Basis Ef0n R0n A convenient route for modeling the chiropti- cal response of moderately flexible molecules consists SVP 316 0.37 −675 in computing the ECD spectra of energetically low- SVPD 320 0.35 −688 lying conformers using quantum chemical methods, TZVP 322 0.35 −700 followed by Boltzmann averaging.97–105 Although this Sadlej 323 0.35 −701 strategy produces satisfactory results in systems with TZVPD 322 0.35 −701 a small number of low-lying conformers separated by − TZVPPD 323 0.35 701 high-potential energy barriers, it is not suitable for − QZVPD 323 0.35 702 floppy molecules with soft or strongly anharmonic Experiment 310 0.4 −400 vibrational modes. The importance of vibronic effects for OR104 and ECD106,107 has recently been emphasized. Gen- sufficient for low-lying excitation energies. For a eral techniques for vibrational averaging are discussed correct description of higher and Rydberg excita- in Refs 108 and 109. Zero-point vibrational cor- tions, diffuse functions need to be included in the rections can account for up to 20% of the optical 109 basis set. Diffuse augmentation is essential for indi- rotation. Vibronic model calculations treat the dis- 106,107,110 vidual transition moments, rotatory strengths, and es- crete vibrational level structure explicitly. pecially for ORs, which can be qualitatively in error However, these models are often based on harmonic with standard polarized basis sets. The most common normal-mode expansions of the nuclear motion and diffuse-augmented basis sets include Dunning’s hier- break down in presence of major changes in structure. archy of augmented correlation-consistent basis sets80 Vibrational averaging methods are thus best suited and Pople-style basis sets, such as 6–31++G∗.81–83 for conformationally restricted molecules. Recently, 111 Sadlej basis sets84,85 are specially designed for polariz- Dierksen et al. investigated the origin of ECD of ability calculations. The recently developed hierarchy isotopically chiral systems. They concluded that vi- of property-optimized Ahlrichs-style basis sets86 uses bronic coupling effects are the main cause for ECD as few diffuse functions as possible and is promising induced by isotopic substitution. for ECD calculations, particularly in larger systems. Molecular dynamics (MD) simulations have Rotatory strengths are quite sensitive to basis successfully been used to quantify dynamic and vi- set quality, especially for smaller systems and weak bronic effects in a classical approximation. These or strongly coupled transitions. Table 1 illustrates the methods involve TDDFT-sampling of the chiroptical basis set convergence for the first strong band in the response over classical nuclear trajectories. In most ECD spectrum of (M)-hexahelicene.87 cases, these trajectories are obtained using force-field 91,112,113 96 Although absolute intensities are commonly re- based MD simulations. Frelek et al. used produced within a factor of two by TDDFT38 (see on-the-fly Born–Oppenheimer MD based on full DFT also Table 1), intensities of individual ECD bands rel- potential energy surfaces in conjunction with TDDFT ative to each other are considerably more accurate. calculations to sample the chiroptical responses of The use of an appropriate basis set is crucial for the flexible molecules, see the example in the section Cir- β correct prediction of relative intensities and hence for cular Dichroism of -lactams. Although this classical the determination of the AC. treatment of the nuclei will at best reproduce a cor- rect spectral envelope,96 it is the method of choice for systems with significant conformational flexibil- Temperature Effects ity that cannot be described in vibronic models. In To date, most routine applications in theoretical ECD light of the sometimes complex relation between ECD spectroscopy neglect all nuclear motion by using static and conformation, the use of force-field-based classi- minimum structures.87,91–94 At low and cal potential energy surfaces instead of the quantum for rigid molecules, the ECD is dominated by the mechanical ones should be carefully validated.

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A simplistic but widely used method to in- clude temperature and effects in ECD spectra qualitatively is empirical line broadening. As estab- lished by Mason et al.,114 Gaussians of constant line width centered at the excitation energies and scaled to integrate to the computed rotatory strengths accord- ing to Eq. (10) may be used to approximate the ECD spectrum. In this procedure, the Gaussian root mean square width is an empirical parameter which is usu- ally chosen to fit the measured spectrum. For ECD spectra in the near UV and visible region, a typical root mean square width is 0.15eV.

Solvent Effects ECD and OR depend sensitively on the presence and nature of solvent interactions.70,115–117 The change in chiroptical response due to solute–solvent interac- tions is sometimes drastic and nonintuitive. A well- known example is the sign inversion in the OR of methyloxirane upon changing the solvent from wa- FIGURE 4| Comparison of TDDFT67 and experimental128 circular 7 ter to benzene.116 Strong solvatochromism in ECD dichroism spectra for (M)- helicene with the three different density 66 spectra may arise from solvent effects on tautomer functionals B2LYP, B2PLYP, and B3LYP along with the TZVP basis set. (Reprinted with permission from Ref 67. Copyright 2007 American equilibria,118 as illustrated in the section Absolute Institute of Physics.) Configuration and Solvatochromism of [2.2]Paracy- lophane Ketimines. Different strategies exist to in- polarized split-valence basis sets is accurate enough clude solvent-solute interactions into chiroptical re- to determine the AC of helicenes, and can even be sponse calculations. Continuum models such as the used to identify substituted helicenes by their ECD 119–121 conductor-like screening model (COSMO) or signature.87 Figure 4 shows a comparison of three 122–124 the polarizable continuum model (PCM) ac- hybrid functionals for [7]-helicene. The B2PLYP dou- count for electrostatic and effects be- ble hybrid outperforms the two other functionals by 125,126 tween a continuous solvent and the molecule. placing bands A and C in the right position, but has Classical force-field MD simulations constitute a pos- deficiencies at higher energies.67 sible route to explicitly account for interactions be- Most higher fullerenes are inherently chiral. It 91,117,127 tween molecule and solvent. Apart from the is very difficult to isolate and separate fullerene enan- high computational cost it might appear most desir- tiomers, and ECD spectroscopy is often the only re- able to treat the solute embedded in a large number liable option to determine their AC. Resolution of of solvent molecules within a first-principles quantum the enantiomers of D2-C84 was first achieved by mechanical MD approach. This might only be pos- Diederich et al.129 in 1999, but existing semiempir- sible with very efficient electronic structure methods ical CNDO/S ECD data130 were not accurate enough such as DFT which on the other hand have deficiencies to pin down the AC. On the other hand, TDDFT using for non-covalent long-range interactions. Semiempir- the BP86 functional reproduces the main features of ical models developed for large scale solvent simula- the experimental spectrum131 (see Figure 5), thus al- tions might be a viable alternative for simulating chi- lowing a definite assignment of the AC. The computed roptical properties in the presence of explicit solvent absolute ECD intensities of D2-C84 are substantially 19 interactions. larger than the measured ones, for reasons not well understood.131

APPLICATIONS Absolute Configuration of Iso-schizogaline Inherently Chiral Chromophores and Iso-schizogamine A classical example of inherently chiral chromophores A paper by Stephens et al. on the two natural products are helicenes. These helical polyacenes exhibit strong iso-schizogaline and iso-schizogamine133 (Figure 6) ECD in the UV and OR values of 1000 ◦/(dm(g/cm3)) illustrates the scope and the limitations of ECD spec- and more. TDDFT using the BP86 functional and troscopy in determining the AC of organic molecules

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FIGURE 7| [2.2]paracyclophane-ketimine derivative featuring planar and central chirality.

129 FIGURE 5| Experimental ECD spectrum of D2-C84 compared to simulations using a semiempirical method130 and TDDFT131(BP86/aug-SVP). The absolute configuration of the fullerene is f A in the old129 and C in the new IUPAC132 nomenclature. ε denotes the molar decadic absorption coefficient, R the rotatory strength. ε Calculated values from TDDFT were scaled by 1/14 to match the FIGURE 8| Structures of the low-energy conformers of the (Rp,R) experimental intensities. (Data reprinted with permission from Ref 131. and (Rp,S) diastereomers. (Reprinted with permission from Ref 118. Copyright 2002 American Chemical Society. CNDO/S data reprinted Copyright 2009 American Chemical Society.) with permission from Ref 130. Copyright 1997 Elsevier.) Absolute Configuration and Solvatochromism of [2.2]Paracylophane Ketimines ECD spectra and the AC of moderately flexible [2.2]paracyclophanes were recently investigated by Warnke et al.118 [2.2]paracyclophanes are highly ef- ficient chiral ligands used in 1,2-additions and conju- gate 1,4-additions of organozinc compounds to alde- hydes and ketones. They feature planar chirality in the paracyclophane moiety as well as an asymmetric carbon atom in the ketimine side chain (see Figure 7). The authors employ ECD and OR as obtained FIGURE 6| Iso-schizogamine and iso-schizogaline. from TDDFT calculations and compare them to ex- perimental data. Geometry optimizations including COSMO for the treatment of solvent effects reveal with several stereocenters. Iso-schizogamine and iso- that diastereomers adopt different conformations (see schizogaline contain four asymmetric carbon atoms, Figure 8), causing differences in their ECD spectra: corresponding to eight enantiomer pairs. To fully The sign of the lowest-energy ECD band at 361 nm resolve the AC of all four stereocenters, the authors is determined by the AC of the planar chiral paracy- compare simulated and experimental chiroptical data clophane moiety. This is not surprising as the low- from three different methods: ECD, OR, and vi- est excitation is characterized by a π → π ∗ transition brational CD. Rotatory strengths, optical rotations, from the highest occupied to the lowest unoccupied and vibrational spectra31 of the low-lying conform- molecular orbital, which is delocalized over the parts ers were obtained using B3LYP and gauge-including of the paracyclophane moiety. On the other hand, atomic orbitals.32 The three methods yield consistent the intensity of the lowest CD band is sensitive to the results and predict the AC of natural iso-schizogaline relative orientation between the paracyclophane and and iso-schizogamine to be (2R,7R,20S,21S). ECD the phenyl group in the ketimine side chain. This ori- spectroscopy alone is not sufficient for a reliable AC entation is different for diastereomers (see Figure 8): assignment here, due to strongly overlapping bands The AC of the asymmetric side-chain carbon is de- in the experimental ECD spectrum. termined by means of the intensity of the ECD band

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FIGURE 10| Computed and measured electronic circular dichroism (ECD) of the tautomeric forms (Rp,S) -arom and (Rp,S)-quino of the studied [2.2]paracyclophane-ketimine derivative (Figure 7). The less polar solvent favors the less polar ortho-hydroquinone-imine whereas the more polar solvent promotes the ortho-quinone-enamine tautomer. Computed spectra were obtained from time-dependent density functional theory (TDDFT) (PBE hybrid/SVP) response calculations. Root mean square linewidths are 0.15eV for the aromatic tautomer and 0.30eV for the quinoidal tautomer (see the section Temperature Effects). Theoretical spectra are scaled by a factor 0.45. (Reprinted with permission from Ref 118. Copyright 2009 American Chemical Society.)

polar tautomer and unpolar cause the aro- matic, less polar tautomer to be more stable. The FIGURE 9| Experimental electronic circular dichroism (ECD) of a two tautomeric forms are spectroscopically distinct pair of diastereomers recorded in toluene is compared to species. The quinoidal, nonaromatic form features corresponding theoretical spectra from time-dependent density substantially red-shifted excitation energies. To re- functional theory (TDDFT) response calculations PBE hybrid produce spectra recorded in solvents of intermediate functional/split-valence plus polarization basis set. Solvent effects are polarity, COSMO calculations were used to deter- included using COSMO (see the section Solvent Effects). Calculated mine the solvent dependent Boltzmann factors for a spectra were scaled with a factor 0.5. Root mean square linewidth for weighted average of the theoretical spectra of both Gaussian broadening is 0.15eV (see section Temperature Effects). The predicted difference in the absolute intensities of the lowest-energy tautomers (see Figure 12). band is present in the measured spectra. (Reprinted with permission from Ref 118. Copyright 2009 American Chemical Society.) Circular Dichroism of β-lactams The experimentally measured ECD is dominated by the chiroptical response of the lowest conformer only at 361 nm. Comparison of experimental and theo- for rigid molecules. Thermal effects can exceed in- retical ECD (Figure 9) confirms the presence of the accuracies inherent to TDDFT and in some cases predicted change in sign and difference in intensities even lead to qualitatively different spectra. A treat- in the experimental data and allows for reliable AC ment of thermal effects becomes necessary for flexible determination. molecules. In a study on models for conformationally The inclusion of solvent effects is vital for this flexible β-lactam antibiotics, Frelek et al.96 explored system: Strong solvatochromism in the experimental a new strategy to systematically address the effects of spectrum (Figure 10) is caused by the influence of nuclear motion: they combined TDDFT calculations the solvent polarity on a tautomer equilibrium be- with full quantum-mechanical Born–Oppenheimer tween an aromatic ortho-hydroquinone-imine form MD using DFT ground-state potential energy sur- and an ortho-quinone enamine form of the molecule faces. This computationally demanding approach in- (Figure 11): Polar solvents favor the quinoidal, more volves sampling over tens of thousands of time-steps

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FIGURE 11| Tautomeric equilibrium explaining the observed solvatochromism. arom is the ortho-hydroquinone-imine form and quino is the ortho-quinone-enamine form.

FIGURE 13| The investigated (6R,7S)-7-((1R)-1-hydroxyethyl)-8- oxo-1-aza-bicyclo[4.2.0]octane.

FIGURE 12| Boltzmann-weighted superposition of the computed ortho-quinone enamine and ortho-hydroquinone-imine electronic circular dichroism (ECD) spectra versus ECD spectrum of the (Rp,S) diastereomer recorded in . Computed circular dichroism spectra from time-dependent density functional theory (TDDFT) (PBE hybrid/SVP) calculations, scaled by a factor 0.5. Conformer energies for Boltzmann-factors were computed including solvent effects through COSMO (see section Solvent Effects). For details see Ref 118. (Reprinted with permission from Ref 118. Copyright 2009 American Chemical Society.) FIGURE 14| Computed structure of the cepham analog (see Figure 13) and its simulated electronic circular dichroism (ECD) spectrum at 0 K and 350 K compared to experiment. The 0 K curve and has become possible with the availability of corresponds to the conformer shown, whereas the 350 K curve is the highly efficient DFT and TDDFT implementations.134 result of MD simulation. ε is the molar decadic absorption coefficient, λ The carbacepham depicted in Figure 13 is the wavelength, and R the computed rotatory strength. The one of the investigated model systems. Figure 14 experimental spectrum was recorded in at room temperature. See Ref 96 for details. (Data reprinted with permission compares the corresponding experimental ECD to the from Ref 96. Copyright 2007 Wiley-VCH Verlag GmbH & Co.) simulated ECD of the lowest conformer at 0 K as well as to the ECD curve resulting from the MD sim- ulation. The qualitative difference between the two tional averaging is not a suitable alternative because theoretical curves is attributed to a complex relation large-amplitude motions and conformational dynam- between ECD and the structure of individual con- ics are present. formers; different conformers of one enantiomer can In combination with MD-based ECD simula- have radically different ECD spectra. The reasonable tions, the temperature-dependence of experimental agreement between experiment and MD result allows ECD spectra can provide detailed insight in confor- for AC assignment and emphasizes the importance of mational dynamics.105 The authors found that a pre- incorporating temperature effects. In this case, vibra- viously developed helicity rule135 for the simple AC

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itself: (S)-pn and (R)-pn ligands preferably adopt δ and λ conformation, respectively. This situation al- lows a systematic study of the influence of conformation on the ECD spectra: By successively replacing, e.g., the en ligands with (S)-pn in a - (λλλ) complex, it is possible to gradually move from a lel3 to an ob3 complex. The authors compare their theoretical results to the available experimental ECD spectra.140,141 They also compare the theoretical spec- 3+ FIGURE 15| -and-configuration of [Co(en)3] . trum of the energetically unfavored ob3 form to the experimental and theoretical ECD of the lel3 form. The energetically low-lying ECD bands are dominated assignment in cephams only holds for conformation- by d → d type transitions, whereas ECD transitions ally averaged spectra. from the central metal ion to the ligands are present in the high-energy region. Based on their TDDFT re- 142 Stereochemistry of Transition Metal sults, the authors discuss a simplified model for Complexes rationalizing qualitative trends in the influence of lig- and conformation on the rotatory strengths of d → d DFT is the workhorse of computational transition and CT transitions. metal (TM) chemistry, although inaccuracies are sub- stantially larger than in main group chemistry.136 ECD spectra of TM complexes are complicated by CONCLUSIONS d → d transitions and metal-to-ligand CT transitions. Wang et al.137 pioneered the theoretical ECD Theory-aided ECD spectroscopy is an inexpensive spectroscopy of octahedral tris-(didentate) complexes and fast tool for determining the AC of a wide range using TDDFT almost 10 years ago. Jorge et al.138 of rigid systems. For a reliable assignment of the AC, explored the possibility of using TDDFT to elucidate ECD spectra should be computed and measured over the role of configurational and conformational effects a range of several hundred nm and should show two on ECD spectra of tris-diamine Co(III) and Rh(III) or more distinct features. If possible, assignments by complexes. Octahedral tris-bidentate TM complexes ECD should be complemented by at least one other occur in two enantiomeric configurations. The con- chiroptical measurement, including OR, vibrational figuration is denoted  if the chelating ligands are CD, or ,9,10 especially if sev- arranged around the TM in form of a right-handed eral chirality elements are present. Moderately flex- helix, and  for a left-handed helix (see Figure 15). ible systems can be treated by Boltzmann-averaging When coordinated to the central TM, each ligand the computed ECD spectra of the energetically low- chain can adopt two different conformations, δ or est conformers. This requires careful conformational λ. The configuration-conformation combinations - analysis, e.g., using force field molecular mechanics. (λλλ) and -(δδδ) are called lel3 and the forms -(δδδ) Although ECD calculations of simple rigid sys- and -(λλλ) are called ob3. Further complications tems can routinely be performed by nonspecialists, the arise from polar and azimutal distortions139 from the simulation of detailed vibronic structure and the treat- ideal octahedral coordination geometry. ment of highly flexible molecules remains difficult and The first step in studying the OA is conforma- computationally demanding. Higher accuracy is of- tional analysis. In agreement with force-field based es- ten required for distinguishing diastereomers or for timations, DFT calculations including solvent effects ECD-based determinations of structural parameters 3+ (COSMO) predict the lel3-forms of the [Co(en)3] such as dihedral angles. TDDFT is the method of first cation to be slightly more stable than the correspond- choice for ECD calculations, but in its present form ing ob3 form. Thus, in the case of ethylene-diamine is not universally accurate, with major limitations in- ligands, the preferred ligand conformation is linked cluding charge transfer and Rydberg excitations, as to the AC of the TM complex. However, the pre- well as double and higher excitations. Despite the re- ferred conformation adopted by the chiral ligand markable progress that theoretical ECD spectroscopy (1,2)-propyl-diamine (pn) is no longer controlled by has seen in the past decade, there is ample room for the AC of the complex, but by the AC of the ligand further developments.

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ACKNOWLEDGMENTS We acknowledge support by the NSF Center for Chemical Innovation ‘Chemistry at the Space-Time Limit’ CaSTL, Grant No. CHE-0802913.

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