## A quantum chemical approach to the design of chiral negative index materials

Optics Express, Vol. 15, Issue 9, pp. 5730-5741 (2007)

http://dx.doi.org/10.1364/OE.15.005730

Acrobat PDF (240 KB)

### Abstract

This paper presents methodology developed for the computational modeling and design of negative refractive index materials (NIMs) based on molecular chirality. An application of the methodology is illustrated by *ab initio* computations on two organometallic molecules which constitute the monomer units of a chiral polymer. Comparisons with experimental data for the polymer are made. Even though the resulting chirality parameter for the pristine material is small, it is shown that negative index can be achieved by introducing sharp plasmonic resonances with metal nanoparticle inclusions.

© 2007 Optical Society of America

## 1. Introduction

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

3. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. **17**, 695–706 (2003). [CrossRef]

## 2. Methodology

11. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. **9**, 432–457 (1937). [CrossRef]

*a*, where λ is the wavelength of the incident light and

*a*is a characteristic size of the material’s constituents. It is also reciprocal because it is assumed linear and non-magnetic (i.e. a magneto-electric effect does not occur). Finally, the bulk material is chiral as consisting of chiral molecules. The bi-isotropic constitutive relations are given as

3. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. **17**, 695–706 (2003). [CrossRef]

*k*

_{±}is the propagation constant of two eigen-waves in the isotropic chiral medium, κ is the chirality parameter, ε denotes the scalar macroscopic dielectric permittivity (for an isotropic material), and μ denotes the macroscopic magnetic permeability.

*ab initio*theoretical modeling of chiral NIMs is of great significance as it can provide a basis to understand the structure-property relations necessary for a rational design of the material’s building blocks. In turn, this can lead to an optimization of structures for producing molecular materials with large chiral parameters. It is worthwhile to note that in the area of computational nanophotonics density functional theory (DFT) is usually preferred to wavefunction based

*ab initio*calculations. This is because DFT combines affordable computational costs and reasonably high accuracy due to effects from electron correlation. We present in this paper a framework to study NIMs which utilizes first principles quantum chemical computations of the mixed electric-magnetic polarizabilities that are responsible for chiral properties of molecules. In this quantum-theoretical approach we start out by building the basic building blocks of the material from atomic nuclei and electrons and express the whole system in terms of the Schrödinger or Dirac equations. These equations can only be solved after using a series of approximations based on the underlying physics and mathematics. The property of interest is then obtained by applying time-dependent variational density-functional response theory to calculate the linear response functions with respect to electric and magnetic dipole fields [12

12. K. Ruud and T. Helgaker, “Optical rotation studied by density-functional and coupled-cluster methods,” Chem. Phys. Lett. **352**, 533–539 (2002). [CrossRef]

19. M. Krykunov and J. Autschbach, “Calculation of origin independent optical rotation tensor components for chiral oriented systems in approximate time-dependent density functional theory,” J. Chem. Phys. **125**, 034102-10 (2006). [CrossRef]

*ab initio*calculations. Such an analysis may be based on analytical models [13

13. J. J. Maki and A. Persoons, “One-electron second-order optical activity of a helix,” J. Chem. Phys. **104**, 9340 (1996). [CrossRef]

14. S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, “Analytical antenna model for chiral scatterers: Comparison with numerical and experimental data,” IEEE Trans. Antennas Propag. **44**, 1006 (1996). [CrossRef]

*ab initio*computations simply by inspecting an optimized geometry (which can be obtained with fast semi-empirical methods, such as AM1 or PM3). A higher value of the ratio of radius to pitch should be expected to indicate an increase in the value of the optical rotatory parameter, in particular when comparing systems that have similar chromophores (meant here as highly polarizable moieties). Other criteria are also of importance. For molecules with chains containing delocalized π-electrons such a criterion is the conjugation length, which influences the magnitude of the molecular polarizability. This and other issues have to be addressed in the design process. We envision that simple models may be used for a fast screening of a large set of potential compounds. However, follow-up

*ab initio*computations need to confirm if the system has a large chirality parameter since the relation between structure and optical activity is complex and influenced by many factors such as functional groups, the exact nature of the chromophore and so on.

*ab initio*computations can be applied to determine the microscopic molecular chirality as well as the polarizability. Various approaches can then be used to assess the macroscopic dielectric properties of a medium composed of chiral molecules. In the simplest case one can assume an oriented gas without interactions between the molecules that constitute the components of the material. Composite materials can be dealt with through an effective medium (EM) approach, for instance the Maxwell-Garnett mixing formula. Within this approach one can also assess the influence of modifications in the local fields encountered by the chiral molecule on the macroscopic chirality parameter. One possible way of obtaining very large chirality parameters is the so-called supramolecular approach where self-assembling helical structures of chiral molecules boost the chirality [15

15. V. Percec, M. Glodde, T. K. Bera, Y. Miura, I. Shiyanovskaya, K. D. Singer, V. S. K. Balagurusamy, P. A. Heiney, I. Schnell, A. Rapp, H. W. Spiess, S. D. Hudson, and H. Duan, “Self-organization of supramolecular helical dendrimers into complex electronic materials,” Nature **419**, 384–387 (2002). [CrossRef] [PubMed]

*ab initio*computations of the optical activity of supramolecular assemblies are far from being straightforward. However, we envision that

*ab initio*computations may be instrumental even for macromolecules in this framework. For instance they may be utilized to determine the transition dipole moments of the electronic excitations for the monomer structures and the influence of coupling between a few oligomer units. Based on this and some basic structural information, a coupling model should be able to yield a reliable estimate of the chirality parameter near a resonance. For wavelengths far away from resonances direct computations of the optical rotatory strength for the monomer and small oligomer units might yield useful insight.

## 3. Estimation of the chirality parameter from an experimental CD spectrum

*n*is the refractive index of the medium. The (

*n*

^{2}+ 2)/3 factor in the last equation may be determined in the

*ab initio*(see below) computations by considering medium effects on

*β*directly, for instance by employing a continuum model. For the purpose of this initial study we have neglected medium effects in the computations.

^{2}/dmol] and the ORD obtained from a numerical Kramers-Kronig transformation of the CD spectrum. Further, we show the corresponding dispersion of the chirality parameter, κ, for the dilute solution used for the measurements. As one can see, the chirality parameter is extremely small, even in the resonant region. The magnitude of κ can readily be increased by about 5 orders of magnitude by using a bulk solid of the chiral polymer instead of a dilute solution. In the solid the concentration of the chiral molecules would be on the order of 1 – 10 M. However, even in the case of a medium composed entirely of the chiral Ni complex polymer the chirality parameter would still be on the order of 10

^{-3}which is significantly lower than what is needed to achieve a negative refractive index in the off-resonant frequency range. This prompts for a modeling and design of molecular structures with significantly higher chirality.

## 4. Estimation of the chirality parameter by means of *ab initio* calculations

*β*computed

*ab initio*according to

*G*′

_{ii}are the diagonal elements of the gyration tensor (mixed electric-magnetic dipole polarizability) which is formally defined as

*and*μ ^

*m̂*are electric dipole and magnetic dipole moment operators, respectively. It is worthwhile to note here that the trace of gyration tensor is gauge origin independent which makes it the only observable quantity for a chiral isotropic medium (solution of chiral molecules). In order to obtain origin-independent optical rotation tensor elements it is necessary to include an electric quadrupole term in addition to

*G*′ [18,19

19. M. Krykunov and J. Autschbach, “Calculation of origin independent optical rotation tensor components for chiral oriented systems in approximate time-dependent density functional theory,” J. Chem. Phys. **125**, 034102-10 (2006). [CrossRef]

20. J. Autschbach and T. Ziegler, “Calculating electric and magnetic properties from time dependent density functional perturbation theory,” J. Chem. Phys. **116**, 891–896 (2002). [CrossRef]

21. J. Autschbach, T. Ziegler, S. Patchkovskii, S. J. A.van Gisbergen, and E. J. Baerends, “Chiroptical properties from time-dependent density functional theory. II. Optical rotations of small to medium sized organic molecules,” J. Chem. Phys. **117**, 581–592 (2002). [CrossRef]

*β*.

22. E. J. Baerends et al., Amsterdam density functional, Theoretical Chemistry, Vrije Universiteit, Amsterdam (URL http://www.scm.com)

22. E. J. Baerends et al., Amsterdam density functional, Theoretical Chemistry, Vrije Universiteit, Amsterdam (URL http://www.scm.com)

*γ*[24

24. J. Autschbach, L. Jensen, G. C. Schatz, Y. C. E. Tse, and M. Krykunov, “Time-dependent density functional calculations of optical rotatory dispersion including resonance wavelengths as a potentially useful tool for determining absolute configurations of chiral molecules,” J. Phys. Chem. A **110**, 2461–2473 (2006). [CrossRef] [PubMed]

16. M. Krykunov, M. D. Kundrat, and J. Autschbach, “Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory,” J. Chem. Phys. **125**, 194110-13 (2006). [CrossRef]

*ab initio*calculations are depicted in Fig. 4. The concentration of chiral molecules was set to 4.4 ×10

^{-6}M in order to compare directly with the experiment. The comparison is obviously somewhat limited in scope because the experimental data have been obtained for a polymer whereas the computation was performed on the monomer unit. Nonetheless, the theoretical data should give a rough estimate of the chirality parameter to be expected as long as the monomer units in the polymer do not couple too strongly. This does not seem to be the case since the theoretically computed κ is of the same order of magnitude as the one computed from the experimental CD spectrum (see Figs. 3 and 4). As already mentioned, the low absolute value of the chirality parameter is in part due to the low concentration of chiral species in solution.

## 5. Tuning the effective refractive index of the chiral medium by applying plasmonic inclusions

25. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B **39**, 9852–9858 (1989). [CrossRef]

*x*= 2

*πa*/λ is the sphere size parameter,

*a*is the radius of the sphere, λ is the wave length in the host medium,

*f*is the volume fraction of nanoparticles, and

*T*

^{E}_{1}is the electric-dipole component of the scattering

*T*-matrix in Mie approximation.

*ε*, and the chirality parameter were obtained by taking into account the local field effects (in the framework of Clausius-Mossotti model corresponding to quasistatic dipolar limit) which are of importance for such a high number density of constituent particles. The results of our modeling are presented in Fig. 5. As one can see, inclusion of gold nanoparticles results in the expected lowering of the real part of the effective dielectric permittivity of the composite material. This, in turn, leads to the negative real part of the effective refractive index in a very narrow band around 676.5 nm. The imaginary part of the effective permittivity is nevertheless large. As the chirality parameter changes sign and reaches a maximum negative value at 550 nm, we could in principle get negative refractivity for the opposite sense of polarization [a positive sign in Eq. (2

_{host}2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

^{-4}). This is not surprising because the molecule has no unpaired electrons.

## 6. Discussion

*a*/ζ (see Figs. 1 and 2). The value of this useful estimative quantity is 0.809 for Co complex and 0.821 for Ni complex. Such a simple estimate does indeed carry some information about the strength of optical rotation and may cautiously be used if rigorous quantum chemical calculations of polarizabilities are not feasible for some reasons. If one is interested in absolute value of the optical rotatory strength, the proportionality factor for a family of compounds must be computed along with the radius-to-pitch ratio [14

14. S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, “Analytical antenna model for chiral scatterers: Comparison with numerical and experimental data,” IEEE Trans. Antennas Propag. **44**, 1006 (1996). [CrossRef]

*ab initio*modeling, there are a number of potential shortcomings arising from the approximations that need to be applied in order to keep the computational effort manageable. For instance, the TDDFT modeling of excitation spectra, CD spectra, and optical rotation is influenced by truncation of the basis set, by approximations in the density functionals, by neglecting or an approximate treatment of the molecule’s chemical environment, or the neglect of vibrational corrections [27]. In principle, these obstacles can be overcome at the expense of computational resources (CPU time and memory). From the comparison of the chirality parameter calculated for a monomer solution with the experimentally derived data we see that the agreement is reasonable for wavelengths below about 600 nm. It appears that the computed excitation energies are somewhat red shifted which, by virtue of the KK relations, rationalizes to some degree the overestimation of κ in the computations. In the long-wavelength regime perfect agreement between the computed chirality parameter and the data derived from experiment should not be expected. The reason lies in the fact that the experimental chirality parameter has been obtained from a numerical KK transformation of the CD spectrum, with the concomitant truncation errors from the finite range of integration. Two of us have shown previously [16

16. M. Krykunov, M. D. Kundrat, and J. Autschbach, “Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory,” J. Chem. Phys. **125**, 194110-13 (2006). [CrossRef]

## 7. Conclusions

*ab initio*calculations of corresponding microscopic polarizabilities is presented. Realization of such a negative refractive index material is shown to be feasible provided molecular entity with high optical activity is synthesized. Plasmonic inclusions are demonstrated to be able to locally lower the real part of dielectric permittivity of a composite material thereby ensuring conditions of chiral nihility within a very narrow frequency range. The need for a three orders of magnitude increase in the value of the macroscopic chirality parameter to obtain a material in which a negative refractive index might be obtainable may be alleviated by the use of inclusions providing sharp resonances. Concomitant use of supramolecular self-assemblies and metal nanoparticles may be a viable pathway to obtaining isotropic negative refractive index materials.

## Acknowledgments

## References and links

1. | A. N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Lett. |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. |

4. | J. B. Pendry, “A chiral route to negative refraction,” Science |

5. | Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express |

6. | J. Q. Shen and S. He, “Backward waves and negative refractive indices in gyrotropic chiral media,” J. Phys. A: Math. Gen. |

7. | J. Q. Shen, M. Norgren, and S. He, “Negative refraction and quantum vacuum effects in gyroelectric chiral mnedium and anisotropic magnetoelectric material,” Ann. Phys. (Leipzig) |

8. | V. M. Agranovich, Yu. N. Gartstein, and A. A. Zakhidov, “Negative refraction in gyrotropic media,” Phys. Rev. B |

9. | Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B |

10. | V. Yannopapas, “Negative index of refraction in artificial chiral materials,” J. Phys.: Condens. Matter |

11. | E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. |

12. | K. Ruud and T. Helgaker, “Optical rotation studied by density-functional and coupled-cluster methods,” Chem. Phys. Lett. |

13. | J. J. Maki and A. Persoons, “One-electron second-order optical activity of a helix,” J. Chem. Phys. |

14. | S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, “Analytical antenna model for chiral scatterers: Comparison with numerical and experimental data,” IEEE Trans. Antennas Propag. |

15. | V. Percec, M. Glodde, T. K. Bera, Y. Miura, I. Shiyanovskaya, K. D. Singer, V. S. K. Balagurusamy, P. A. Heiney, I. Schnell, A. Rapp, H. W. Spiess, S. D. Hudson, and H. Duan, “Self-organization of supramolecular helical dendrimers into complex electronic materials,” Nature |

16. | M. Krykunov, M. D. Kundrat, and J. Autschbach, “Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory,” J. Chem. Phys. |

17. | C. R. Jeggo, “Nonlinear optics and optical activity,” J. Phys. C: Solid State Physics |

18. | A. D. Buckingham and M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A |

19. | M. Krykunov and J. Autschbach, “Calculation of origin independent optical rotation tensor components for chiral oriented systems in approximate time-dependent density functional theory,” J. Chem. Phys. |

20. | J. Autschbach and T. Ziegler, “Calculating electric and magnetic properties from time dependent density functional perturbation theory,” J. Chem. Phys. |

21. | J. Autschbach, T. Ziegler, S. Patchkovskii, S. J. A.van Gisbergen, and E. J. Baerends, “Chiroptical properties from time-dependent density functional theory. II. Optical rotations of small to medium sized organic molecules,” J. Chem. Phys. |

22. | E. J. Baerends et al., Amsterdam density functional, Theoretical Chemistry, Vrije Universiteit, Amsterdam (URL http://www.scm.com) |

23. | M. Krykunov and J. Autschbach, “Calculation of optical rotation with time-periodic magnetic field-dependent basis functions in approximate time-dependent density functional theory,” J. Chem. Phys. |

24. | J. Autschbach, L. Jensen, G. C. Schatz, Y. C. E. Tse, and M. Krykunov, “Time-dependent density functional calculations of optical rotatory dispersion including resonance wavelengths as a potentially useful tool for determining absolute configurations of chiral molecules,” J. Phys. Chem. A |

25. | W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B |

26. | S. Tretyakov, A. Sihvola, and L. Jylhä, “Backward-wave regime and negative refraction in chiral composites,” arXiv:cond-mat/0509287, 1 (2005). |

27. | J. Autschbach, “Density Functional Theory applied to calculating optical and spectroscopic properties of metal complexes: NMR and Optical Activity,” Coord. Chem. Rev., in press. |

**OCIS Codes**

(160.5470) Materials : Polymers

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 5, 2007

Revised Manuscript: April 17, 2007

Manuscript Accepted: April 17, 2007

Published: April 25, 2007

**Citation**

Alexander Baev, Marek Samoc, Paras N. Prasad, Mykhaylo Krykunov, and Jochen Autschbach, "A quantum chemical approach to the design of chiral negative index materials," Opt. Express **15**, 5730-5741 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5730

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### References

- A. N. Grigorenko, "Negative refractive index in artificial metamaterials," Opt. Lett. 31, 2483-2485 (2006). [CrossRef] [PubMed]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, "Waves and energy in chiral nihility," J. Electromagn. Waves Appl. 17, 695-706 (2003). [CrossRef]
- J. B. Pendry, "A chiral route to negative refraction," Science 306, 1353-1355 (2004). [CrossRef] [PubMed]
- Y. Jin and S. He, "Focusing by a slab of chiral medium," Opt. Express 13, 4974-4979 (2005). [CrossRef] [PubMed]
- J. Q. Shen and S. He, "Backward waves and negative refractive indices in gyrotropic chiral media," J. Phys. A: Math. Gen. 39, 457-466 (2006). [CrossRef]
- J. Q. Shen, M. Norgren, and S. He, "Negative refraction and quantum vacuum effects in gyroelectric chiral mnedium and anisotropic magnetoelectric material," Ann. Phys. (Leipzig) 15, 894-910 (2006). [CrossRef]
- V. M. Agranovich, Yu. N. Gartstein, and A. A. Zakhidov, "Negative refraction in gyrotropic media," Phys. Rev. B 73, 045114 (2006). [CrossRef]
- Q. Cheng and T. J. Cui, "Negative refractions in uniaxially anisotropic chiral media," Phys. Rev. B 73, 113104 (2006). [CrossRef]
- V. Yannopapas, "Negative index of refraction in artificial chiral materials," J. Phys.: Condens. Matter 18, 6883-6890 (2006). [CrossRef]
- E. U. Condon, "Theories of optical rotatory power," Rev. Mod. Phys. 9, 432-457 (1937). [CrossRef]
- K. Ruud and T. Helgaker, "Optical rotation studied by density-functional and coupled-cluster methods," Chem. Phys. Lett. 352, 533-539 (2002). [CrossRef]
- J. J. Maki and A. Persoons, "One-electron second-order optical activity of a helix," J. Chem. Phys. 104, 9340 (1996). [CrossRef]
- S. A. Tretyakov, F. Mariotte, C. R. Simovski, T. G. Kharina, and J.-P. Heliot, "Analytical antenna model for chiral scatterers: Comparison with numerical and experimental data," IEEE Trans. Antennas Propag. 44, 1006 (1996). [CrossRef]
- V. Percec, M. Glodde, T. K. Bera, Y. Miura, I. Shiyanovskaya, K. D. Singer, V. S. K. Balagurusamy, P. A. Heiney, I. Schnell, A. Rapp, H. W. Spiess, S. D. Hudson, and H. Duan, "Self-organization of supramolecular helical dendrimers into complex electronic materials," Nature 419, 384-387 (2002). [CrossRef] [PubMed]
- M. Krykunov, M. D. Kundrat, and J. Autschbach, "Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory," J. Chem. Phys. 125,194110-13 (2006). [CrossRef]
- C. R. Jeggo, "Nonlinear optics and optical activity," J. Phys. C: Solid State Physics 5, 330-337 (1972). [CrossRef]
- A. D. Buckingham and M. B. Dunn, "Optical activity of oriented molecules," J. Chem. Soc. A1988 (1971).
- M. Krykunov and J. Autschbach, "Calculation of origin independent optical rotation tensor components for chiral oriented systems in approximate time-dependent density functional theory," J. Chem. Phys. 125, 034102-10 (2006). [CrossRef]
- J. Autschbach and T. Ziegler, "Calculating electric and magnetic properties from time dependent density functional perturbation theory," J. Chem. Phys. 116, 891-896 (2002). [CrossRef]
- J. Autschbach, T. Ziegler, S. Patchkovskii, S. J. A. van Gisbergen, and E. J. Baerends, "Chiroptical properties from time-dependent density functional theory. II. Optical rotations of small to medium sized organic molecules," J. Chem. Phys. 117, 581-592 (2002). [CrossRef]
- E. J. Baerends et al., Amsterdam density functional, Theoretical Chemistry, Vrije Universiteit, Amsterdam (URL http://www.scm.com).
- M. Krykunov and J. Autschbach, "Calculation of optical rotation with time-periodic magnetic field-dependent basis functions in approximate time-dependent density functional theory," J. Chem. Phys. 123, 114103-10 (2005). [CrossRef]
- J. Autschbach, L. Jensen, G. C. Schatz, Y. C. E. Tse, and M. Krykunov, "Time-dependent density functional calculations of optical rotatory dispersion including resonance wavelengths as a potentially useful tool for determining absolute configurations of chiral molecules," J. Phys. Chem. A 110, 2461-2473 (2006). [CrossRef] [PubMed]
- W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989). [CrossRef]
- S. Tretyakov, A. Sihvola, and L. Jylhä, "Backward-wave regime and negative refraction in chiral composites," arXiv:cond-mat/0509287, 1 (2005).
- J. Autschbach, "Density Functional Theory applied to calculating optical and spectroscopic properties of metal complexes: NMR and Optical Activity," Coord. Chem. Rev., in press.

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