## Microscopic cascading of second-order molecular nonlinearity: new design principles for enhancing third-order nonlinearity

Optics Express, Vol. 18, Issue 8, pp. 8713-8721 (2010)

http://dx.doi.org/10.1364/OE.18.008713

Acrobat PDF (1334 KB)

### Abstract

Herein, we develop a phenomenological model for microscopic cascading and substantiate it with *ab initio* calculations. It is shown that the concept of local microscopic cascading of a second-order nonlinearity can lead to a third-order nonlinearity, without introducing any new loss mechanisms that could limit the usefulness of our approach. This approach provides a new molecular design protocol, in which the current great successes achieved in producing molecules with extremely large second-order nonlinearity can be used in a supra molecular organization in a preferred orientation to generate very large third-order response magnitudes. The results of density functional calculations for a well-known second-order molecule, (*para*)nitroaniline, show that a head-to-tail dimer configuration exhibits enhanced third-order nonlinearity, in agreement with the phenomenological model which suggests that such an arrangement will produce cascading due to local field effects.

© 2010 OSA

## 1. Introduction

1. K. Rustagi and J. Ducuing, “Third-order optical polarizability of conjugated organic molecules,” Opt. Commun. **10**(3), 258–261 (1974). [CrossRef]

2. A. Adronov, J. M. J. Fréchet, G. S. He, K.-S. Kim, S.-J. Chung, J. Swiatkiewicz, and P. N. Prasad, “Novel Two-Photon Absorbing Dendritic Structures,” Chem. Mater. **12**(10), 2838–2841 (2000). [CrossRef]

3. C. B. Gorman and S. R. Marder, “An investigation of the interrelationships between linear and nonlinear polarizabilities and bond-length alternation in conjugated organic molecules,” Proc. Natl. Acad. Sci. U.S.A. **90**(23), 11297–11301 (1993). [CrossRef] [PubMed]

4. I. Albert, T. J. Marks, and M. A. Ratner, “Rational design of molecules with large hyperpolarizabilities. electric field, solvent polarity, and bond length alternation effects on merocyanine dye linear and nonlinear optical properties,” J. Phys. Chem. **100**(23), 9714–9725 (1996). [CrossRef]

4. I. Albert, T. J. Marks, and M. A. Ratner, “Rational design of molecules with large hyperpolarizabilities. electric field, solvent polarity, and bond length alternation effects on merocyanine dye linear and nonlinear optical properties,” J. Phys. Chem. **100**(23), 9714–9725 (1996). [CrossRef]

*α*is microscopic molecular polarizability,

_{ij}*β*is molecular first hyperpolarizability, and

_{ijk}*γ*is molecular second hyperpolarizability. The local field factors are introduced when transitioning from microscopic properties to macroscopic linear,

_{ijkl}*χ*, second- order,

^{(1)}*χ*, and third-order,

^{(2)}*χ*, response of the medium. In this work we refer to

^{(3)}*β*or

*χ*as second-order nonlinearity, and to γ or

^{(2)}*χ*as the third-order nonlinearity. Hence, all known approaches to enhance molecular nonlinearities rely on structural changes of molecular moieties, resulting in corresponding changes of (hyper)polarizabilites in each given order. New, radically different approaches are needed to create a major advancement in this field.

^{(3)}*ab initio*computations, using time-dependent density functional theory (TDDFT), along with a phenomenological model. In particular, we demonstrate the numerical observation of strongly enhanced second hyperpolarizabilies,

*γ*(the microscopic counterpart of

*χ*) generated by means of cascading processes quadratic in the first hyperpolarizability,

^{(3)}*β*(the microscopic counterpart of

*χ*).

^{(2)}5. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. **18**(1), 13–15 (1993). [CrossRef] [PubMed]

*χ*nonlinear response from a

^{(3)}*χ*material by means of propagation effects, and therefore requires stringent phase-matching conditions. Microscopic cascading makes use of local field effects and dipole-dipole interactions to allow neighboring molecules to mimic a gamma-type response [6

^{(2)}6. J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Pair correlations, cascading, and local-field effects in nonlinear optical susceptibilities,” Phys. Rev. A **46**(7), 4172–4184 (1992). [CrossRef] [PubMed]

7. K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. **103**(11), 113902 (2009). [CrossRef] [PubMed]

8. B. Bedeaux and N. Bloembergen, “On the relation between microscopic and microscopic nonlinear susceptibilities,” Physica **69**(1), 57–66 (1973). [CrossRef]

*L =*(

*ε*+ 2)/3. Here, the first term represents the usual direct contribution to

*χ*and the second term yields the cascaded contribution. Note that the cascading term scales quadratically with the molecular number density

^{(3)}*N*. For number densities approaching those of condensed matter, the cascading term can yield the dominant contribution to the nonlinear susceptibility, as was recently demonstrated [7

7. K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. **103**(11), 113902 (2009). [CrossRef] [PubMed]

*χ*.

^{(5)}9. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A **76**(6), 063806 (2007). [CrossRef]

13. C. Kolleck, “Cascaded second-order contribution to the third-order nonlinear susceptibility,” Phys. Rev. A **69**(5), 053812 (2004). [CrossRef]

## 2. Theory

*p*is the modulus of

_{2}*E*is the modulus of

_{0}*β*=

_{1}*β*=

_{2}*β*. If they are oriented parallel to each other, we find thatwhereas if they are antiparallel to each other we obtainAs a different special case, let us now assume that

*p*vanishes. More general situations can be modeled by means of direct numerical evaluation of Eqs. (3) through (5).

_{2}## 3. Computations

*para*)nitroaniline (PNA) “push-pull” molecule as it is known to have relatively large

*β*and has been a convenient benchmarking system for many studies [14

14. K. V. Mikkelsen, Y. Luo, H. Ågren, and P. Jørgensen, “Solvent induced polarizabilities and hyperpolarizabilities of para-nitroaniline studied by reaction field linear response theory,” J. Chem. Phys. **100**, 8240 (1994). [CrossRef]

15. A. Ye, S. Patchkovskii, and J. Autschbach, “Static and dynamic second hyperpolarizability calculated by time-dependent density functional cubic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **127**(7), 074104 (2007). [CrossRef] [PubMed]

16. A. Ye and J. Autschbach, “Study of static and dynamic first hyperpolarizabilities using time-dependent density functional quadratic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **125**(23), 234101 (2006). [CrossRef] [PubMed]

17. E. J. Baerends, J. Autschbach, A. Berces, F. M. Bickelhaupt, C. Bo, P. M. Boerrigter, L. Cavallo, D. P. Chong, L. Deng, R. M. Dickson, D. E. Ellis, M. van Faassen, L. Fan, T. H. Fischer, C. Fonseca Guerra, S. J. A. van Gisbergen, J. A. Groeneveld, O. V. Gritsenko, M. Gruning, F. E. Harris, P. van den Hoek, C. R. Jacob, H. Jacobsen, L. Jensen, G. van Kessel, F. Kootstra, E. van Lenthe, D. A. McCormack, A. Michalak, J. Neugebauer, V. P. Osinga, S. Patchkovskii, P. H. T. Philipsen, D. Post, C. C. Pye, W. Ravenek, P. Ros, P. R. T. Schipper, G. Schreckenbach, J. G. Snijders, M. Solà, M. Swart, D. Swerhone, G. te Velde, P. Vernooijs, L. Versluis, L. Visscher, O. Visser, F. Wang, T. A. Wesolowski, E. van Wezenbeek, G. Wiesenekker, S. K. Wolff, T. K. Woo, A. L. Yakovlev, and T. Ziegler, Amsterdam Density Functional, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands.

18. J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-correlation hole of a many-electron system,” Phys. Rev. B **54**(23), 16533–16539 (1996). [CrossRef]

19. C. Adamo and V. Barone, “Toward reliable density functional methods without adjustable parameters: the PBE0 model,” J. Chem. Phys. **110**(13), 6158 (1999). [CrossRef]

20. D. Bryce and J. Autschbach, “Relativistic hybrid density functional calculations of indirect nuclear spin-spin coupling tensors. comparison with experiment for diatomic alkali metal halides,” Can. J. Chem. **87**(7), 927–941 (2009). [CrossRef]

15. A. Ye, S. Patchkovskii, and J. Autschbach, “Static and dynamic second hyperpolarizability calculated by time-dependent density functional cubic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **127**(7), 074104 (2007). [CrossRef] [PubMed]

16. A. Ye and J. Autschbach, “Study of static and dynamic first hyperpolarizabilities using time-dependent density functional quadratic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **125**(23), 234101 (2006). [CrossRef] [PubMed]

15. A. Ye, S. Patchkovskii, and J. Autschbach, “Static and dynamic second hyperpolarizability calculated by time-dependent density functional cubic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **127**(7), 074104 (2007). [CrossRef] [PubMed]

16. A. Ye and J. Autschbach, “Study of static and dynamic first hyperpolarizabilities using time-dependent density functional quadratic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. **125**(23), 234101 (2006). [CrossRef] [PubMed]

21. B. Jansik, P. Salek, D. Jonsson, O. Vahtras, and H. Ågren, “Cubic response function in time-dependent density functional theory,” J. Chem. Phys. **122**(5), 054107 (2005). [CrossRef]

**127**(7), 074104 (2007). [CrossRef] [PubMed]

*γ(ω;ω,−ω,ω)*at telecommunication wavelengths. Our interest in degenerate second hyperpolarizability is motivated by a particular application – all-optical switching – where the real part of the non-resonant degenerate third-order susceptibility determines the Intensity Dependent Refractive Index (IDRI) of a medium.

*D*, between PNA molecules was initially set to 3 Angstroms. We did not further optimize the geometry of the dimer, keeping both PNA molecules planar. For both non-hybrid and hybrid TDDFT)computations on PNA dimer in an anti-parallel (head-to-tail, HT) configuration (Fig. 2), we found a substantial enhancement of the degenerate

*γ(ω;ω,−ω,ω)*at the telecommunication wavelength of 1300 nm - to about three times the monomer value for the hybrid DFT computations, indicating a significant contribution from microscopic cascading (see Table 1 ). In a parallel (head-to-head, HH) arrangement, a significant suppression of

*γ*was found, confirming the expectations from our model and validating the sign of the cascading term in Eqs. (11) and 12. Moreover, the simple phenomenological model for cascading yields a similar magnitude for the enhancement when we use the

*β*tensor components, computed for an isolated PNA molecule to estimate the cascading term in Eq. (12) (the results are collected in Table 1). This means that in order to investigate cascading mechanisms, relatively cheaper computations of the

*β*tensors of monomers, instead of computing

*γ(ω;ω,−ω,ω)*for a compound system, might already point one’s efforts in the right direction. For example, recently synthesized large

*β*twisted chromophores [22

22. I. D. L. Albert, T. J. Marks, and M. A. Ratner, “Remarkable NLO response and infrared absorption in simple twisted molecular π-chromophores,” J. Am. Chem. Soc. **120**, 11174 (1998). [CrossRef]

*χ*.

^{(3)}23. J. Autschbach, “Charge-transfer excitations and time-dependent density functional theory: problems and some proposed solutions,” Chem. Phys. Chem. **10**(11), 1757–1760 (2009). [CrossRef] [PubMed]

25. A. Dreuw and M. Head-Gordon, “Single-reference ab initio methods for the calculation of excited states of large molecules,” Chem. Rev. **105**(11), 4009–4037 (2005). [CrossRef] [PubMed]

*γ(ω;ω,−ω,ω)*of the PNA head-to-tail dimer in terms of localized molecular orbitals was performed [15

**127**(7), 074104 (2007). [CrossRef] [PubMed]

*π*orbitals can be associated with very nearly the complete overall enhancement of

*γ*. This pair of orbitals is shown in Fig. 3 . Their contribution to the averaged

*γ*mostly stems from the

**zzzz**component of the tensor (i.e. along the normal vector

*D*, between the chromophores predicted by the model, we calculated

*γ*of the HT dimer and of the isolated PNA molecule as a function of

*D*. The cascading term,

*γ*

_{cascade}, was computed by subtracting two times the orientationally averaged

*γ*of the isolated PNA molecule from

*γ*calculated for the dimer at corresponding values of

*D*. The results are presented in Fig. 4 . A fit over the range of distances studied here shows that

*γ*

_{cascade}is roughly inversely proportional to the power 2.5 of the distance between the molecules. This slight disagreement with the model, which affords an exact inverse cube behavior (Eq. (12), can be attributed to a number of reasons. First, TDDFT with exchange-correlation potentials as used in this work is not expected to describe correctly the behavior at large distances due to a rapid decay of the local exchange contributions with

*D*. On the other hand, the model does not take into account effects that arise when the orbitals of the monomers begin to overlap in the dimer at shorter distances; a regime where TDDFT on the other hand is expected to generate accurate results. Taking these factors into consideration, the rapid decay of the cascading effect calculated from

*γ(ω;ω,−ω,ω)*for the dimer at larger separations is consistent with the predictions from the phenomenological model.

*γ*of the dimer grows faster than

*γ*of isolated PNA molecule as the wavelength decreases. Below 1064 nm the cascading contribution becomes larger than

*γ*of the isolated PNA molecule, and below about 900 nm an anomalous dispersion indicates that the influence of electronic excitations becomes dominating. This trend suggests that the cascaded

*γ*of the dimer is spectrally red shifted. The reason for the red shift is most likely attributable to the enhanced π-delocalization, which further supports the results of NMLO analysis.

*para*)nitroaniline molecules linked to a tetrahedral Au

_{20}cluster delivered up to two orders of magnitude enhancement of the second hyperpolarizability. Although this enhancement is possibly overestimated in these computations, these results suggest that a combination of microscopic cascading with electromagnetic and chemical enhancement from a metal cluster has the potential to push

*χ*to magnitudes far exceeding current limits. These computations are on-going; the results will be published elsewhere [27].

^{(3)}## 4. Conclusion

*b initio*computations of first and second hyperpolarizabilies of para-nitroaniline and dimers thereof substantiated the model and suggest a design protocol for obtaining materials with particularly large values of the real part of third-order nonlinearity. The most obvious way to increase the real part of the direct (as opposed to cascaded) contribution to the second polarizability,

*γ*, is to use resonant enhancement. Unfortunately, the imaginary part will inevitably grow as one approaches one- or two-photon resonance of a molecule. Our design protocol allows for using molecules with a large real part of the first hyperpolarizability,

*β*, to generate a large real part of second hyperpolarizability,

*γ*, without introducing extra loss, because molecular resonances, in particular two-photon resonance, are not involved.

## Acknowledgments

## References and links

1. | K. Rustagi and J. Ducuing, “Third-order optical polarizability of conjugated organic molecules,” Opt. Commun. |

2. | A. Adronov, J. M. J. Fréchet, G. S. He, K.-S. Kim, S.-J. Chung, J. Swiatkiewicz, and P. N. Prasad, “Novel Two-Photon Absorbing Dendritic Structures,” Chem. Mater. |

3. | C. B. Gorman and S. R. Marder, “An investigation of the interrelationships between linear and nonlinear polarizabilities and bond-length alternation in conjugated organic molecules,” Proc. Natl. Acad. Sci. U.S.A. |

4. | I. Albert, T. J. Marks, and M. A. Ratner, “Rational design of molecules with large hyperpolarizabilities. electric field, solvent polarity, and bond length alternation effects on merocyanine dye linear and nonlinear optical properties,” J. Phys. Chem. |

5. | G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. |

6. | J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Pair correlations, cascading, and local-field effects in nonlinear optical susceptibilities,” Phys. Rev. A |

7. | K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. |

8. | B. Bedeaux and N. Bloembergen, “On the relation between microscopic and microscopic nonlinear susceptibilities,” Physica |

9. | K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A |

10. | G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B |

11. | Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. |

12. | Ch. Bosshard, “Cascading of second-order nonlinearities in polar materials,” Adv. Mater. |

13. | C. Kolleck, “Cascaded second-order contribution to the third-order nonlinear susceptibility,” Phys. Rev. A |

14. | K. V. Mikkelsen, Y. Luo, H. Ågren, and P. Jørgensen, “Solvent induced polarizabilities and hyperpolarizabilities of para-nitroaniline studied by reaction field linear response theory,” J. Chem. Phys. |

15. | A. Ye, S. Patchkovskii, and J. Autschbach, “Static and dynamic second hyperpolarizability calculated by time-dependent density functional cubic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. |

16. | A. Ye and J. Autschbach, “Study of static and dynamic first hyperpolarizabilities using time-dependent density functional quadratic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. |

17. | E. J. Baerends, J. Autschbach, A. Berces, F. M. Bickelhaupt, C. Bo, P. M. Boerrigter, L. Cavallo, D. P. Chong, L. Deng, R. M. Dickson, D. E. Ellis, M. van Faassen, L. Fan, T. H. Fischer, C. Fonseca Guerra, S. J. A. van Gisbergen, J. A. Groeneveld, O. V. Gritsenko, M. Gruning, F. E. Harris, P. van den Hoek, C. R. Jacob, H. Jacobsen, L. Jensen, G. van Kessel, F. Kootstra, E. van Lenthe, D. A. McCormack, A. Michalak, J. Neugebauer, V. P. Osinga, S. Patchkovskii, P. H. T. Philipsen, D. Post, C. C. Pye, W. Ravenek, P. Ros, P. R. T. Schipper, G. Schreckenbach, J. G. Snijders, M. Solà, M. Swart, D. Swerhone, G. te Velde, P. Vernooijs, L. Versluis, L. Visscher, O. Visser, F. Wang, T. A. Wesolowski, E. van Wezenbeek, G. Wiesenekker, S. K. Wolff, T. K. Woo, A. L. Yakovlev, and T. Ziegler, Amsterdam Density Functional, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands. |

18. | J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-correlation hole of a many-electron system,” Phys. Rev. B |

19. | C. Adamo and V. Barone, “Toward reliable density functional methods without adjustable parameters: the PBE0 model,” J. Chem. Phys. |

20. | D. Bryce and J. Autschbach, “Relativistic hybrid density functional calculations of indirect nuclear spin-spin coupling tensors. comparison with experiment for diatomic alkali metal halides,” Can. J. Chem. |

21. | B. Jansik, P. Salek, D. Jonsson, O. Vahtras, and H. Ågren, “Cubic response function in time-dependent density functional theory,” J. Chem. Phys. |

22. | I. D. L. Albert, T. J. Marks, and M. A. Ratner, “Remarkable NLO response and infrared absorption in simple twisted molecular π-chromophores,” J. Am. Chem. Soc. |

23. | J. Autschbach, “Charge-transfer excitations and time-dependent density functional theory: problems and some proposed solutions,” Chem. Phys. Chem. |

24. | S. Grimme and M. Parac, “Substantial errors from time-dependent density functional theory for the calculation of excited states of large pi systems,” ChemPhysChem |

25. | A. Dreuw and M. Head-Gordon, “Single-reference ab initio methods for the calculation of excited states of large molecules,” Chem. Rev. |

26. | F. Weinhold, ‘Natural bond orbital methods’. In Encyclopedia of computational chemistry, von Rague Schleyer, P., Ed. John Wiley & Sons: Chichester, 1998; pp 1792–1811. |

27. | H. Ågren, J. Autschbach, A. Baev, M. Swihart, and P. N. Prasad, “Rational Design of Organo-Metallic Complexes for Enhanced Third-Order Nonlinearity,”manuscript in preparation. |

**OCIS Codes**

(000.1570) General : Chemistry

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4710) Nonlinear optics : Optical nonlinearities in organic materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 25, 2010

Manuscript Accepted: March 28, 2010

Published: April 9, 2010

**Citation**

Alexander Baev, Jochen Autschbach, Robert W. Boyd, and Paras N. Prasad, "Microscopic cascading of second-order molecular nonlinearity: new design principles for enhancing third-order nonlinearity," Opt. Express **18**, 8713-8721 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8713

Sort: Year | Journal | Reset

### References

- K. Rustagi and J. Ducuing, “Third-order optical polarizability of conjugated organic molecules,” Opt. Commun. 10(3), 258–261 (1974). [CrossRef]
- A. Adronov, J. M. J. Fréchet, G. S. He, K.-S. Kim, S.-J. Chung, J. Swiatkiewicz, and P. N. Prasad, “Novel Two-Photon Absorbing Dendritic Structures,” Chem. Mater. 12(10), 2838–2841 (2000). [CrossRef]
- C. B. Gorman and S. R. Marder, “An investigation of the interrelationships between linear and nonlinear polarizabilities and bond-length alternation in conjugated organic molecules,” Proc. Natl. Acad. Sci. U.S.A. 90(23), 11297–11301 (1993). [CrossRef] [PubMed]
- I. Albert, T. J. Marks, and M. A. Ratner, “Rational design of molecules with large hyperpolarizabilities. electric field, solvent polarity, and bond length alternation effects on merocyanine dye linear and nonlinear optical properties,” J. Phys. Chem. 100(23), 9714–9725 (1996). [CrossRef]
- G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18(1), 13–15 (1993). [CrossRef] [PubMed]
- J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Pair correlations, cascading, and local-field effects in nonlinear optical susceptibilities,” Phys. Rev. A 46(7), 4172–4184 (1992). [CrossRef] [PubMed]
- K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009). [CrossRef] [PubMed]
- B. Bedeaux and N. Bloembergen, “On the relation between microscopic and microscopic nonlinear susceptibilities,” Physica 69(1), 57–66 (1973). [CrossRef]
- K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007). [CrossRef]
- G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24(10), 5522–5532 (1981). [CrossRef]
- Ch. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74(14), 2816–2819 (1995). [CrossRef] [PubMed]
- Ch. Bosshard, “Cascading of second-order nonlinearities in polar materials,” Adv. Mater. 8(5), 385–397 (1996). [CrossRef]
- C. Kolleck, “Cascaded second-order contribution to the third-order nonlinear susceptibility,” Phys. Rev. A 69(5), 053812 (2004). [CrossRef]
- K. V. Mikkelsen, Y. Luo, H. Ågren, and P. Jørgensen, “Solvent induced polarizabilities and hyperpolarizabilities of para-nitroaniline studied by reaction field linear response theory,” J. Chem. Phys. 100, 8240 (1994). [CrossRef]
- A. Ye, S. Patchkovskii, and J. Autschbach, “Static and dynamic second hyperpolarizability calculated by time-dependent density functional cubic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. 127(7), 074104 (2007). [CrossRef] [PubMed]
- A. Ye and J. Autschbach, “Study of static and dynamic first hyperpolarizabilities using time-dependent density functional quadratic response theory with local contribution and natural bond orbital analysis,” J. Chem. Phys. 125(23), 234101 (2006). [CrossRef] [PubMed]
- E. J. Baerends, J. Autschbach, A. Berces, F. M. Bickelhaupt, C. Bo, P. M. Boerrigter, L. Cavallo, D. P. Chong, L. Deng, R. M. Dickson, D. E. Ellis, M. van Faassen, L. Fan, T. H. Fischer, C. Fonseca Guerra, S. J. A. van Gisbergen, J. A. Groeneveld, O. V. Gritsenko, M. Gruning, F. E. Harris, P. van den Hoek, C. R. Jacob, H. Jacobsen, L. Jensen, G. van Kessel, F. Kootstra, E. van Lenthe, D. A. McCormack, A. Michalak, J. Neugebauer, V. P. Osinga, S. Patchkovskii, P. H. T. Philipsen, D. Post, C. C. Pye, W. Ravenek, P. Ros, P. R. T. Schipper, G. Schreckenbach, J. G. Snijders, M. Solà, M. Swart, D. Swerhone, G. te Velde, P. Vernooijs, L. Versluis, L. Visscher, O. Visser, F. Wang, T. A. Wesolowski, E. van Wezenbeek, G. Wiesenekker, S. K. Wolff, T. K. Woo, A. L. Yakovlev, and T. Ziegler, Amsterdam Density Functional, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands.
- J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-correlation hole of a many-electron system,” Phys. Rev. B 54(23), 16533–16539 (1996). [CrossRef]
- C. Adamo and V. Barone, “Toward reliable density functional methods without adjustable parameters: the PBE0 model,” J. Chem. Phys. 110(13), 6158 (1999). [CrossRef]
- D. Bryce and J. Autschbach, “Relativistic hybrid density functional calculations of indirect nuclear spin-spin coupling tensors. comparison with experiment for diatomic alkali metal halides,” Can. J. Chem. 87(7), 927–941 (2009). [CrossRef]
- B. Jansik, P. Salek, D. Jonsson, O. Vahtras, and H. Ågren, “Cubic response function in time-dependent density functional theory,” J. Chem. Phys. 122(5), 054107 (2005). [CrossRef]
- I. D. L. Albert, T. J. Marks, and M. A. Ratner, “Remarkable NLO response and infrared absorption in simple twisted molecular π-chromophores,” J. Am. Chem. Soc. 120, 11174 (1998). [CrossRef]
- J. Autschbach, “Charge-transfer excitations and time-dependent density functional theory: problems and some proposed solutions,” Chem. Phys. Chem. 10(11), 1757–1760 (2009). [CrossRef] [PubMed]
- S. Grimme and M. Parac, “Substantial errors from time-dependent density functional theory for the calculation of excited states of large pi systems,” ChemPhysChem 4(3), 292–295 (2003). [CrossRef] [PubMed]
- A. Dreuw and M. Head-Gordon, “Single-reference ab initio methods for the calculation of excited states of large molecules,” Chem. Rev. 105(11), 4009–4037 (2005). [CrossRef] [PubMed]
- F. Weinhold, ‘Natural bond orbital methods’. In Encyclopedia of computational chemistry, von Rague Schleyer, P., Ed. John Wiley & Sons: Chichester, 1998; pp 1792–1811.
- H. Ågren, J. Autschbach, A. Baev, M. Swihart, and P. N. Prasad, “Rational Design of Organo-Metallic Complexes for Enhanced Third-Order Nonlinearity,”manuscript in preparation.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.