Benchmarking Post-HartreeâFock Methods To Describe the Nonlinear

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Benchmarking Post-Hartree Fock Methods to Describe the Nonlinear Optical Properties of Polymethines: An investigation of the accuracy of Algebraic Diagrammatic Construction (ADC) approaches Stefan Knippenberg, Rebecca L. Gieseking, Dirk R. Rehn, Sukrit Mukhopadhyay, Andreas Dreuw, and Jean-Luc Bredas J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00615 • Publication Date (Web): 07 Oct 2016 Downloaded from http://pubs.acs.org on October 8, 2016

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Journal of Chemical Theory and Computation

Benchmarking Post-Hartree Fock Methods to Describe the Nonlinear Optical Properties of Polymethines: An investigation of the accuracy of Algebraic Diagrammatic Construction (ADC) approaches

Stefan Knippenberg,a,* Rebecca L. Gieseking,b,c Dirk R. Rehn,d Sukrit Mukhopadhyay,b,$ Andreas Dreuw,e and Jean-Luc Brédasb,f

a

Division of Theoretical Chemistry and Biology, KTH Royal Institute of Technology, Roslagstullsbacken 15, S-106 91 Stockholm, Sweden b

School of Chemistry and Biochemistry and

Center for Organic Materials for All-Optical Switching (COMAS), Georgia Institute of Technology, Atlanta, Georgia 30332-0400 c

Department of Chemistry, Northwestern University, 2145 Sheridan Rd, Evanston, Illinois 60208

d

Department of Physics, Chemistry and Biology, Linköping University, S-581 83 Linköping, Sweden e

Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

f

Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi-Arabia

* [email protected] $

Present address: The Dow Chemical Company, Midland, Michigan 48674.

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Abstract

Third-order nonlinear optical (NLO) properties of polymethine dyes have been widely studied for applications such as all-optical switching. However, the limited accuracy of the current computational methodologies has prevented a comprehensive understanding of the nature of the lowest excited states and their influence on the molecular optical and NLO properties. Here, attention is paid to the lowest excited-state energies and their energetic ratio, as these characteristics impact the figure-of-merit for all-optical switching. For a series of model polymethines, we compare several Algebraic Diagrammatic Construction (ADC) schemes for the polarization propagator with approximate second-order Coupled Cluster (CC2) theory, the widely used INDO/MRDCI approach and the SAC-CI algorithm incorporating singles and doubles linked excitation operators (SAC-CI SD-R). We focus in particular on the ground-toexcited state transition dipole moments and the corresponding state dipole moments, since these quantities are found to be of utmost importance for an effective description of the third-order polarizability  and two-photon absorption spectra. A sum-over-states expression has been used, which is found to quickly converge. While ADC(3/2) has been found to be the most appropriate method to calculate these properties, CC2 performs poorly.

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1. Introduction

Polymethines with an odd number of methine groups have been widely applied in nonlinear optics (NLO) over the past several decades.1-5 Unlike most π-conjugated systems, these positively or negatively charged closed-shell compounds inherently have a molecular thirdorder polarizability γ with a very large negative real part and a small imaginary part (Re(γ) and Im(γ), respectively).1-3,5 This makes them particularly suitable for applications such as alloptical switching (AOS), which requires molecules with a figure-of-merit (FOM), defined as |Re(γ)/Im(γ)| > 4π.6 However, the optical and NLO properties of polymethines have proven particularly challenging to compute accurately.3,7,8

An important characteristic of polymethines is a large energy gap between the first excited state e, which has a large one-photon absorption (OPA) cross-section, and the second excited state e’, which has a significant two-photon absorption (TPA) cross section.1,2,9 If the energy ħω of the incoming light signal is tuned such that Ege < 2ħω < Ege’, the magnitude of Re(γ) is significantly increased due to pre-resonant enhancement because of the small energy difference between ħω and Ege. Also, since 2ħω is energetically located between states e and e’, Im(γ) is expected to be small.5 The required FOM for all-optical switching has recently been achieved in thiopyrylium and selenopyrylium polymethines by leveraging these unique molecular properties.5,10 In contrast, only a very small FOM is generally possible for polyenes and most other π-conjugated systems because the first two excited states e and e’ are much more similar in energy,11-13 and as a result, 2ħω must be < Ege to avoid TPA losses. In these systems, the magnitude of Re(γ) is limited because little pre-resonant enhancement can be achieved without incurring substantial TPA losses (ħω has to be ] 2𝑛 𝑐 𝜖0 ℏ

(2)

where L denotes a local field factor; n, the refractive index; c, the speed of light in vacuum; and 𝜖0 , the permittivity of free space. Since typical formulations of the local field factor for solvated chromophores can lead to systematic errors,55 here we focus on simulations in vacuum, and thus L and n are set to 1. For more background to these formulas, we refer to the discussion in the Supplementary Information.

3. Computational details

The geometries of the four model streptocyanines with three, five, seven, and nine carbon atoms along the backbone and two dimethylamino end groups (Figure 1), are optimized using a density functional theory (DFT) approach with the long-range corrected B97x functional56 and Pople's 6-31G** basis set.57,58 While most standard DFT functionals over-delocalize the ground-state wavefunction in π-conjugated systems, the long-range corrected functionals typically provide a more accurate description of the ground-state wavefunction and degree of bond-length alternation (BLA);45 the B97x functional in particular has been shown to provide reliable geometries.59 We have recently shown that LRC functionals are in excellent agreement with CCSD(T) geometries for small polymethines,45 and the geometries used here are consistent with those previously used.20 We recall that the streptocyanines under investigation are closedshell singlet cations, with a C2v molecular point group symmetry.

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Using these geometries, the excited-state properties were computed at several levels of theory. The ADC computations were performed with Pople's 6-31G and 6-31G* basis sets as well as Dunning's cc-pVDZ basis set.60 Since ADC(2)-s calculations with the aug-cc-pVDZ basis set could not be completed due to convergence problems, the slightly smaller cc-pVDZ++ basis set, which excludes d-type diffuse functions on heavy atoms, was used instead. The difference between double and triple zeta basis sets is expected to be rather limited for the smaller closedshell cation compounds, while the computational demands become prohibitive towards the larger compounds.61-63 In this study, we do not aim to describe excited states with a profound Rydberg character. Focusing upon a comparison between different methods, the chosen moderate basis sets bear insight into the relevant →* excitations of the polymethines.

The electronic structure and absorption spectra of the streptocyanines have also been investigated using the symmetry adapted cluster configuration interaction (SAC-CI) SD-R method64,65 with the 6-31G* basis set. In our study, the maximum excitation level of the linked excitation operators was two. Neither the 9, 11, 13 and 15 core orbitals nor the corresponding number of uppermost virtual orbitals for 3C, 5C, 7C and 9C, respectively, are included in the active space.

Calculations have in addition been performed using the Intermediate Neglect of Differential Overlap (INDO) Hamiltonian66,67 with multireference determinant single- and doubleexcitation configuration interaction (MRDCI). Five reference determinants have been included, as well as an active space of 20 orbitals.21,68 The standard Mataga and Nishimoto parameterization69 was applied to express the Coulomb repulsion term; for the sake of comparison, the Ohno parameterization70 was also used. The nature of the excited states

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computed with INDO/MRDCI has been visualized with the ZOA program.71 Finally, RICC2/TZVP and CCSD/6-31G* calculations have also been performed.

For the ADC(2)-s, ADC(2)-x and ADC(3/2) calculations, a development version of the Q-Chem package has been used,72 while Gaussian09 has been employed for the SAC-CI calculations and the geometry optimizations.73 The INDO/MRDCI calculations were performed using the ZINDO code.74 The RI-CC2 calculations were performed with the Turbomole program 6.3.1,75 while the CCSD calculations were carried out using Dalton 2013.76,77

For the third-order polarizability , 300 excited states have been taken into account in the Sumover-States expression at the INDO/MRDCI level. For SAC-CI, as well as for the ADC(2)-s and ADC(2)-x levels, 5 excited states have been used. For ADC(3/2), the number of states has been truncated to the one-photon allowed first excited state and the next two two-photon allowed ones. Only the lowest excited states are found to be prominent contributors in the sumover-states expression of equation (1) through their coupling to the first excited state e, which is in itself strongly coupled to the ground state. In the static limit or at optical frequencies of light, this expression converges quickly because the denominator contains a product of three energy differences. In addition, a response theory approach has been used to compute the TPA cross-sections at the ADC(2)-s and ADC(2)-x levels. To compute the response-theory values, the two-photon absorption cross-section is expressed in terms of an inversion of the ADC matrix (see equation S4 in SI); the resulting equations are solved using iterative solvers for systems of linear equations and the DIIS (Direct Inversion of the Iterative Subspace) formalism as detailed in the SI.78

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4. Results and discussion

4.1. Excited-state energies

As can be inferred by examining the essential-state model for γ, an accurate evaluation of the NLO properties of polymethine dyes requires a proper evaluation of, at least, the lowest two excited states. The nature of the lowest excited states in streptocyanines is very similar for all computational approaches considered: the first excited state e has B2 symmetry and primarily involves a H → L transition, while the second and third excited states, e’ and e”, have A1 symmetry and involve linear combinations of the H-1 → L, H → L+1, and H2 → L2 electronic transitions. All of the orbitals involved in these transitions, see Figure 2, are π orbitals that extend across the entire length of the molecule. The relative contributions of these transitions to states e’ and e” vary among the levels of theory under consideration (see Table 1 for 5C and Table S1 in the Supplementary Information, SI, for data concerning the other three compounds). At the ADC(2)-x, the ADC(3/2), and INDO/MRDCI levels, the e’ state has a significant doubleexcitation character. For ADC(2)-s and CC2, these contributions are largely absent, which is not unexpected since these methodologies only incorporate zeroth-order orbital differences on the diagonal of the 2p-2h block. The e” state still has a considerable amount of doubleexcitation character, although it is less significant than in the e’ state. Among the methods studied, only SAC-CI yields a comparable amount of double excitation character for states e’ and e”.

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Figure 2. Frontier molecular orbitals of the four streptocyanines under consideration (HF/631G* data).

Table 1: Analysis and decomposition of the 𝒆, 𝒆′ and 𝒆" excited states for 𝟓𝑪, obtained at different levels of theory and the 6-31G* basis set. 𝐚 𝑒: 0.65 |𝐻 → 𝐿〉 𝑒′: 0.64 |𝐻 − 1 → 𝐿〉 𝑒": 0.64 |𝐻 → 𝐿 + 1〉 ADC(2)-x 𝑒: 0.64 |𝐻 → 𝐿〉 𝑒′: 0.47 |𝐻 − 1 → 𝐿〉 + 0.26 |𝐻 2 → 𝐿2 〉 𝑒" b : 0.50 |𝐻 → 𝐿 + 1〉 ADC(3/2) 𝑒: 0.64 |𝐻 → 𝐿〉 𝑒′: 0.42 |𝐻 − 1 → 𝐿〉 + 0.31 |𝐻 2 → 𝐿2 〉 + 0.27 |𝐻 → 𝐿 + 1〉 𝑒": 0.48 |𝐻 → 𝐿 + 1〉 + 0.41 |𝐻 − 1 → 𝐿〉 CC2 𝑒: 0.98 |𝐻 → 𝐿〉 𝑒′: 0.91 |𝐻 − 1 → 𝐿〉 + 0.39 |𝐻 → 𝐿 + 1〉 𝑒": 0.86 |𝐻 → 𝐿 + 1〉 + 0.44 |𝐻 − 1 → 𝐿〉 SAC-CI 𝑒: 0.94 |𝐻 → 𝐿〉 𝑒′: 0.66 |𝐻 − 1 → 𝐿〉 + 0.22 |𝐻 2 → 𝐿2 〉 + 0.15 |𝐻 → 𝐿 + 1〉 𝑒": 0.91 |𝐻 → 𝐿 + 1〉 + 0.22 |𝐻 − 1 → 𝐿〉 + 0.17 |𝐻 2 → 𝐿2 〉 INDO/MRDCI 𝑒: 0.90 |𝐻 → 𝐿〉 𝑒′: 0.33 |𝐻 − 1 → 𝐿〉 + 0.66|𝐻 → 𝐿 + 1〉 + 0.54|𝐻 2 → 𝐿2 〉 𝑒": 0.75 |𝐻 − 1 → 𝐿〉 + 0.49|𝐻 → 𝐿 + 1〉 ADC(2)-s

a

𝐻 and 𝐿 denote the highest occupied and the lowest unoccupied orbitals, respectively. The INDO/MRDCI results are obtained with the standard Mataga-Nishimoto parameterization. The INDO/MRDCI orbitals are found to be identical to the ones obtained by HF/6-31G*. b At the ADC(2)-x level, the 𝑒" excited state consists of 0.48|𝐻 → 𝐿 + 1〉 + 0.11|𝐻2 → 𝐿2 〉 for 7𝐶, while it amounts to 0.46|𝐻 → 𝐿 + 1〉 + 0.12|𝐻2 → 𝐿2 〉 for 9𝐶.

We now turn to the excited-state energies (see Table S2) and first focus on the energy Ege of state e, which decreases as molecular length and conjugation increase (Figure 3). We compare 13



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the computed excited-state energies to the experimental absorption maxima in dilute solution.16,79 Although in general comparing the computed Ege to the experimental absorption spectra requires accounting for vibronic effects,80-82 the streptocyanine absorption spectra have narrow peaks where the 0-0 vibronic peaks dominate due to the non-bonding character of the frontier molecular orbitals and the resulting small geometric changes between the ground and excited states.3,16,46 Thus, a direct comparison of Ege to the absorption maxima is expected to be reasonable for these systems. We note that experimental absorption spectra have been measured for dilute solutions of the dyes in dichloromethane, methanol, and trifluoroethanol, and the absorption maxima vary by no more than 0.1 eV among these solvents; test calculations on 3C and 5C taking some of these solvents into account at ADC(2)-s/ptSS-PCM(PTD*) level83 along with the 6-31G* basis set show negligible shifts with respect to the gas phase values.

Among the methods under consideration, substantial variation in the Ege values is observed. Whereas CC2 and CCSD overestimate the experimental Ege by 0.4-0.5 eV as has been previously observed,8 INDO/MRDCI dramatically underestimates Ege, particularly for the shortest streptocyanines. SAC-CI yields more accurate Ege values, although the decrease in Ege with increasing molecular length is steeper than is seen experimentally in contrast with the other methods considered. Among the ADC approaches, the ADC(3/2) and ADC(2)-s results are within about 0.2 eV of the experimental Ege (4.01 [3.97], 3.02 [3.01], 2.44 [2.44] and 2.03 [1.96] eV for the experimental absorption maxima in methanol [trifluoroethanol])8,16,79, whereas ADC(2)-x underestimates Ege by about 0.5-0.6 eV. The Ege values obtained using polarized basis functions are up to 0.1 eV higher than those obtained using unpolarized ones, especially for SAC-CI and ADC(3/2). From these results, the conclusion can be drawn that a more complete treatment of double excitations does not necessarily result in a more accurate Ege: since the first excited state e can be described within the singly excited manifold, the accuracy 14


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of the ADC approaches does not correlate with the treatment of double excitations. Furthermore, the accuracy is not significantly improved when going from CC2 to CCSD.

Figure 3. Comparison between the lowest excited-state energies (Ege) of the four cyanines, as obtained by various methods (in eV). Solid lines correspond to the 6-31G* basis set, dashed lines correspond to the 6-31G basis set, and dotted lines correspond to the TZVP basis set. The experimental data in trifluoroethanol are taken from Refs 16 and 17.

Because of experimental challenges, to the best of our knowledge, the second excited state energy Ege’ has only been measured for 3C and 7C.46,47 Since the TPA of state e’ is close to the onset of OPA into state e, it is hard to observe the full vibronic structure of the TPA peak. However, the relatively sharp onset of the TPA peaks46,47 and the non-bonding character of the orbitals involved in state e’ suggest that the absorption is sufficiently narrow that a direct 15



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comparison between Ege’ and the TPA peak is valid. For the 7C molecule, the ADC(3/2) excitation energy is within 0.3 eV of the experimental value; in the 3C molecule, however, the value is overestimated by more than 1 eV. An inspection of the energy differences Ege’ [Ege”] between the ground state and the second [third] excited state e’ [e”] reveals the relative overestimation of the CC2 energies (see Figure 4), which is consistent with the overestimation of Ege at this level.

Figure 4: The comparison between the (a) second (Ege’) and (b) third excited state (Ege”) energies of the four cyanines obtained by various methods (in eV). Solid lines correspond to the 6-31G* basis set and dashed lines correspond to the 6-31G basis set. The experimental data in trifluoroethanol are taken from Ref. 46. 16


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As was seen for Ege, the INDO/MRCI second and third excited-state energies exhibit not only the lowest values but also a different slope. The slopes of the ADC(3/2) and ADC(2)-x excitedstate energy curves are found to be largely parallel. With respect to ADC(2)-s and CC2, however, the slopes of the former higher-order methods are found to be steeper with the elongation of the cyanine backbone. Especially for Ege’, the differences between the SAC-CI values and the ADC(2)-x results increase with the length of the cyanine backbone. This is a consequence of the large double-excitation character of the A1 states, which is somewhat more pronounced for e’ than for e”. In the case of Ege”, this effect is therefore less pronounced when considering the energy differences among the different ADC(2)-s, SAC-CI, and ADC(3/2) results for the two longest compounds. This finding is in line with our work on linear polyacenes, where it was found that the evolution of the 21Ag state energy is much steeper at the ADC(2)-x level than at the ADC(2)-s level, owing to the pronounced double excitation character of this state in the longer polyacenes.37

As described in the Introduction, the properties of the second excited state, in particular the energy gap between the first and second excited states, are key in evaluating the third-order NLO properties of the specific compound. The ratio Ege’/Ege of the first two excited-state energies is experimentally observed to be 1.71 for the 7C streptocyanine; ratios in the range 1.67-1.73 have also been measured for polymethines with thiazolium (5-9 carbons) and indolium (7-11 carbons) end groups. The consistency in this ratio across a range of lengths and end groups suggests that this ratio should be applicable to the longer streptocyanines. For shorter polymethines, the ratio may be smaller, with values of 1.45 observed for the 3C streptocyanine and 1.55 for the 5C polymethine with indolium end groups.

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At the INDO/MRDCI level, Ege’/Ege is quite large, particularly for the longer cyanines; in fact, Ege’ is underestimated relative to the experimental values despite this large ratio that comes about because Ege is much smaller than the experimental value. Although SAC-CI predicts the closest ratio compared to experiment for the 3C streptocyanine, the ratios for the larger cyanines are substantially overestimated due to the pronounced underestimation of Ege. In contrast, CC2 provides Ege’/Ege values close to 1.7 for the range of lengths considered but only partially captures the experimental decrease in the ratio for the shortest cyanines.

Figure 5. The ratio of the first and second excited state energies of the four cyanines. Solid lines correspond to the 6-31G* basis set and dashed lines correspond to the 6-31G basis set. The shown experimental values in trifluoroethanol are taken from Ref. 46; for longer cyanine molecules, the experimentally extrapolated ratio is ~1.7 (see Ref. 9).

Among the ADC approaches, there is likewise significant variation in the Ege’/Ege ratios. ADC(3/2)/6-31G* provides ratios very close to 1.7 value for the shorter cyanines, whereas both ADC(2) approaches overestimate the ratio. The ADC(2)-s ratios start to plateau for longer cyanines, while the ADC(2)-x values continue to increase nearly linearly. Within each scheme,

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the ratios are smaller (and closer to 1.7) when polarization functions are considered: Ege slightly increases and Ege’ slightly decreases upon inclusion of polarization functions.

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4.2. Excited-state properties The third-order NLO properties depend critically on the transition dipole moments and state dipole moments among the lowest several streptocyanine electronic states (see equation (1)). The difference Δμeg between the state dipole moments of states e and g appears in the D term; due to the C2v symmetry, the streptocyanine Δμeg can only be non-zero along the short axis of the π system. As Δμeg is negligibly small, we will focus on the transition dipole moments μge, which dominate the N term, and the transition dipole moments between excited states e and e’ (μee’) and e” (μee”), which are important for the T term.

The transition dipole moment μge from state g to state e is roughly the same at all levels of calculation for each molecule. The absolute value of the x-component is  7.6 Debye for 3C,  10.7 Debye for 5C,  13.4 Debye for 7C, and 16.0 Debye for 9C at the ADC(3/2) level. The INDO/MRCI and CC2 methods slightly overestimate the μge values relative to ADC(3/2), while SAC-CI underestimates them especially for the longest compounds. Since this transition predominantly involves a H → L excitation and the spatial distributions of the HOMO and LUMO wavefunctions are consistent among all levels of theory, the small variations in μge among the applied methods are unsurprising (see Figure 6a and Table S2).

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Figure 6. Transition dipole moments (in Debye) from: (a) the ground state g to the first excited state e; (b) between the first excited state e and the second excited state e’; and (c) between the first excited state e and the third excited state e”, for the four cyanines. Solid lines correspond to the 6-31G* basis set and dashed lines correspond to the 6-31G basis set. 21



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From Figure 6 (b), it can be seen that at all levels of theory, ee’ increases with increasing streptocyanine length. The ee’ values strongly depend on the level of theory and, in particular, on the treatment of double excitations. The methods that do not describe double excitations beyond zeroth order, like SAC-CI and CC2, provide generally large values of ee’, whereas INDO/MRDCI, which treats single and double excitations on equal footing within the chosen active space, yields much smaller ee’ values and a very weak length dependence. A similar pattern can be observed when comparing the ADC methodologies. Within the ADC series, the value of ee’ sharply decreases at all molecular lengths as the treatment of double excitations improves from ADC(2)-s to ADC(2)-x and ADC(3/2). These observations are consistent with our recent work demonstrating that the double excitation has a critical role in decreasing the magnitude of ee’.21 Since the magnitude of Im(γ) and thus the TPA cross-section of the second excited state strongly depend on ee’ as will be detailed in the following section, these results highlight the importance of adequately treating double excitations to accurately model the streptocyanine NLO properties.

The transition dipole moments obtained at the CC2 level are determined through first order, while the ones based upon the ADC(2)-s, ADC(2)-x and ADC(3/2) methodologies are calculated through second order in perturbation theory. This is illustrated in Figure 6 (c), where CC2 is seen to exhibit a different pattern compared to both ADC(2) methods and to ADC(3/2). This evolution contrasts with that of the energies, which are consistently obtained through second order for CC2 as well as both ADC(2) methods; for ADC(3/2), the energy is obtained through third order. For the discussion of the third-order molecular polarizability  and the twophoton absorption cross-section, which strongly depend on the quality of the description of the 22


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first excited state dipole moments, it is thus more appropriate to focus on the performance of the ADC methods.

The transition dipole moments ee’ and ee” at the ADC(2)-s and ADC(2)-x levels exhibit the same characteristics, in the sense that both gradually increase as the streptocyanine backbone is extended (Figure 6). In contrast, the ADC(3/2) level, which gives a rather flat curve for ee’ around ~2.5-3 Debye, gives rise to a considerably steep curve and higher values when ee” is considered for the longer cyanines. The values for ee’ at the ADC(3/2) level are considerably smaller than both the ADC(2)-s and ADC(2)-x ones; however, for ee”, the ADC(3/2) results are intermediate.

4.3 Third-order polarizabilities (;-,,)

We now turn to a discussion of the third-order polarizabilities of the streptocyanines. We first consider the static (ħω = 0) values. As Im(γstat) is negligible, we focus here on Re(γstat) (see Table 2). Since the SOS expressions contain a product of four (state or transition) dipole moments in the numerator and three energies in the denominator, the value for Re(γstat) is expectedly very sensitive to the accuracy of the excited-state properties described in the previous section. Re(γ) converges rapidly with the number of excited states included in the summation within the SOS formalism. For ADC, differences due to varying the number of excited states (5 or 3, see computational details) are only of marginal importance. A consideration of 5 excited states at the INDO/MRDCI level introduces no more than a 25% error relative to a summation over 300 excited states. With an increasing molecular length, this error decreases and does not change the trends in the magnitude of Re(γ). This convergence is 23



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due to the form of the SOS expression and the unique excited-state structure of the polymethines. Because the denominator consists of a product of three energy differences, the magnitudes of the contributions to Re(γstat) decrease rapidly with increasing excited-state energy. Since the first excited state is energetically well-separated from the higher excited states and is more strongly optically allowed than any of the other low-lying states, its contribution typically dominates the magnitude of Re(γstat) and only a few two-photon-absorbing states have to be considered.

To rationalize the magnitude of Re(γstat), we consider the magnitude of the terms appearing in the essential-state model (Equation (1)). As we mentioned earlier, due to the C2v molecular symmetry, the state dipole moments can only be non-zero along the short axis of the π system. Thus, there is little difference in state dipole moments among the different states and the D term is negligibly small. We note that in longer polymethines where ground-state symmetrybreaking occurs, the D term has a much more significant contribution to Re(γ).44,84 For polymethines, the N term typically dominates; as μge falls within a narrow range for all of the levels of theory considered, the magnitude of the N term is primarily dependent on Ege. Since μge increases and Ege decreases with increasing molecular length, the evolution of the N term causes Re(γstat) to increase by more than two orders of magnitude as the molecular backbone is extended from three to nine carbons. The T term is usually smaller in magnitude than the N term; the relative magnitudes of these two terms depend on the energies and transition dipole moments of the TPA-active excited states. We note that the experimental Re(γstat) value is small and positive for the 3C streptocyanine with a perchlorate anion in trifluoroethanol.46 However, among our calculations, all approaches give negative Re(γstat) values, and only the SAC-CI value is comparable in magnitude to the experimental value (see Table 2); the difference in sign may be related to the essential-state approximation, which considers only a handful of excited 24


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states; we also neglect factors such as vibronic effects and the distribution of geometries present experimentally. For streptocyanine 5C, all methods give the same negative sign, but only the SAC-CI value is on the same order of magnitude as the experimental one.

Table 2: Real parts of the static third-order polarizability [𝜸𝒔𝒕𝒂𝒕 ] for the 3C, 5C, 7C and 9C cyanines. 𝐚 ADC(2)-s ADC(2)-x CC2 SAC-CI ADC(3/2) INDO/MRDCI Experiment c

6-31G*

3C -31 -81 -43 -2 -45 -141 +5

5C -274 -827 -267 -34 -404 -1360 b -53

7C -1337 -4957 -795 -565

9C -4582 -21302 +2443 -3333

-6885 -1224

-23609 -5082

a

𝛾𝑠𝑡𝑎𝑡 is given in 10−36 esu. All experimental and computed values are reported using a Taylor series expansion for 𝛾, which yields values a factor of 6 larger than the power series expansion commonly used in experimental studies.3,49 b Standard (Mataga-Nishimoto) parametrisation; the values for the Ohno parametrization is: 𝛾𝑠𝑡𝑎𝑡 = −384 ⋅ 10−36 esu. c In trifluoroethanol; taken from Ref. 46.

The relative magnitudes of Re(γstat) at the various levels of theory (Table 2) can be rationalized in terms of the excited-state properties discussed in the previous Section. The magnitude of Ege plays a particularly significant role in determining the magnitude of Re(γstat): INDO/MRDCI and ADC(2)-x underestimate Ege and thus substantially overestimate Re(γstat) relative to experiment. This difference is primarily due to the enhancement of the N term when Ege is small. The accuracy of description of the higher-lying excited-state properties has a smaller but still noticeable effect via the T term. For example, when comparing Re(γstat) for the 3C cyanine at the ADC(2)-s/6-31G and ADC(3/2)/6-31G levels, the first excited-state properties and thus the magnitudes of the N term are similar. However, since μee’ and μee” are several times larger

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at the ADC(2)-s/6-31G level, the T term is larger and overall Re(γstat) is less negative in the latter case.

Especially for the larger cyanines 7C and 9C, the CC2-calculated Re(stat) values deviate severely from the experimental ones and the ones obtained with all other considered methods. The unphysical, large positive value for 9C is a direct consequence of the dramatic overestimation of ee”. It can be seen that the quality of the transition dipole moments among the excited states e, e’, and e” are decisive for the third-order polarizability: the CC2 energies Ege, Ege’ and Ege” largely agree with the ADC(2)-s ones; however, the transition dipole moments and the Re(stat) values are markedly different.

4.4

Two-photon absorption

As described in the Introduction, the TPA cross-section δ at a photon energy ħω is directly proportional to Im(γ) at that energy. Im(γ) is negligibly small except at energies such that 2ħω approaches resonance with a (two-photon allowed) excited state. At these energies, the denominators (𝐸ge − 2ℏ𝜔 − 𝑖Γge ) in the D term or (𝐸ge′ − 2ℏ𝜔 − 𝑖Γge′ ) in the T term become vanishingly small. Here, we calculate the TPA cross-section δ for the lowest streptocyanine excited states (for each state x, at an energy ħω = 𝐸gx /2).

Because δ is exquisitely sensitive to the detuning of ħω from 𝐸ge , even slight deviations in the ratio 𝐸ge′ /𝐸ge can lead to substantial errors in δ; we recall that 𝐸ge′ /𝐸ge is expected to be close to 1.7 across a range of polymethine lengths. As a result, when 𝐸ge′ /𝐸ge is very close to 2, the computed δ is dramatically enhanced, which is not physically meaningful; if this ratio was 26


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observed experimentally, one-photon absorption would dominate the signal and TPA would be unobservable. Therefore, the exceptionally large δ values obtained using several of the computational approaches for the polymethines of 5 carbons and longer must be interpreted with care (see Tables 3-4). The chosen line broadening value  of 0.1 eV may also affect the accuracy; as the experimental OPA peaks have widths close to 0.1 eV and the TPA peaks are likewise comparably narrow as detailed in the previous Section,9,47 this value of  is expected to be physically reasonable. Since the computed values of δ span several orders of magnitude, small inaccuracies in  will not significantly affect the comparisons made here.

Table 3: Results of the two-photon absorption calculations obtained using response theory and the three-state model for 3C and 5C. 𝐚 Method Exc state ADC(2)-s/6-31G* 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 ADC(2)-x/6-31G* 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 ADC(3/2)/6-31G 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 ADC(3/2)/6-31G* 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 CC2/6-31G* 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 SAC-CI/6-31G* 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 INDO/MRDCI b 𝑒′ 𝐴1 𝑒" 𝐴1 a

3C Energy Resp. theo Three-state TPA Exc. state 4.18 (0.946) 1 0 𝑒 𝐵1 7.05 (0.037) 1186 1188 𝑒′ 𝐴1 7.38 (0.007) 842 849 𝑒" 𝐴1 3.46 (0.758) 1 0 𝑒 𝐵1 5.81 (0.027) 430 436 𝑒′ 𝐴1 54 56 6.57 c (0.000) 𝑒" 𝐴1 4.14 (0.914) 0 𝑒 𝐵1 6.92 (0.036) 168 𝑒′ 𝐴1 7.51 (0.001) 64 𝑒" 𝐴1 4.24 (0.907) 0 𝑒 𝐵1 6.89 (0.032) 115 𝑒′ 𝐴1 7.45 (0.001) 23 𝑒" 𝐴1 4.48 (1.014) 0 𝑒 𝐵1 7.23 (0.042) 117 𝑒′ 𝐴1 7.56 (0.000) 5 𝑒" 𝐴1 4.18 (0.829) 0 𝑒 𝐵1 6.21 (0.025) 613 𝑒′ 𝐴1 7.44 (0.012) 536 𝑒" 𝐴1 2.84 (0.674) 0 𝑒 𝐵1 4.84 (0.016) 94 𝑒′ 𝐴1 711 5.90 c (0.015) 𝑒" 𝐴1

5C Energy Resp. theo Three-state TPA 3.24 (1.424) 2 0 5.81 (0.028) 11168 11168 125647 6.36 c (0.005) 125651 2.51 (1.066) 1 0 4.53 (0.018) 3606 3638 5.46 (0.000) 105 3.12 (1.361) 0 5.49 (0.023) 774 9506 6.32 c (0.000) 3.24 (1.353) 0 5.44 (0.020) 296 2598 6.29 c (0.000) 3.54 (1.593) 0 5.97 (0.030) 2945 6.54 (0.000) 5479 3.10 (1.161) 0 5.01 (0.020) 4959 6.35 c (0.006) 39416 2.10 (0.990) 0 15796 4.11 c (0.015) 4.84 (0.002) 257

The excitation energies are given in eV, along with their oscillator strengths (in parentheses); the results of the TPA are given in Göppert-Mayer (GM). For 3C, at the CASPT2 level of theory and an ANO-S basis set with a contraction scheme of 4s3p1d/2s, for 𝑒 [ 1 𝐵2 ] and 𝑒′ [ 1 𝐴1 ] energies of 4.17 eV and 6.12 eV are reported. For 5C, using the same method and basis set, for 𝑒 [ 1 𝐵2 ] and 𝑒′ [ 1 𝐴1 ] energies of 2.95 eV and 4.68 eV are reported.19 For the e’ state of 3C, an experimental TPA cross section of 10 GM has been reported in trifluoroethanol.46 b Standard (Mataga-Nishimoto) parametrisation. c One half of the energy of this state is relatively close to the energy of the first excited state (ratio 1.9-2.1).

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Table 4: Results of the two-photon absorption calculations obtained using response theory and the three-state model for 7C and 9C. 𝐚 Method

Exc state

ADC(2)-s/6-31G*

𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1

ADC(2)-x/6-31G*

ADC(3/2)/6-31G

CC2/6-31G*

SAC-CI/6-31G* INDO/MRDCI b a

7C Resp. theo Three-state TPA Exc state

Energy 2.60 4.80 5.41 c 1.87 3.58 c 4.43 2.40 4.39 5.24 2.89 4.98 5.59 c 2.24 4.20 5.31 1.66 3.49 c 4.11

(1.789) (0.012) (0.015) (1.242) (0.011) (0.002) (1.651) (0.016) (0.001) (2.095) (0.019) (0.007) (1.291) (0.013) (0.013) (1.221) (0.015) (0.001)

3 53391 93496 2 50734 399

0 53522 93405 0 50976 0 3785 4672 0 13662 305380 0 157058 1542 0 4768 1024

𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1 𝑒 𝐵1 𝑒′ 𝐴1 𝑒" 𝐴1

9C Resp. Three-state TPA theo (2.111) 4 0 (0.003) 202831 203480 (0.027) 47642 47531 (1.353) 3 0 (0.006) 886215 886126 (0.007) 966 (1.863) 0 (0.012) 28253 (0.005) 8166 (2.588) 0 (0.010) 13153 (0.019) 442346102 (1.362) 0 (0.007) 444088 (0.027) 1318 (1.435) 0 (0.013) 3342 (0.008) 2863

Energy 2.17 4.09 4.73 1.44 2.93 c 3.69 1.91 3.63 c 4.44 2.46 4.28 4.90 c 1.71 3.61 4.56 1.38 3.06 3.64

The excitation energies are given in eV, along with their oscillator strengths (in parentheses); the results of the TPA are given in Göppert-Mayer (GM). For 7C, at the CASPT2 level of theory and an ANO-S basis set with a contraction scheme of 4s3p1d/2s, for 𝑒 [ 1 𝐵2 ] and 𝑒′ [ 1 𝐴1 ] energies of 2.32 eV and 3.56 eV are reported. For 9C, using the same method and basis set, for 𝑒 [ 1 𝐵2 ] and 𝑒′ [ 1 𝐴1 ] energies of 1.93 eV and 2.93 eV are reported.19 For the e’ state of 7C, an experimental TPA cross section of 557 GM has been reported in trifluoroethanol.46 b Standard (Mataga-Nishimoto) parametrisation. c One half of the energy of this state is relatively close to the energy of the first excited state (ratio 1.9-2.1).

Experimentally, the TPA values of the e’ states of the 3C and 7C systems have been reported as 10 GM and 557 GM, respectively.46 However, the computed values consistently overestimate the TPA cross-sections, in many cases by multiple orders of magnitude. This is due to the limitations of these methodologies in predicting the ratios of the excited-state energies and the transition dipole moments between excited states. Although ADC(3/2) still overestimates both of the experimentally reported values by roughly an order of magnitude, its reasonable performance for both excited-state energies and transition dipole moments makes it currently the best performer among the methods studied. In contrast, ADC(2)-s, ADC(2)-x, SAC-CI, and 28


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CC2 all yield unphysically large TPA cross-sections in the majority of cases due to their limitations discussed in the previous Section. Interestingly, INDO/MRDCI provides relatively small TPA cross-sections for all of the cyanines except for 5C; although this approach correctly predicts small transition dipole moments among the excited states, the Ege’/Ege ratio is unphysically larger than 2 for the longer streptocyanines. These results highlight once again the significant challenges in accurately modeling the polymethine TPA properties, as minor inaccuracies in the excited-state characteristics can lead to substantial changes in the accuracy of the NLO properties.

For states e’ and e”, results from the simplified three-state model and the response theory approach are consistently in excellent agreement, which demonstrates that the TPA into these states is predominantly due to coupling through the first excited state as a virtual state. Due to the excited-state spacing in the polymethines, which have a low lying one-photon allowed state and a strongly coupled higher-lying second excited state which is two-photon allowed, the SOS expression converges much more quickly with the number of excited states in the summation than has previously been observed for ethylene using ADC/ISR approaches.38 As shown in Tables S4 and S5, the excellent agreement between the three-state model and response theory approach holds even for excited states up to several eV higher in energy than states e’ and e”.

5. Conclusions and outlook

Accurate modeling of the polymethine excited-state properties, in particular, energies and transition dipole moments among the first several excited states, is critical to evaluate their third-order NLO properties. Here, we have focused on the first several excited states of four streptocyanine molecules with varying backbone length and the resulting description of the 29



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two-photon absorption and the third-order polarizability . In particular, we compared the strict and extended second-order Algebraic Diagrammatic Construction schemes along with the intermediate state representation to a few benchmark ADC(3/2) calculations to investigate the influence of doubly excited configurations. These results have also been compared to other computational methods like CC2, SAC-CI SD-R, and the semi-empirical INDO/MRDCI algorithm, which have previously been used to study polymethine NLO properties. For the simulation of two-photon absorption spectra, an essential three-state model has been compared with a methodology based on matrix inversion and the direct inversion of the iterative subspace (DIIS) algorithm; the results are found to be very comparable and validate the sum-over-states expression, which quickly converges.

The linear and non-linear optical properties of polymethines are notoriously hard to simulate as the energies as well as the characteristics of the lowest excited states need to be properly described. Highly accurate methods must be applied to obtain chemically relevant results. In our analysis, ADC(3/2) has been found to be the most appropriate method that is consistently and reasonably accurate across all properties. Although the Re() calculated by means of a limited essential-state formalism somewhat overestimates the experimental values, the ADC(3/2) TPA cross sections are found to be more reasonable than those from any of the other methodologies studied. However, its computational cost prohibits application to larger polymethines that are of experimental interest.

Comparison of the ADC(3/2) results to the ADC(2)-s and ADC(2)-x schemes reveals that each of the latter two methods has strengths and marked weaknesses. For an appropriate choice of the second-order ADC methodology, it is important to consider the relevant properties of interest. ADC(2)-x is known to underestimate excited-state energies; however, it has a clear 30


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advantage with respect to ADC(2)-s in the description of the transition dipole moments ee’ and

ee” because of its more complete treatment of doubly excited states. Thus, Re() is better described by ADC(2)-s, since its magnitude depends primarily on the first excited-state properties.

Among the non-ADC methodologies, the excited-state energies obtained by CC2 are consistent with ADC(2)-s and show an Ege’/Ege ratio very close to 1.7, but the transition dipole moments between excited states are greatly overestimated due to the limited treatment of double excitations; thus, CC2, which describes response properties only through first order, performs poorly for both Re() and TPA (i.e., Im()). The properties calculated by SAC-CI SD-R are consistently among the less accurate, while the excited-state energies decrease too rapidly with increasing polymethine length; to describe streptocyanines, other methods bearing a similar computational cost could therefore be selected. We note that ADC(2)-x, CCSD and SAC-CI SD-R scale like N6, while the computational effort for CC2 and ADC(2)-s follows a N5 scaling, with N the number of orbitals. INDO/MRDCI, which allows for a proper treatment of double excitations at a very low computational cost, bears reasonable transition dipole moments and a Re() which shows an acceptable trend; however, the excited-state energies are only moderately reliable.

The benchmarking performed here can be used to inform the choice of computational methodologies to understand the properties of new polymethines of experimental interest. In addition, since the methods described here are still limited in their accuracy, this study suggests that further work to accurately compute the nonlinear optical properties of polymethines is needed. One natural extension is an improved description of the third-order polarizability  for 31



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the various ADC schemes by taking into account transition dipole moments between the ground state and higher-lying excited states. Further on, complex response theory can be considered, as it has been done for applications in Hartree-Fock or DFT calculations with the conjugate residual and preconditioned iterative subspace algorithms,85 as well as in coupled cluster theory with the complex polarization propagator approaches.86

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Acknowledgments S.K. is grateful to the Georgia Institute of Technology for their hospitality in Fall 2012. The authors thank Dr. Paul Winget for computational assistance. J.L.B. acknowledges generous support of this work by KAUST.

Supplementary information The Supporting Information is available free of charge on the ACS Publications website at DOI:… Description of the calculation of excitations, TPA spectra and third-order polarizabilities by means of ADC; decomposition of the e, e’, and e” excited states for 3C, 7C, and 9C at various levels of theory; extended tables of the computed vertical excitation energies of the first excited states of the four cyanines, together with the transition dipole moments from the ground to the first, second, and third excited states; real parts of the static and dynamic third-order polarizabilities; results of the TPA calculations obtained using response theory and the threestate model for the four investigated compounds.

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TOC 221x155mm (150 x 150 DPI)


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