Vibrational Sampling and Solvent Effects on the ... - ACS Publications

Subscriber access provided by AUSTRALIAN NATIONAL UNIV

Spectroscopy and Excited States

Vibrational Sampling and Solvent Effects on the Electronic Structure of the Absorption Spectrum of 2-Nitronaphthalene J. Patrick Zobel, Moritz Heindl, Juan José Nogueira, and Leticia Gonzalez J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00198 • Publication Date (Web): 25 Apr 2018 Downloaded from http://pubs.acs.org on April 26, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

Vibrational Sampling and Solvent Effects on the Electronic Structure of the Absorption Spectrum of 2-Nitronaphthalene J. Patrick Zobel, Moritz Heindl, Juan J. Nogueira,∗ and Leticia González∗ Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Währinger Straße 17, A-1090 Vienna, Austria E-mail: [email protected]; [email protected] Abstract The influence of vibrational motion on electronic excited state properties is investigated for the organic chromophore 2-nitronaphtalene in methanol. Specifically, the performance of two vibrational sampling techniques –Wigner sampling and sampling from an ab initio molecular dynamics trajectory– is assessed, in combination with implicit and explicit solvent models. The effects of the different sampling/solvent combinations on the energy and electronic character of the absorption bands are analyzed in terms of charge transfer and exciton size, computed from the electronic transition density. The absorption spectra obtained using sampling techniques and its underlying properties are compared to those of the electronic excited states calculated at the Franck-Condon equilibrium

1 Plus Environment ACS Paragon

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

geometry. It is found that the absorption bands of the vibrational ensembles are red-shifted compared to the Franck-Condon bright states, and this red-shift scales with the displacement from the equilibrium geometry. Such displacements are found larger and better described when using ensembles from the harmonic Wigner distribution than snapshots from the molecular dynamics trajectory. Particularly relevant is the torsional motion of the nitro group that quenches the charge transfer character of some of the absorption bands. This motion, however, is better described in the molecular dynamics trajectory. Thus, none of the vibrational sampling approaches can satisfactorily capture all important aspects of the nuclear motion. The inclusion of solvent also red-shifts the absorption bands with respect to the gas phase. This red-shift scales with the charge-transfer character of the bands and is found larger for the implicit than for the explicit solvent model. The advantages and drawbacks of the different sampling and solvent models are discussed to guide future research on the calculation of UV-VIS spectra of nitro-aromatic compounds.

Keywords: 2-nitronaphtalene, UV absorption spectrum, vibrational sampling, solvent effects, charge transfer

1

Introduction

The simplest approach to calculate UV-vis absorption spectra is to compute vertical energies of the electronically excited states at the ground-state minimumenergy or Franck-Condon (FC) geometry. In this static description the computational effort is greatly reduced by neglecting the vibrational motion of the molecule. The success of this approach is usually assessed by comparing the obtained excitation energies of the bright electronic states with the energies of 2 Plus Environment ACS Paragon

Page 2 of 46

Page 3 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the experimental absorption band maxima. A good agreement between these quantities is taken as indicator of an adequate description of the excited states, although in some unfortunate cases, it can lead to wrong conclusions; 1,2 in such cases, it is mandatory to include the effects of nuclear motion. Experimentally measured observables always involve an ensemble of molecules possessing different geometries due to vibrational motion. Calculations done on a single representative geometry assume that this structure reflects the average properties of the statistical ensemble. For small molecules, this is generally the case. However, in molecules with certain flexibility, the averaged geometry can be meaningless (as e.g., in the case of torsions that connect a set of two enantiomers) or possess wrong properties (as in the case of some nitro-aromatic molecules, which although planar at the FC geometry, possesses in average a non-zero nitro group torsional angle. 1,3 ) To alleviate such situations, molecular properties should be computed for an ensemble of geometries generated by a vibrational sampling technique. While this is routinely done in the study of large molecules such as proteins, lipid membranes, DNA, or synthetic polymers, where a large number local minima are thermally accessible, it is usually neglected when dealing with small molecules. There exist two common sampling approaches: quantum (zero-temperature) sampling and thermal (finite-temperature) sampling. 4,5 In thermal sampling, the system is given a finite temperture T and the phase space is explored either by following a molecular dynamics (MD) trajectory 6 or using a stochastic Monte Carlo (MC) method, 7 the former being more frequently employed for subsequent electronically excited-state calculations. In these approaches, the system is assumed to be in thermal equilibrium, i.e., all degrees of freedom possess in average an equal amount of energy hi i = kB T . The potential in which the MD trajectory 3 Plus Environment ACS Paragon


is propagated can be computed using quantum mechanics (ab initio MD), empirical force fields (classical MD), or employing a hybrid version of both where one part of the system is described by quantum mechanics and the rest by molecular mechanics using a force field (QM/MM-MD). To generate an ensemble, uncorrelated snapshots from a sufficiently long MD trajectory are sampled. In quantum sampling, one represents a quantum mechanical vibrational state Ψ by a Wigner function, which maps the quantum-mechanical probability distribution to the classical phase space. 8 For simplicity, a harmonic-oscillator Wigner distribution, where the vibrational nuclear degrees of freedom are represented by harmonic oscillators, 9 is usually employed. Here, the system is given its zero-point energy P P hωi , (ZPE) for each vibrationally degree of freedom, i.e., E = i=1 i = i=1 1/2¯ where i and ωi are the vibrational energy and frequency of each normal mode. In addition, temperature effects can be added to the quantum sampling by calculating the Wigner distribution of a canonical ensemble, 10 although this is only rarely done. 11 Both sampling approaches differ in a few important aspects. The ZPE in quantum sampling is usually much larger than the average energy of each degree of freedom in thermal sampling at room temperature. For example, if in average ωi = 500 cm−1 , this corresponds to a thermal energy at a temperature of 720 K. Furthermore, the distribution of the total energy is also different for both sampling techniques. While in thermal sampling, each degree of freedom has the same (average) energy, in quantum sampling each vibrational mode gets its own ZPE, i.e., high-frequency modes receive much more energy then low-frequency modes. 12 Therefore, based on an energetic criterion, (zero-temperature) Wigner sampling accounts for a larger fraction of the total energy than thermal sampling. However, an important restriction of the harmonic Wigner sampling is 4 Plus Environment ACS Paragon

Page 4 of 46

Page 5 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the negligence of all effects of the anharmonicity of the potential-energy surface. This is especially relevant for large systems or chromophores embedded in explicit environment, possessing a large number of low-frequency degrees of freedom with an important amount of anharmonicity. In contrast, thermal sampling is able to describe anharmonic motions, especially if the potential-energy gradients along the dynamics are computed quantum mechanically or by QM/MM schemes. Therefore, thermal sampling should perform better when low-frequency motions dominate the dynamics of the system. Thermal and quantum sampling also differ in their computational cost. Generating an ensemble for a small-size system from a harmonic-oscillator Wigner distribution is computationally less expensive than thermal sampling at a given level of theory for the electronic-structure part. Generating the Wigner distribution requires only a geometry optimization to find the minimum-energy structure and a subsequent frequency calculation at the stationary point, in contrast to the many calculations performed during the MD simulation. For large systems, however, the generation of a Wigner-based ensemble is not feasible because the frequency calculation becomes unaffordable. In these cases, thermal sampling is the method of choice. There exists also a hybrid approach combining quantum and thermal sampling. 13 In this approach, a Wigner distribution is generated for the solute, and the resulting geometries are placed in an environment and kept frozen during a subsequent MD simulation to allow the environment to adapt to the individual structures of the solute. The description of the electronic properties of a system becomes further involved in the presence of an environment, e.g., in solution. This interaction between solute and solvent can be well described with the so-called continuum models, and more specifically with models based on apparent surface charges. 14–16 In continuum models, the electrostatic interaction of the bulk of solvent molecules 5 Plus Environment ACS Paragon


with the solute is modeled by the electrostatic potential created by the solute and a set of point charges that are placed on the made-up surface of a cavity that contains the charge density of the solvent. When solvent molecules should be explicitly considered, a cluster model 17–19 or QM/MM approaches 1,17,20–22 are in order. In QM/MM, the solvent molecules are described using MM which allows inclusion of a large number of solvent molecules in the calculation due to the low computational cost. In cluster models, a small number of solvent molecules is included explicitly in the QM calculation, while QM/MM allows the inclusion of a large number of solvent molecules. Additionally, the cluster approach may be combined with the usage of QM/continuum or a QM/MM descriptions to model the rest of the bulk solvent. The choice of the solvent model also has an influence on the approach used for sampling. For small systems, if a continuum model is employed, one can usually do both, quantum sampling as well as thermal sampling. For larger systems, e.g., when one uses a QM/MM model to describe the solvent effects, one is usually restricted to thermal sampling due to the drawbacks and difficulties of a Wigner sampling –as discussed above. In this work, we want to investigate the effects of both nuclear motion and interaction with a solvent environment on the electronic excited states of 2nitronaphthalene (2NN) in methanol, using different sampling and environmental methods. Because of the presence of the nitro group, it is expected that the UVvis absorption spectrum of 2NN is strongly affected by the inclusion of vibrational motion, as it was bare nitrobenzene. 1 In particular, vibrational motion is considered by thermal sampling from a MD trajectory and by Wigner sampling. In addition, for each sampling approach, solvent effects are described by both explicit and implicit models. The importance of vibrational sampling and interaction with the environment is discussed based not only in terms of the energy of 6 Plus Environment ACS Paragon

Page 6 of 46

Page 7 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the absorption bands but also on their electronic features, such a charge-transfer character and exciton size, computed from the electronic transition density. The rest of the paper is organized as follows. The computational details of all calculations are listed in Section 2 and Sections 3 collect all the results. As a reference, the electronic excited states of 2NN are computed using only the FC geometry. In Sections 3.1 and 3.2, the excitation energies obtained at the FC geometry are compared with the experiment and then with the results obtained including sampling. Section 3.3 and 3.4 analyse the character of the excited states contributing to the absorption bands calculated including the sampling and those obtained only at the FC geometry, respectively. Finally, sections 3.5 and 3.6 investigate the influence of the nuclear motion and more specifically the torsion of the nitro group, on the position and composition of the bands, and section 7 is focused on the effect of the solvent against gas phase.

2

Computational Details

2.1

Excited-State Calculations at the FC Geometry

The structures of 2NN in methanol and n-heptane have been optimized at the B3LYP 23–26 /def2-TZVP 27 level of theory using the conductor-like screening model (COSMO) 15 with dielectric constants of εMeOH = 32.66 and εn−heptane = 1.92, respectively. The coordinates of the optimized geometries are reported in Table S1 in the SI. Using those geometries, the 20 lowest-lying excited singlet states were calculated using COSMO in combination with time-dependent densityfunctional theory (TDDFT) and the second-order algebraic diagrammatic construction [ADC(2)] scheme 28,29 using both, the def2-SVP and def2-TZVP basis sets. 27 7 Plus Environment ACS Paragon


The TDDFT calculations using the B3LYP, PBE, 30 and PBE0 31 functionals employed the Tamm-Dancoff approximation (TDA), as implemented in Turbomole 7.0. 32 In the SCF calculations the convergence was set to 10−7 a.u. and for the TDDFT calculations, the convergence (Euclidean residual norm of the TDA vectors) was set to 10−6 . All other parameters are used as default in Turbomole.

2.2

Simulation of the Absorption Spectra and Sampling techniques

The absorption spectrum of 2NN in MeOH was simulated using four different sampling techniques, as described below and summarized in Table 1. From each ensemble of 2NN in MeOH, 100 random geometries were selected on which excited-state calculations were carried out as for the Franck-Condon geometry (see Section 2.1). The final absorption spectra for each ensemble was obtained by convoluting the 100 stick spectra with Lorentzian functions using FWHM = 0.1 eV. The resulting spectra were analyzed using the TheoDORE package. 33–36 Wigner/COSMO Sampling. Here, the vibrational ensemble was obtained from a harmonic Wigner distribution. The geometry optimization and corresponding frequency calculation was carried out at the B3LYP 23–26 /def2-TZVP 27 level of theory using the COSMO model to describe MeOH. MD/COSMO Sampling. Here, the vibrational ensemble was obtained from an ab initio molecular dynamics (AIMD) trajectory computed at the B3LYP 23–26 / def2-SVP 27 level of theory with the TERACHEM 1.9 progran packate 37,38 using the COSMO model to describe MeOH. The trajectory was run at a temperature


Page 8 of 46

Page 9 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


of 300 K using the Langevin thermostat for a simulation time of 5 ps with time steps of 0.5 fs. MD/MM Sampling. Here, the vibrational ensemble was obtained from a QM/MM AIMD trajectory carried out with AMBER14/Gaussian09, 39,40 where 2NN is in the QM part described at the B3LYP 23–26 /def2-SVP 27 level of theory and the classical MM part consist of 592 MeOH molecules. 41 The QM/MM trajectory was run at a temperature of 300 K using the Langevin thermostat and a pressure of 1 bar for 10 ps using a time step of 1 fs. The SHAKE algorithm was employed to constrain the movement of the MeOH molecules and allow for 1 fs time steps. Wigner/MM Sampling. Here, 100 structures of 2NN from the Wigner/COSMO sampling were taken and placed in an environment of MeOH molecules described using classical MM. For each of the 100 systems, then a classical MD trajectory is run at 300 K and 1 bar for a time of 10 ps with a time step of 2 fs, with 2NN frozen and the motion of the MeOH molecules constrained using the SHAKE algorithm. Table 1: Description of the system and the environment in the different sampling approaches. Sampling Excited States Method System Environment System Environment Wigner/COSMO Wigner COSMO QM COSMO MD/COSMO AIMD COSMO QM COSMO MD/MM AIMD MM QM MM Wigner/MM Wigner (frozen) MM QM MM



2.3

Scan Along the Nitro Group Torsion

A relaxed potential energy scan of the nitro group torsion of 2NN in MeOH was performed at the B3LYP 23–26 /def2-SVP 27 level of theory using the Gaussian09 program package. 40 Solvent effects were modeled implicitly using the polarizable continuum model (IEFPCM). 14,42 The nitro group torsion was defined by the dihedral angle of the atoms C11 -C14 -N18 -O20 , see Figure S3 in the SI. For the scan, the previously optimized geometry with an initial angle of 0◦ was used, and the angle was increased in steps of 10◦ up to a value of 90◦ . At each structure, excitedstate calculations were carried out at the PBE0/def2-TZVP level of theory using TURBOMOLE7.0 32 and the same parameters as described above.

2.4

Gas-Phase Calculations

The excited states/absorption spectra of 2NN were also calculated in gas phase at the PBE0/def2-TZVP level of theory using the solvent-relaxed geometries from the ensembles and the FC geometry in MeOH. Thus, we can directly evaluate the effect on the solvent of the electronic structure of the excited states by neglecting any further structural effects.

3 3.1

Results and Discussion Excited States at the Franck-Condon Geometry

We start our investigation of the absorption spectrum of 2NN in MeOH by analyzing the excited states at the FC geometry. Here, we will discuss only the excitedstate results obtained at the PBE0/def2-TVZP level of theory, as it yielded the best agreement with the experimental absorption spectrum; results obtained at

10Plus Environment ACS Paragon

Page 10 of 46

Page 11 of 46

the other levels of theory (B3LYP, PBE, and ADC2) can be found in Section S1 of the SI.

Intensity [arb. units]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 16 12 8 4 0 16 12 8 4 0

MeOH

Band 4 Band 3

(a)

Band 2 Band 1 S14 PBE0/def2-TVZP S10

S5

(b) S3

S1

200 250 300 350 400 450 Wavelength [nm]

Figure 1: (a) Experimental absorption spectrum 43 and (b) excited-state stick spectra at PBE0/def2-TZVP level of theory of 2NN in MeOH at the FC geometry. The experimental absorption spectrum of 2NN in MeOH, 43 see Figure 1a, shows four broad bands in the range of 200-400 nm that we label as band 1-4 in order of increasing energy. Bands 1, 2, and 4 possess their absorption maximum at wavelengths of 347 nm (3.57 eV), 301 nm (4.12 eV), and 211 nm (5.88 eV), respectively. Band 3 has a maximum at 261 nm (4.75 eV) and presents a shoulder at 255 nm (4.86 eV). The ratio between the intensities of the band maxima is approximately 1:3:9:17 for bands 1, 2, 3, and 4. Figure 1b shows the PBE0/def2-TZVP stick spectra of 2NN in MeOH calculated at the FC geometry with COSMO; corresponding excitation energies, oscillator strengths, and associated characters are in Table S2. For PBE0/def2TZVP, three bright excited states (S1 , S3 , and S5 ) are found with energies (3.38, 4.19 and 4.97 eV) and intensity ratios (1:2.7:6.1) similar to those of the three lowest-lying experimental absorption band maxima. This is best appreciated in 11Plus Environment ACS Paragon


Fig. 2, which compares the energies and intensity ratios of the bright states with the experimental data. The values of FC/COSMO show that the computed S1 7 E [eV] 6 5 4

0.56

Band 4

Band 3 Band 2 Band 1

0.22 0.09 −0.16

0.38

0.07 −0.11 −0.33

0.57

0.38

0.48

0.22

0.11

0.12

0.07

0.02

0.00

−0.18

−0.11

−0.11

3 Experiment

FC COSMO

Wigner Wigner MD COSMO MM COSMO

MD MM

Figure 2: Comparison of excitation energies and intensity ratios of the experimental absorption band maxima, FC bright states, and calculated absorption band maxima computed at the PBE0/def2-TZVP level of theory. The height of the boxes corresponds to the ratio of the intensity at the maxima and the oscillator strengths of the absorption bands and bright states, respectively; the heights of lowest-energy band/bright state are used as a reference. The difference of the calculated excited-state energies and calculated absorption band maxima to the experiment absorption maxima is indicated by the numbers above the boxes (see Tables S4 and S5 in the SI). state is underestimated by −0.16 eV, while the following two bands are only slightly overestimated by the computed S3 and S5 states by 0.09 and 0.22 eV, respectively. As the deviations between the experimental and computed values are small, PBE0 seems to be able to provide a good description of the lower-energy excited states of 2NN in MeOH at the FC geometry. However, the shoulder of band 3 cannot be explained, as there is only one excited state with notable oscillator strength, the S5 , in the energy range of band 3 (recall Fig. 1b). Thus, one may assume that the experimentally observed shoulder is due to vibronic transitions rather than due to another electronic state. According to Fig. 1b, the highest-energy band 4 should be assigned to the S14 state at the FC geometry. Notably, the energetic difference between the S14 and


Page 12 of 46

Page 13 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the maximum of the experimental band 4 (0.56 eV) is larger than the error of the other computed states. At slightly lower energy than the S14 , the S10 state is found with an oscillator strength half that of the S14 state. This finding may suggest two different scenarios. One is that the S10 may be a minor contribution to band 4. This contribution, however, is not directly visible in the spectrum, e.g., manifesting itself as a lower-energy shoulder of band 4. Alternatively, the S10 may be the state responsible for the higher-energy shoulder of experimental absorption band 3. However, this explanation is not very satisfactory either, given the good agreement between the energies of the other bright states and the experimental absorption maxima: being this scenario the case, the error of the S10 state with the shoulder would amount to 1.41 eV –which is abnormally large. Interestingly, when changing the solvent from MeOH to n-heptane, the experimental absorption band 3 splits into two distinct maxima, see Fig. S3a. The calculated excited states of 2NN at the FC geometry in n-heptane are displayed in Fig. 3b (see also Table S3). Intriguingly, in n-heptane the S14 state (corresponding to the S10 in MeOH of the same character), which could be potentially involved in the shoulder of band 3 in MeOH, has an oscillator strength only a tenth of that of the bright S12 state, thereby questioning its relevance in the absorption spectrum. One then concludes that the analysis of the excited states at the FC geometry cannot resolve the question about the nature of the shoulder of band 3, which could be then only attributed to vibronic transitions. The absence of a bright state in n-heptane corresponding to the S10 in MeOH at the FC geometry suggests that the S10 state in MeOH contributes to band 4 and not to the shoulder of band 3. Before discussing absorption spectra including sampling, we briefly comment on the results obtained at the PBE, B3LYP, and ADC(2) levels of theory (see 13Plus Environment ACS Paragon

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Intensity [arb. units]


20 16 12 8 4 0 16 12 8 4 0

Band 4 n-Heptane Bands 3a/b

(a)

Band 2 Band 1 S12 PBE0/def2-TVZP S5 S14

(b) S3

S1

200 250 300 350 400 450 Wavelength [nm]

Figure 3: (a) Experimental absorption spectrum 43 and (b) excited-state stick spectra at PBE0/def2-TZVP level of theory of 2NN in n-heptane at the FC geometry. also Section S1 in the SI). In general, B3LYP yields results similar to PBE0. The B3LYP wavefunction character and oscillator strengths are similar to that of PBE0, albeit energies are consistently red-shifted by ca. 0.1-0.2 eV. PBE and ADC(2) deviate more from the PBE0 results. The PBE bright states are redshifted from the PBE0 bright states by 0.4-0.6 eV. In contrast, the three lower ADC(2) bright states are blue shifted 0.1-0.5 eV; unlike the TDDFT calculations, ADC(2) does not predict any state with large oscillator strength corresponding to absorption band 4, but rather three states (S7 /S9 /S11 ) with medium intensity, which are found at similar energies as the PBE0 bright state S14 . From all these results, we conclude that the best description of the absorption spectrum of 2NN based on calculations at the FC geometry is given by PBE0, whose excitation energies deviate from the experimental bands by only 0.1-0.2 eV for the three lowest-energy bright states and 0.5 eV for band 4.


Page 14 of 46

Page 15 of 46

3.2

Absorption Spectra Using Vibrational Sampling

After investigating the excited states of 2NN in MeOH (and n-heptane) at the FC geometry, we simulated the absorption spectrum of 2NN in MeOH employing four different sampling approaches for the generation of vibrational ensembles (see Section 2.2). From the different levels of theory employed above (PBE0, B3LYP, PBE and the ADC(2)), here we only discuss the PBE0 results, as for the FC calculations PBE0 provided the most reasonable agreement with the experimental data; the results obtained at the other levels of theory can be found in Section S1.2 in the SI. The experimental and PBE0-calculated absorption spectra are superimposed in Figure 4. In all cases, the calculated absorption spectra exhibit four bands with maxima with energies and intensity ratios similar to the experimental ones, as also appreciated in Fig. 2. Intensity [arb. units]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


120 100 80 60 40 20 0

Wigner/COSMO Wigner/MM MD/COSMO MD/MM Exp

200 250 300 350 400 450 500 Wave Length [nm]

Figure 4: Experimental 43 and calculated absorption spectra of 2NN in MeOH using different sampling and solvent models. A clear trend is visible in the energies, regarding the description of the solvent effects: The maxima obtained with MM sampling appear at energies 0.10.2 eV higher than those with COSMO, for a given vibrational sampling method (compare Wigner/MM vs. Wigner/COSMO and MD/MM vs. MD/COSMO). By comparing with experiment, bands 1 and 2 are better described by MM, while bands 3 and 4 are better described by COSMO. In contrast, when comparing the 15Plus Environment ACS Paragon


sampling approaches for a given solvent model, no trend in energies is apparent. While the Wigner-based excitation energies are higher than the MD-based ones for the MM solvent model, the opposite is true for the COSMO solvent model. Yet, an important feature that can be observed in Fig. 2 is that the excited-state energies of the different ensembles are red-shifted with respect to the excitedstate energies computed at the FC geometry. This behaviour agrees with the general vibrationally-induced red-shift of the absorption spectrum observed in other chromophores. 5 In comparison to the experiment and considering all bands, the smallest unsigned mean error of the energies of the maxima is found for the MD/MM spectrum (0.17 eV). In addition, when comparing the intensity ratios, better agreement is achieved using the explicit MM solvent model versus COSMO. Therefore, the MD/MM combination seems to be slightly superior to the others in this case, although is fair to note that all four sampling/solvent combinations perform similarly and reasonably well regarding energies and intensities. Conspicuously, despite the small differences in the energies and intensity ratios, there are notable differences in the electronic characters of the states comprising the absorption bands; these will be discussed in detail in the next section. In passing we note that the trends identified for the PBE0 spectra also appear in most of the absorption spectra computed at the other levels of theory (B3LYP, PBE, and ADC(2), see Section S1.2 in the SI).

3.3

Characterization of the Absorption Bands

To characterize the excited states in the absorption bands of 2NN, we analyze the properties of the one-electron density matrix that connects the ground and excited states, using the TheoDORE program package. 33–36 Figure 5 shows the averaged 16Plus Environment ACS Paragon

Page 16 of 46

Page 17 of 46

10

Band 4

Band 3

Band 2

Band 1

(a) Wigner/COSMO

8

Hole Population

6 Electron Population

4 2

Differences

0 (b) Wigner/MM

8

Hole Population


4 Intnesity [arb. units ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

Differences

0 (c) MD/COSMO

8

Hole Population


4 2

Differences

0 (d) MD/MM

8

Hole Population


4 2

Differences

0 150

200

250 300 350 400 Wave Length [nm]

450

Figure 5: Average weighted atomic electron (blue) and hole (red) populations as well as electron/hole difference populations of the excitations contributing to the absorption bands. Size of the circles corresponds to the size of the populations, see also Tables S6-S10.



atomic hole and electron populations of the different electronic transitions for all the absorption bands, as well as the differences between the electron and hole populations. The averaging was performed in two steps: first, all states computed for 100 snapshots from each ensemble were assigned to one of the four absorption bands based on energetic criteria (see Table S6 in the SI). Then, the averaged hole and electron populations were computed as an oscillator-strength-weighted average including all states belonging to one band. Both, the individual populations and their difference are useful to describe different types of excitations. The difference is particularly useful to identify intramolecular charge-transfer (CT) excitations, where the hole and electron populations are located at different sites of the molecule. A significant hole/electron population difference indicates charge flow, while excitonic transitions show small differences close to zero. Note that an excitonic transition can be localized, with a hole and electron located at a specific site of the molecule (e.g., the nitro group), or delocalized with hole and electron spread throughout the whole molecule. With the help of such analysis one can immediately see that band 1 is described by a CT excitation from the aromatic ring (large hole population represented by blue circles) to the nitro group (large electron population represented by red circles). This picture is consistent along all sampling and solvent approaches (Figures 5a-d). Bands 2 and 3 also exhibit a certain degree of CT character but less significantly than that for band 1. In case of band 4, both the hole and electron populations are delocalized over the whole molecule resulting in a state with negligible charge-transfer contribution (delocalized excitonic transition). In order to quantify the characteristics of the absorption bands, we have also calculated a number of descriptors 33–36 and tabulated them in Figure 6: (i) the average CT numbers (labeled as CTN), (ii) the root-mean-square electron-hole 18Plus Environment ACS Paragon

Page 18 of 46

Page 19 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


separation (EH), and (iii) participation ratios (PR) of the states contributing to the different bands. The CTN measures the fraction of electron that is transferred between different fragments of the molecule and ranges from 0 (pure excitonic transition) to 1 (pure CT). To perform the analysis of the CTNs, we divide the molecule into two fragments consisting of the nitro group and the aromatic ring, respectively. 44 The root-mean-square electron-hole separation provides the size of the exciton formed after excitation by quantifying the extension of the charge displacement. Finally, the PR describes the number of fragments involved in the excited electron and ranges from 1 (localized excitation) to 2 (delocalized excitation). All CTN/EH/PR descriptors were weighted by the oscillator strengths of the corresponding states to account for the transition probability. Further details are given in Section S2. As can be seen in Figure 6(a), the weighted CTN (gray boxes) agree with the analysis presented in Figure 5, i.e., band 1 presents important CT character, with CTN ranging between 0.53-0.60 for the different samplings and solvent models. The pronounced CT character of band 1 is also manifested in the largest weighted exciton size (EH, blue boxes) of all sampling and solvent approaches, which lies between 4.0 and 4.2 ˚ A. Band 2 possesses similar features as band 1 and is also described by a ring-to-nitro-group CT excitation, recall Figure 5. However, the overall CT and EH separation in band 2 are smaller compared to those of band 1. For band 2 the weighted CTN decrease in average by about 0.14 and also the weighted exciton sizes are smaller by 0.3-0.5 ˚ A than the corresponding EH separation values of band 1 (Figure 6). This decrease can be understood by looking again at Figure 5: as the electron populations in the excitations of both bands are very similar, the CT decrease observed in band 2 is mainly due to larger hole populations at the nitro group and the adjacent carbon atom. 19Plus Environment ACS Paragon


(a) Absorption Bands in MeOH 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3

EH [˚ A]

CTN

PR

(i.) Wigner/COSMO

CTN = 0.59

CTN = 0.49 CTN = 0.31 CTN = 0.31

(ii.) Wigner/MM CTN = 0.53 CTN = 0.36 CTN = 0.37

CTN = 0.41

(iii.) MD/COSMO

CTN = 0.33 CTN = 0.31

CTN = 0.60

CTN = 0.41

(iv.) MD/MM

CTN = 0.56

CTN = 0.40 CTN = 0.36 CTN = 0.36

Band 4

Band 3

Band 2

Band 1

(b) Absorption Bands in Gas Phase 4 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0

(c) FC Bright States in MeOH (COSMO) EH [˚ A]

CTN

PR

4.6 4.4 CTN = 0.62 CTN = 0.49 4.2 4 CTN = 0.37 3.8 CTN = 0.27 3.6 3.4 3.2 3 S14 S5 S3 S1

Page 20 of 46

4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3 4.4 4.2 4 3.8 3.6 3.4 3.2 3

EH [˚ A]

CTN

PR

(i.) Wigner/COSMO CTN = 0.50 CTN = 0.35

CTN = 0.40 CTN = 0.40

(ii.) Wigner/MM CTN = 0.50 CTN = 0.35

CTN = 0.40 CTN = 0.40

(iii.) MD/COSMO

CTN = 0.35

CTN = 0.40

CTN = 0.45 CTN = 0.30

(iv.) MD/MM CTN = 0.49 CTN = 0.35

Band 4

CTN = 0.39

Band 3

CTN = 0.34

Band 2

Band 1

4 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0 3.5 3 2.5 2 1.5 1 0.5 0

(d) FC Bright States in Gas Phase 4 3.5 3 2.5 2 1.5 1 0.5 0

EH [˚ A]

CTN PR 4.6 4.4 4.2 CTN = 0.52 4 CTN = 0.42 3.8 CTN = 0.31 CTN = 0.31 3.6 3.4 3.2 3 S13 S5 S4 S1

4 3.5 3 2.5 2 1.5 1 0.5 0

Figure 6: Root-mean-square electron-hole separation (EH, blue boxes, left axis), charge-transfer numbers (CTN, gray boxes, labels), and fragment participation ratios (PR, red boxes, right axis) of the absorption bands and FC bright states in MeOH (a,c) and gas phase (b,d), respectively. See sections S2 and S4 in the SI.


Page 21 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Band 3 also possesses partial CT character, albeit much less pronounced than that of bands 1 and 2. The electron population in band 3 is spread over the whole molecule while the hole is only sited at the aromatic ring, resulting only in a small difference population (Figure 5). Compared to band 2, the weighted CTN of band 3 decrease further to 0.31-0.37, while the weighted exciton sizes EH are approximately the same with the exception of the Wigner/COSMO ensemble (Figure 6a). For the Wigner/COSMO ensemble, EH of band 3 decreases by ca. 0.3 ˚ A with respect to band 2. This large difference is not found for the other three ensembles. As all four ensembles show similar EH values for band 3, the large difference in the Wigner/COSMO ensemble is due to the large EH value of band 2. For band 4, Figure 6a shows CTN and exciton sizes that are very similar to those of band 3 independent of the sampling and solvent methods. As a notable difference between the characters of bands 3 and 4, we mention, however, the smaller EH difference populations in band 4, which are due to the fact, that for band 4, both, the electron and the hole populations are spread over the whole molecule. We now briefly discuss the PR (red boxes in Figure 6a), which, as mentioned above, quantify the degree of delocalization of the excited electron. All absorption bands present very similar PRs regardless the sampling and solvent approaches employed. Specifically, the PR values range from 1.42 to 1.63, indicating that at least half of the contributing transitions to the bright states are delocalized over the two fragments of the molecule (nitro group and aromatic ring). The same can be concluded by looking at the electron populations shown in Figure 5.



3.4

Characterization of the Excited States Calculated at the Franck-Condon Geometry

Here we analyze the character of the excited states calculated at the FC geometry and compare them with the results obtained from the absorption spectra when sampling is considered (Section 3.3). To this aim, the electron/hole populations as well as the CTN/EH/PR descriptors were calculated for the excited states at the FC geometry. The results are presented in Figure 7 –alongside with that obtained with Wigner/COSMO, as it uses the same solvent model (COSMO) as in the FC calculation. Thus, each box of Figure 7 shows the electronic features of the different absorption bands of the Wigner/COSMO spectrum next to the corresponding features of the electronic states involved in this band at the FC geometry. For example, the red box shows the electronic features of band 2 of the Wigner/COSMO absorption spectrum and that of the S2 and S3 states at the FC geometry. In the energy range of band 1 (blue box in Figure 7 and Table S6), there is only one state at the FC geometry, the S1 . Its properties resemble closely the averages of band 1, with CTN and exciton sizes of ca. 0.6 and 4.2 ˚ A, respectively. Thus, the absorption band 1 obtained with Wigner/COSMO seems well described by the S1 at the FC geometry, although its maximum is 0.16 eV lower in energy than the S1 . For band 2, there are two states in the corresponding energy range, S2 and S3 –the S4 is at an energy between bands 2 and 3 and we assign it to band 3. Inspecting the hole-electron difference population, one easily notices the close resemblance between the average of band 2 and the S3 state. This is not surprising as the oscillator strength of the S3 is much larger than that of the S2 (and the


Page 22 of 46

Page 23 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Band 1

S1

CTN = 0.62 CTN = 0.59 EH = 4.12 ˚ A EH = 4.23 ˚ A PR = 1.48 PR = 1.46 ∆E = 3.22 eV ∆E = 3.38 eV fosc = 0.091 Band 3

S4

Band 2

S2

S3

CTN = 0.23 CTN = 0.49 CTN = 0.49 EH = 3.83 ˚ A EH = 2.36 ˚ A EH = 3.98 ˚ A PR = 1.58 PR = 1.28 PR = 1.55 ∆E = 4.01 eV ∆E = 3.94 eV ∆E = 4.19 eV fosc = 0.000 fosc = 0.241 S5

S6

S7

CTN = 0.29 CTN = 0.27 CTN = 0.34 CTN = 0.45 CTN = 0.32 EH = 3.50 ˚ A EH = 2.37 ˚ A EH = 3.46 ˚ A EH = 3.11 ˚ A EH = 4.22 ˚ A PR = 1.42 PR = 1.41 PR = 1.37 PR = 1.41 PR = 1.50 ∆E = 4.82 eV ∆E = 4.56 eV ∆E = 4.97 eV ∆E = 5.11 eV ∆E = 5.20 eV fosc = 0.000 fosc = 0.527 fosc = 0.040 fosc = 0.023 Band 4

S8

S9

S10

S11

CTN = 0.29 CTN = 0.08 CTN = 0.43 CTN = 0.47 CTN = 0.31 EH = 3.58 ˚ A EH = 3.97 ˚ A EH = 3.41 ˚ A EH = 4.06 ˚ A EH = 4.32 ˚ A PR = 1.40 PR = 1.50 PR = 1.08 PR = 1.56 PR = 1.55 ∆E = 5.88 eV ∆E = 5.86 eV ∆E = 6.03 eV ∆E = 6.27 eV ∆E = 6.31 eV fosc = 0.000 fosc = 0.022 fosc = 0.000 fosc = 0.554 S12

S13

S14

S15

S16

CTN = 0.89 CTN = 0.48 CTN = 0.37 CTN = 0.17 CTN = 0.62 A A A A A EH = 3.50 ˚ EH = 4.65 ˚ EH = 3.83 ˚ EH = 3.54 ˚ EH = 2.89 ˚ PR = 1.16 PR = 1.88 PR = 1.59 PR = 1.24 PR = 1.53 ∆E = 6.44 eV ∆E = 6.47 eV ∆E = 6.52 eV ∆E = 6.66 eV ∆E = 6.85 eV fosc = 0.031 fosc = 0.987 fosc = 0.000 fosc = 0.062 fosc = 0.000 S17

S18

S19

S20

CTN = 0.15 CTN = 0.23 CTN = 0.14 CTN = 0.48 EH = 4.14 ˚ A EH = 3.21 ˚ A EH = 2.60 ˚ A EH = 4.19 ˚ A PR = 1.19 PR = 1.29 PR = 1.18 PR = 1.56 ∆E = 6.86 eV ∆E = 6.92 eV ∆E = 6.96 eV ∆E = 7.14 eV fosc = 0.154 fosc = 0.000 fosc = 0.093 fosc = 0.000

Figure 7: Electron/hole difference population and CTN/EH/PR descriptors of the computed absorption bands of 2NN in MeOH computed with Wigner/COSMO and for the excited states at the FC geometry that lie in the energetic range of the respective bands. Excitation energies (∆E) and oscillator strengths (fosc ) are added for the individual excited states.



S4 ). As for band 1 and S1 , also band 2 and its main contribution, the S3 state, share similar CTN, EH, and PR numbers, despite that the maximum of the band again is found at lower energies (0.18 eV) than the S3 . Belonging to the band 3, we assign the S4 , S5 , S6 , and S7 states at the FC geometry. From them, only the S5 is bright, and again is ca. 0.15 eV lower in energy than the maximum of band 3. While the hole-electron-difference population plots of band 3 and the S5 state are similar, their resemblance is less pronounced as for bands 1 and 2, indicating that other excited states than the S5 contribute to band 3. Interestingly, the CTN of S5 is smaller than that of band 3, while their exciton sizes are similar (around 3.5 ˚ A). As the CTN of the S5 is the smallest of all four states (S4 -S7 ), band 3 cannot be described solely by the S5 , but it needs also contributions from the other states with higher CTN. Thus, the similarity of the EH values of band 3 and S5 is just coincidental, as the EH of the S5 lies in between the EH values of the other states. Finally, we attribute the remaining 13 states (S8 -S20 ) computed at the FC geometry to the band 4. From them, only the S10 and S14 possess a notable oscillator strength. However, when comparing their properties to the average properties of band 4, no similarities can be found, indicating that there is a strong mixing of the characters of the states that contribute to the absorption band. Moreover, the maximum of the Wigner/COSMO sampling-based band is red shifted by 0.39 and 0.64 eV with respect to S10 and S14 at the FC geometry. We therefore conclude that in some cases the properties of the bright states in the band differ considerably from the bright states at the FC geometry. This clearly indicates that vibrational sampling modifies not only the excitation energies but also the electronic properties of the electronically excited states and it needs be taken into account for a correct description of the absorption spectrum. 24Plus Environment ACS Paragon

Page 24 of 46

Page 25 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.5

Influence of Nuclear Motion on the Position of the Absorption Bands

In the previous section, we have seen that bands 1 and 2 –and to a lesser extent band 3 –in the Wigner/COSMO sampling display a similar character to that of the bright states computed at the FC geometry. However, the maxima of the absorption bands are red-shifted compared to the FC bright states by 0.17 eV in average (Figure 2). Also for MD/COSMO, an average red shift of 0.07 eV for bands 1-3 compared to the FC bright states is found. For MD/MM, this red-shift decreases to 0.05 eV while the Wigner/MM bands in average are found at virtually the same energies than the FC bright states. Therefore, in general, a red-shift of the absorption bands is present when vibrational sampling is introduced. We now discuss this vibrationally-induced red-shift and its origin in detail. For this, we compare the Wigner/COSMO and MD/COSMO ensembles with the results at the FC geometry. As both ensembles and the calculations at the FC geometry employ the same solvent model (COSMO), their comparison allows to discuss solely the vibrational effects. The average ground-state energies for both ensembles with respect to the ground-state energy of the FC geometry is displayed in Figure 8a. As can be seen, when going from the FC geometry to the vibrational ensembles the ground-state energy increases. This increase is smaller for the MD/COSMO ensemble (1.26 eV) than for the Wigner/COSMO ensemble (1.76 eV). Also the energy of the excited states is increased in the ensembles, albeit in smaller amount, thereby resulting in the red-shift of the absorption bands. Accordingly, as shown in Figure 8a, the maximum of band 1 is found at 3.20 and 3.05 eV for the MD/COSMO and Wigner/COSMO ensembles, respectively, i.e., red-shifted with respect to the S1 state at the FC geometry (at



(a) Effect of Nuclear Motion

E

Band 1 S1 3.20 eV

hBand 1i

3.05 eV hS0i

3.38 eV

S0

S0

1.26 eV

FC COSMO

MD COSMO

0

1.76 eV Wigner COSMO

(b) Energy vs. RMSD (Wigner/COSMO)

7.5 7.0

Band 1 Band 2

Band 3 Band 4

6.5 Energy [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6.0 5.5 5.0 4.5 4.0 3.5 3.0 0.005

0.015 0.025 0.035 0.045 RMSD per Atom [A]

0.055

Figure 8: (a) Effect of nuclear motion on the averaged energy of absorption band 1 for the Wigner/COSMO and MD/COSMO ensembles. (b) Average energy of the states in the absorption bands 1-4 as a function of the RMSD of the molecule. Averaged energies in (a/b) weighted by the oscillator strength


Page 26 of 46

Page 27 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.38 eV) by 0.18 and 0.33 eV. Interestingly, there is also a correlation between the increase of the groundstate energy and the red-shift of the absorption band for each ensemble. This is shown for the Wigner/COSMO ensemble in Figure 8b, which plots the excitation energies of the four lowest-energy absorption bands as a function of the root-mean square deviation (RMSD) of the geometries of the ensemble with respect to the FC geometry. Clearly, the excitation energies of the absorption bands decrease with the increase of the RMSD, i.e., the larger the motion in the ensemble, the more red-shifted the absorption bands will be.

3.6

Influence of the Nitro Group Torsion on the Character of the Absorption Bands

In the previous section, we have explained the effects of nuclear motion on the energies of the absorption bands. In this section, we address the effects of nuclear motion on the electronic character of the absorption bands. To start, we show the superpositions of the solute geometries for each ensemble in Figure 9. The structures in each ensemble are almost planar, the largest deviation from the FC geometry being the out-of-plane torsion of the nitro group. Indeed, while displacements within the aromatic ring are very similar among the ensembles, the amount of torsion of the nitro group differs considerably between the ensembles. In order to illustrate these differences, we show a pictorial representation of RMSD of the whole molecule and the fragments in Figure 9, where the RMSD is represented by ellipses and the area of each ellipse corresponds to the size of the RMSD (see also Section S3.2). The RMSD of the ring system is ca. 0.02 ˚ A per atom for all ensembles. For the nitro group, the RMSD of the Wigner and



(a) Wigner/(COSMO/MM)

RMSD [˚ A] 0.028 0.020 0.021

(b) MD/COSMO 0.088 0.019 0.030 (c) MD/MM 0.072 0.021 0.029

Figure 9: Left-hand side: Superposition of the solute geometries of each ensemble. Right-hand side: pictorial representation of the RMSD of the structures in the ensembles with respect to the FC geometry of all atoms (gray), the nitro group (red), and the ring system (blue), respectively. The area inside each ellipse corresponds to the RMSD value per atom of all atoms/the fragments (see Table S13 in the SI). MD-based ensembles is ca. 0.03 and 0.08 ˚ A per atom, respectively. Thus, the RMSD values reflect the structural features already seen in the superpositions of the geometries: the motion of the nitro group is larger than that of the atoms of the aromatic ring, and the motion of the nitro group is larger for the MD ensemble than for the Wigner ensemble. The large motion of the atoms of the nitro group is easily explained considering that this motion is given mainly by the torsion of the whole nitro group, which is described by a low-energy mode with an harmonic frequency of 56 cm−1 at the FC geometry. As the displacement of a harmonic normal mode is inversely proportional to its frequency (e.g., the coordinate of the harmonic oscillator turning point xt is proportional to xt ∼ µ ω −1/2 ), the torsion of the nitro group leads to quite large displacements compared to the other normal modes of 2NN –which possess an average frequency of hωi = 1240 cm−1 . The low frequency of the nitro torsion also accounts the smaller motion in the Wigner ensemble: this frequency 28Plus Environment ACS Paragon

Page 28 of 46

Page 29 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


corresponds to a temperature of only TNO2 = h ¯ ωNO2 /kB = 80 K. Thus, the nitro group torsion in the Wigner ensemble possesses only about a fourth of the energy of the torsion in the MD ensemble (with a temperature of 300 K). Gratefully, the underestimation of the motion of the nitro group in the Wigner sampling does not have a large impact on the energies of the absorption bands as the absorption spectra computed for the different ensembles were quite similar (see, e.g., Figure 4). However, it has a substantial impact on the character of certain bands. In Figure 6, we show the CT analysis of the absorption bands for all ensembles. Comparing the Wigner/COSMO and MD/COSMO ensembles, we see that the properties of bands 1 and 3 are very similar in both ensembles but the CT character of band 2 (see CTN and exciton sizes) is considerably smaller for MD/COSMO than for Wigner/COSMO. In order to understand this behavior we calculated the CTNs and exciton sizes of the bright states (contributing to absorption bands 1-3) as a function of the NO2 torsion, see Figure 10. As can be seen in Figure 10a, the variation of the CTN and EH of the bright states is different for each state and depends on the region of the nitro torsional angle τ . The CTN of the S1 is almost constant for small torsional angles (ca. τ < 25◦ ) before increasing slowly with larger τ , while its exciton size remains constant throughout the complete torsion. Both, the CTN and exciton size of the S3 and S5 states are also nearly constant for small torsional angles, but around τ ≈ 10◦ both properties concomitantly decrease with increasing τ for the S3 . Figure 10c shows the distribution of the nitro group torsion τ obtained for the Wigner/COSMO and MD/COSMO ensembles. The Wigner/COSMO ensemble displays a narrow distribution with an average of hτ i = 7.8◦ and all τ < 25◦ . As expected, the distribution of the thermal MD/COSMO ensemble is broader and possesses a larger average torsional angle of hτ i = 11.7◦ with ca. 20 % of the 29Plus Environment ACS Paragon


1.0

S1

S3

S5

CTN

0.8 0.6 0.4

(a)

EH [A]

0.2

Nr. of Geometries

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.0 4.5 4.0 3.5 3.0 2.5 2.0 50 40 30 20 10 0

S1

S3

S5

(b) Wigner/COSMO MD/COSMO = 7.8° = 11.7° (c) 5 15 25 35 45 55 65 75 85 Nitro Group Torsion Angle τ [°]

Figure 10: (a) CT numbers and (b) exciton sizes of the FC bright states as a function of the nitro group torsion τ (taken from a relaxed potential energy surface scan). (c) Distribution of the nitro group torsion τ for the geometries in the WIGNER/COSMO and MD/COSMO ensembles together with average torsion angles hτ i.


Page 30 of 46

Page 31 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


structures showing τ > 25◦ . These facts allow to explain when the nitro group torsion will have an impact on the character of an absorption band, and when it will leave the character unaltered. On the one hand, the properties of the bright states S1 and S5 do not vary much for small torsional angles τ . Therefore, the larger torsion in the MD/COSMO ensemble does not have an impact on the character of the absorption bands 1 and 3, which are mainly described by these states. On the other hand, the larger nitro group torsion will have an impact on absorption band 2 that is mainly composed by the bright state S3 . This is consistent with the fact that the CT character of band 2 is much smaller in the MD/COSMO ensemble than in the Wigner/COSMO one, corresponding to the decrease in CT character of the S3 state with larger nitro group torsion angles. Further analysis of this scan for all the states is shown in Section S3.3. For the sake of completeness we also show the nitro group torsional angle distribution for the MD/MM ensemble alongside the other ensembles in Figure S8 in the SI, but we will only discuss the effects of the different solvent models in Section 3.7.

3.7

Solvent Effects on the Electronic Structure

So far, we have characterized the absorption bands of the different ensembles in terms of their average electronic-structure properties and discussed the effects of nuclear motion by comparing the results with the excited states at the FC geometry. This comparison was focused on contrasting the FC results with the Wigner/COSMO and MD/COSMO ones, since these descriptions employed the same solvent model and one could capture solely the effects of nuclear motion. In this section, we focus on the effects of the solvent on the electronic structure in the absorption spectrum of 2NN. Specifically, we address the differences introduced when going from gas phase to solution as well as when changing the 31Plus Environment ACS Paragon


solvent model from implicit (COSMO) to explicit (MM). To this aim, we have also calculated the excited states of 2NN in gas phase, where we have used the geometries from the ensembles in solution and the FC geometry in MeOH, thus allowing us to disentangle the solvent effects on the electronic structure of the excited states from geometrical effects. The results obtained for the FC geometry

Insensity [arb. Units]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

20 COSMO 16 S13 S14 GAS 12 S5 S5 8 (a) FC Geometry S4 S3 4 S1 S1 0 COSMO 16 MM GAS 12 8 (b) WIGNER 4 0 COSMO 16 GAS 12 8 (c) MD/COSMO 4 0 MM 16 GAS 12 (d) MD/MM 8 4 0 150 200 250 300 350 400 450 Wave Length [nm]

Figure 11: Exicted states and absorption spectra calculated at the MeOHoptimized Franck-Condon geometry (a) and using the geometries from the ensembles (b-d). TDDFT calculations were performed using a solvent model (COSMO, MM) or for the molecule in gas phase (GAS). is shown in Figure 11a. The absorption spectra in gas phase and in MeOH –using 32Plus Environment ACS Paragon

Page 32 of 46

Page 33 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


both MM and COSMO solvent models –for the Wigner ensemble are shown in Figure 11b, while the corresponding spectra of the MD/COSMO and MD/MM vibrational ensembles in gas phase and MeOH are shown in Figures 11c and 11d, respectively. Additionally, in Section S4, the energies and intensity ratios of the FC bright states and band maxima (Figure S9) as well as a detailed analysis of all FC excited states in the gas phase (Figure S10) are presented. Figure 11 illustrates that, in general, the FC bright states and the absorption bands are red-shifted when going from gas phase (red lines) to solution (blue lines). This red-shift is in average 0.23 eV when using COSMO but smaller for MM (0.06 eV); it depends, however, on the specific bands. It is largest for the bright state S1 and band 1, and it decreases for the higher-energy bright states and absorption bands, even leading to a small blue shift of 0.02-0.03 eV for the bands 4 in the Wigner and MD/MM ensembles. To understand the differences in solvent-induced red-shifts on the absorption bands, we can consider the following simplistic model. In general, CT states are energetically stabilized in polar media, and the stabilization is proportional to the amount of CT. In a rather conceptional picture, this also applies to single electronic configurations, so that CT configurations move to lower energies in more polar media, while the energy of configurations with small CT character stays constant. Thus, upon solvation in a polar medium, the CT character of the low-lying states increases even more at the expense of the CT character of the high-lying states. To see how the amount of CT character changes from gas phase to solution, the CTN and exciton sizes of all bands in gas phase and solution are collected in Figure 6a,b. Indeed, it can be seen that the CT numbers and exciton sizes increase for the low-energy bands 1 and 2 when going from gas phase to solution 33Plus Environment ACS Paragon


while they decrease for the high-energy bands 3 and 4. This behavior is found not only considering the complete absorption spectrum, but it is also manifested for the states within each absorption band. In the absorption bands, the lower-energy part is comprised by the states with higher CT character while the high-energy part contains the states with lower CT character. We show this in Figure S11 exemplarily for the Wigner/COSMO obtained bands. As pointed out above, the energetic differences between gas phase and solution are larger when using the COSMO solvent model than when using the MM solvent model. The same trend is also found for the state characters, i.e., the CTNs and exciton sizes differ more between gas phase and COSMO than between gas phase and MM. As an example, the average difference between the CTNs of the COSMO and gas-phase absorption bands is ∆CT = 0.09, while this difference is only ∆CT = 0.03 comparing the MM and gas phase results. In QM/continuum models there is a mutual polarization between the QM part and the (classical) solvent, while in electrostatic-embedding QM/MM only the QM region is polarized by the (classical) solvent. Thus, the energy shift and change of character of the absorption bands induced by electrostatic interactions with the bulk solvent should be described better by QM/COSMO than by electrostatic-embedding QM/MM. It is interesting to note that the difference between COSMO and gas phase results is even more pronounced for the bright states (∆CT = 0.12) at the FC geometry (Figure 6c,d) than it is between the absorption bands (Figure 6a,b) –although the energetic differences between gas phase and COSMO results are very similar for the FC bright states and the absorption bands (Figure S9). Thus, a simplistic analysis of the bright states at the FC geometry overestimates the effect of the solvent on the character of the absorption band despite providing the 34Plus Environment ACS Paragon

Page 34 of 46

Page 35 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


correct solvatochromic energy shift. Similarly, it has been previously shown by us that, also for nitrobenzene, the analysis of the electronic structure at the FC geometry provided right energetics but wrong CT character when compared with an ensemble of geometries. 1 Finally, we also analyzed the effect of explicit interactions with solvent molecules located in the first solvation shell in our calculated absorption spectra. However, the effect of such explicit interactions on both the energies and the characters of the absorption bands is very small and, therefore, negligible compared to the differences introduced by changing the solvent model (see Section S4).

4

Summary and Conclusion

A comprehensive study of the effects of vibrational sampling, solvent model, and their interplay on the absorption spectrum of the organic compound 2nitronaphthalene in methanol is reported. Initially, a number of quantum chemical methods was investigated using the equilibrium geometry to find that the PBE0/def2-TZVP level of theory was best suited to describe the absorption spectrum in the energy range between 200-400 nm. Four electronic excited states were found with energies and oscillator strengths in very good agreement with the positions and intensity ratios of four experimental absorption band maxima. Only the appearance of a shoulder on absorption band 3, likely caused by vibronic transitions, could not be explained by the excited state calculations performed at a single geometry. The good performance of PBE0/def2-TZVP was then validated by simulating the complete absorption spectrum using different vibrational sampling techniques (Wigner distribution and MD) as well as solvent models (implicit solvent with



COSMO and explicit solvent via a force field). We have shown that while the two lower-energy bands possess very similar electronic properties compared to the lower-energy bright states at the FC geometry, the higher-energy absorption bands showed pronounced state mixing caused by vibrational motion. We have found that the absorption bands of the vibrational ensembles are red-shifted compared to the FC bright states, and that this red-shift is due to vibrational motion, with the size of the red-shift scaling directly with the displacement from the FC geometry. This displacement, and thus the red-shift of the bands, is larger when we use a harmonic Wigner distribution based sampling than snapshots from a molecular dynamics trajectory. Since the zero-point energy in Wigner sampling captures a larger amount of the total energy than the thermal energy in the MD sampling, Wigner sampling gives a better account of the extent of displacements and, thus, the red-shift of the bands. An important motion of the vibrational ensemble is the low-frequency nitro group torsion. This torsion affects the character of some absorption bands by quenching the charge-transfer character, which in turn depends on the extent of the torsion. The influence of the torsion is not homogeneous along the different bands and does not have a large effect on the energetics, so it is easy to miss its importance for the correct description of the character of the absorption bands. In contrast to high-frequency modes, the extent of the low-frequency nitrogroup torsion is underestimated in the Wigner ensemble and, as a consequence, the charge-transfer character of the absorption bands is slightly overestimated. Therefore, none of the vibrational sampling approach –Wigner and MD-based sampling –is able to capture all important features introduced by the nuclear motion in the absorption spectrum. In combination with the two vibrational sampling approaches, we have stud36Plus Environment ACS Paragon

Page 36 of 46

Page 37 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ied the performance of implicit and explicit solvent models. We find that both COSMO and force field solvent models induce a red-shift of the absorption bands compared to the gas phase spectrum. This red-shift is larger for the implicit COSMO solvent model than for the explicit MM solvent model, and it depends on the character of the absorption band: the larger the charge-transfer character, the larger the red-shift. Compared to experiment, we found that the lowerenergy bands 1 and 2 are better described using the MM solvent model, while the higher-energy bands 3 and 4 show better agreement with experiment when using COSMO. This is unexpected considering the character of these bands: the lowlying energy bands possess a more pronounced charge-transfer character than the high-lying ones. The larger charge redistribution in an excited state with larger charge-transfer character should be better described with the polarizable COSMO solvent model than by using a fixed-charge force field. Considering all four sampling combinations, the best agreement with the energies and intensities of the experimental absorption bands is achieved using molecular dynamics in MM solvent as sampling technique and QM/MM for computing the excitation energies. This is a curious conclusion since we have seen that MD underestimates the red-shift of the bands induced by vibrational motion and MM also underestimates the red-shift of charge-transfer states induced by interactions with the polar solvent. Therefore, the good performance of MD sampling in MM solvent followed by QM/MM excited-state calculations relies on strong error cancellation using the PBE0/def2-TZVP level of theory. The challenge in capturing all features when simulating an absorption spectrum thus lies in choosing the best combination of the electronic structure method, vibrational sampling technique, and solvent model, to properly and simultaneously describe the energies, intensities and electronic properties of different ab37Plus Environment ACS Paragon


sorptions bands. As we have seen for the case of 2-nitronaphthalene, this choice is complicated as regrettably depend on the particular absorption band. The insights gained in the present work are expected to guide making educated choices in future studies for related nitro-aromatic compounds.

Supporting Information Further results obtained at the FC geometry with different levels of theory (Section S1), TheoDORE analysis of the sampling results (Section S2), analysis of the effects of nuclear motion (Section S3), and analysis of solvent effects (Section S4) can be found in the Supporting Information. This Supporting Information is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement JPZ is a recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Theoretical Chemistry at the University of Vienna. The computational results presented have been partially achieved using the Vienna Scientific Cluster (VSC). The authors also thank the University of Vienna for financial support.

References (1) Zobel, J. P.; Nogueira, J. J.; González, L. Quenching of Charge Transfer in Nitrobenzene Induced by Vibrational Motion. J. Phys. Chem. Lett. 2015, 6, 3006–3011.


Page 38 of 46

Page 39 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(2) Nogueira, J. J.; Plasser, F.; González, L. Electronic Delocalization, Charge Transfer and Hypochromism in the UV Absorption Spectrum of Polyadenine Unravelled by Multiscale Computations and Quantitative Wavefunction Analysis. Chem. Sci. 2017, 8, 5682–5691, DOI: 10.1039/C7SC01600J. (3) He, Y.; Gahlmann, A.; Feenstra, J. S.; Park, S. T.; Zewail, A. H. Ultrafast Electron Diffraction: Structural Dynamics of Molecular Rearrangement in the NO Release from Nitrobenzene. Chem. Asian J. 2006, 1, 56–63. (4) Barbatti, M.; Sen, K. Effects of Different Initial Condition Sampling on Photodynamics and Spectrum of Pyrrole. Int. J. Quant. Chem. 2016, 116, 762–771. (5) Nogueira, J. J.; González, L. Computational Photophysics in the Presence of an Environment. Annu. Rev. Phys. Chem. 2018, in press, DOI: 10.1146/annurev-physchem-050317-021013. (6) Adcock, S. A.; McCammon, J. A. Molecular Dynamics: Survey of Methods for Simulating the Activity of Proteins. Chem. Rev. 2006, 106, 1589–1615. (7) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys. 1953, 21, 1087–1092. (8) Wigner, E. On the Quantum Correction for Thermodynamic Equilibrium. Phys. Rev. 1932, 40, 749–759. (9) Dahl, J. P.; Springborg, M. The Morse oscillator in position space, momentum space, and phase space. J. Chem. Phys. 1988, 88, 4535–4547.



(10) Hillery, M.; O’Connell, R. F.; Scully, M. O.; Wigner, E. P. Distribution Functions in Physics: Fundamentals. Physics Reports 1984, 106, 121–167. (11) Bonaˇcić-Kouteck´ y, V.; Mitrić, R. Theoretical Exploration of Ultrafast Dynamics in Atomic Clusters: Analysis and Control. Chem. Rev. 2005, 105, 11–65. (12) Klaffki, N.; Weingart, O.; Garavelli, M.; Spohr, E. Sampling excited state dynamics: influence of HOOP mode excitations in a retinal model. Phys. Chem. Chem. Phys. 2012, 14, 14299–14305. (13) Ruckenbauer, M.;

Barbatti, M.;

M¨ uller, T.;

Lischka, H. Nona-

diabatic Excited-State Dynamics with Hybrid ab Initio QuantumMechanical/Molecular-Mechanical Methods: Solvation of the Pentadieniminium Cation in Apolar Media. J. Phys. Chem. A 2010, 114, 6757–6765. (14) Miertuˇs, S.; Scrocco, E.; Tomasi, J. Electrostatic interaction of a solute with a continuum. A direct utilizaion of ab initio molecular potentials for the prevision of solvent effects. Chem. Phys. 1981, 55, 117–129. (15) Klamt, A.; Sch¨ uu ¨rmann, G. COSMO: a new approach to the dielectric screening in solvents with explicit expressions for the screening energy and its gradient. J. Chem. Soc. Perkin Trans. 2 1993, 799–805. (16) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Models. Chemical Reviews 2005, 105, 2999–3094. (17) Nogueira, J. J.; Corani, A.; El Nahhas, A.; Pezzella, A.; d’Ischia, M.; González, L.; Sundström, V. Sequential Proton-Coupled Electron Transfer Mediates Excited-State Deactivation of a Eumelanin Building Block. J. Phys. Chem. Lett. 2017, 8, 1004–1008. 40Plus Environment ACS Paragon

Page 40 of 46

Page 41 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(18) Barbatti, M. Photorelaxation Induced by Water-Chromophore Electron Transfer. J. Am. Chem. Soc. 2014, 136, 10246–10249. ˇ (19) Szabla, R.; Sponer, J.; Góra, R. W. Electron-Driven Proton Transfer Along H2 O Wires Enables Photorelaxation of States in Chromophore Water Clusters. J. Phys. Chem. Lett. 2015, 6, 1467–1471. (20) Nogueira, J. J.; Oppel, M.; González, L. Enhancing Intersystem Crossing in Phenotiazinium Dyes by Intercalation into DNA. Angew. Chem. Int. Ed. 2015, 54, 4375–4378. (21) Nogueira, J. J.; Meixner, M.; Bittermann, M.; González, L. Impact of Lipid Environment on Photodamage Activation of Methylene Blue. ChemPhotoChem 2017, 1, 178–182. (22) Etienne, T.; Very, T.; Perpète, E. A.; Monari, A.; Assfeld, X. A QM/MM Study of the Absorption Spectrum of Harmane in Water Solution and Interacting with DNA: The Crucial Role of Dynamic Effects. J. Phys. Chem. B 2013, 117, 4973–4980. (23) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652. (24) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlationenergy formula into a functional of the electron density. Physical Review B 1988, 37, 785–789. (25) Vosko, S. H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Cand. J. Phys. 1980, 58, 1200–1211. 41Plus Environment ACS Paragon


(26) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623–11627. (27) Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297–3305. (28) Schirmer, J. Beyond the random-phase approximation: A new approximation scheme for the polarization propagator. Phys. Rev. A 1981, 26, 2395– 2416. (29) Trofimov, A. B.; Schirmer, J. An efficient polarization propagator approach to valence electron excitation spectra. J. Phys. B 1995, 28, 2299–2324. (30) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (31) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158– 6170. (32) TURBOMOLE V7.0 2015, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007; available from http://www.turbomole.com. (33) Plasser, F.; Lischka, H. Analysis of Excitonic and Charge Transfer Interactions from Quantum Chemical Calculations. J. Chem. Theory Comput. 2012, 8, 2777–2789.


Page 42 of 46

Page 43 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(34) Plasser, F.; Wormit, M.; Dreuw, A. New tools for the systematic analysis and visualization of electronic excitations. I. Formalism. J. Chem. Phys. 2014, 141, 024106. (35) Mewes, S. A.; Mewes, J.-M.; Dreuw, A.; Plasser, F. Excitons in poly(para phenylene vinylene): a quantum-chemical perspective based on high-level ab initio calculations. Phys. Chem. Chem. Phys. 2016, 18, 2548–2563. (36) Plasser, sity,

F.

orbital

TheoDORE relaxation,

1.4: and

a

package

exciton

for

theoretical

den-

analysis;

available

from

http://theodore-qc.sourceforge.net. (37) Ufimtsev, I. S.; Mart´ınez, T. J. Quantum Chemistry on Graphical Processing Units. 3. Analytical Energy Gradients, Geometry Optimization, and First Principles Molecular Dynamics. J. Chem. Theory Comput. 2009, 5, 2619– 2628. (38) Titov, A. V.; Ufimtsev, I. S.; Luehr, N.; Mart´ınez, T. J. Generating Efficient Quantum Chemistry Codes for Novel Architectures. J. Chem. Theory Comput. 2013, 9, 213–221. (39) Case, D. A.;

Betz, R. M.;

Botello-Smith, W.;

Cerutti, D. S.;

Cheatham III, T. E.; Darden, T. A.; Duke, R. E.; Giese, T. J.; Gohlke, H.; Goetz, A. W.; Homeyer, N.; Izadi, S.; Janowski, P.; Kaus, J.; Kovalenko, A.; Lee, T. S.; LeGrand, S.; Li, P.; Lin, C.; Luchko, T.; Luo, R.; Madej, B.; Mermelstein, D.; Merz, K. M.; Monard, G.; Nguyen, H.; Nguyen, H. T.; Omelyan, I.; Onufriev, A.; Roe, D. R.; Roitberg, A.; Sagui, C.; Simmerling, C. L.; Swails, J.; Walker, R. C.; Wang, J.; Wolf, R. M.; Wu, X.;



Xiao, L.; York, D. M.; Kollman, P. A. AMBER 2016, University of California, San Francisco. 2016. (40) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; ¨ Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Daniels, A. D.; Farkas, O.; Fox, D. J. Gaussian 09, Revision E.01, Gaussian, Inc., Wallingford CT. 2009. (41) Caldwell, J. W.; Kollman, P. A. Structure and Properties of Neat Liquids Using Nonadditive Molecular Dynamics: Water, Methanol, and NMethylacetamide. J. Phys. Chem. 1995, 99, 6208–6219. (42) Mennucci, B. Polarizable continuum model. WIREs Comput. Mol. Sci. 2012, 2, 386–404. (43) Kojima, M.; Nagakura, S. Electronic Spectra and Electronic Structures of


Page 44 of 46

Page 45 of 46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


α- and β-Nitronaphthalenes and 1,8-Dinitronapthalene. Bull. Chem. Soc. Jpn. 1966, 39, 1262–1269. (44) Mai, S.; Plasser, F.; Dorn, J.; Fumanal, M.; Daniel, C.; González, L. Quantitative wave function analysis for excited states of transition metal complexes. Coord. Chem. Rev. 2018, 361, 74–97.



Graphical TOC Entry


Page 46 of 46

Vibrational Sampling and Solvent Effects on the ... - ACS Publications

Recommend Documents