Article pubs.acs.org/JPCB
Interaction of Dark Excited States. Comparison of Computational Approaches Alexander A. Voityuk* Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain Institut de Química Computacional i Catàlisi (IQCC), Universitat de Girona, 17071 Girona, Spain ABSTRACT: A systematic theoretical study of the electronic interaction of dark excited states in a model system, formaldehyde dimer is reported. Using the fragment transition density scheme, we estimate the excitonic interaction in different configurations of the dimer. The excited state properties of the system are computed with several quantum mechanical methods. We show that the orbital interaction of the monomers rather than Coulomb interaction of their transition quadrupoles gives the major contribution to the coupling at intermolecular distances shorter than 5 Å. It is found that the exitonic interaction alters drastically by conformational changes. Benchmark couplings computed with EOM CCSD, MS-CASPT2, CASSCF, TD DFT, CIS, and INDO/S and different basis sets are provided. The evaluation of the calculations shows that the TD cam-B3LYP scheme performs best, giving good estimates for all considered structures. In contrast, the TD B3LYP scheme leads to drastically overestimated values. The data obtained using the Tamm−Dancoff approximation are similar to the TD DFT results. CASSCF and CIS calculations underestimate the coupling, indicating that dynamic electron correlation may have a large effect on the short-range coupling. The INDO/S method fails to describe the excited state interaction both at short and long distances.
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INTRODUCTION Excitation energy transfer (EET) is one of the important processes in biological systems and in nanomaterials.1,2 The rate of EET is controlled by the electronic interaction of excited states involved in the process. Usually one considers “bright” excited states which are easily accessible by standard spectroscopic methods. Their excitonic coupling is determined by Coulomb interaction of the transition dipole moments of the chromophores. The theory of Förster resonance energy transfer was recently extended for multichromophoric systems.3 The Coulomb interaction of dark excited states is determined by transition quadrupoles and higher multipoles and can be estimated using transition charges.1 Such an approach can be applied when the intermolecular distance is quite large and the orbital interaction between the chromophores is negligibly small. In many interesting cases, however, the distance between molecules involved in EET is shorter than 5 Å, and the electrostatic model becomes inaccurate.1,4,5 To estimate the coupling that includes both the electrostatic and orbital terms, quantum mechanical (QM) calculations of the whole system should be performed. Then, the fragment transition density (FTD) scheme6 can be applied to derive the excitonic coupling from the QM results. FTD can be used in combination with different quantum mechanical methods to treat systems containing two or several chromophores. The scheme is based on the orthogonal transformation of adiabatic states {ψ} to diabatic states {φ} introduced by London7 and employed later by other researchers to explore ET and EET processes.8−16 The nature of the diabatic states {φ} that correspond to initial and final states in ET and EET as well as © XXXX American Chemical Society
the relation of these states to the dynamic diabatic states {Φ}, which minimize the coupling ⟨Φi|∇R|Φj⟩, are considered in the literature.10,17−19 The application of accurate ab initio methods such as EOM CCSD and MS-CASPT2 is still limited in practice to small systems (for instance, CCSD formally scales as N6 with the number of orbitals N). Therefore, approximate methods should be applied to larger systems. In this work, the FTD scheme is used together with EOM CCSD (equation of motion couple cluster method for singles and doubles), MS-CASPT2 (multistate complete-active-space second-order perturbation theory), CASSCF, TD DFT, CIS and INDO/S calculations to obtain benchmark couplings for dark excited states in the formaldehyde dimer as the test case. The evaluation of present calculations suggests that that TD DFT approach with the CAM-B3LYP functional performs best providing good estimates for all considered structures. In contrast, TD DFT calculations with the uncorrected B3LYP functional give drastically overestimated coupling values. The CASSCF and CIS methods underestimate the coupling. The INDO/S method fails to describe the coupling of the dark states both at short and long distances. Model. The planar and crossed conformations of the formaldehyde dimer considered in the paper are shown in Figure 1. Both structures belong to the C2V symmetry group (Z Special Issue: John R. Miller and Marshall D. Newton Festschrift Received: November 4, 2014 Revised: December 13, 2014
A
DOI: 10.1021/jp511035p J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B
⎛T A T BA ⎞ T = ⎜⎜ ⎟⎟ ⎝T AB T B ⎠
(2) 6
More details are provided in ref 6. The scheme can easily be implemented and used in combination with any QM method. Our approach differs radically from the method of Hsu et al.26 where the transition density of the system AB is approximated by the block-diagonal matrix constructed from the transition densities of noninteracting chromophores and the coupling is approximated by the first-order correction. The scheme26 was successfully applied in several works including the study of solvation effects on excitonic coupling.27,28 However, it cannot provide accurate estimates for systems where the orbital and exchange interactions dominate the coupling (in particular, for dark states).
Figure 1. Planar and crossed conformations of the formaldehyde dimer.
is the C2 axis, all atoms of the planar dimer are in YZ plane). The calculations were performed for the intermolecular distance R of 4.0, 5.0, and 6.0 Å. In formaldehyde, the lowest 1A2 excited state is a real dark state (the A1 → A2 transition is forbidden by symmetry) and cannot be populated from the ground state by light absorption. This state is well-separated from higher excited states, the energy gap is 5 eV. Because of that, clusters containing two or more formaldehyde molecules are good models to study electronic interaction of dark states. In particular, the dimer can be properly treated within a twostate model that includes the 1A2 state of each monomer. Method. Quantum Mechanical Calculations. The model systems were computed by Gaussian 0920 using several quantum mechanical methods EOM CCSD, TD DFT, TDA DFT, CIS, and INDO/S. Three basis sets 6-31G*, cc-pVTZ, and aug-cc-pVTZ were employed. The CASSCF and MSCASPT2 calculations with the ANO-s basis sets were performed with the MOLCAS program.21 Note that the MSCASPT2 energies are calculated at the PT2 level, while the transition densities do not contain explicit PT2 corrections (some effects of electron correlation are accounted for by the MS scheme). The TD DFT calculations were carried out using the B3LYP22 and CAM-B3LYP functionals.23 B3LYP shows a better performance as compared to other functionals when calculating excited state properties of individual molecules.24 However, the energy of charge transfer states in molecular complexes is significantly underestimated. The situation may be improved by using the long-range corrected functional, CAMB3LYP. In addition to the standard TD DFT, several calculations were carried out using the Tamm−Dancoff approximation (TDA), which neglects contributions to the excitation energies coming from the correlated ground state and is comparable to the CIS scheme. The TDA can improve the stability of the TDDFT calculations.25 Estimation of Exitonic Coupling. Within the FTD method,6 the electronic interaction between two chromophores A and B can be estimated as
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RESULTS AND DISCUSSION Let us briefly consider excited states of the formaldehyde molecule computed with different QM methods. The formaldehyde molecule belongs to the C2V symmetry group. The lowest excited state 1A2 is well-described by single excitation n → π*; n and π* MO are of B1 and B2 symmetry. The dark state 11A2 cannot be populated by light absorption (in the dipole approximation, the transition A1 → A2 is forbidden by symmetry). QM calculations predict a gap of c.a. 5 eV between the first excited state A2(n→π*) and higher states (Table 1). Because of the large gap, electronic interaction Table 1. Singlet Excitation Energies in Formaldehyde (eV) method EOM CCSDa CISa CISb TD B3LYPa TD B3LYPb TD CAM-B3LYPa TD CAM-B3LYPb INDO/S a
(T2A T1A − T2BT1B) [(T2A T2A
−
T2BT2B‐T1A T1A
2A1(π→π*)
3.94 4.51 4.56 3.85 3.99 3.84 3.96 3.11
8.07 9.44 10.01 7.33 9.62 7.81 9.71 8.62
1B1(σ→π*) 9.24 9.64 9.69 8.82 8.90 8.92 9.06 8.82
aug-cc-pVTZ basis set. b6-31G* basis set.
between monomers in formaldehyde clusters should not lead to contamination of this state by states of higher energy and a cluster containing two or more formaldehyde molecules appears to be a good model to study electronic interaction of dark excited states in molecular complexes. Different from triplet excited states where all S0 → Ti transition multipoles are zero, the coupling of singlet dark states includes two contributions (1) the orbital + exchange interaction, which decays exponentially with the distance and becomes small at intermolecular distances, R, longer than 5 Å, and (2) the electrostatic interaction of transition quadrupoles (and higher multipoles), which is proportional to 1/R5. In formaldehyde, the 1A1 → 1A2 transition quadrupole moment has only one nonzero component Qxy. Its value increases by extension of the basis set used in the calculation. For instance, the CIS calculations with the 6-31G* and aug-cc-pVTZ basis sets give 1.052 and 1.253 D Å, respectively. Thus, the computed quadrupole−quadrupole coupling, which is proportional to the square of Qxy, will significantly increase by including diffuse functions in the basis set. On the other hand, Qxy is rather insensitive to the QM method. The TD B3LYP
V = (E2 − E1) ×
1A2(n→π*)
+ T1BT1B)2 + 4(T2A T1A − T2BT1B)2 ]1/2
(1)
where the E1 and E2 are excitation energies of the systems, and vectors TXi (X = A, B) are constructed using the transition density matrix T for excitations ψ0 → ψ1 and ψ0 → ψ2. In the dimer AB, TAi and TBi are the diagonal blocks of Ti containing N2A and N2B elements (NX is the number of AOs in the fragment X, X = A or B) extracted from the transition density computed for the entire system B
DOI: 10.1021/jp511035p J. Phys. Chem. B XXXX, XXX, XXX−XXX
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Table 2. Exitonic Coupling of the 1A2 Dark States in the Planar (P) and Crossed (C) Conformations of the Formaldehyde Dimer (in meV) R = 4.0 Å
R = 5.0 Å
R = 6.0 Å
method
P
C
P
C
P
C
EOM-CCSD/aug-cc-pVTZ EOM-CCSD/cc-pVTZ MS-CASPT2/ANO-s CASSCF/ANO-s CIS/aug-cc-pVTZ CIS/cc-pVTZ CIS/6-31G* TD B3LYP/aug-cc-pVTZ TD B3LYP/cc-pVTZ TD B3LYP/6-31G* TDA B3LYP/aug-cc-pVTZ TD CAM-B3LYP/aug-cc-pVTZ TD CAM-B3LYP/cc-pVTZ TD CAM-B3LYP/6-31G* TDA CAM-B3LYP/aug-cc-pVTZ TDA CAM-B3LYP/6-31G* INDO/S
37.3 30.9 38.0 22.7 24.7 22.6 19.3 140 129 111 145 53.0 46.3 38.7 54.3 38.8 2.52
6.41 5.52 4.18 4.05 6.42 5.82 4.58 4.81 4.17 3.19 5.70 4.91 4.25 4.01 6.12 4.05 4.0 × 10−3
2.65 1.75 2.68 1.55 1.98 1.56 1.15 5.56 3.54 2.15 5.86 3.18 2.10 1.48 3.38 1.47 1.3 × 10−2
1.45 1.10 0.99 0.83 1.41 1.17 0.91 1.33 1.04 0.80 1.51 1.33 1.04 0.89 1.53 0.89 0
0.478 0.327 0.426 0.253 0.430 0.343 0.269 0.593 0.352 0.250 0.652 0.495 0.326 0.268 0.554 0.271 2.2 × 10−7
0.424 0.312 0.388 0.305 0.411 0.337 0.267 0.419 0.308 0.240 0.474 0.412 0.305 0.262 0.470 0.267 0
of B2 symmetry. The orbital interaction between the monomers results in the short-range term of the excited state coupling. The situation is quite different for the crossed dimer. Although both n → π* excited states are of A2 symmetry (as in the planar system), the n orbitals of the monomers are of different symmetry (B1 and B2) and cannot mix (Figure 2 right panel). Also, π* MOs of the molecules differ by symmetry (B2 and B1) and do not combine. As a result, the short-range term of the coupling considerably decreases by passing from the planar (P) conformation to the crossed (C) one (Table 2). As the orbital interaction dominates the coupling at R = 4.0 Å, a large difference in the coupling is found for the P and C dimers. By contrast, at R = 6.0 Å, the quadrupole−quadrupole interaction gives the main contribution to the coupling. The magnitude of the orbital interaction can be estimated by comparing coupling values V in the planar and crossed dimers. The difference of V(P) and V(C) provides a reasonable guess of the orbital contribution. From EOM-CCSD/aug-cc-pVTZ data (Table 2), V(P) − V(C) amounts to 33.2, 1.20, and 0.054 meV at 4.0, 5.0, and 6.0 Å, respectively. As the orbital interaction decays exponentially with the intermolecular distance, exp(−σR), σ ∼ 3.2 Å−1, the conformational dependence of the excitonic coupling will drastically increase when the distance between monomers becomes shorter. Comparison of Computational Methods. In the further discussion, the EOM-CCSD/aug-cc-pVTZ data are considered as reference values for the excited state coupling. As already noted, the calculated transition quadrupole moment of the monomer becomes smaller by excluding diffuse functions from the basis leading to a decrease of the quadrupole−quadrupole interaction. This effect shows up clearly at R = 5 and 6 Å (Table 2) when instead of the aug-cc-pVTZ basis one uses cc-pVTZ. In the planar dimer, the MS-CASPT2 couplings are similar to the reference values. In the crossed conformations, however, where the quadrupole−quadrupole interaction gives the main contribution, the computed coupling is smaller than the reference values because of lacking diffuse functions in the ANO-s basis. Comparison of the CASPT2 and CASSCF couplings suggests that to obtain accurate values one take into account the effects of dynamic electron correlation.
values, 0.982 and 1.240 D Å, obtained with the 6-31G* and aug-cc-pVTZ basis sets, are close to the corresponding CIS values. The electrostatic contribution to the coupling can be directly estimated from the transition multipoles or from the transition atomic charges computed for individual monomers. To accurately estimate the short-range term; however, the quantum-mechanical treatment of the entire bichromophore system is required. Absolute coupling values calculated in 6 structures of the formaldehyde dimer are listed in Table 2. As seen, the interaction of the dark excited states is very sensitive to the mutual position of the monomers. In particular, passing from the planar conformation (P) to the crossed one (C) in the dimer with R = 4 Å leads to a drastic decrease of the coupling. At larger distance, R = 6 Å, however, the values found for the P and C dimers are similar. Note that the interaction of transition quadrupoles Qxy is equal to zero when the twist angle between the molecular planes is 45°. The estimated coupling depends also on the QM method employed for the calculation. This issue will be discussed in the next section. In the planar dimer, all MOs can be divided into pairs of orbitals that are of the same symmetry and stemming from different monomers. For instance, HOMO and HOMO−1 are of B1 symmetry and correspond to the n state of the first and second molecules. The electronic interaction leads to some mixture of the orbitals (Figure 2, left panel). Similarly, the unoccupied π* MOs of each monomer form two combinations
Figure 2. HOMO and HOMO−1 in the planar and crossed dimer at R = 4.0 Å (HF/6-31G* data). C
DOI: 10.1021/jp511035p J. Phys. Chem. B XXXX, XXX, XXX−XXX
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computational modeling of electron transfer. Financial support from MICINN (Ministry of Science and Innovation, Spain) was provided by Grant CTQ 2011-26573.
CIS calculations underestimate the coupling as compared to the EOM CCSD results. In particular, the CIS/aug-cc-pVTZ value computed for the planar complex with R = 4.0 Å is by a factor of 1.5 smaller than the reference value. As expected, the coupling in the crossed conformation agrees perfectly with the EOM CCSD data. As seen from Table 2, the TD B3LYP calculation drastically overestimate the orbital interaction. The estimated coupling value at R = 4.0 Å, 140 meV, is almost 4 times bigger than the reference. Also in the planar dimer with R = 5.0 Å, the TD B3LYP estimate is twice as large as the reference value. Obviously, this approach cannot be recommended for estimating the excitonic coupling. The use of the CAM-B3LYP functional radically improves the performance of the TD DFT calculation. In particular, the computationally efficient scheme cam-B3LYP/6-31G* provides reasonable estimates of the coupling for all structures. The use of the Tamm−Dancoff approximation (TDA) instead of the TD method does not practically change the results; the TDA B3LYP values are consistently somewhat larger than related TD estimates. The TDA and TD results obtained with camB3LYP/6-31G* are very similar. Finally, the data collected in Table 2 imply that the semiempirical method INDO/S fails to correctly describe the interaction of the excited state. The computed matrix elements are too small. For instance, in the planar dimer with R = 4.0 Å, the INDO/S value is by an order of magnitude smaller than the reference values; at R = 5.0 and 6.0 Å, the errors are much larger. The main reason for that is neglecting two-center two-electron integrals which describe the quadrupole interaction within the INDO approximation.
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(1) Scholes, G. D. Long-Range Resonance Energy Transfer in Molecular Systems. Annu. Rev. Phys. Chem. 2003, 54, 57−87. (2) Bredas, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J. ChargeTransfer and Energy-Transfer Processes in π-Conjugated Oligomers and Polymers: A Molecular Picture. Chem. Rev. 2004, 104, 4971− 5003. (3) Jang, S.; Newton, M. D.; Silbey, R. J. Multichromophoric Förster Resonance Energy Transfer. Phys. Rev. Lett. 2004, 92, 218301. (4) Beljonne, D.; Curutchet, C.; Scholes, G. D.; Silbey, R. J. Beyond Forster Resonance Energy Transfer in Biological and Nanoscale Systems. J. Phys. Chem. B 2009, 113, 6583−6599. (5) Kistler, K. A.; Spano, F. C.; Matsika, S. A. Benchmark of Excitonic Couplings Derived from Atomic Transition Charges. J. Phys. Chem. B 2013, 117, 2032−2044. (6) Voityuk, A. A. Fragment Transition Density Method to Calculate Electronic Coupling for Excitation Energy Transfer. J. Chem. Phys. 2014, 140, 244117. (7) London, F. On the Theory of Non-adiabatic Chemical Reactions. Z. Phys. 1932, 74, 143−148. (8) Werner, H.; Meyer, W. MCSCF Study of The Avoided Curve Crossing of the Two Lowest 1σ+ States of LiF. J. Chem. Phys. 1981, 74, 5802−5807. (9) Cederbaum, L. S. The Multistate Vibronic Coupling Problem. J. Chem. Phys. 1983, 78, 5714−5728. (10) Mead, C. A.; Truhlar, D. G. Conditions for the Definition of a Strictly Diabatic Electronic Basis for Molecular Systems. J. Chem. Phys. 1982, 77, 6090−6098. (11) Cave, R. J.; Newton, M. D. Calculation of Electronic Coupling Matrix Elements for Ground and Excited State Electron Transfer Reactions: Comparison of the Generalized Mulliken−Hush and Block Diagonalization Methods. J. Chem. Phys. 1997, 106, 9213−9226. (12) Voityuk, A. A.; Rosch, N. Fragment Charge Difference Method for Estimating Donor-Acceptor Electronic Coupling: Application to DNA π-Stacks. J. Chem. Phys. 2002, 117, 5607−5616. (13) Hsu, C.-P.; You, Z.-Q.; Chen, H.-C. H. Characterization of the Short-Range Couplings in Excitation Energy Transfer. J. Phys. Chem. C 2008, 112, 1204−1212. (14) Vura-Weis, J.; Newton, M. D.; Wasielewski, M. R.; Subotnik, J. E. Characterizing the Locality of Diabatic States for Electronic Excitation Transfer By Decomposing the Diabatic Coupling. J. Phys. Chem. C 2010, 114, 20449−20460. (15) Voityuk, A. A. Electronic Coupling for Charge Transfer in Donor-Bridge-Acceptor Systems. Performance of the Two-State FCD Model. Phys. Chem. Chem. Phys. 2012, 14, 13789−13793. (16) Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. Constructing Diabatic States From Adiabatic States: Extending Generalized Mulliken−Hush to Multiple Charge Centers with Boys Localization. J. Chem. Phys. 2008, 129, 244101. (17) Newton, M. D. Quantum Chemical Probes of Electron-Transfer Kinetics: The Nature of Donor-Acceptor Interactions. Chem. Rev. 1991, 91, 767−792. (18) Subotnik, J. E.; Cave, R. J.; Steele, R. P.; Shenvi, N. The Initial and Final States of Electron and Energy Transfer Processes: Diabatization as Motivated by System-Solvent Interactions. J. Chem. Phys. 2009, 130, 234102. (19) Pavanello, M.; Neugebauer, J. Linking the Historical and Chemical Definitions of Diabatic States for Charge and Excitation Energy Transfer Reactions in Condensed Phase. J. Chem. Phys. 2011, 135, 134113. (20) 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.; et al. Gaussian 09, revision D 01; Gaussian, Inc.: Wallingford, CT, 2009.
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CONCLUSIONS The interaction of dark excited states in different structures of the formaldehyde dimer has been studied using the EOM CCSD, CASPT2, CIS, TD DFT, and INDO/S quantum mechanical methods and the FTD scheme. The benchmark coupling values have been computed. The excitonic coupling is shown to be quite sensitive to structural changes in the dimer. The orbital interaction of the monomers rather than Coulomb interaction of their transition quadrupoles gives the major contribution to the coupling at intermolecular distances shorter than 5 Å. The evaluation of the present benchmark calculations shows that TD cam-B3LYP performs best providing good estimates for all considered structures. Because this approach is computationally efficient, it may be applied to relatively large multichromophore systems. In contrast, the TD B3LYP scheme leads to drastically overestimated values. The data obtained using the Tamm-Dancoff approximation are similar to the TD DFT results. CASSCF and CIS calculations underestimate the coupling, indicating that dynamic electron correlation should be accounted for. The INDO/S method fails to provide reasonable estimates.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS It was a great pleasure to work with Marshall during his longterm visit to Technical University of Munich. I am very grateful for his inspiring and helpful insights into the theoretical and D
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The Journal of Physical Chemistry B (21) Aquilante, F.; De Vico, L.; Ferre, N.; Ghigo, G.; Malmqvist, P.A.; Neogrady, P.; Pedersen, T. B.; Pitonak, M.; Reiher, M.; Roos, B. O.; et al. MOLCAS 7: The Next generation. J. Comput. Chem. 2010, 31, 224−247. (22) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (23) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid ExchangeCorrelation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−57. (24) Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks for Electronically Excited States: Time-Dependent Density Functional Theory and Density Functional Theory Based Multireference Configuration Interaction. J. Chem. Phys. 2008, 129, 104103. (25) Casida, M. E.; Huix-Rotllant, M. Progress in Time-Dependent Density-Functional Theory. Annu. Rev. Phys. Chem. 2012, 63, 287− 323. (26) Hsu, C. P.; Fleming, G. R.; Head-Gordon, M.; Head-Gordon, T. Excitation Energy Transfer in Condensed Media. J. Chem. Phys. 2001, 114, 3065−3072. (27) Iozzi, M. F.; Mennucci, B.; Tomasi, J.; Cammi, R. Excitation Energy Transfer (EET) between Molecules in Condensed Matter: A Novel Application of the Polarizable Continuum Model. J. Chem. Phys. 2004, 120, 7029−7040. (28) Andreussi, O.; Biancardi, A.; Corni, S.; Mennucci, B. PlasmonControlled Light-Harvesting: Design Rules for Biohybrid Devices via Multiscale Modeling. Nano Lett. 2013, 13, 4475−4484.
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DOI: 10.1021/jp511035p J. Phys. Chem. B XXXX, XXX, XXX−XXX
Interaction of Dark Excited States. Comparison of Computational Approaches Alexander A. Voityuk* Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain Institut de Química Computacional i Catàlisi (IQCC), Universitat de Girona, 17071 Girona, Spain ABSTRACT: A systematic theoretical study of the electronic interaction of dark excited states in a model system, formaldehyde dimer is reported. Using the fragment transition density scheme, we estimate the excitonic interaction in different configurations of the dimer. The excited state properties of the system are computed with several quantum mechanical methods. We show that the orbital interaction of the monomers rather than Coulomb interaction of their transition quadrupoles gives the major contribution to the coupling at intermolecular distances shorter than 5 Å. It is found that the exitonic interaction alters drastically by conformational changes. Benchmark couplings computed with EOM CCSD, MS-CASPT2, CASSCF, TD DFT, CIS, and INDO/S and different basis sets are provided. The evaluation of the calculations shows that the TD cam-B3LYP scheme performs best, giving good estimates for all considered structures. In contrast, the TD B3LYP scheme leads to drastically overestimated values. The data obtained using the Tamm−Dancoff approximation are similar to the TD DFT results. CASSCF and CIS calculations underestimate the coupling, indicating that dynamic electron correlation may have a large effect on the short-range coupling. The INDO/S method fails to describe the excited state interaction both at short and long distances.
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INTRODUCTION Excitation energy transfer (EET) is one of the important processes in biological systems and in nanomaterials.1,2 The rate of EET is controlled by the electronic interaction of excited states involved in the process. Usually one considers “bright” excited states which are easily accessible by standard spectroscopic methods. Their excitonic coupling is determined by Coulomb interaction of the transition dipole moments of the chromophores. The theory of Förster resonance energy transfer was recently extended for multichromophoric systems.3 The Coulomb interaction of dark excited states is determined by transition quadrupoles and higher multipoles and can be estimated using transition charges.1 Such an approach can be applied when the intermolecular distance is quite large and the orbital interaction between the chromophores is negligibly small. In many interesting cases, however, the distance between molecules involved in EET is shorter than 5 Å, and the electrostatic model becomes inaccurate.1,4,5 To estimate the coupling that includes both the electrostatic and orbital terms, quantum mechanical (QM) calculations of the whole system should be performed. Then, the fragment transition density (FTD) scheme6 can be applied to derive the excitonic coupling from the QM results. FTD can be used in combination with different quantum mechanical methods to treat systems containing two or several chromophores. The scheme is based on the orthogonal transformation of adiabatic states {ψ} to diabatic states {φ} introduced by London7 and employed later by other researchers to explore ET and EET processes.8−16 The nature of the diabatic states {φ} that correspond to initial and final states in ET and EET as well as © XXXX American Chemical Society
the relation of these states to the dynamic diabatic states {Φ}, which minimize the coupling ⟨Φi|∇R|Φj⟩, are considered in the literature.10,17−19 The application of accurate ab initio methods such as EOM CCSD and MS-CASPT2 is still limited in practice to small systems (for instance, CCSD formally scales as N6 with the number of orbitals N). Therefore, approximate methods should be applied to larger systems. In this work, the FTD scheme is used together with EOM CCSD (equation of motion couple cluster method for singles and doubles), MS-CASPT2 (multistate complete-active-space second-order perturbation theory), CASSCF, TD DFT, CIS and INDO/S calculations to obtain benchmark couplings for dark excited states in the formaldehyde dimer as the test case. The evaluation of present calculations suggests that that TD DFT approach with the CAM-B3LYP functional performs best providing good estimates for all considered structures. In contrast, TD DFT calculations with the uncorrected B3LYP functional give drastically overestimated coupling values. The CASSCF and CIS methods underestimate the coupling. The INDO/S method fails to describe the coupling of the dark states both at short and long distances. Model. The planar and crossed conformations of the formaldehyde dimer considered in the paper are shown in Figure 1. Both structures belong to the C2V symmetry group (Z Special Issue: John R. Miller and Marshall D. Newton Festschrift Received: November 4, 2014 Revised: December 13, 2014
A
DOI: 10.1021/jp511035p J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B
⎛T A T BA ⎞ T = ⎜⎜ ⎟⎟ ⎝T AB T B ⎠
(2) 6
More details are provided in ref 6. The scheme can easily be implemented and used in combination with any QM method. Our approach differs radically from the method of Hsu et al.26 where the transition density of the system AB is approximated by the block-diagonal matrix constructed from the transition densities of noninteracting chromophores and the coupling is approximated by the first-order correction. The scheme26 was successfully applied in several works including the study of solvation effects on excitonic coupling.27,28 However, it cannot provide accurate estimates for systems where the orbital and exchange interactions dominate the coupling (in particular, for dark states).
Figure 1. Planar and crossed conformations of the formaldehyde dimer.
is the C2 axis, all atoms of the planar dimer are in YZ plane). The calculations were performed for the intermolecular distance R of 4.0, 5.0, and 6.0 Å. In formaldehyde, the lowest 1A2 excited state is a real dark state (the A1 → A2 transition is forbidden by symmetry) and cannot be populated from the ground state by light absorption. This state is well-separated from higher excited states, the energy gap is 5 eV. Because of that, clusters containing two or more formaldehyde molecules are good models to study electronic interaction of dark states. In particular, the dimer can be properly treated within a twostate model that includes the 1A2 state of each monomer. Method. Quantum Mechanical Calculations. The model systems were computed by Gaussian 0920 using several quantum mechanical methods EOM CCSD, TD DFT, TDA DFT, CIS, and INDO/S. Three basis sets 6-31G*, cc-pVTZ, and aug-cc-pVTZ were employed. The CASSCF and MSCASPT2 calculations with the ANO-s basis sets were performed with the MOLCAS program.21 Note that the MSCASPT2 energies are calculated at the PT2 level, while the transition densities do not contain explicit PT2 corrections (some effects of electron correlation are accounted for by the MS scheme). The TD DFT calculations were carried out using the B3LYP22 and CAM-B3LYP functionals.23 B3LYP shows a better performance as compared to other functionals when calculating excited state properties of individual molecules.24 However, the energy of charge transfer states in molecular complexes is significantly underestimated. The situation may be improved by using the long-range corrected functional, CAMB3LYP. In addition to the standard TD DFT, several calculations were carried out using the Tamm−Dancoff approximation (TDA), which neglects contributions to the excitation energies coming from the correlated ground state and is comparable to the CIS scheme. The TDA can improve the stability of the TDDFT calculations.25 Estimation of Exitonic Coupling. Within the FTD method,6 the electronic interaction between two chromophores A and B can be estimated as
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RESULTS AND DISCUSSION Let us briefly consider excited states of the formaldehyde molecule computed with different QM methods. The formaldehyde molecule belongs to the C2V symmetry group. The lowest excited state 1A2 is well-described by single excitation n → π*; n and π* MO are of B1 and B2 symmetry. The dark state 11A2 cannot be populated by light absorption (in the dipole approximation, the transition A1 → A2 is forbidden by symmetry). QM calculations predict a gap of c.a. 5 eV between the first excited state A2(n→π*) and higher states (Table 1). Because of the large gap, electronic interaction Table 1. Singlet Excitation Energies in Formaldehyde (eV) method EOM CCSDa CISa CISb TD B3LYPa TD B3LYPb TD CAM-B3LYPa TD CAM-B3LYPb INDO/S a
(T2A T1A − T2BT1B) [(T2A T2A
−
T2BT2B‐T1A T1A
2A1(π→π*)
3.94 4.51 4.56 3.85 3.99 3.84 3.96 3.11
8.07 9.44 10.01 7.33 9.62 7.81 9.71 8.62
1B1(σ→π*) 9.24 9.64 9.69 8.82 8.90 8.92 9.06 8.82
aug-cc-pVTZ basis set. b6-31G* basis set.
between monomers in formaldehyde clusters should not lead to contamination of this state by states of higher energy and a cluster containing two or more formaldehyde molecules appears to be a good model to study electronic interaction of dark excited states in molecular complexes. Different from triplet excited states where all S0 → Ti transition multipoles are zero, the coupling of singlet dark states includes two contributions (1) the orbital + exchange interaction, which decays exponentially with the distance and becomes small at intermolecular distances, R, longer than 5 Å, and (2) the electrostatic interaction of transition quadrupoles (and higher multipoles), which is proportional to 1/R5. In formaldehyde, the 1A1 → 1A2 transition quadrupole moment has only one nonzero component Qxy. Its value increases by extension of the basis set used in the calculation. For instance, the CIS calculations with the 6-31G* and aug-cc-pVTZ basis sets give 1.052 and 1.253 D Å, respectively. Thus, the computed quadrupole−quadrupole coupling, which is proportional to the square of Qxy, will significantly increase by including diffuse functions in the basis set. On the other hand, Qxy is rather insensitive to the QM method. The TD B3LYP
V = (E2 − E1) ×
1A2(n→π*)
+ T1BT1B)2 + 4(T2A T1A − T2BT1B)2 ]1/2
(1)
where the E1 and E2 are excitation energies of the systems, and vectors TXi (X = A, B) are constructed using the transition density matrix T for excitations ψ0 → ψ1 and ψ0 → ψ2. In the dimer AB, TAi and TBi are the diagonal blocks of Ti containing N2A and N2B elements (NX is the number of AOs in the fragment X, X = A or B) extracted from the transition density computed for the entire system B
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Table 2. Exitonic Coupling of the 1A2 Dark States in the Planar (P) and Crossed (C) Conformations of the Formaldehyde Dimer (in meV) R = 4.0 Å
R = 5.0 Å
R = 6.0 Å
method
P
C
P
C
P
C
EOM-CCSD/aug-cc-pVTZ EOM-CCSD/cc-pVTZ MS-CASPT2/ANO-s CASSCF/ANO-s CIS/aug-cc-pVTZ CIS/cc-pVTZ CIS/6-31G* TD B3LYP/aug-cc-pVTZ TD B3LYP/cc-pVTZ TD B3LYP/6-31G* TDA B3LYP/aug-cc-pVTZ TD CAM-B3LYP/aug-cc-pVTZ TD CAM-B3LYP/cc-pVTZ TD CAM-B3LYP/6-31G* TDA CAM-B3LYP/aug-cc-pVTZ TDA CAM-B3LYP/6-31G* INDO/S
37.3 30.9 38.0 22.7 24.7 22.6 19.3 140 129 111 145 53.0 46.3 38.7 54.3 38.8 2.52
6.41 5.52 4.18 4.05 6.42 5.82 4.58 4.81 4.17 3.19 5.70 4.91 4.25 4.01 6.12 4.05 4.0 × 10−3
2.65 1.75 2.68 1.55 1.98 1.56 1.15 5.56 3.54 2.15 5.86 3.18 2.10 1.48 3.38 1.47 1.3 × 10−2
1.45 1.10 0.99 0.83 1.41 1.17 0.91 1.33 1.04 0.80 1.51 1.33 1.04 0.89 1.53 0.89 0
0.478 0.327 0.426 0.253 0.430 0.343 0.269 0.593 0.352 0.250 0.652 0.495 0.326 0.268 0.554 0.271 2.2 × 10−7
0.424 0.312 0.388 0.305 0.411 0.337 0.267 0.419 0.308 0.240 0.474 0.412 0.305 0.262 0.470 0.267 0
of B2 symmetry. The orbital interaction between the monomers results in the short-range term of the excited state coupling. The situation is quite different for the crossed dimer. Although both n → π* excited states are of A2 symmetry (as in the planar system), the n orbitals of the monomers are of different symmetry (B1 and B2) and cannot mix (Figure 2 right panel). Also, π* MOs of the molecules differ by symmetry (B2 and B1) and do not combine. As a result, the short-range term of the coupling considerably decreases by passing from the planar (P) conformation to the crossed (C) one (Table 2). As the orbital interaction dominates the coupling at R = 4.0 Å, a large difference in the coupling is found for the P and C dimers. By contrast, at R = 6.0 Å, the quadrupole−quadrupole interaction gives the main contribution to the coupling. The magnitude of the orbital interaction can be estimated by comparing coupling values V in the planar and crossed dimers. The difference of V(P) and V(C) provides a reasonable guess of the orbital contribution. From EOM-CCSD/aug-cc-pVTZ data (Table 2), V(P) − V(C) amounts to 33.2, 1.20, and 0.054 meV at 4.0, 5.0, and 6.0 Å, respectively. As the orbital interaction decays exponentially with the intermolecular distance, exp(−σR), σ ∼ 3.2 Å−1, the conformational dependence of the excitonic coupling will drastically increase when the distance between monomers becomes shorter. Comparison of Computational Methods. In the further discussion, the EOM-CCSD/aug-cc-pVTZ data are considered as reference values for the excited state coupling. As already noted, the calculated transition quadrupole moment of the monomer becomes smaller by excluding diffuse functions from the basis leading to a decrease of the quadrupole−quadrupole interaction. This effect shows up clearly at R = 5 and 6 Å (Table 2) when instead of the aug-cc-pVTZ basis one uses cc-pVTZ. In the planar dimer, the MS-CASPT2 couplings are similar to the reference values. In the crossed conformations, however, where the quadrupole−quadrupole interaction gives the main contribution, the computed coupling is smaller than the reference values because of lacking diffuse functions in the ANO-s basis. Comparison of the CASPT2 and CASSCF couplings suggests that to obtain accurate values one take into account the effects of dynamic electron correlation.
values, 0.982 and 1.240 D Å, obtained with the 6-31G* and aug-cc-pVTZ basis sets, are close to the corresponding CIS values. The electrostatic contribution to the coupling can be directly estimated from the transition multipoles or from the transition atomic charges computed for individual monomers. To accurately estimate the short-range term; however, the quantum-mechanical treatment of the entire bichromophore system is required. Absolute coupling values calculated in 6 structures of the formaldehyde dimer are listed in Table 2. As seen, the interaction of the dark excited states is very sensitive to the mutual position of the monomers. In particular, passing from the planar conformation (P) to the crossed one (C) in the dimer with R = 4 Å leads to a drastic decrease of the coupling. At larger distance, R = 6 Å, however, the values found for the P and C dimers are similar. Note that the interaction of transition quadrupoles Qxy is equal to zero when the twist angle between the molecular planes is 45°. The estimated coupling depends also on the QM method employed for the calculation. This issue will be discussed in the next section. In the planar dimer, all MOs can be divided into pairs of orbitals that are of the same symmetry and stemming from different monomers. For instance, HOMO and HOMO−1 are of B1 symmetry and correspond to the n state of the first and second molecules. The electronic interaction leads to some mixture of the orbitals (Figure 2, left panel). Similarly, the unoccupied π* MOs of each monomer form two combinations
Figure 2. HOMO and HOMO−1 in the planar and crossed dimer at R = 4.0 Å (HF/6-31G* data). C
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computational modeling of electron transfer. Financial support from MICINN (Ministry of Science and Innovation, Spain) was provided by Grant CTQ 2011-26573.
CIS calculations underestimate the coupling as compared to the EOM CCSD results. In particular, the CIS/aug-cc-pVTZ value computed for the planar complex with R = 4.0 Å is by a factor of 1.5 smaller than the reference value. As expected, the coupling in the crossed conformation agrees perfectly with the EOM CCSD data. As seen from Table 2, the TD B3LYP calculation drastically overestimate the orbital interaction. The estimated coupling value at R = 4.0 Å, 140 meV, is almost 4 times bigger than the reference. Also in the planar dimer with R = 5.0 Å, the TD B3LYP estimate is twice as large as the reference value. Obviously, this approach cannot be recommended for estimating the excitonic coupling. The use of the CAM-B3LYP functional radically improves the performance of the TD DFT calculation. In particular, the computationally efficient scheme cam-B3LYP/6-31G* provides reasonable estimates of the coupling for all structures. The use of the Tamm−Dancoff approximation (TDA) instead of the TD method does not practically change the results; the TDA B3LYP values are consistently somewhat larger than related TD estimates. The TDA and TD results obtained with camB3LYP/6-31G* are very similar. Finally, the data collected in Table 2 imply that the semiempirical method INDO/S fails to correctly describe the interaction of the excited state. The computed matrix elements are too small. For instance, in the planar dimer with R = 4.0 Å, the INDO/S value is by an order of magnitude smaller than the reference values; at R = 5.0 and 6.0 Å, the errors are much larger. The main reason for that is neglecting two-center two-electron integrals which describe the quadrupole interaction within the INDO approximation.
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CONCLUSIONS The interaction of dark excited states in different structures of the formaldehyde dimer has been studied using the EOM CCSD, CASPT2, CIS, TD DFT, and INDO/S quantum mechanical methods and the FTD scheme. The benchmark coupling values have been computed. The excitonic coupling is shown to be quite sensitive to structural changes in the dimer. The orbital interaction of the monomers rather than Coulomb interaction of their transition quadrupoles gives the major contribution to the coupling at intermolecular distances shorter than 5 Å. The evaluation of the present benchmark calculations shows that TD cam-B3LYP performs best providing good estimates for all considered structures. Because this approach is computationally efficient, it may be applied to relatively large multichromophore systems. In contrast, the TD B3LYP scheme leads to drastically overestimated values. The data obtained using the Tamm-Dancoff approximation are similar to the TD DFT results. CASSCF and CIS calculations underestimate the coupling, indicating that dynamic electron correlation should be accounted for. The INDO/S method fails to provide reasonable estimates.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS It was a great pleasure to work with Marshall during his longterm visit to Technical University of Munich. I am very grateful for his inspiring and helpful insights into the theoretical and D
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