Laureates of the sCs faLL Meeting 2013 CHIMIA 2014, 68, Nr. 4 227 brought to you by View metadata, citation and similardoi:10.2533/chimia.2014.227 papers at core.ac.uk Chimia 68 (2014) 227–230 © Schweizerische Chemische Gesellschaft CORE provided by Infoscience - École polytechnique fédérale de Lausanne A Vibronic Hamiltonian to Describe the Ultrafast Dynamics of a Cu(i)-Phenanthroline Complex

Gloria Capano§ab, Thomas J. Penfold*c, Ursula Röthlisbergera, and Ivano Tavernellia

§SCS-Metrohm Foundation Award for best oral presentation

Abstract: We present a model Hamiltonian to study the nonadiabatic dynamics of photoexcited [Cu(dmp) ]+, (dmp 2 = 2,9-dimethyl-1,10-phenanthroline). The relevant normal modes, identified by the magnitude of the first order coupling constants, correspond closely to those observed experimentally. The potential energy surfaces (PES) and nonadiabatic couplings for these modes are computed and provide a first interpretation of the nonadiabatic relaxation mechanism. The Hamiltonian incorporates both the low lying singlet and triplet states, which will make it possible to follow the dynamics from the photoexcitation event to the initial stages of intersystem crossing.

Keywords: Cu(i)-phenanthroline complex · Multi-configuration time-dependent Hartree · Nonadiabatic · Vibronic coupling Hamiltonian · Time-dependent density functional theory

1. Introduction solvent-dependent excited-state lifetimes other processes of ≈660 fs and ≈7.4 ps, which are significantly quenched in elec- which were assigned to a PJT distortion Transition metal complexes play a cen- tron-donating solvents.[3] This has been at- (ligand flattening) and intersystem cross- tral role as photocatalysts and as sensitiz- tributed to the complexation of a solvent ing, respectively. In a later study,[7b] they ers in dye-sensitized solar cells.[1] Thus to the metal center in the excited also probed the time-evolution in the low- understanding their photodynamic proper- state,[3c] which becomes possible due to the est singlet Metal-Ligand Charge Transfer ties is of fundamental as well as practical pseudo Jahn-Teller (PJT) distortion of the (MLCT) state, using the observed dynam- importance. Cu(i)-phenanthroline com- ligands, exposing the copper ion to the sol- ics to predict the most important normal plexes are a class of systems that has re- vent. However, using time-resolved X-ray modes activated during the excited-state cently received increasing attention. These absorption ,[4] combined with dynamics. Interestingly, they also report- compounds exhibit many properties simi- first-principles molecular dynamics simu- ed that the S state exhibits a small ener- 1 lar to the popular ruthenium polypyridines, lations in explicit solvent, we have recently getic barrier leading to the PJT distortion. but have a lower coordination number of shown that this is not the case.[5] Instead, However, such a barrier is at odds with the 4, which permits larger structural distor- the solvent interaction is transient and aris- spontaneous structural instability usually tions in the excited state. While this offers es from a particular solvent structure that is associated with JT type effects. greater flexibility to fine tune their pho- also present in the electronic ground state. For a full understanding of the excited- tophysical properties, it also gives rise to To reduce the influence of the surround- state dynamics of such complexes, simula- strong structure-dependent energetics and ing solvent and to prolong the excited-state tions provide an important tool. In this con- susceptibility to solvent effects, which has lifetime, it is important that structural mod- tribution we present a Vibronic Coupling so far hampered their development.[2] ifications of the phenanthroline-derived Hamiltonian suitable for use in quantum Previous studies of the excited-state ligands are performed in such a way that nuclear dynamics to study the excited- properties of Cu(i)-phenanthrolines have they are able to disrupt the structure of the state properties of [Cu(dmp) ]+. We iden- 2 focused upon understanding their strongly first solvation shell.[6] tify the normal modes which are most rel- Importantly, these modifications not evant and discuss the calculated PES along only affect the interaction with the solvent, these modes in relation to the excited-state but also the femtosecond (fs) dynamics dynamics. The Hamiltonian incorporates that follow photoexcitation, characterized both the low lying singlet and triplet states, by couplings between multiple excited which makes it possible to probe the entire *Correspondence: Dr. T. J. Penfoldc states leading to strong nonadiabatic ef- dynamics during the first picosecond (ps) E-mail: [email protected] aLaboratory of and fects. Ultrafast absorption and emission after photoexcitation. Biochemistry studies used to probe these photodynamics Ecole Polytechnique Fédérale de Lausanne have focused upon the prototypical Cu(i)- CH-1015 Lausanne phenanthroline complex, [Cu(dmp) ]+.[7] 2. Theory bLaboratory of Ultrafast Spectroscopy 2 Ecole Polytechnique Fédérale de Lausanne Tahara and co-workers[7a] concluded that CH-1015 Lausanne upon photoexcitation, decay of the ini- To describe the nonadiabatic dynamics c SwissFEL tially populated state occurs with a time of [Cu(dmp) ]+ we use the Vibronic Coup- Paul Scherrer Institute 2 CH-5232 Villigen constant of ≈45 fs. This is followed by two ling Hamiltonian, described in ref. [8]. 228 CHIMIA 2014, 68, Nr. 4 Laureates of the sCs faLL Meeting 2013

Briefly, an N state Hamiltonian, is ex- the D point group relevant for the copper 3.1 Zeroth-order Expansion 2 pressed using an NxN matrix and expand- phenanthroline complexes considered here Coefficients ed as a Taylor series around the Franck- we can write, The zeroth-order coefficients (Wˆ (0)) are Condon (FC) point Q : reported in Table 1. All of these states are 0 composed of an excitation from a metal d- Γ ⇥ Γ ⇥ Γ 0 ⊃ Γ (5) n Qi n A orbital to a ligand π* orbital (Fig. 1) and are Hˆ = Hˆ + Wˆ (0) + Wˆ (1) + Wˆ (2) (1) 0 therefore MLCT states.[12] where Γ and Γ are the irreducible rep- The lowest four triplet states are all n n’ The first term includes the kinetic energy resentations of the states n and n’ and Γ below the S state and all lie in a close Qi 1 operator and a harmonic term represent- is the irreducible representation of normal energy range of ≈0.1 eV. The lowest two ing the ground state Hamiltonian. Wˆ (0) is mode Q .[8] singlet states (S and S ) have excitation i 1 2 the zeroth order diagonal coupling matrix The expansion coefficients expressed energies of 2.44 and 2.51 eV, respectively, which contains the vertical excited state in Eqns (2)–(4) have been obtained by at the ground state optimized geometry. In energies at the FC geometry. The third performing a fit to previous simulations which constrained term, Wˆ (1), contains the linear coupling el- points calculated at nuclear geometries the structure to a D symmetry,[7d,12] these 2d ements expressed as: along the relevant normal modes, using states are degenerate. However, no such DFT. Couplings between the vibrational constraints were used in this study and we ˆ DOFs were calculated using diagonal ˆ (1) ∂Hel Wnn = hφn| |φniQi (2) cuts through pairs of normal modes. The ∂Qi ground and excited state energies were cal- culated using DFT/TDDFT as implement- and ed in Gaussian09[9] within the approxima- LUMO+1 (B3) tion of the B3LYP functional.[10] For all π * ˆ calculations a TZVP basis set was used for 2 ˆ (1) ∂Hel Wnn0 = hφn| |φn0iQi (3) the copper atom and an aug-SVP basis set ∂Qi for N, C and H. The fit of the PES was performed using the VCHAM program, LUMO (A) where φ are the electronic wavefunctions. distributed with the Heidelberg MCTDH n The quantities within the Dirac brackets package.[11] π1* are the on- (Eqn. (2)) and off-diagonal (Eqn. (3)) coupling constants, usually rep- resented by κ (n) and λ (n,n’), respectively. 3. Results and Discussion i i The on-diagonal terms are related to the HOMO (A) derivative of the adiabatic PES with re- [Cu(dmp) ]+ has 57 atoms and there- 2 spect to the coordinates and represent the fore 165 normal modes. Consequently, d forces acting on the diabatic surface. The calculating the full PES is unrealistic. To xz off-diagonal terms are the nonadiabatic reduce the computational effort we use a couplings. model Hamiltonian that includes only the ˆ (2) The second order (W ) nonadiabatic most important vibrational DOFs for the HOMO-1 (B3) coupling terms are generally small and ultrafast dynamics under study. In fact, usually neglected, however the on-diago- while this approximated Hamiltonian will dyz nal terms can play an important role and be unable to capture longer time effects, [7d] are expressed: such as vibrational cooling (≈10 ps ), the Fig. 1. Molecular orbital diagram and char- approximations made will have little influ- acteristics of the HOMO-1, HOMO, LUMO, X 2 ˆ ence on the dynamics during the first ps.[7d] (2) 1 ∂ Hel and LUMO+1 orbitals involved in the low lying Wn = hφn| |φniQiQj (4) excited states of [Cu(dmp) ]+. Contour levels 2 ∂Qi∂Qj 2 i,j drawn at 0.02 a.u.

In this case, the quantity within the Dirac brackets is usually referred to as Table 1. Excitation energies (eV) and oscillator strength of the singlet and triplet states of [Cu(dmp) + at the ground-state equilibrium geometry calculated using TD-DFT with the B3LYP γ (n). When Q = Q the coupling occurs 2] i,j i j functional. within a nuclear degree of freedom (DOF) and causes a change of frequency of the State Symmetry Energy [eV] Oscillator Descriptiona excited state potential. For Q ≠ Q (bilin- i j Strength ear), coupling occurs between two nuclear S 1B 2.44 0.0006 d →π *, d →π * DOFs and is responsible for intramolecu- 1 3 yz 1 xz 2 lar vibrational redistribution. S 1A2.51 0.0000 d →π *, d →π * In the Hamiltonian in Eqn. (1), the 2 yz 2 xz 1 S 1B 2.70 0.1608 d →π *, d →π * number of expansion coefficients (i.e. κ, λ 3 3 xz 1 xz 2 and γ) can quickly become very large as T 3A2.27 - d →π * the number of DOFs increases. To reduce 1 yz 2 T 3A2.31 - d →π * computational cost, symmetry constraints 2 yz 1 may be used. For the linear terms, the T 3B 2.33 - d →π * 3 3 xz 1 expansion coefficients are non-zero only T 3B 2.37 - d →π * when the product of the irreducible repre- 4 3 xz 2 sentation of the electronic states and of the aSee Fig. 1 for a representation of the orbitals. The dominant characters of the electronic vibrational DOF is totally symmetric. For transitions are indicated. Laureates of the sCs faLL Meeting 2013 CHIMIA 2014, 68, Nr. 4 229

find that the lowest energy structure has D Table 2. Symmetry and frequency (cm–1) of the selected normal modes. The normal modes were 2 symmetry, giving rise to a small but siz- calculated using DFT within the approximation of the B3LYP functional. The experimental values able splitting of these states. The S state, are taken from ref. [6a]. 3 whose transition from the ground state is Mode Symmetry Theory [cm–1]Expt. [cm–1]Description dipole-allowed, has an energy of 2.70 eV ν a99.16 125 Breathing and closely corresponds to the maximum 8 observed in the experimental absorption ν b 193.61 191 Rocking 19 3 spectra of [Cu(dmp) ]+, 2.73 eV.[7c] 2 ν b 247.61 240 Rocking 21 3 ν a270.85 290 Twist 3.2 First-order Expansion 25 Coefficients ν b 420.77 438 Bending The first-order terms arise from cou- 31 3 ν b 502.65 –Bending pling of the electronic states to a specific 41 3 ν b 704.75 704 Rocking nuclear DOF (Eqns (2) and (3)). Based 55 3 on the magnitude of the linear coupling ν b 790.76 –Rocking constants we have identified eight normal 58 3 modes that are likely to be dominant in the initial photoexcited dynamics (Table 3.0 3.0 ν ν 2). Although this clearly represents a sig- (a) 8 (b) 19 nificant reduction in the dimensionality of the investigated configuration space, the modes included closely correspond to those identified in the emission study 2.8 2.8 of ref. [7b]. Cuts through the PESs along some of the most important modes (ν , ν , 8 19 ν and ν ) are depicted in Fig. 2. 21 25

Owing to symmetry, only modes ν and V

8 /e

ν can yield non-zero on-diagonal linear gy 2.6 2.6 25

coupling coefficients (κ). Indeed, both of Ener these modes exhibit excited-state minima that are shifted with respect to the ground- state equilibrium position (an effect of on-diagonal linear coupling). For ν , this 8 2.4 2.4 shift reflects a strengthening of the Cu–N bonds in the excited state, which is sup- ported by experimental observations[6] and is due to the π back-donation character of the ligands and the enhanced electrostatic 2.2 2.2 interaction between metal and ligands fol- -4 -2 0 2 4 -4 -2 0 2 4 lowing the charge transfer excitation. Nuclear Displacement Nuclear Displacement Along ν the S and S states also ex- 25, 1 2 3.0 3.0 hibit a shortening of the Cu–N bonds. In (c) ν21 (d) ν contrast the triplet states exhibit a profile 25 more reminiscent of a PJT distortion with an asymmetric double minimum profile. This behavior is due to linear off-diagonal coupling between the T /T and T /T states 2.8 2.8 1 2 3 4 (Table 3). The modes ν and ν have b sym- 19 21 3 metry and correspond to a flattening mo- tion of the two ligands and to an off center movement of the Cu atom, respectively. 2.6 2.6 The on-diagonal linear coupling coeffi- cients (κ) are always zero by symmetry, however the off-diagonal expansion coef- ficients (λ) can be non-zero between states S /S and S /S . As shown in Table 3 these 2.4 2.4 3 2 2 1 modes strongly couple the S and S sur- 1 2 faces and are responsible for the PJT ef- fects in [Cu(dmp) ]+. This effect is stron- 2 gest in ν where both the lowest singlet 21 and triplet states are strongly characterized 2.2 2.2 by double minima profiles located sym- -4 -2 0 2 4 -4 -2 0 2 4 metrically with respect to the ground state Nuclear Displacement Nuclear Displacement equilibrium position, which arises from Fig. 2. Cuts through the PES along a) ν , b) ν , c) ν and d) ν . The dots are results from the 8 19 21 25 the aforementioned coupling. The effect is quantum chemistry calculations for the singlet (red) and triplet (blue) states. The lines correspond weaker for ν Finally, as shown in Table to their fit from which the expansion coefficients are determined. 19. 230 CHIMIA 2014, 68, Nr. 4 Laureates of the sCs faLL Meeting 2013

Table 3. On-diagonal (κ) and off-diagonal (λ) linear coupling constants in eV for the selected dynamics simulations, to shed new insights vibrational modes of [Cu(dmp) ]+. 2 into the ultrafast excited state dynamics of [Cu(dmp) ]+. ν ν ν ν ν ν ν 2 19 21 25 31 41 55 58 κ –– 0.0053 –––– Acknowledgements S We thank the Swiss National Science κ–– 0.0070 –––– Foundation Grant 200021-137717 and the S2 κ–––0.0031 –––– NCCR MUST interdisciplinary research pro- S3 gram. λ0.0027 0.0695 –0.0463 0.0156 0.0218 0.0417 S1-S2 λ ––0.0275 –––– Received: January 17, 2014 S1-S3 λ0.0520 0.0178 -0.0145 0.0077 ––0.0020 S2-S3 [1] a) B. Kumar, M. Llorente, J. Froehlich, T. λ ––0.0491 –––– T1-T2 Dang, A. Sathrum, C. P. Kubiak, Ann. Rev. λ–0.0033 0.0512 –0.0243 0.0860 0.0136 0.0136 Phys. Chem. 2012, 63, 541; b) B. O’Regan, M. T1-T3 Grätzel, Nature 1991, 353, 24. λ 0.0267 0.0137 –0.0243 0.0073 0.0126 0.0310 [2] N. Armaroli, Chem. Soc. Rev. 2001, 30, 113. T1-T4 [3] a) D. R. McMillin, J. R. Kirchhoff, K. V. λ –0.0296 0.0051 –0.0323 0.0115 0.0107 0.0342 Goodwin, Coord. Chem. Rev. 1985, 64, 83; T2-T3 b) C. T. Cunningham, K. L. H. Cunningham, λ 0.0011 –––0.0068 0.0137 0.0042 T2-T4 J. F. Michalec, D. R. McMillin, Inorg. Chem. λ ––0.0441 –––– 1999, 38, 4388; c) L. X. Chen, G. Shaw, I. T3-T4 Novozhilova, T. Liu, G. Jennings, K.Attenkofer, G. Meyer, P. Coppens, J. Am. Chem. Soc. 2003, 125, 7022. 3 we also find an off-diagonal expansion with vibrational cooling, such effects are [4] T. J. Penfold, C. J. Milne, M. Chergui, Adv. coefficient (λ) along the ν coupling the beyond the scope of our investigation since Chem. Phys. 2013, 153, 1. 25 S and S states. This offers the possibility their effect on the short time (<1 ps) dy- [5] T. J. Penfold, S. Karlsson, G. Capano, F. A. 3 1 for the wavepacket to relax directly from namics is expected to be small. Lima, J. Rittmann, M. Reinhard, H. Rittmann- Frank, O. Bräm, E. Baranoff, R. Abela, I. the initially populated state into the low- Tavernelli, U. Rothlisberger, C. J. Milne, M. est singlet state, however given the large Chergui, J. Phys. Chem. A 2013, 117, 4591. energy separation and the strong coupling 4. Conclusions and Outlook [6] C. E. McCusker, F. N. Castellano, Inorg. Chem. between the S and S states, this pathway 2013, 52, 8114. 1 2 will likely provide only these a minor con- Using TDDFT, we have calculated [7] a) M. Iwamura, S. Takeuchi, T. Tahara, J. Am. Chem. Soc. 2007, 129, 5248; b) M. Iwamura, tribution to the overall dynamics. a model Hamiltonian based upon the H. Watanabe, K. Ishii, S. Takeuchi, T. Tahara, Vibronic Coupling ansatz. Our calculated J. Am. Chem. Soc. 2011, 133, 7728; c) Z. 3.3 Second-order Expansion PES show that at the equilibrium geom- Siddique, Y. Yamamoto, T. Ohno, K. Nozaki, Coefficients etry, which possesses D symmetry, the Inorg. Chem. 2003, 42, 6366; d) G. Shaw, C. 2 All of the modes considered in this optically bright state is the S MLCT state. Grant, H. Shirota, E. Castner, G. Meyer, L.X. 3 Chen, J. Am. Chem. Soc. 2007, 129, 2147. work require on-diagonal second-order Importantly, the two excited states which [8] a) C. Cattarius, G. A. Worth, H.-D. Meyer, J. coupling terms (Table 4), which account lie below S are very close in energy, and Chem. Phys. 2001, 131, 2088; b) T. J. Penfold, 3 for a change of frequency of the excited are strongly coupled. This means that the G. A. Worth, J. Chem. Phys. 2009, 131, 064303; state potentials compared to the ground rate of internal conversion into the S PJT c) T. J. Penfold, R. Spesyvtsev, O. M. Kirkby, R. 1 S. Minns, D. S. N. Parker, H. H. Fielding, G. A. state. However we find that they are all minimum, and ultimately the triplet states, Worth, J. Chem. Phys. 2012, 137, 204310. relatively small, meaning that their effect will be strongly determined by the rate of [9] M. J. Frisch, G. W. Trucks, H. B. Schlegel, is expected to be negligible. nonadiabatic relaxation between the S and G. E. Scuseria, M. Robb, J. R. Cheeseman, 3 In addition, small bilinear terms were S states, which are weakly coupled and G. Scalmani, V. Barone, B. Mennucci, G. A. 2 Petersson, H. Nakatsuji, M. Caricato, X. Li, also fitted between each pair of modes. are separated by a larger energy gap. In ad- H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. These terms, which are responsible dition, contrary to previous observations[7a] Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, for redistribution of vibrational energy the present PESs calculated in this work K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, formed during the electronic relaxation, do not exhibit a barrier leading to the T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. were found to be smaller than <0.005 eV. PJT distortion in the excited state. The Vreven, J. A. A. Montgomery, J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, Although these terms will have an impor- Hamiltonian derived in this work can be K. N. Kudin, V. N. Staroverov, R. Kobayashi, tant role for long time dynamics associated used, in conjunction with nuclear quantum J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. Table 4. On-diagonal (λ) second-order coupling constants in eV for the selected vibrational modes B. Cross, V. Bakken, C. Adamo, J. Jaramillo, of [Cu(dmp) ]+. R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. 2 Austin, R. Cammi, C. Pomelli, J. W. Ochterski, ν ν ν ν ν ν ν ν R. L. Martin, K. Morokuma, V. G Zakrzewski, 8 19 21 25 31 41 55 58 G. A. Voth, P. Salvador, J. J Dannenberg, γ 0.0027 0.0010 0.0035 –0.0183 0.0124 0.0020 0.0024 0.0034 S. Dapprich, A. D. Daniels, V. Farkas, J. B S1 Foresman, J. V. Ortiz, J. Cioslowski, D. J. Fox, γ 0.0015 0.0004 0.0032 –0.0069 –0.0011 0.0027 –0.0035 S2 Gaussian Inc., Wallingford, CT, 2009. γ –0.0011 –0.0029 0.0054 0.0067 –0.0029 0.0017 0.0022 0.0029 [10] a) A. D. Becke, J. Chem. Phys. 1993, 98, 5648; S3 b) C. Lee, W. Yang, R. G. Parr, Phys. Rev. γ –––0.0009 –0.0099 0.0016 0.0020 0.0020 0.0021 B 1988, 37, 785; c) S. H. Vosko, L. Wilk, M. T1 Nusair, Can. J. Phys. 1980, 58, 1200; d) P. J. γ ––0.0041 0.0038 –0.0035 0.0022 0.0021 0.0016 0.0047 Stephens, F. J. Devlin, C. F. Chabalowski, M. J. T2 Frisch, J. Phys. Chem. 1994, 98 11623. γ –0.0007 –0.0006 –0.0079 0.0027 0.0022 0.0021 0.0018 T3 [11] M. Beck, A. Jackle, G. A. Worth, H.-D. Meyer, γ –0.0001 0.0051 –0.0036 0.0051 0.0020 0.0016 0.0052 Phys. Rep. 2000, 324, 1. T4 [12] M. Zgierski, J. Chem. Phys. 2003, 118, 4045.