Bridging Scales Through Wavefunction Analysis Felix Plasser Institute for Theoretical Chemistry, University of Vienna Excited States Bridging Scales Marseille, November 7 10, 2016 F. Plasser Wavefunction Analysis 1 / 50
Introduction Excited state quantum chemistry Accurate computations - Computational methods: Semi-emp., TDDFT, CC, ADC, CASSCF, DMRG, CASPT2, MR-CI,... - Algorithmic efforts: Linear scaling, resolution-of-the-identity,... - Parallelization Analysis and interpretation of the results - Looking at the orbitals F. Plasser Wavefunction Analysis 2 / 50
Introduction Excited states at increasing length scales New scales - new problems More low-lying excited states Sampling of geometries More work Orbitals of mixed character Many interacting configurations Analysis becomes ambiguous and affected by personal bias F. Plasser Wavefunction Analysis 3 / 50
Introduction Excited states at increasing length scales New scales - new physics Small molecules - Molecular orbitals contain all relevant information - nπ, ππ,... states Interacting chromophores and extended systems - Excitonic effects - Excited state collectivity and correlation F. Plasser Wavefunction Analysis 4 / 50
Introduction How can we understand excited states for large systems? Tedious analysis work Ambiguous results Challenging physics Problems can be solved through systematic wavefunction analysis F. Plasser Wavefunction Analysis 5 / 50
DNA Task: Understand the UV absorption of DNA Local excitations Delocalized excitations - excitons Charge transfer states F. Plasser Wavefunction Analysis 8 / 50
DNA QM/MM calculation 4 nucleobases in the QM region ADC(2) excitation energies - 20 states Sampling of intra- and intermolecular motions - 300 snapshots How do we analyze 6000 excited states? F. Plasser Wavefunction Analysis 9 / 50
DNA 16 Possibilities - 4 local transitions - 12 charge transfer transitions T 1 A 2 A 1 A 2 T 2 A 2 A 2 A 2 T 1 T 2 A 1 T 2 T 2 T 2 A 2 T 2 T 1 A 1 A 1 A 1 T 1 A 1 A 2 A 1 T 1 T 1 A 1 T 1 T 2 T 1 A 2 T 1 Where can this information be found? Transition density matrix F. Plasser Wavefunction Analysis 10 / 50
Transition Density Matrix 1-Electron transition density matrix D 0I D 0I µν = Ψ 0 â µâ ν Ψ I µν Matrix representation of the 1TDM Ψ 0, Ψ I Ground and excited state wavefunctions â µ, â ν Creation and annihilation operators Connection to physical observables through transition properties - Rigorous meaning - Well-defined independent of the computational method - No explicit dependence on the orbitals Approximation: CI vector / response vector F. Plasser Wavefunction Analysis 11 / 50
Charge Transfer Numbers Summation over squared 1TDM elements - For two nucleobases A and B Correction for non-orthogonality of the AOs Charge transfer numbers Ω AB = 1 [ (D 0I S ) 2 µν Ω AA µ A ν B ( SD 0I ) µν + ( D0I µν SD 0I S ) µν] Weight of local excitations on nucleobase A Ω AB, A B Amount of charge transfer from A to B 1 FP, H. Lischka JCTC 2012, 8, 2777. 2 FP, M. Wormit, A. Dreuw JCP 2014, 141, 024106. F. Plasser Wavefunction Analysis 12 / 50
Charge Transfer Numbers T 1 A 2 A 1 A 2 T 2 A 2 A 2 A 2 T 1 T 2 A 1 T 2 T 2 T 2 A 2 T 2 Ω AB - pseudocolor matrix plots T 1 A 1 A 1 A 1 T 1 A 1 A 2 A 1 T 1 T 1 A 1 T 1 T 2 T 1 A 2 T 1 0.9 0.8 0.7 0.6 S 1 (4.42 ev) S 2 (4.46 ev) S 3 (4.56 ev) electron hole 0.5 0.4 0.3 0.2 0.1 0.0 S 4 (4.74 ev) S 5 (4.82 ev) S 6 (4.99 ev) F. Plasser Wavefunction Analysis 13 / 50
DNA Additional steps: Extract the essential information and classify the states - Charge transfer character - Delocalization Do this for all 300 geometries 20 states Decompose the absorption spectrum into different classes of states F. Plasser Wavefunction Analysis 14 / 50
DNA UV absorption spectrum Black: full spectrum Red: CT states Blue: deloc. at least 1.5 bases Green: deloc. at least 2.5 bases 1 FP, A. J. A. Aquino, W. L. Hase, H. Lischka JPCA 2012, 116, 11151. F. Plasser Wavefunction Analysis 15 / 50
Conjugated Polymers Poly(para phenylene vinylene) ADC(2)/SV(P) Cut into pieces (formally) Same analysis as before 1 A. Panda, FP, A. J. A. Aquino, I. Burghardt, H. Lischka JPCA 2013, 117, 2181. 2 S. A. Mewes, J.-M. Mewes, A. Dreuw, FP PCCP 2016, 18,2548. F. Plasser Wavefunction Analysis 17 / 50
Exciton Analysis Wannier excitons Hydrogen atom in a box Particle-in-a-box states Hydrogenic states F. Plasser Wavefunction Analysis 18 / 50
Exciton Analysis Wannier excitons - singlet Singlet 1 1 B u - W(1,1) 2 1 A g - W(1,2) 2 1 B u - W(1,3) 3 1 A g - W(1,4) 7 1 B u - W(1,5) 10 1 A g - W(1,6) 4 1 A g - W(2,1) 3 1 B u - W(2,2) 8 1 A g - W(2,3) 9 1 B u - W(2,4) 11 1 A g - W(2,5) 10 1 B u - W(3,1) Triplet F. Plasser Wavefunction Analysis 19 / 50
Exciton Analysis 10 Wannier 1 B u - W(3,1) excitons - triplet Triplet 1 3 B u - W(1,1) 1 3 A 3 3 g - W(1,2) 2 3 B 2 3 A g - W(1,4) 3 3 u - W(1,3) B u - W(1,5) A g - W(1,6) 4 3 B u - W(1,7) 4 3 A g - W(1,8) 5 3 B u - W(1,9) 6 3 A g - W(1,10) 5 3 A g - W(2,1) 9 3 B u - W(2,2) F. Plasser Wavefunction Analysis 20 / 50
Conjugated Polymers Problems with this analysis Results depend on fragmentation scheme chosen Plots have to be inspected manually Can we do better? F. Plasser Wavefunction Analysis 21 / 50
Transition Density Matrix Coordinate representation of the 1TDM 1-Electron transition density matrix (1TDM) γ 0I (x h, x e ) = n... Ψ 0 (x h, x 2,..., x n )Ψ I (x e, x 2,..., x n )dx 2... dx n γ 0I (x h, x e ) Coordinate representation of the 1TDM x h, x e Coordinates of the excitation hole and excited electron 1TDM in second quantization γ 0I (x h, x e ) = µν D 0I µνχ µ (x h )χ ν (x e ) D 0I µν Matrix representation of the 1TDM F. Plasser Wavefunction Analysis 22 / 50
Exciton Analysis Exciton analysis Interpret the 1TDM as the wavefunction χ exc of the electron-hole pair Use as a basis for analysis Exciton wavefunction χ exc (x h, x e ) = µν D 0I µνχ µ (x h )χ ν (x e ) Operator expectation value Ô = χ exc Ô χ exc χ exc χ exc. 1 S. A. Bäppler, FP, M. Wormit, A. Dreuw Phys. Rev. A 2014, 90, 052521. F. Plasser Wavefunction Analysis 23 / 50
Exciton Analysis Exciton size Exciton size d exc 2 = (r e r h ) 2 Average separation of the electron and hole quasi-particles Static and dynamic charge transfer effects No fragment definition required Evaluation through multipole AO integrals 1 S. A. Bäppler, FP, M. Wormit, A. Dreuw Phys. Rev. A 2014, 90, 052521. F. Plasser Wavefunction Analysis 24 / 50
Conjugated Polymers Exciton size / excitation energy - 20 singlet and 20 triplet states compressed into one plot Formation of different Wannier exciton bands Clustered Frenkel excitons Comparison with size of the molecule F. Plasser Wavefunction Analysis 25 / 50
Conjugated Polymers Exciton size with increasing size of the system - n = 2,..., 8 TDDFT/CAM-B3LYP Exciton size quickly levels off Orbitals stay delocalized How is this possible? Exciton size (Å) 7.0 6.0 5.0 4.0 2 3 4 5 6 7 8 (a) Singlet Triplet 1 FP JCP 2016, 144, 194107. 0.4 F. Plasser Wavefunction Analysis 26 / 50 elation coe cient 1.0 0.8 (b) 0.6
Exciton Analysis Quantify correlations between the electron and hole in analogy to Pearson s correlation coefficient Correlation coefficient R eh = 0 No correlation R eh = r h r e r h r e σ h σ e 1 R eh 1 R eh > 0 Positive correlation - exciton binding R eh < 0 Negative correlation - dynamic repulsion F. Plasser Wavefunction Analysis 27 / 50
Conjugated Polymers Exciton size and correlation coefficient - n = 2,..., 8 Correlation coefficient goes up as exciton size levels off Correlation in TDDFT? 1 FP JCP 2016, 144, 194107. F. Plasser Wavefunction 4.0 (c) Analysis 28 / 50 Exciton size (Å) Correlation coe cient ates 7.0 6.0 5.0 4.0 1.0 0.8 0.6 0.4 0.2 0.0 2 3 4 5 6 7 8 (a) (b) Singlet Triplet
Natural Transition Orbitals Singular value decomposition of the 1TDM Natural transition orbitals D 0I = U diag( λ1, ) λ 2,... V T U Hole orbital coefficients λ i Transition amplitudes V Electron orbital coefficients Compact representation of the excitation 1 R. L. Martin J. Chem. Phys. 2003, 11, 4775. F. Plasser Wavefunction Analysis 29 / 50
Conjugated Polymers NTOs of the S 1 state Hole Particle λ 1 = 0.703 λ 1 = 0.703 λ 2 = 0.192 λ 2 = 0.192 λ 3 = 0.061 λ 3 = 0.061 F. Plasser Wavefunction Analysis 30 / 50
Conjugated Polymers NTOs of the T 1 state Hole Particle λ 1 = 0.573 λ 1 = 0.573 λ 2 = 0.208 λ 2 = 0.208 λ 3 = 0.090 λ 3 = 0.090 λ 4 = 0.045 λ 4 = 0.045 Look similar to S 1 but different singular values F. Plasser Wavefunction Analysis 31 / 50
Collectivity Interpretation of the NTO singular value spectrum within quantum information theory Electron-hole entanglement entropy S H E = i λ i log 2 λ i Number of entangled states Z HE = 2 S H E = 1/ i λ λi i Z HE Number of configurations involved 1 FP J. Chem. Phys. 2016, 144, 194107. F. Plasser Wavefunction Analysis 32 / 50
Conjugated Polymers Exciton size (Å) 7.0 6.0 5.0 4.0 2 3 4 5 6 7 8 (a) Singlet Triplet Exciton size, correlation coefficient, number of entangled states Correlation due to multiconfigurational character of the excited state Correlation coe cient Nr. of entangled states 1.0 0.8 0.6 0.4 0.2 0.0 4.0 3.0 2.0 1.0 (b) (c) 2 3 4 5 6 7 8 Number of phenyl rings 1 FP JCP 2016, 144, 194107. F. Plasser Wavefunction Analysis 33 / 50
Natural Transition Orbitals Classification of excited states λ 1 = 1 Simple transition between two orbitals λ 1... λ k 1/k Collective single-electron excited state i λ i 1 Multiple excitation F. Plasser Wavefunction Analysis 34 / 50
Conjugated Polymers Next step - More molecules - More functionals S (a) (b) H H (c) n S S S S HC H H (d) (e) (f) H H n H n H H H N N O N N O H H n 1 S. Kraner, R. Scholz, FP, C. Koerner, K. Leo JCP 2015, 143, 244905. 2 S. A. Mewes, FP, A. Dreuw, in preparation F. Plasser Wavefunction Analysis 35 / 50
Conjugated Polymers Plot exciton size / molecular size Universal trends among different conjugated polymers Strong difference among functionals - Not only energies but overall description - More exchange strong exciton binding 1 S. A. Mewes, FP, A. Dreuw, in preparation F. Plasser Wavefunction Analysis 36 / 50
Conjugated Polymers Plot correlation coefficient / molecular size Negative correlation for local PBE functional Weak positive correlation for global hybrids B3LYP and PBE0 Strong positive correlation for range-separated, M06-2X, and ADC(2) - ADC(2) CAM-B3LYP - 100% long-range exchange overshoots 1 S. A. Mewes, FP, A. Dreuw, in preparation F. Plasser Wavefunction Analysis 37 / 50
Outlook Transition metal complexes D I0 Division into central metal (M) and ligands (L1, L2, L3) 1TDM blocks naturally correspond to types of states MC, MLCT, LMCT,... Automatic assignment No problems due to mixed orbital characters Hole L3 L2 L1 M LMCT MC M LLCT LC LC LC LLCT MLCT L1 L2 L3 Electron 1 FP, A. Dreuw JPCA 2015, 119, 1023. F. Plasser Wavefunction Analysis 39 / 50
Transition metal complex Example: iridium complex CASSCF(12/12) ADC(3) State E LC MLCT LLCT E LC MLCT LLCT 2 1 A 3.56 0.12 0.74 0.10 3.91 0.50 0.31 0.07 3 1 A 3.93 0.05 0.58 0.19 4.35 0.08 0.50 0.27 4 1 A 5.36 0.15 0.47 0.20 4.72 0.11 0.36 0.32 Easy identification of state character Comparison between methods possible F. Plasser Wavefunction Analysis 40 / 50
Outlook Characterization of intramolecular excitations - Automatic assignment of nπ, ππ, Rydberg etc. character Proof-of-principle application 1 1 FP, B. Thomitzni, S. A. Bäppler et al. JCC 2015, 36, 1609. F. Plasser Wavefunction Analysis 42 / 50
Outlook Different physics Unpaired electrons 1 Orbital relaxation 2 x Electron correlation x Two-electron excitations x Plasmons 1 FP, H. Pasalic et al. Angew. Chem., Int. Ed. 2013, 52,2581. 2 FP, S. A. Bäppler, M. Wormit, A. Dreuw JCP 2014, 141, 024107. F. Plasser Wavefunction Analysis 43 / 50
Outlook Model building Use energies and wavefunction information for model building Better transferability Property based diabatization Application to a charge transfer model system 1 Application to DNA 2 x Application to conjugated polymers 1 FP, H. Lischka JCP 2011, 134, 034309. 2 A. A. Voityuk JCP 2014, 140, 244117. F. Plasser Wavefunction Analysis 44 / 50
Charge transfer model system - Ethylene dimer cation - Fragment charge differences (FCD) Nonadiabatic couplings computed from FCDs Property based diabatization works 1 FP, H. Lischka JCP 2011, 134, 034309. F. Plasser Wavefunction Analysis 45 / 50
TheoDORE TheoDORE - Theoretical Density, Orbital Relaxation and Exciton analysis Program package for wavefunction analysis Interfaces to various quantum chemistry programs: Columbus, Molcas, Turbomole, Orca, GAMESS, Gaussian,... Open-source: http://theodore-qc.sourceforge.net Analysis functionalities Enhanced post-processing and plotting capabilities Utility functions F. Plasser Wavefunction Analysis 47 / 50
libwfa libwfa - An open-source wavefunction analysis tool library 1 State/transition/difference DM analysis methods Orbitals + densities Different population analyses Multipole analysis in coordinate space Available in Q-Chem: TDDFT, EOM-CC, ADC Interface to MOLCAS in progress: CASSCF, CASPT2 Interface to COLUMBUS will come soon... 1 https://github.com/libwfa/libwfa F. Plasser Wavefunction Analysis 48 / 50
Conclusions Analysis of electronic wavefunctions - why? 1 Make things easier Compact orbital representations Automatization 2 Give specific quantitative results Charge transfer character Delocalization Double excitation character 3 Provide new physical insight Exciton correlation Orbital relaxation There is more to wavefunctions than meets the eye! F. Plasser Wavefunction Analysis 49 / 50
Acknowledgements Heidelberg S. A. Mewes M. Wormit A. Dreuw Vienna/Lubbock/Tianjin H. Lischka Frankfurt I. Burghardt F. Plasser Wavefunction Analysis 50 / 50