Special 5: Wavefunction Analysis and Visualization

Transcription

1 Special 5: Wavefunction Analysis and Visualization Felix Plasser Institute for Theoretical Chemistry, University of Vienna COLUMBUS in China Tianjin, October 10 14, 2016 F. Plasser Wavefunction Analysis 1 / 47

2 Wavefunction Analysis How to move from a computation to a phenomenological description? Rigorous, independent of the wavefunction model Compact visual representation Quantitative information Detailed understanding F. Plasser Wavefunction Analysis 2 / 47

3 Many-electron wavefunctions Starting point: Electronic Schrödinger Equation ĤΨ 0 (x 1, x 2,...) = E 0 Ψ 0 (x 1, x 2,...) ĤΨ I (x 1, x 2,...) = E I Ψ I (x 1, x 2,...) Where are the orbitals? How to describe the many-electron function Ψ 0 (x 1, x 2,...)? How to discuss changes between Ψ 0 (x 1, x 2,...) and Ψ I (x 1, x 2,...)? Systematic complexity reduction through reduced density matrices F. Plasser Wavefunction Analysis 3 / 47

4 Density Matrices Integrate out all the coordinates except for one 1-Electron reduced density matrix (1DM) γ(x, x ) = n... Ψ(x, x 2,..., x n )Ψ(x, x 2,..., x n )dx 2... dx n 1DM in second quantization D µν = Ψ â µâ ν Ψ γ(x, x ) = D µν χ µ (x)χ ν (x ) µν γ(x, x ) Coordinate representation of the 1DM D pq Matrix representation of the 1DM χ µ, χ ν Atomic orbitals â µ, â ν Creation and annihilation operators F. Plasser Wavefunction Analysis 6 / 47

5 Density Matrix Physical meaning Expectation value of a 1-electron operator Ψ Ô1 Ψ = n... Ψ(x 1, x 2,..., x n )Ô1Ψ(x 1, x 2,..., x n )dx 1 dx 2... dx n Dipole moment, angular momentum, kinetic energy,... Using the 1DM Ψ Ô1 Ψ = µν D µν χ µ (x 1 )Ô1χ ν (x 1 )dx 1 Many-electron integral reduced to a summation over 1-electron integrals All wavefunction-specific information contained in the 1DM Logical starting point for wavefunction analysis F. Plasser Wavefunction Analysis 7 / 47

6 Electron Density Electron Density ρ(x) = γ(x, x) ρ(x) = n... Ψ(x, x 2,..., x n ) 2 dx 2... dx n Description of the overall electron distribution Adenine Ground state nπ state ππ (L b ) state ππ (L a ) state Not really helpful... F. Plasser Wavefunction Analysis 8 / 47

7 Natural Orbitals Diagonalization of the 1DM Natural orbitals D = T diag (n 1, n 2,...) T T T Natural orbital coefficients n i Occupation numbers Closed shell orbitals (n i = 2) Open shell / active orbitals (0 < n i < 2) Unoccupied orbitals (n i = 0) F. Plasser Wavefunction Analysis 9 / 47

8 Natural orbitals Natural orbitals: Graphene substructure n i = 1.28 n i = 0.72 n i = 1.89 n i = 0.11 F. Plasser Wavefunction Analysis 10 / 47

9 Unpaired electrons Quantify how much the eigenvalues differ from 2 Unpaired electrons n u = i n u,nl = i min(n i, 2 n i ) n i 2 (2 n i ) 2 Simple measure of correlation n u all types of correlation n u,nl only nondynamic/strong correlation F. Plasser Wavefunction Analysis 11 / 47

10 Unpaired electrons Visualization: Density with scaled eigenvalues Density ρ(x) = i n i ψ NO i (x) 2 ψi NO (x) Natural orbital Unpaired density ρ u (x) = i ρ u,nl (x) = i min(n i, 2 n i )ψ NO i (x) 2 n i 2 (2 n i ) 2 ψ NO i (x) 2 F. Plasser Wavefunction Analysis 12 / 47

11 Unpaired electrons Unpaired density: Graphene substructure F. Plasser Wavefunction Analysis 13 / 47

12 Transition Density Matrices Comparison of two wavefunctions Ψ 0 and Ψ I 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 1TDM in second quantization D 0I µν = Ψ 0 â µâ ν Ψ I γ 0I (x h, x e ) = µν D 0I µνχ µ (x h )χ ν (x e ) γ 0I (x h, x e ) Coordinate representation of the 1TDM D 0I µν Matrix representation of the 1TDM x h, x e Coordinates of the excitation hole and excited electron F. Plasser Wavefunction Analysis 15 / 47

13 Transition Density Matrix Physical meaning Transition property of a 1-electron operator Ψ 0 Ô1 Ψ I = µν D 0I µν χ µ Ô1 χ ν Transition moments, oscillator strength F. Plasser Wavefunction Analysis 16 / 47

14 Transition Density Transition Density Adenine ρ 0I (x) = γ 0I (x, x) nπ state ππ (L b ) state ππ (L a ) state Physical meaning: transition moments Interaction with light But: no intuitive interpretation F. Plasser Wavefunction Analysis 17 / 47

15 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 Important for large basis sets 1 R. L. Martin J. Chem. Phys. 2003, 11, F. Plasser Wavefunction Analysis 18 / 47

16 Natural Transition Orbitals Adenine: ADC(3)/aug-cc-pVDZ S 4 state Canonical orbitals, 5 dominant transitions LUMO+6 LUMO+11 LUMO+13 LUMO+10 LUMO Low energy virtual orbitals are very diffuse (quasi-continuum) F. Plasser Wavefunction Analysis 19 / 47

17 Natural Transition Orbitals Individual orbitals do not have a physical meaning Canonical orbitals can be misleading Decisive: 1TDM γ 0I (x h, x e ) Optimal representation through natural transition orbitals F. Plasser Wavefunction Analysis 20 / 47

18 Natural Transition Orbitals Adenine: ADC(3)/aug-cc-pVDZ Natural transition orbitals One dominant transition - Compact valence state Diffuse orbitals were only artifacts! 97% F. Plasser Wavefunction Analysis 21 / 47

19 Natural Transition Orbitals Linear combination of the orbitals = F. Plasser Wavefunction Analysis 22 / 47

20 Two-Electron Excitations One-electron character Ω = γ 0I (x h, x e ) 2 dx h dx e = i λ i Ω = 1 One-elctron excited state Ω < 1 Contributions of higher excitations Ω = 0 No one-electron contributions All transition properties of one-electron operators vanish Method-independent definition of double (and higher) excitation character F. Plasser Wavefunction Analysis 23 / 47

21 Two-Electron Excitations Example: hexatriene molecule ADC(3) MR-CISD(6/6) State E T 2 n u n u,nl Ω E+P n u n u,nl Ω 2 1 A g B u Bu F. Plasser, S. A. Bäppler, M. Wormit, A. Dreuw J. Chem. Phys. 2014, 141, F. Plasser Wavefunction Analysis 24 / 47

22 Statistical Analysis Analysis of the distribution of the hole/electron in space Hole density ρ h (x h ) = γ 0I 2 (x h, x e )dx e (Excess) electron density ρ e (x e ) = γ 0I 2 (x h, x e )dx h Analysis of statistical moments - Centroids - Root-mean-square size σ h, σ e F. Plasser Wavefunction Analysis 25 / 47

23 Statistical Analysis Adenine, ADC(3)/aug-cc-pVDZ Statistical moments E(eV) Osc. Str. σ h (σ h,z ) σ e Character S (0.75) 2.27 ππ S (0.73) 2.39 ππ S (0.75) 4.11 Rydberg S (0.43) 2.23 nπ S (0.75) 4.73 Rydberg F. Plasser Wavefunction Analysis 26 / 47

24 Difference Density Matrices Subtract the density matrices of the ground state and excited state 1-Electron difference density matrix (1DDM) 0I = D II D 00 Adenine nπ state ππ (L b ) state ππ (L a ) state Physical meaning: changes in one-electron properties during the excitation process But: Not very intuitive F. Plasser Wavefunction Analysis 27 / 47

25 Attachment/detachment analysis Natural difference orbitals D II D 00 = W diag (κ 1, κ 2,...) W T W Natural difference orbital coefficients κ i < 0 Detachment eigenvalues, d i κ i > 0 Attachment eigenvalues, a i Weighted sum over all positive (negative) eigenvalues leads to the attachment (detachment) densities 1 1 M. Head-Gordon, A. M. Grana, D. Maurice, C. A. White J. Chem. Phys. 1995, 99, F. Plasser Wavefunction Analysis 28 / 47

26 Attachment/Detachment Densities Adenine, ADC(2)/cc-pVDZ nπ state ππ (L b ) state ππ (L a ) state F. Plasser Wavefunction Analysis 29 / 47

27 1DM 1TDM/1DDM Exciton Practical C Natural Difference Orbitals I Adenine, ADC(2)/cc-pVDZ, nπ state I Comparison of transition and difference DMs NTOs NDOs λ1 = 0.84 a1 = 0.94 a2 = 0.07 a3 = 0.06 λ1 = 0.84 d1 = 0.93 d2 = 0.07 d3 = 0.06 I Main transition the same I But additional contributions in the 1DDM I Orbital relaxation F. Plasser Wavefunction Analysis 30 / 47

28 Exciton Analysis Main idea 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 ) F. Plasser Wavefunction Analysis 32 / 47

29 Exciton Analysis Charge transfer numbers Ω AB = dx h dx e χ exc (x h, x e ) 2 Ω AB A B A Fragment where the hole is localized B Fragment where the electron is localized Probability Pseudocolor matrix plot of the Ω AB matrix _ r e r h r e (a) (b) (c) r e + χ exc r h r h F. Plasser Wavefunction Analysis 33 / 47

30 Exciton Analysis Example: Poly(para phenylene vinylene) ADC(2)/SV(P) 1 S. A. Mewes, J.-M. Mewes, A. Dreuw, F. Plasser PCCP 2016, 18,2548. F. Plasser Wavefunction Analysis 34 / 47

31 Exciton Analysis 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 35 / 47

32 Exciton Analysis 10 1 B u - W(3,1) 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 36 / 47

33 Exciton Analysis Exciton size d exc 2 = Ω 1 γ 0I (x h, x e )(r e r h ) 2 γ 0I (x h, x e )dx h dx e Description of dynamic and static charge transfer effects No fragment definition required F. Plasser Wavefunction Analysis 37 / 47

34 Exciton analysis Exciton size / excitation energy Formation of different exciton bands Comparison with size of the molecule F. Plasser Wavefunction Analysis 38 / 47

35 Practical Aspects Wavefunction Analysis in COLUMBUS Integrated methods External TheoDORE package F. Plasser Wavefunction Analysis 40 / 47

36 Integrated Methods Mulliken populations Electron density Natural orbitals. Computed automatically - Conversion to Molden format possible Unpaired densities - Available via colinp and potential.x see Tutorial (Section 2.4) Natural difference orbitals and attachment/detatchment densities - Available via colinp F. Plasser Wavefunction Analysis 41 / 47

37 TheoDORE TheoDORE - Theoretical Density, Orbital Relaxation and Exciton analysis Program package for wavefunction analysis Interfaces to a number of quantum chemistry programs: Columbus, Molcas, Turbomole, Orca, GAMESS,... Open-source: F. Plasser Wavefunction Analysis 42 / 47

38 TheoDORE Main scripts theoinp Input generation analyze_sden.py Analysis of state/difference density matrices - Analysis of unpaired electrons - Attachment/detachment analysis - Population analysis (density, unpaired density, A/D density) analyze_tden.py Analysis of transition density matrices Natural transition orbitals Electron-hole correlation plots F. Plasser Wavefunction Analysis 43 / 47

39 libwfa 1DM 1TDM/1DDM Exciton Practical C libwfa - A wavefunction analysis library State/transition/difference DM analysis methods Orbitals + densities Different population analyses Also tools in coordinate space Available in Q-Chem: TDDFT, EOM-CC, ADC Interface to MOLCAS in progress: CASSCF, CASPT2 Interface to COLUMBUS will come soon... F. Plasser Wavefunction Analysis 44 / 47

40 Conclusions Analysis of electronic wavefunctions - why? 1 Make things easier Compact orbital representations Automatization 2 Give specific quantitative results Charge transfer character Delocalization Spatial extent Double excitation character 3 Provide new physical insight Orbital relaxation Exciton correlation There is more to wavefunctions than meets the eye! F. Plasser Wavefunction Analysis 46 / 47

41 Literature Analysis techniques F. Plasser, H. Lischka J. Chem. Theory Comput. 2012, 8, F. Plasser, M. Wormit, A. Dreuw J. Chem. Phys. 2014, 141, Density matrices A. Szabo and N. Ostlund "Modern Quantum Chemistry" 1989, McGraw-Hill, Inc. Other textbooks... F. Plasser Wavefunction Analysis 47 / 47