Excited Electronic States, Study notes of Geometry

Download Excited Electronic States and more Study notes Geometry in PDF only on Docsity! Excited Electronic States Configuration Interaction Video VII.iv Excited Electronic States •  We usually write the Schrödinger equation as HΨ = EΨ •  However, that obscures the reality that there are infinitely many solutions to the Schrödinger equation, so it is better to write HΨn = EnΨn •  Hartree-Fock theory provides us a prescription to construct an approximate ground-state wave function (as a single Slater determinant) •  How do we build from there to construct an excited- state wave function? CI in a Nutshell EHF 0 0 0 0dense d e n s e sparse sparse extremely sparse dense sparse very sparse very sparse extremely sparse extremely sparse ΨHF ΨHF Ψi a Ψi a Ψij ab Ψijk abc Ψij ab Ψijk abc The bigger the CI matrix, the more electron correlation can be captured. The CI matrix can be made bigger either by increasing basis-set size (each block is then bigger) or by adding more highly excited configurations (more blocks). The ranked eigenvalues correspond to the electronic state energies. Most common compromise is to include only single and, to lower ground state, double excitations (CISD)— not size extensive. CI Singles (CIS) There are m x n singly excited configurations where m and n are the number of occupied and virtual orbitals, respectively. Diagonalization gives excited-state energies and eigenvectors containing weights of singly excited determinants in the pure excited state Quality of excited-state wave functions about that of HF for ground state. Efficient, permits geometry optimization; semiempirical levels (INDO/S) optimized for CIS method. EHF 0 0 dense Ψi a Ψi a ΨHF ΨHF CI Singles (CIS) — Acrolein Example Excited State 1: Singlet-A" 4.8437 eV 255.97 nm f=0.0002 14 -> 16 0.62380 3.0329 408.79 14 -> 17 0.30035 3.73 Excited State 2: Singlet-A' 7.6062 eV 163.01 nm f=0.7397 15 -> 16 0.68354 6.0794 203.94 6.41 Excited State 3: Singlet-A" 9.1827 eV 135.02 nm f=0.0004 11 -> 16 -0.15957 6.6993 185.07 12 -> 16 0.55680 14 -> 16 -0.19752 14 -> 17 0.29331 Excited State 4: Singlet-A" 9.7329 eV 127.39 nm f=0.0007 9 -> 17 0.19146 10 -> 16 0.12993 11 -> 16 0.56876 12 -> 16 0.26026 12 -> 17 -0.11839 14 -> 17 -0.12343 Eigenvectors CIS/6-31G(d) and INDO/S LUMO+1: π4* LUMO: π3* HOMO: π2 HOMO–1: nO O H Expt Time-Dependent Perturbation Theory Consider the time-dependent Schrödinger equation −  i ∂Ψ ∂t = HΨ with eigenfunctions Ψj = e − iE jt / ( ) Φ j where Φj is an eigenfunction of the time- independent Schrödinger equation Perturb the Hamiltonian with a radiation field H = H0 + e0rsin 2πνt( ) The wave function evolves in the presence of the perturbation and may be expressed as a linear combination of the complete set of solutions to H Ψ = cke − iEkt / ( )Φk k ∑ Termination of the radiation field will cause the wave function to collapse (upon sampling) to a stationary state with probability |ck|2. The ck will evolve according to −  i ∂ ∂t cke − iEkt /( )Φk k ∑ = H0 + e0rsin 2πνt( )[ ] ck e − iEkt / ( )Φk k ∑ Time-Dependent Perturbation Theory (cont.) Taking the time derivative on the left and expanding on the right Which simplifies to Left multiplication by state of interest and integration yields −  i ∂ ∂t cke − iEkt /( )Φk k ∑ = H0 + e0rsin 2πνt( )[ ] ck e − iEkt / ( )Φk k ∑ −  i ∂ck ∂t e− iEkt / ( )Φk k ∑ + ckEke − iEkt / ( )Φk k ∑ = ckEke − iEkt /( )Φk k ∑ + e0rsin 2πνt( ) cke − iEkt / ( )Φk k ∑ −  i ∂ck ∂t e− iEkt / ( )Φk k ∑ = e0r sin 2πνt( ) cke − iEkt /( )Φk k ∑ € −  i ∂ck ∂t e− iEk t / ( )δmk k ∑ = e0 sin 2πνt( ) cke − iEk t / ( ) Φm rΦk k ∑ Time-Dependent Perturbation Theory (cont.) Evaluate Kronecker delta, rearrange, and assume perturbation is small, so ground state can be used for right-hand-side coefficients Integrating over time of perturbation where € −  i ∂ck ∂t e− iEk t / ( )δmk k ∑ = e0 sin 2πνt( ) cke − iEk t / ( ) Φm rΦk k ∑ ∂cm ∂t = − i  e0 sin 2πνt( )e− i Em−E0( )t /[ ] Φm rΦ0 cm τ( ) = − i  e0 sin 2πνt( )e− i Em−E0( )t / [ ] Φm rΦ00 τ ∫ dt = 1 2i e0 ei ωm0 +ω( )τ −1 ωm0 +ω − ei ωm0−ω( )τ −1 ωm0 − ω ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Φm rΦ0 ω = 2πν ωm0 = Em − E0  TDDFT — Acrolein Example Excited State 1: Singlet-A" 4.8437 eV 255.97 nm f=0.0002 14 -> 16 0.62380 14 -> 17 0.30035 Excited State 2: Singlet-A' 7.6062 eV 163.01 nm f=0.7397 15 -> 16 0.68354 Excited State 3: Singlet-A" 9.1827 eV 135.02 nm f=0.0004 11 -> 16 -0.15957 12 -> 16 0.55680 14 -> 16 -0.19752 14 -> 17 0.29331 Excited State 4: Singlet-A" 9.7329 eV 127.39 nm f=0.0007 9 -> 17 0.19146 10 -> 16 0.12993 11 -> 16 0.56876 12 -> 16 0.26026 12 -> 17 -0.11839 14 -> 17 -0.12343 Eigenvectors PBE1/6-31G(d) LUMO+1: π4* LUMO: π3* HOMO: nO HOMO–1: π2 O H 3 7829 327 75 0 5 7412 5 1 54 6 714 84 66 3785 4 0530 14 -> 17 0.12143 Excited State 3: Singlet-A" 7.2723 eV 170.49 nm f=0.0004 3 18077 5 1 86 5 66 06 7 8041 58 87 6 13 -> 16 0.67088 5 7 - 8640 : π2 HOMO–1: nO CI Singles (CIS) — Acrolein Example Recall expt 3.73 / 6.41 eV CIS/6-31G(d) Excited Electronic States Conical Intersections and Dynamics Video VII.vi Avoided Crossings and Conical Intersections | H – ES | = 0 € H11− E H12 H 21 H 22 − E = 0 € E = H11 + H22( ) ± H11 −H22( )2 + 4H12 2 2 Can two states have the same energy E? Requires H11 = H22 and H12 = 0 This restricts two degrees of freedom and is thus not possible in a diatomic (avoided crossing rule) but it is possible for larger molecules (conical intersection) and indeed multiple electronic states can be degenerate provided sufficient numbers of degrees of freedom are available to satisfy the necessary constraints. Probability of Surface Hopping—Landau-Zener Model E ψ1 ψ2 Q V12U1 U2 € P = e − πV12 2 / hv ˙ ψ 1− ˙ ψ 2( ) € ˙ ψ i = dUi dQ What If Two States Have Different Spin Multiplicity? •  In non-relativistic quantum mechanics, transitions between two states of different spin multiplicity are strictly forbidden (although it is mildly paradoxical to refer to spin at all if one is imagining non-relativistic QM) •  However, a relativistic Hamiltonian includes operators that affect spin, including the spin-orbit operator, the spin-spin dipole operator (coupling two electrons) and the hyperfine operator (coupling electronic and nuclear spins) •  Spin-orbit coupling increases with the 4th power of the atomic number, so with heavier nuclei, this process can be very efficient € HSO ≈ 1 c 2 Zk 4 rik 3 k=1 M ∑ i=1 N ∑ l j • s j Nondynamical Photophysical Processes for a Single Geometry Dynamics adds substantial complication by changing relative state energies. Solvation compounds the difficulty by changing state energies in a time-dependent fashion as non- equilibrium solvation decays to equilibrium solvation (b) PHOTOISOMERIZATION QUANTUM YIELDS TRANS-SIDE Nn! * Bigger arrows show the most favorite paths. The red color identifies 7* decay paths, while the blue color identifies n7* decay paths. Excited Electronic States Solvatochromism Video VII. vii Solvatochromism of Dye ET30 (S1 – S0) N O + – Solvent Color λmax, nm anisole yellow 769 acetone green 677 2-pentanol blue 608 ethanol violet 550 methanol red 515 QM "SC"RF for Excited State Ψ*,"SC"RF ε, n Ψ∗ minimizes a non-equilibrium quantity Ground State Solvation Free Energy Polarization Component Generalized Born Approach ΔGP GS = − 1 2 1− 1 ε ⎛ ⎝ ⎞ ⎠ qk GSqk ʹ′ GS k,kʹ′ atoms ∑ γ kk ʹ′ is the bulk dielectric constant of the medium q is a partial atomic charge (from SCRF) Excited State Polarization Free Energy ΔGP* = − 1 2 1− 1 n2 ⎛ ⎝ ⎞ ⎠ qk*qkʹ′ * k,kʹ′ atoms ∑ γ kkʹ′ (electronic SCRF) − 1 n2 − 1 ε ⎛ ⎝ ⎞ ⎠ qk *qkʹ′ GS k,kʹ′ atoms ∑ γ kkʹ′ (elec/orient inter) + 1 2 1 n2 − 1 ε ⎛ ⎝ ⎞ ⎠ qkGSqk ʹ′ GS k,k ʹ′ atoms ∑ γ kk ʹ′ (orient cost) + 1 2 1− 1 n 2 ⎛ ⎝ ⎞ ⎠ 1 n2 − 1 ε ⎛ ⎝ ⎞ ⎠ qk * − qk GS( )qkʹ′GS k,kʹ′ atoms ∑ γ kk ʹ′ (cross term) Li et al. Int. J. Quantum Chem. 2000, 77, 264 Marenich et al. Chem. Sci. 2011, 2, 2143 Other Solvation Components! 1) Dispersion (largely responsible for red shifts in non-polar solvents) ΔνD = D n2 −1 2n2 +1 Optimized value for D = 3448 cm–1 2) Hydrogen bonding (explicit solvation effect) ΔνH = Hα Optimized value for H = –1614 cm–1 Solvatochromism of Acetone n→ * , cm–1 Solvent n EPDH Experiment (gas phase) (1.0) (1.0) (0.0) (36,165) Heptane 1.91 1.3878 0.0 417 195 Cyclohexane 2.04 1.4266 0.0 440 440 CCl4 2.23 1.4601 0.0 447 440 Diethyl Ether 4.24 1.3526 0.0 84 65 Chloroform 4.71 1.4459 0.15 –31 –125 Ethanol 24.85 1.3611 0.37 –708 –680 Methanol 32.63 1.3288 0.43 –880 –880 Acetonitrile 37.5 1.3442 0.07 –273 –335 Water 78.3 1.3330 0.82 –1522 –1670 Mean unsigned error 65 cm–1