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? Correlated Methods. I. Configuration Interaction

A Hartree-Fock one-electron orbital (wave function) is expressed as a linear combination of basis functions with expansion coefficients optimized according to a variational principle (where S is the overlap matrix) N | F – ES | = 0 φ = ∑aiϕi i=1 The HF many-electron wave function is the Slater determinant formed by occupation of lowest possible energy orbitals, but, the HF orbitals are not “perfect” because of the HF approximation

So, one way to improve things would be to treat the different Slater determinants that can be formed from any occupation of HF orbitals to themselves be a basis set to be used to create an improved many-electron wave function

occ. vir. occ. vir. r r rs rs | H – ES | = 0 Ψ = a0ΨHF +∑∑ai Ψi +∑∑aij Ψij + i r i

€ Configuration Interaction (CI) Example:

Minimal Basis H2

occ. vir. occ. vir. σ* r r rs rs ΨCI = a0ΨHF + ∑∑ai Ψi + ∑∑aij Ψij + i r i< j r

σ H11 − E H12 € | H – ES | = 0 = 0 H 21 H 22 − E

H H 2 2 E − (H11 + H 22)E +(H11H 22 − H12) = 0 € Lowest energy eigenvalue 2 2 (H11 + H22 ) ± (H11 − H22 ) + 4H12 is lower than EHF if H12 is E = positive (as it is) € 2 Recall: Higher energy eigenvalue corresponds to excited € Hab = Ψa H Ψb , e.g., H11 = E HF, H12 = Kσσ * electronic state

€ CI in a Nutshell

The bigger the CI matrix, a ab abc ΨHF Ψi Ψij Ψijk the more electron

ΨHF EHF 0 dense 0 correlation can be captured.

a sparse very sparse Ψi 0 dense The CI matrix can be made bigger either by increasing basis-set size (each block is d then bigger) or by adding e ab n sparse sparse extremely sparse Ψij more highly excited s e configurations (more blocks).

The ranked eigenvalues correspond to the electronic state energies. very extremely abc extremely sparse Ψijk 0 sparse sparse 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 a ΨHF Ψi excited configurations where m and n are the ΨHF EHF 0 number of occupied and virtual orbitals, respectively.

Diagonalization gives excited-state energies and eigenvectors containing weights of singly excited a determinants in the pure Ψ 0 dense i 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. CI Singles (CIS) — Acrolein Example

Excited State 1: Singlet-A" 4.8437 eV 255.97 nm f=0.0002 O 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 H 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 LUMO+1: π4* 14 -> 16 -0.19752 14 -> 17 0.29331 LUMO: π3* HOMO: π2 Excited State 4: Singlet-A" 9.7329 eV 127.39 nm f=0.0007 9 -> 17 0.19146 HOMO–1: nO 10 -> 16 0.12993 11 -> 16 0.56876 12 -> 16 0.26026 12 -> 17 -0.11839 14 -> 17 -0.12343 Expt Eigenvectors CIS/6-31G(d) and INDO/S CI: Thème et Variation

frozen HF If one chooses not to include all excited virtual orbitals configurations (full CI) perhaps one should enforced empty reoptimize the basis-function coefficients of the most important orbitals instead of using no more than n excitations their HF values in permitted Maybe more excitations into complete active space E restricted lower-energy orbitals is a all possible better option than any occupation active shemes allowed space excitations into higher- energy orbitals

no more than The general term for this class of n excitations calculations is multiconfiguration self- out permitted consistent field (MCSCF)—special cases are CASSCF and RASSCF— enforced doubly CASPT2 adds accuracy occupied Orbital optimization can be for an average frozen HF of state energies so as not to bias the occupied orbitals orbitals to any one state Excited Electronic States

Perturbation Theory and TD-DFT

Video VII.v Time-Dependent Perturbation Theory

Consider the time-dependent Schrödinger equation

where Φj is an − iEjt /   ∂Ψ with eigenfunctions Ψ = e ( )Φ eigenfunction of the time- − = HΨ j j i ∂t independent Schrödinger equation Perturb the Hamiltonian with a radiation field 0 H = H + 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

−(iEkt / ) Ψ = ∑cke Φk k Termination of the radiation field will cause the wave function to collapse (upon 2 sampling) to a stationary state with probability |ck| . The ck will evolve according to  ∂ −(iEkt /) 0 −(iEkt / ) − ∑cke Φk = [H + e0rsin(2πνt)]∑ck e Φk i ∂t k k Time-Dependent Perturbation Theory (cont.)

 ∂ −(iEkt /) 0 −(iEkt / ) − ∑cke Φk = [H + e0rsin(2πνt)]∑ck e Φk i ∂t k k Taking the time derivative on the left and expanding on the right

 ∂ck −(iEkt / ) −(iEkt / ) − ∑ e Φk + ∑ck Eke Φk i k ∂t k

−(iEkt /) −(iEkt / ) = ∑ck Eke Φk + e0rsin(2πνt)∑cke Φk k k Which simplifies to

 ∂ck −(iEkt / ) −(iEkt /) − ∑ e Φk = e0r sin(2πνt)∑cke Φk i k ∂t k Left multiplication by state of interest and integration yields

 ∂ck −(iEk t / ) −(iEk t / ) − ∑ e δ mk = e0 sin(2πνt)∑cke Φm r Φk i ∂t k k

€ Time-Dependent Perturbation Theory (cont.)

 ∂ck −(iEk t / ) −(iEk t / ) − ∑ e δ mk = e0 sin(2πνt)∑cke Φm r Φk i ∂t k k Evaluate Kronecker delta, rearrange, and assume perturbation is small, so ground state can be used for right-hand-side coefficients € ∂cm i −[i(Em −E0 )t /] = − e0 sin(2πνt)e Φm r Φ0 ∂t  Integrating over time of perturbation

i τ −[i(Em − E0 )t / ] cm (τ) = − e0∫ sin(2πνt)e Φm r Φ0 dt  0 i ω +ω τ i ω −ω τ 1 ⎡ e ( m0 ) −1 e ( m0 ) −1⎤ = e0⎢ − ⎥ Φm r Φ0 2i ωm0 +ω ωm0 − ω ⎣ ⎦ where Em − E0 ω = 2πν ωm0 =  Time-Dependent Perturbation Theory (cont.)

Adding Franck-Condon overlap for vibrational wave functionsexcited st a(assumingte little interaction with high frequency field)

⎡ i(ωm0 +ω)τ i(ωm0 −ω)τ ⎤ 1 e −1 e −1 m 0 cm,n(τ) = e0 ⎢ − ⎥ Φm r Φ0 Ξn Ξ0 2i ωm0 + ω ωm0 − ω E ⎣ ⎦

Em − E0

ω = 2ν πν ωm0 = h  ground state

Qualitative points: =

s b a -

The second term in brackets becomest large (but remains well behaved based on r e

series expansion of the exponential)v when the radiation frequency comes into resonance with the state-energyE separation Δ

The transition-dipole moment expectation value differentiates the absorption probability of one state from another

Excited vibrational states should have turning points at the ground-state equilibrium geometry for maximum overlap Generic Coordinate Time-Dependent Density Functional Theory

A similar mathematical formalism applied to density functional theory shows that excitation energies can be determined as poles of the polarizability matrix

⎡ 2 2 ⎤ Φm r Φ0 Φm r Φ0 α = ∑ ⎢ ± ⎥ ω ⎢ ω + ω ω − ω ⎥ m>0⎣ m0 m0 ⎦

Em − E0 ω = 2πν ωm0 =  Qualitative points:€ TD DFT tends to be more accurate than CIS but this is sensitive to choice of functional and certain special situations

Charge-transfer transitions are particularly problematic

No wave function is created, but eigenvectors analogous to those predicted by CIS are provided CI SinglesTDDFT (CIS) — Acrolein — Acrolein Example Example

Excited State 1: Singlet-A" 3.78294.8437 eV 327.75255.97 nm f=0.0000f=0.0002 O 1514 -> 16 0.674120.62380 1514 -> 17 0.105450.30035

Excited State 2: Singlet-A' 6.71427.6062 eV 184.66163.01 nm f=0.3785f=0.7397 1415 -> 16 0.605300.68354 H 14 -> 17 0.12143 Excited State 3: Singlet-A" 9.1827 eV 135.02 nm f=0.0004 Excited 11 -> State 16 3: -0.15957 Singlet-A" 7.2723 eV 170.49 nm f=0.0004 1312 -> 16 0.180770.55680 1514 -> 16 -0.11786-0.19752 LUMO+1: π4* 1514 -> 17 0.663060.29331 LUMO: π3* Excited State 4: Singlet-A" 7.80419.7329 eV 158.87127.39 nm f=0.0006f=0.0007 HOMO:HOMO: nπO2 13 9 -> -> 17 16 0.19146 0.67088 1510 -> 1716 -0.18640 0.12993 HOMO–1:HOMO–1: nπO2 11 -> 16 0.56876 12 -> 16 0.26026 12 -> 17 -0.11839 14 -> 17 -0.12343

Eigenvectors PBE1/6-31G(d)CIS/6-31G(d) Recall expt 3.73 / 6.41 eV Excited Electronic States

Conical Intersections and Dynamics

Video VII.vi Avoided Crossings and Conical Intersections

H − E H | H – ES | = 0 11 12 = 0 H 21 H 22 − E

2 2 (H11 + H22 ) ± (H11 − H22 ) + 4H12 E = € 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. Conical Intersection Example (NO2)

Note that 3 atoms leaves one final degree of freedom, so the CI is not a point, but a “seam” (that has a minimum)

Professor Carlo Petrongolo Conical Intersection Example 1D Projection

CIs permit radiationless transitions from one state to another. Kasha’s rule says that such internal conversions among excited states will be very fast until one

reaches S1 (the first state above the

ground state S0) Probability of Surface Hopping—Landau-Zener Model

− πV 2 / hv ψ˙ −ψ˙ P = e ( 12 1 2 )

U dU 2 ψ˙ = i i dQ € U 1 V12

E € ψ2

ψ1

Q 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

1 N M Z 4 H ≈ k l • s SO c 2 ∑ ∑ 3 j j i=1 k=1 rik

€ 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 Conti, I.; Marchioni, F.; Credi, A.; Orlandi, G.; Rosini, G.; Garavelli, M. JACS 2007, 129, 3198 Dynamics Occurs in All Degrees of Freedom

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 Solvatochromism Redux Equilibrium vs. Nonequilibrium Solvation gas Ψ* solution

ΔG*sol GS ΔEsol = ΔEgas + ΔG*sol – ΔG sol

solvatochromic effect ΔEsol = hνsol ΔEgas = hνgas (Stokes shift)

ΨGS

GS ΔG sol QM Self-Consistent Reaction Field (SCRF)

Ψ minimizes Hgas + Vint + Gcost — equilibrium quantity

ΨGS,SCRF

ε QM "SC"RF for Excited State Ψ∗ minimizes a non-equilibrium quantity

Ψ*,"SC"RF

ε, n Ground State Solvation Free Energy Polarization Component

Generalized Born Approach

atoms GS 1 ⎛ 1⎞ GS GS ΔGP = − 1− ∑ qk qk ʹ γ kkʹ 2 ⎝ ⎠ ε k,kʹ

is the bulk dielectric constant of the medium

q is a partial atomic charge (from SCRF) Excited State Polarization Free Energy

1 1 atoms ΔG* = − ⎛ 1− ⎞ q*q* γ (electronic SCRF) P ⎝ 2 ⎠ ∑ k kʹ kkʹ 2 n k,kʹ atoms ⎛ 1 1 * GS − − ⎞ q q γ (elec/orient inter) ⎝ 2 ⎠ ∑ k kʹ kkʹ n ε k,kʹ 1 ⎛ 1 1 atoms ⎞ qGSqGS (orient cost) + 2 − k k ʹ γ kkʹ 2 ⎝ n ε⎠ ∑ k,k ʹ atoms 1 ⎛ 1 ⎛ 1 1 * GS GS + 1− ⎞ − ⎞ q − q q γ (cross term) ⎝ 2 ⎠ ⎝ 2 ⎠ ∑( k k ) kʹ kkʹ 2 n n ε k,kʹ

Li et al. Int. J. Quantum Chem. 2000, 77, 264 Marenich et al. Chem. Sci. 2011, 2, 2143

Solvatochromism of Acetone n→ *

, cm–1 Solvent n VEM42 Experiment (gas phase) (1.0) (1.0) (36,165)

Heptane 1.91 1.3878 –241 195 Cyclohexane 2.04 1.4266 –264 440

CCl4 2.23 1.4601 –294 440 Diethyl Ether 4.24 1.3526 –530 65 Chloroform 4.71 1.4459 –515 –125 Ethanol 24.85 1.3611 –736 –680 Methanol 32.63 1.3288 –769 –880 Acetonitrile 37.5 1.3442 –763 –335 Water 78.3 1.3330 –787 –1670

Not so exciting... Other Solvation Components!

1) Dispersion (largely responsible for red shifts in non-polar solvents)

n2 −1 ΔνD = D 2n 2 +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