1. Electronic Excited
States!
Todd J Martinez
!
03/06/13!

2. Energy From Light
Light è e- + h+èCurrent Light è e- + h+èChemistry
Balzani, Stoddart, Flood PNAS (2006)
Light èMechanical Motion
Gust and coworkers, Nature Nano (2008)

3. Energy from Light
Gratzel Cell

4. Light in Biology
• Light detection / signalling
Photoactive
Yellow
Protein
Rhodopsin
• Fluorescence / chemiluminsecence / bioimaging
Fire7ly
Luciferase
Green
Fluorescent
Protein

5. Basic Principles
• Ground state chemical reactions
– Generally concerned with near-equilibrium properties
– Reaction rates well described with statistical theories
rate ! " attempt e # $E † /kT
!E †
Probability to cross barrier
“attempt frequency”
reactant
Potential energy surface (PES)
product
Where does the potential energy surface come from? PES
Byproduct of the Born-Oppenheimer approximation
! ( R,r ) " # nuc ( R )$ el ( r; R ); H el ( r; R )$ el ( r; R ) = E ( R )$ el ( r; R )
ˆ
BOA generally valid for ground state reactions at low (< 5000K) T

6. Excited State Reactions
• Generalization of BOA can be entertained:
! ( R,r ) " # nuc ( R )$ iel ( r; R ); H el ( r; R )$ iel ( r; R ) = Ei ( R )$ iel ( r; R )
ˆ
S1
PES for ith electronic state
S0
Sn is nth
singlet spin
electronic
state
This makes sense if we ignore all other electronic states.
But if electronic gap gets small, this will not make sense…
And classical mechanics will become problematic:
!Ei ! ! Only know how to solve this with one
" F = ma potential surface, i.e. one of the Ei
!R

7. Excited State Reactions
• Light absorption is near-instantaneous (Franck-Condon principle)
Thus, excited state dynamics is often initiated far from equilibrium
– Statistical theories may fail dramatically
– In many cases, need dynamics
– Reactions can be very fast (< 1 picosecond)
• Cartoon picture of excited state reaction:
Akin to two-slit
S1 experiment – wavepacket
breaks into two parts
S0
hvabs hvfl
“Avoided Crossing”
BOA and classical mechanics fail
Radiative decay (fluorescence)
Typically nanoseconds…

8. Adiabatic and Diabatic Representations
• Electronic transitions are promoted by off-diagonal elements of total
(nuclear and electronic) Hamiltonian
• Adiabatic representation
– Born-Oppenheimer states that diagonalize the electronic Hamiltonian
– Coupling terms are in kinetic energy
! Tˆ Mv i d12 $ ! V1 ( R ) 0 $
# & +# &
# Mv i d21
" Tˆ & #
% " 0 V2 ( R ) &
%
• Diabatic representation
– Electronic states are chosen to minimize kinetic couplings
– Coupling terms are in potential energy
– Can be proven that strictly diabatic states only exist for diatomics…
– But nearly diabatic states can always be obtained (means there will be
small residual couplings in kinetic energy)
! T 0 $ ! V11 ( R ) V12 ( R )
ˆ $
# & +# &
" 0 T % # V12 ( R ) V22 ( R )
ˆ
" &
%

9. Adiabatic and Diabatic Representations
V2
V11 V22
V1
Covalent
Covalent - AB
Ionic
Ionic – A+B-
Adiabatic Diabatic
These are the states which come out Need to construct these states by
of an electronic structure code – unique, Rotating adiabatic states to minimize
but rapidly changing electronic character kinetic coupling terms. Not unique,
near crossings. but state labels correspond to electronic
character

10. Adiabatic and Diabatic Representations
• Electronic transitions promoted by:
– Diabatic: V12(R)
! !
– Adiabatic: Mv i d12
Electronic “velocities” – how fast is electronic
Nuclear velocities wavefunction changing?
ˆ
" H el
!el
! !2
! el "
1
"R
d12 = !1 ! ! 2 =
el
"R V1 ( R ) # V2 ( R )
Large near avoided crossings

11. Nonadiabatic Transitions
• For avoided crossing of two states in one dimension, transition
probabilities given by Landau-Zener formula (in diabatic
representation):
$ '
& !2" V12
2
) V12àinfinity; PLZà0
PLZ = exp & )
#V #V vàinfinity; PLZà1
& !v 1 #R ! 2 #R )
% (
Phop =
• PLZ is probability to stay on the same surface
• Assumes linear diabats with constant coupling and constant nuclear
velocities
V1 V2
V12

12. Pictures of Internal Conversion
S1
$ '
2 #E12
2
hν
P LZ
= exp & !2" ! ! )
hop
&
% h vid12 )
(
! "
S0 d12 = ! 1 ( r; R ) ! 2 ( r; R )
0o 90o 180o "R r
trans cis
“Nonadiabatic Coupling”
Avoided Crossings Conical Intersections
For many years, it was thought that avoided
crossings were the whole story…
Now it is known that CIs are the rule, not the
exception: e.g., Michl, Yarkony, Robb, …

13. Are CIs and ACs Different?
• CIs
• Many avoided crossings in the neighborhood
• Many CIs in neighborhood (N-2 dimensional seams)
• Energy gap lifted linearly around CI
• Geometric phase
• ACs ! electronic " #! electronic
• Energy gap lifted quadratically around AC
• Isolated from other ACs X
No CI
CI
! electronic " ! electronic
Geometric (Berry’s) Phase

14. Refined Cartoon Picture of Excited State Dynamics
Conical Intersection

15. Do CIs Matter?
MECI – Minimal Energy CIs –
lowest energy point (locally) along seam

16. Conical Intersection Topography
“Sloped”
“Peaked”
Does topography affect transition rate?

17. Limiting Scenarios
Lifetime determined Lifetime determined Barrier-like, but no
by dynamics by barrier crossing well-defined transition state
Simulate dynamics directly Rate theory?
Simulate dynamics directly
Use transition state theory to address barrier crossing

18. Obstacles in Excited State Simulations
• Need electronic structure methods that can describe excited
electronic states…
• Difficult to use empirical force fields – often insufficiently flexible to
describe excited states
• Need to describe nonadiabatic effects (curve/surface crossings) –
some form of quantum dynamics is needed

19. Excited States
• DFT is a ground state theory – does this mean we cannot
access excited electronic states?
• Not really – excitation energies are a property of the
ground state…
fI Frequency-dependent
! (" ) = $
" I2 # " 2
I
polarizability
! I = E I " E0
2
( ˆ
fI = ! I # 0 x # I
3
2
ˆ
+ #0 y #I
2
ˆ
+ #0 z #I
2
)
If we know FDP, look for poles and these are excitation
energies…
19

20. TDDFT
• Runge-Gross Theorem – analog of Hohenberg-Kohn for
time-dependent system
• There is a time-dependent potential that maps the density
of a noninteracting (Kohn-Sham-like) system onto the
true time-dependent density
• New wrinkles:
– The RG potential can depend on the initial wavefunction
– The RG potential can be nonlocal in time
• Common approximations
– Ignore dependence on initial state
– Assume RG potential has form of Vxc (adiabatic approximation)
• Now, can calculate response properties of molecule to
time-varying electric field, e.g. FDP
20

21. TDDFT
Adiabatic approximation: ignore ω dependence
Tamm-Dancoff approximation: ignore B
Closely related to CI restricted to single excitations…
See Chem. Rev. 105 4009 (2005)
21

22. TDDFT - Example
Some functionals are good, some are not
Unfortunately, different ones are good
22
for different problems

23. Failures of TDDFT
Polarizability should scale linearly with size of chain…
Derivative discontinuity again, i.e. problem from DFT… 23

24. Conical Intersections
Degeneracy should be lifted
along two coordinates
Is this true in TDDFT?
24

25. Conical Intersection Branching Plane
Why are there two directions which break the degeneracy?
Can it be one? Can it be three or more?
Electronic Hamiltonian in diabatic representation:
! V ! ! $
! 11 ( )
R V12 ( )
R !
&= E 0 $ ! '( V12 $
H el ( )
R =# ! ! # & +# V &
# V
" 12 ( )
R V22 ( )
R & " 0
%
E % # 12 (
" &
%
!
( )
E ± R = E ± V12 + "2
2
Conical intersection only if:
!
V12 ( R) = 0 Each equation defines an N-1 dimensional surface
! " Intersection of two N-1 dimensional surfaces has
( )
" R =0 dimension N-2
Two independent functions – two degrees of freedom to satisfy two equations
Only by accident or miracle!
!
25

26. CIs in TDDFT?
• First, consider Single Excitation CI
AX = ! X; Aia, jb ˆ "b
= " H j a
i
or
! E0 0 $ ! c0 $ ! c0 $
# &# & = E# &
" 0 A %" X % " X %
Vanishes at ALL geometries – Brillouin s Thm
Only ONE condition to satisfy – E0 = Elowest excited
Does this matter?
26
Mol Phys 104 1039 (2006)

27. Example Conical Intersection
CASSCF
H2 – O – H1
27

28. Conical Intersection in CIS
No conical intersections b/t S0 and S1
Infinitely many more intersections b/t S0 and S1
28

29. Does TDDFT Solve the Problem?
No… Lesson is that DFT and TDDFT as usually practiced cannot
solve problems with underlying wavefunction ansatz…
29

30. Excited State Electronic Structure
• Need to be able to describe multiple degenerate states
– Without this, intersections will always be incorrect…
• Need dynamic electron correlation
– Electron correlation effects are very different on different
electronic states; thus excitation energies are sensitive to this
• CIS
– Assumes ground state is nondegenerate; no dynamic correlation
• TDDFT
– Assumes ground state is nondegenerate; up to 1000 atoms
• MCSCF
– No dynamic correlation
• Multireference Perturbation Theory
– currently best option, but not feasible for large molecules