Electronic Excited States!

Todd J Martinez!03/06/13!

Energy From Light

Balzani, Stoddart, Flood PNAS (2006)

Light è e- + h+èChemistry Light è e- + h+èCurrent

Light èMechanical Motion Gust and coworkers, Nature Nano (2008)

Energy from Light

Gratzel Cell

Light in Biology

•  Light detection / signalling

•  Fluorescence / chemiluminsecence / bioimaging

Photoactive  Yellow  Protein  

Green  Fluorescent  Protein  

Rhodopsin  

Fire7ly  Luciferase  

Basic Principles

•  Ground state chemical reactions –  Generally concerned with near-equilibrium properties –  Reaction rates well described with statistical theories

rate !"attempte#$E†/kT

“attempt frequency” Probability to cross barrier

reactant

product

!E†

Potential energy surface (PES)

Where does the potential energy surface come from?

Byproduct of the Born-Oppenheimer approximation

! R,r( ) " #nuc R( )$el r;R( ); Hel r;R( )$el r;R( ) = E R( )$el r;R( )

PES

BOA generally valid for ground state reactions at low (< 5000K) T

Excited State Reactions

•  Generalization of BOA can be entertained:

! R,r( ) " #nuc R( )$iel r;R( ); Hel r;R( )$i

el r;R( ) = Ei R( )$iel r;R( )

PES for ith electronic state

S0

S1

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

!R"!F = m!a Only know how to solve this with one

potential surface, i.e. one of the Ei

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:

S0

S1

“Avoided Crossing” BOA and classical mechanics fail

hvabs hvfl

Radiative decay (fluorescence) Typically nanoseconds…

Akin to two-slit experiment – wavepacket breaks into two parts

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

•  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 00 T

!

"#

$

%& +

V11 R( ) V12 R( )V12 R( ) V22 R( )

!

"##

$

%&&

T Mv i d12Mv i d21 T

!

"##

$

%&&+

V1 R( ) 0

0 V2 R( )!

"##

$

%&&

Adiabatic and Diabatic Representations

V1

V2

V11 V22

Adiabatic Diabatic

These are the states which come out of an electronic structure code – unique, but rapidly changing electronic character near crossings.

Ionic – A+B-

Covalent - AB

Need to construct these states by Rotating adiabatic states to minimize kinetic coupling terms. Not unique, but state labels correspond to electronic character

Ionic

Covalent

Adiabatic and Diabatic Representations

•  Electronic transitions promoted by: –  Diabatic: V12(R) –  Adiabatic: M

!v i!d12

Nuclear velocities Electronic “velocities” – how fast is electronic wavefunction changing?

!d12 = !1

el ""!R!2

el =!1

el "H"!R!2

el

V1 R( )#V2 R( )

Large near avoided crossings

Nonadiabatic Transitions

•  For avoided crossing of two states in one dimension, transition probabilities given by Landau-Zener formula (in diabatic representation):

•  PLZ is probability to stay on the same surface •  Assumes linear diabats with constant coupling and constant nuclear

velocities

Phop =

PLZ = exp!2"V12

2

!v #V1 #R ! #V2 #R

$

%

&&&

'

(

)))

V1 V2

V12

V12àinfinity; PLZà0 vàinfinity; PLZà1

hν

trans cis

S1

S0 0o 90o 180o

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, …

Pictures of Internal Conversion

PhopLZ = exp !2" 2 #E12

2

h !vi!d12

$

%&&

'

())

!d12 = ! 1 r;R( ) "

"R! 2 r;R( )

r

“Nonadiabatic Coupling”

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 •  Energy gap lifted quadratically around AC •  Isolated from other ACs X

! electronic " #! electronic

CI

! electronic "! electronic

No CI

Geometric (Berry’s) Phase

Refined Cartoon Picture of Excited State Dynamics

Conical Intersection

Do CIs Matter?

MECI – Minimal Energy CIs – lowest energy point (locally) along seam

Conical Intersection Topography

“Peaked” “Sloped”

Does topography affect transition rate?

Limiting Scenarios

Lifetime determined by dynamics

Lifetime determined by barrier crossing

Barrier-like, but no well-defined transition state

Rate theory? Simulate dynamics directly Simulate dynamics directly

Use transition state theory to address barrier crossing

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

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…

19

! "( ) = fI" I2 #" 2

I$ Frequency-dependent

polarizability ! I = EI " E0

fI =23! I # 0 x # I

2+ # 0 y # I

2+ # 0 z # I

2( )

If we know FDP, look for poles and these are excitation energies…

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

TDDFT

21

Adiabatic approximation: ignore ω dependence Tamm-Dancoff approximation: ignore B Closely related to CI restricted to single excitations…

See Chem. Rev. 105 4009 (2005)

TDDFT - Example

22

Some functionals are good, some are not Unfortunately, different ones are good for different problems

Failures of TDDFT

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

Conical Intersections

24

Degeneracy should be lifted along two coordinates

Is this true in TDDFT?

Conical Intersection Branching Plane

25

Why are there two directions which break the degeneracy? Can it be one? Can it be three or more? Electronic Hamiltonian in diabatic representation:

Hel

!R( ) =

V11!R( ) V12

!R( )

V12!R( ) V22

!R( )

!

"

##

$

%

&&= E 0

0 E

!

"#

$

%& +

'( V12V12 (

!

"##

$

%&&

!

E±

! R ( ) = E ± V12

2 + "2

Conical intersection only if:

!

V12! R ( ) = 0

"" R ( ) = 0

Two independent functions – two degrees of freedom to satisfy two equations

Each equation defines an N-1 dimensional surface Intersection of two N-1 dimensional surfaces has

dimension N-2

Only by accident or miracle!

CIs in TDDFT?

•  First, consider Single Excitation CI

26

E0 00 A

!

"#

$

%&

c0X

!

"#

$

%& = E

c0X

!

"#

$

%&

Vanishes at ALL geometries – Brillouin’s Thm

AX =!X; Aia, jb = " ia H " j

b

or

Only ONE condition to satisfy – E0 = Elowest excited Does this matter?

Mol Phys 104 1039 (2006)

Example Conical Intersection

27

H2 – O – H1

CASSCF

“Conical Intersection” in CIS

28

No conical intersections b/t S0 and S1 Infinitely many more intersections b/t S0 and S1

Does TDDFT Solve the Problem?

29

No… Lesson is that DFT and TDDFT as usually practiced cannot solve problems with underlying wavefunction ansatz…

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