bios 203 lecture 5: electronic excited states
TRANSCRIPT
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)
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
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 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)
“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