Ab Initio Multiple Spawning DynamicsTodd J. Martínez
Department of Chemistry & The Beckman InstituteUniversity of Illinois at Urbana-Champaign
Ab Initio Molecular Dynamics Terminology
ClassicalNoAnyDFT Ab initio
BOMDBorn-Oppenheimer Molecular Dynamics
ClassicalNoAnyDFT Ab initioSemiempirical
Direct Dynamics
ClassicalYesGaussianDFT or SCFADMPAtom-Centered Density Matrix Propagation
ClassicalYesPlane WaveDFTCPMDCar-Parrinello Molecular Dynamics
DynamicsExtended Lagrangian?
Electronic Basis
Electronic Structure
Electronic Basis Sets
Plane waves -– Natural for periodic systems– Hellman-Feynman Forces are exact– Very inefficient for nonperiodic systems (large empty spaces between periodic
replicas; in almost all cases, this empty space is filled with basis functions)– Obvious, well-defined convergence hierarchy
Gaussian –– Periodicity is more difficult to incorporate (but not impossible, e.g. Crystal)– Generally atom-centered (“floating” Gaussians also possible)
Electrons like to be near nuclei!– Pulay terms in forces for atom-centered Gaussians– Convergence hierarchy more subtle, but now well-understood (Dunning)
2re α−
ikre
……Empty space
“Pulay” Terms
( ) ( )
( ) ( )
( ) ( ) ( ) ( ); ;ˆ ˆ; ;
ˆ; ;
ˆ; ;elec
el
elec elecelec elec elec ele
elec el
ec elec
r
ec elec
r
r
cr
Hr R r R
E r R H r
r R r RH r R r R
R
R
R
R
R
H
Rψ ψ
ψ ψψ ψ
ψ ψ=
∂ ∂+
∂ ∂=
∂
∂
∂
+∂
∂
∂
“Hellmann-Feynman Force”
“Pulay Force”
Pulay force apparently requires derivatives of electronic wavefunctions!Actually, often it does not, but it is always more expensive than HF Force
Extended Lagrangian TechniqueWhat is it?
– Electronic degrees of freedom are propagated under a new Lagrangian, instead of using the variational principle to determine them at each new geometry
Benefits– Wavefunction does not need to be converged at each time step
“1” iteration vs. typical 10-15 iterations (apparently)But, good BOMD codes require < 5 iter b/c geometry change in one time step is small
Disadvantages– Need to ensure that the electronic coefficients are always “cold” and nuclei are
always “hot”Energy transfer between ions and “fictitious” degrees of freedom is unphysical
– Time step needs to be smaller than in BOMD – fictitious degrees of freedom are high-frequency relative to nuclei
Bottom line (according to TJM)– EL schemes have ½ time step to save 5 iterations/time step. Works out to a 2.5x
speed advantage in favor of EL– But need to be careful to maintain adiabatic separation of nuclei and electrons. – Both are reasonable and choice depends on taste
EL expands dynamic range of frequencies to reduce SCF iterations
Electronic Excited States
C C
H
Ph H
Ph
(CH3)2C C(CH3)2hν
Ph
Ph
CH3
CH3
CH3
CH3
+
N N
N N
Mg
O
R
CO2CH3
N
Opsin
H
Chemical Synthesis
Energy Conversion
Vision
Light-Driven Molecular Devices
F
hν
F
Switch electron transport on and off
Light → Mechanical Work
Other Applications…Optical MemoryMolecular LogicLocal control of pH –
“chemical circuits”Drug delivery
Yu, et al, Nature (2003)Hugel, et al, Science (2002)
More Light-Driven Devices
Tour, Org. Lett. (2006)
Balzani, Stoddart, PNAS (2006)
Breakdown of Born-Oppenheimer Approximation
Avoided CrossingsDegeneracy lifted inN-dimensional space
Conical IntersectionsDegeneracy lifted in 2-dimensional space
CIs are the rule, not the exception: e.g., Michl, Yarkony, Robb, …
Including Quantum Effects
Classical with “ad hoc corrections”– Not guaranteed to converge to the right answer…– No well-defined nuclear wavefunction
Mixed Quantum/Classical (See Iyengar Talk)– Based on time-dependent self-consistent field approximation– Not obvious that QM and CM are strictly compatible!
Fully Quantum Mechanical– Approach is well-defined– But difficult (in fact, exponentially difficult in general case)– Basis set and path integral approaches
Feynman Path integrals – Imaginary time (quantum statistics) is easy– “Sign” problem in real time – progress, but still unsolved
Basis set– What kind of functions?– How to choose them?
Nonadiabatic Dynamics
Ehrenfest Methods
( ) ( )*ave I J IJ
IJV R c c V R= ∑
r r( )I IJ J
Jc i H R c= − ∑
r&
Vave is unphysical asymptotically!Why? - Wavepacket splits into two parts
and averaging becomes senseless
Hamilton’s Eqns with Vave for R and P
• Surface Hopping (Tully, Truhlar, many more…)
Avoid unphysical averagingStochastic “hops”No systematic hierarchy
to improve approximations
• Ab Initio Quantum Dynamics → “On-the-fly” solution of electronic and nuclear Schrödinger equations
• Multiple electronic states/Tunneling → Need quantum nuclear dynamics• Bond rearrangement → Need to solve electronic Schrödinger equation
•Exact numerical solution of the nuclear Schrödinger equation is impractical for large systems.
• Ab initio quantum chemistry is local• Nuclear Schrödinger equation is global • Not a problem in Car-Parrinello b/c Newtonian mechanics is local…
Further considerations: • Need PESs for both ground and excited states.• Matrix elements which couple electronic states • Minimize the number of PESs evaluations b/c of extreme expense.
Tailor the requirements of quantum dynamics to quantum chemistryand vice-versa!
ψ ψIelectronic
Jelectronic
R∂
∂
Ab initio Quantum Dynamics
Basis Set Approaches to Quantum Dynamics
( )1xδ( )2xδ Orthogonal basis of δ functions ( ) ( )
( )
i ii
i i
x c x
c x
ψ δ
ψ
≈
=
∑
Determine ci(t) using FFT method, as described by Batista yesterday…
For cognoscenti – the basis functions in the FFTmethod are actually sinc functions
Nonorthogonal basis( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )1
,
,0 0 0
0 0
i ii
j i j i i jii i
x t c t x
x x dx c x x dx c S
c
ψ χ
χ ψ χ χ
ψ−
≈
= =
=
∑
∑ ∑∫ ∫S
r r
Insert into t-Dep Schrödinger Equation:d idt
−= − 1c S Hc
Set of ODE’s – integrate as discussed by Martyna
Basis Set Approaches to Quantum Dynamics
χ1(x,t) χ2(x,t) χ3(x,t)t-dependent nonorthogonal basis set
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )1
, ,
,0 ,0 0 ,0 ,0 0 0
0 0 0
i ii
j i j i i jii i
x t c t x t
x x dx c x x dx c S
c
ψ χ
χ ψ χ χ
ψ−
≈
= =
=
∑
∑ ∑∫ ∫S r r
Insert ansatz into t-Dep Schrödinger Equation: ( )1i i− ⎡ ⎤= − −⎣ ⎦C S H S C& &
( )( ) ( ) ( ), ,i jijt x t x t dx
tχ χ∂
=∂∫S& Note this is NOT the time derivative
of the overlap matrix!
Ideal Basis Set
Ideally, our basis set would be– Compact– Accurately describe the wavefunction at all times– Easily determined at any point in time
Fixed basis sets (orthogonal and nonorthogonal)– Must span all Hilbert space which is sampled by the wavefunction from
beginning to end of simulation!– Recall Batista’s example of a Gaussian basis function evolving in a
harmonic potential – it does not deform and its center follows classical mechanics
– With a fixed basis set, we must cover the entire configuration space accessed
– Even in 1D, the problem is much worse if wavefunction is unbound – for free particle, we MUST have basis functions covering ALL space uniformly
Time Evolution
Batista already showed us one way to get around this – adaptively expanding a Gaussian basis set at each point in timeCould we make a good guess at the evolution of these basis functions?If we really believe that classical mechanics is usually sufficient, there should be a way to take advantage of it!Why Gaussians (“coherent states”)?– Exhibit classical motion without deformation in harmonic potential– Minimum uncertainty states – as p/q localized as possible in QM
Localization is good for scaling – analogy to local bases in quantum chemistryLocalization lends itself well to numerical approximations (e.g. integrals)Localization is needed for interface to electronic structure!Overcomplete if all p,q allowed – this will be good for us!
Would a basis set of Gaussians with classical EOM be sufficient in principle?– Need to have classical energy width!– Otherwise, energy conservation will restrict allowed space– If no hidden conserved quantities and all energies are allowed, infinite
basis by this prescription will be overcomplete, hence the answer is YES
•Wavefunction ansatz: Ψ R R; ;t C t t IjI
jI
jIa f a f a f= ∑∑ χ
1 244 344
Nuclear wavefunction Electronic state
• Nuclear wavefunction on each electronic state is a product of 3Nfrozen Gaussian basis functions:
χ χ αγρ ρ ρ ρ ρ
ρjI i t
jI
j j jN
t e R R t P tjI
R; ~ ( ; , , )a f a f a fa f= ∏=1
3
Semiclassical phaseCartesian degrees of freedom
Position, momentum, width
• Classical evolution for R(t) and P(t).•Variational principle for coefficients:
Nuclear overlap matrix Hamiltonian matrix
( )1I II II II I IJ Ji i− ⎡ ⎤= − − +⎣ ⎦C S H S C H C& &
The Full Multiple Spawning (FMS) Method
( )( )Re ,Ij R tχr
Principle to Practice
In principle, we do not need to do anything special – just to have enough “trajectory basis functions”But who cares if the basis set prescription would be sufficient in principle?!Our computers have limited memory and we have limited lifespans…Our basis set needs to be able to adapt to the problem!
01234567
80 90 100 110 120 130Effe
ctiv
e no
n-ad
iaba
tic c
oupl
ing
Time / fs
crossing time
crossing time
spawningthreshold
Adaptive Basis Set
New Basis Functions added to augment classical mechanics and accelerate convergence
R(t)
time
“Spawned” Basis Functions
Nonadiabatic Coupling Regions
|C(t)
|2
Total
Individual basisfunctions
RS|T|
Time Discretization
• Basis set is small because crossing time is typically short.
• Each basis function is “responsible” for a portion of the population transfer.
Time Discretization
Cou
plin
g
Time
Central discretizationpoint should be chosen at maximum coupling, not midway in time interval
Other discretization points chosen to be equispaced in time
MultiSpawn Number of discretization points per event
Time Discretization
Start of “event” CSThresh > CFThresh
Usually chosen to be very close to each other
TimeCFThresh
End of “event”
Avoid many small spawning regions
CSThresh
Cou
plin
g
Time Discretization – Long Events
Time
Each trajectory basis function can be in “spawning” mode In this case, it knows not to spawn again until a certain time is reached.
Spawn separately in each time interval
Cou
plin
g Maximum time for discretization
More Sophisticated Spawning Schemes C
oupl
ing
Time
“Constant Area” Spawning Invariant to time step; Under development
Spawning Regions
Where to Spawn?
P
X
momentumjumpposition
jump
E1
P
X
positionjump
E2
•Want largest matrix element connecting new and old basis functions•Demand Classical Energy Conservation – still have freedom
-0.01
-0.005
0
0.005
0.01
-3 -2 -1 0 1 2 3
Ener
gy
X
E2
E1
Energy for Spawned Trajectories
k=4
Energy for Spawned Trajectories
k=10
k
x
Wigner view of Spawning
“position jump” type of behavior
k=4
Wigner view of Spawning
x
k
“momentum jump” type of behaviork=10
“Frustrated” Spawns
EShellOnly=False
SteepestDescent=l
E
Steepest Descent does not always find a solution within allotted time. In this case, best solution is used iff EShellOnly=False
Linear Dependence and Matrix Inversion
When many gaussian basis functions are used, linear dependence may become an issueControlled by regularization
– Singular value decomposition of Overlap Matrix– Singular values below a threshold are discarded– Norm conservation is affected
Overlap matrix does not have to be inverted. Solve by conjugate gradient (nice regularizing properties!):
( )i i⎡ ⎤= − −⎣ ⎦SC H S C& &
Cost is N2 instead of N3, but sparsity makes it O(N)
Diabatic vs Adiabatic Representations
Diabatic– Smoother, therefore easier to integrate classical equations of motion for
trajectories– Couplings can be large – much spawning required– If diabatic model is purely quadratic, benefit from exact propagation when
couplings are small– Also, may be able to avoid integral approximations– Requires global information about PES – not suitable for ab initio MD
Adiabatic– Nonadiabatic events are spatially/temporally localized – few spawning
events lasting for short time– Classical mechanics is more nearly correct
because electronic states are usually independent
Adiabatic
Diabatic
Running A Simulation – FMS Dynamics
FMS Dynamics– Widths need to be determined – these need to be tested for robustness –
25% changes should not affect amount of population transfer– Need to determine appropriate values for CSThresh and CFThresh – this
needs to be checked for every new molecule– Run several trajectories without spawning, monitor nonadiabatic coupling,
choose thresholds low enough to capture all population transfer but otherwise as large as possible
Running a Simulation – AIMS Dynamics
Ab initio multiple spawning– Need to ensure that electronic structure method is accurate– This inolves iteration between AIMS and high-level electronic structure – Example with CASSCF – need to choose an active space and state-
averagingState ordering and S2/S1 gap at Franck-Condon pointPreliminary dynamics calculationCalculate CASPT2 PESs along CAS trajectories – demand consistencyMinimize snapshots from dynamics to find “attractor” minima or MECIsVerify energetics of these points with
CASPT2 / MS-CASPT2If inconsistent, dynamics results are discarded –
continue search for active spaceCAS
CASPT2
• Two Diabatic Surfaces (A+BC→AB+C collinear model)• Triangles denote centroids of Gaussians – not all are populated!• Linear dependence “prunes” number of Gaussians spawned• “Back-spawning” is allowed
Initial Basis Functions
Spawned Basis FunctionsNonadiabatic Coupling Region
Multiple Spawning
VA+BC
VAB+C
Initial Conditions
Sampling Points
p
q
FirstGauss=TrueGuarantees that the center of the Wigner is first basis function
OMaxI if overlap between anytwo initial conditions > OMaxI,one is rejected. Prevents excessive linear dependence at t=0
Initial Conditions
Wigner distribution – only obtainable in practice within harmonic approximationProblematic above v=0 because of negative regions; difficult to sampleNot appropriate for condensed phase because of many low frequency vibrationsQuasiclassical + Wigner – Use Wigner for chromophore and quasiclassical for remainder – for systems where chromophore/solvent decomposition is obvious
σ ω ω φ φ ωA I I i i IC t i t dtFH IK−∞
+∞FH IK= z e j exp ~
φ μ φ φi i ex ii t iH t= = −FHGIKJRe j e j; exp $
σ ω ω ω φ φ ωRi f
I I s f i IC t i t dt→ FH IK FH IK∞
= ′ z30
2
e j exp ~
Time-domain Formulation of Spectroscopy (Heller)
Electronic Absorption Spectrum:
Resonance Raman Excitation Profiles:
•Anharmonicity and Duschinsky rotation included
•Coordinate dependence of transition dipole can be included
Absorption and Resonance Raman Spectra
0
0.2
0.4
0.6
0.8
1
0 1000 2000 3000 4000 5000 6000 7000 8000
Time / atu
-1
-0.5
0
0.5
1
0 1000 2000 3000 4000 5000 6000 7000 8000
Time / atu
Correlation Functions
Correlation Function Spectrum
-1.5 -1 -0.5 0 0.5 1 1.5
Energy / eV
0
0.2
0.4
0.6
0.8
1
0 1000 2000 3000 4000 5000 6000 7000 8000
Time / atu
Convergence
-1 -0.5 0 0.5 1
Energy / eV-1 -0.5 0 0.5 1
Energy / eV
1 BFxn
10 BFxns
10 BFxns
20 BFxns
Ethylene V State, Vibronic Coupling Model
Pyrazine Electronic Absorption Spectra
S1 S2
N
N
(1) (1)1 0 1 1 6a 6a 10a
(2) (2)10a 1 0 1 1 6a 6a
E H Q Q QH
Q E H Q Q⎡ ⎤+ + κ + κ λ
= ⎢ ⎥λ + + κ + κ⎣ ⎦
22
0 i i2i i
1H ( Q )2 Q
∂= ω − +
∂∑
• Structureless S2 spectrum implying fast S2→S1 internal conversion• 3D/4D/24D vibronic coupling (Domcke/Cederbaum/Meyer/Koppel)
Pyrazine Electronic Absorption Spectra Effect of Nonadiabaticity on Spectrum
MCTDHNonadiabatic
Adiabatic150fs
• C(t) limited to 150fs – still leads to significant structure w/o nonadiabaticity• MCTDH for comparison
Pyrazine: 3 dimensional model
Pyrazine:4 dimensional model
Pyrazine:24 dimensions
Results for Electronic Spectra
• Prespawning used in all of these calculations• Final number of basis functions ranges from 100-900• Somewhat better agreement from IVR methods w/ 106 trajectories
Thoss, Miller and Stock, JCP (2000)• Fewer basis functions may suffice in adiabatic representation
Results for Electronic Spectra Dependence on Spawning Threshold
λ0=2.5 λ
0=5.0 λ
0=20.0
• Large spawning thresholds get basic envelope correct
• Small thresholds required to reproduce detailed structure
• Smooth convergence observed
Dependence on Spawning Threshold
Saddle-Point Approximations
Integrals which are difficult to evaluate in AIMS are treated with saddle point approximation:
( )I J I Ji IJ j i j IJV V Rχ χ χ χ≈
Centroid of Product Gaussian
Minimal effect on spectrum
w/SPA
w/o SPA
(Higher order variants possible, but require Hessian)
Accuracy of FMS Method: The Spin Boson Model
Two level system linearly coupled to a Harmonic bath:
In the diabatic representation:
-- vibrational frequency of the jth oscillator-- dimensionless linear coupling of the jth oscillator-- electronic coupling
-- dimensionless position and momentum
BS S BH H H H −= + +
{ }
( )
1 21,2
2 20 0
. .
1 ; 2
k k kk
k j j j j jj j
H h c c
h h C X h P X
ϕ ϕ ϕ ϕ
ω
=
= + Δ +
= + = +
∑
∑ ∑jωjC
Δ,j jX P
In the form of Domcke-Cederbaum-Koppel vibronic coupling modelsOften used as a model for conical intersection problems
Bath properties are determined by the spectral density:
The case of an Ohmic bath:
Exact Results:400 dimensional model
– C. H. Mak and D. Chandler, Phys. Rev. A 44, 2352 (1991)– D. E. Makarov and N. Makri, Chem. Phys. Lett. 221, 482 (1994)– R. Egger and C. H. Mak, Phys. Rev. B 50, 15210 (1994)
4 dimensional model– G. Stock, J. Chem. Phys. 103, 1561 (1995) – Multiconfigurational Time-Dependent Hartree (MCTDH)
Manthe, Meyer, Cederbaum Chem Phys Lett 165, 163 (1990)
( ) ( )2
2 j jj
J Cπω δ ω ω= −∑
The Spin Boson Model
( )2
cJ e ω ωπω α −= ⋅ ( )1 22
j jC Jω ωπ
⎡ ⎤= Δ⎢ ⎥⎣ ⎦
Extended Coupling Regions
Adaptive expansion is tailored to time/space-local breakdowns of the Born-Oppenheimer approximationWhat about cases where the BOA breaks down completely?Continuous spawning – ensure that basis functions always have “partners” on other electronic states when they are coupled. Why is this even practical? PESs tend to be nearly parallel in extended coupling regions!
12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0
Spin Boson Tests – 4 DOF
FMS-CS Numerically Exact
Surface-Hopping
Ehrenfest
Spin Boson Tests – 400 DOF
Surface Hopping
In high dimensionality and finite T, convergence is actually faster –basis function separation is physically correct.
(See early work by Neria & Nitzan, Rossky, Bittner)
Time →
Multiple Spawning
• FMS is a hierarchy of methods • Dynamics on a single electronic state→coupled frozen Gaussians (Heller,Metiu)• Useful Approximations in Ab Initio Dynamics
• Independent First Generation Approximation• Different initial Gaussian wavepackets are uncoupled• Includes coherences between basis function and its “children”• Neglects coherence between initial basis functions
• Saddle-Point Approximation for Integrals• Use locality of Gaussians to evaluate integrals using local information
( ) ci j i jR f R R dR f R R R dRχ χ χ χ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
≈∫ ∫Centroid of χiχj
Independent First-Generation Approximation
*ˆ ˆi
i ii
i ii
c
O c c Oψ ψ
ψ ψ=
≈
∑
∑IFG
Run 1
Run 2
Saddle-Point Approximations
( ) ( ) ( ) ( )2
2
2
( )
c c
c c c cR R
f ff R R R R R RR R
i j
i j i j
R f R R dR
R R dR f R R
χ χ
χ χ χ χ
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
=
≈
∫
∫ ∫L
•Taylor expansion of potential around maximum of product•First-order term requires gradient at maximum of product•Second-order term can be included if Hessian of potential available
AIMS-EOM-CCSD
Expt
R3s
V
f=0.15
f=0.38
EOM-CCSD Spectra for Ethylene
R3s and V spectra shifted to large basis MRSDCI vertical excitation energies
•Prototype for cis-trans isomerization •Conventional Picture:
φ →cis trans
φ
• One-dimensional nature - emphasizing torsion• Chemistry and Isomerization occur on S0 • Experimental Photoproducts: cis-trans isomers, HC≡CH, H2, H• Excited State Lifetime from Absorption Spectrum: < 50 fs• No fluorescence
Photochemistry /Photophysics of Ethylene
hν 0 fs 50 fs 110 fs 150 fs
170 fs
AIMS Excited State Dynamics of Ethylene
CAS(2/2)*SCI/DZV0
0 .2
0 .4
0 .6
0 .8
1
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 00 50 100 150 200 250 300 350
|Δq|
/ el
ectro
ns( ) ( )2 2
Left Rightel elq q CH q CHΔ = −
Time / fs
Ethylene Dynamics Overview
Charge Transfer State Critical to S1 DecayTorsion alone is insufficient – in contrast
to textbook picture
Do Conical Intersections Matter?
Minimum Energy Gap (eV)0.00 0.05 0.10 0.15 0.20
Pop
ulat
ion
Tran
sfer
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Regions near intersections dominate nonadiabatic events
Do Minimal Energy Conical Intersections Matter?
Energy above MECI (eV)0 1 2 3 4 5
Popu
latio
n
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Minimal energy point along seam may have little significance!
Still useful in cartoon pictures of mechanisms
FC Point
Photoinduced cis-trans Isomerization - Butadiene
0 100 175time / fs
0.0
0.6
-0.8
char
ge /
q e
0 100 175 0 100 175
time / fs
0
5 0
1 0 0
1 5 0
0 5 0 1 0 0 1 5 0 2 0 0
C3
C2 C1H1
H2
Twis
t Ang
le /
deg
Terminal
Central
Again, charge transfer stateplays key role!
SA-3-CAS(4/3)/6-31G*
S2 S1 S0
q(C
2H3le
ft )–
q(C
2H3rig
ht)
Excited State Intramolecular Proton Transfer
O
CC
C
OH
H H
H
hν←⎯→O
CC
C
O
H
H
H
H
Time / psLochbrunner, JCP (2001)Experiments: Lochbruner, Stolow, Riedle, Zewail …
No H/D isotope effect!?
Methyl Salicylate Energy Diagram
FC S1 keto S1 enol
3.87
4.05
-0.21-0.36
-0.33
-2.03
-2.82
-0.79
1.78
1.56
SA2-CASPT(2/2)
Experiment
-0.48
-3.03
H-Atom Transfer in Methyl Salicylate
H-Atom Transfer TimeExpt: 60±10 fs (Herek)
AIMS: ≈50 fs
S0 Keto
S1 Enol
SA-2-CAS(2/2)-PT2 AIMS Dynamics
CASPT2 Gradients: Celani and Werner, JCP 119 5044 (2003)
Isotope Effects in MS ESIPT
No significant isotope effect!
Bond Alternation “Gates” H-Atom Motion
S0 Keto
S1 Enol
Wavepacket Dynamics on S1
Reduced densities determined by Monte Carlo integrationusing FMS wavefunction
( ) ( ) ( ) ( )( )*r dx x x r x rρ ψ ψ δ∝ −∫r r r r
C7-OaAH-DH
Time/Energy-Resolved Fluorescence Spectra
AIMSExp
MS-HMS-D
Expt: Herek, Pederesen, Banares, and Zewail, JCP 97 9046 (1992)
Intersections from Semiempirical CASCI
Semiempirical analog of state-averaged multi-reference ab initio methodsGood agreement with ab initio CASSCF results
– Qualitative intersection geometries– Coordinates which lift electronic degeneracy
BUT, energetics are qualitatively incorrectSuggests feasibility of reparameterization
Comparison of intersection geometries and modes which lift degeneracy inretinal protonated Schiff base analog
RPSB Analog Reparameterization
Multireference semiempirical model – Fractional Occ MO*CAS(6/6)-CICombine with QM/MM model to represent solventAb initio data included in reparameterization
– SA-2-CAS(10/10)/6-31G*– S0/S1 excitation energy at S0 minimum– Relative energies of MECIs for 11-12 and 13-14 twisted
RPSB Analog – RPSE-CAS(6/6)
0
0.5
1
1.5
2
2.5
3
3.5
40 60 80 100 120 140 160 180
Ene
rgy
/ eV
C11
=C12
Twist Angle
0
0.5
1
1.5
2
2.5
3
3.5
40 60 80 100 120 140 160 180
1
C11
=C12
Torsion Angle / Degrees
S0 S1
MS-CAS(10/10)-PT2/6-31G
RPSE
RPSB Analog – RPSE-CAS(6/6)
0
0.5
1
1.5
2
2.5
3
3.5
0 40 80 120 160
Ene
rgy
/ eV
C9=C
10 Twist Angle
0
0.5
1
1.5
2
2.5
3
3.5
0 40 80 120 160
1
C9=C10 Twist Angle
S0 S1
MS-CAS(10/10)-PT2/6-31G
RPSE
Comparisons for torsion around bonds not included in reparameterization
QM/MM Semiempirical FOMO-CAS CI
/ˆ ˆ ˆ ˆ
TOT QM QM MM MMH H H H= + +
12 6
/ˆ 4m m m m
QM MM mi m m mim m m m
Z Z ZH
r R R Rα α α
αα αα α α
σ σε
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= − + + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
∑∑ ∑∑ ∑∑
QM electrons / nuclei MM nuclei
Solvated GFP ChromophoreConical Intersection
RPSB Isomerization QY / Lifetime
0
0.2
0.4
0.6
0.8
1
0 1000 2000 3000 4000 5000Time/fs
Popu
latio
n ra
tio
MeOH, S1
Gas Phase, S1
Isomerized, S0
all-trans, S0
Isomerization QY: 13%-20% (Expt)24% (Theory, MeOH) 38% (Theory, Isolated)
Lifetime: 2-12 ps (Expt)Photoproduct branching: 1:6:4 (13-cis:11-cis:9-cis) MeOH
1:3:2 Isolation
RPSB / MeOH Fluorescence
τ1=35fsτ2=2.5ps
Kandori & Sasabe, CPL 1993
τ1 ≈ 90fsτ2 ≈ 3-4ps
QM/MM Model80 OPLS/AA MeOH
Isolated RPSB Fluorescence
τ=205fsMuch shorter lifetime in isolationDominant product 11-cis in both
MeOH and gas phase
RPSB Isomerization BarrierC-backbone fixed
0
0.5
1
1.5
2
2.5
90 110 130 150 170
C11=C12 torsion angle
eV RPSB, isolated RPSB/MeOH
No barrier in gas phaseWhat causes barrier in solution?
RPSB Free Energy Surface on S1
C11=C12
0.010
0.012
0.014
0.0140.016
0.01
4
0.008
0.008
0.012
0.01
2
0.010
0.006
0.006
0.010
0.008
0.008
0.008
0.008
0.010
0.006
0.00
8
0.010
0.012
0.012
X Data
1.35 1.40 1.45 1.50
Y D
ata
80
100
120
140
160
180
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
1.35
1.40
1.45
1.50
80100
120140
160
Z D
ata
X Data
Y Data
C11=C12
( )( , ) ln ,G r kT P rθ θ∝ −
Tors
ion
Ang
le θ
Bond Alternation r
Torsion Angle θ
Bond A
ltern
ation
r
Need free energy surfaces to understand behavior in solution!
Free Energy Barrier at θ ≈ 120°
Origin of RPSB/MeOH Isomerization Barrier
0
0.2
0.4
0.6
0.8
1
90 100 110 120 130 140 150 160 170 180
C11=C12 torsion angle
char
ge
qright - Isolatedqright - MeOH
Conical intersection Topography
“Peaked” “Sloped”
Reparameterized Semiempirical CASCI
Green Fluorescent Protein
CASPT2
CASSCF RPSE
RPSE fit to EOM-CCSD in FC region
Toniolo, TJM and coworkers: JPC 106A 4679 (2002); JPC 107A 3822 (2003); Chem Phys 304 133 (2004); Far. Disc. 127 149 (2004)
Reparameterized SE S1 Dynamics
100fs to reach twisted geometryNo quenching at 250fsInitial dynamics dominated by H motion
Reparameterized SE QM/MM S1 Dynamics
60fs to reach twisted geometryQuenching begins at 60fsSolvent stabilizes conical intersectionInitial dynamics similar to gas phase150 TIP3P Water Molecules
GFP neutral GFP neutral chromophorechromophore lifetimelifetime
Water
Vacuum
Solvent Effects on Conical Intersections
S0 minimum I-Twisted MECIS1 minimum
1
2
3
4
5
0
Ener
gy /
eV
Intersection becomes absolute minimum!
Vacuum
Hydrated
Solvent Effect on Topography
CTψ
CT solvatedψ
covalentψ
When one state is preferentially stabilized, intersections become 1) Peaked2) Absolute minima
Faster decay than in vacuum…
Time, fs0 200 400 600 800 1000 1200 1400
Pop
ulat
ion
0.0
0.2
0.4
0.6
0.8
1.0
S0
S1
S2
Time, fs0 200 400 600 800 1000 1200 1400
Angl
e, d
egre
es
0
20
40
60
80
100
120
140
160
180
IP
Neutral in the ProteinNeutral in the Protein
Anion in the ProteinAnion in the Protein
Tims, fs0 200 400 600 800 1000 1200 1400
Ang
les,
deg
rees
0
20
40
60
80
IP
Time, fs0 200 400 600 800 1000 1200 1400
Pop
ulat
ion
0.0
0.2
0.4
0.6
0.8
1.0
S0
S1
S2
Tunneling via Multiple Spawning Dynamics
Can tunneling be included in multiple spawning?• Trivial case: Tunneling in a multi-electronic state framework• But we do not want to include more electronic states!
Single-Surface Spawning• Label atoms which may tunnel• Label sites where atoms may tunnel• Spawn basis functions where needed
Spawned Basis Function
single state multi-state
Single-Surface Spawning Test
• 2-D Model for Proton Transfer 2
210 2 2
( )( , ) ( ) f sV s Q V s m Qm
ωω
⎛ ⎞= + −⎜ ⎟⎝ ⎠
2 41 10 2 4( )V s as bs= − +
( )f s cs= 2( )f s cs=
• 15 Basis Functions to represent Initial State• Wavefunction propagated for up to 4 ps• Up to 30 final basis functions (15 spawned)
Single-Surface Spawning Results
-1
-0.5
0
0.5
1
0 200 400 600 800 1000Time / fs
QuantumFMS
1
0
S t tFH IKFHGIKJFH IK= ψ ψ0
Re S tFH IKFHGIKJ
Im S tFH IKFHGIKJ
-0.8-0.6-0.4-0.2
00.20.40.60.8
0 200 400 600 800 1000 1200
Quantum FMS
Time / fs
Average Position
Note agreement for S(t) -This requires correct phase interference!
Single-Surface Spawning Spectrum
Tunneling SplittingFMS: 91.89 cm-1
Quantum: 91.90 cm-1
Single-Surface Spawning for Quadratic Coupling
750
800
850
900
950
1000
1050
1050
1100
1150
1200
1250
1300
1350
0 0.002 0.004 0.006 0.008 0.01
Fre
quen
cy /
cm-1
Coupling / a. u.
FMSQuantum
v=0v=1
Single-Surface Spawning for Linear Coupling
750
800
850
900
1050
1100
1150
0 0.002 0.004 0.006
Fre
quen
cy /
cm-1
Coupling / a. u.
FMSQuantum
0
20
40
60
80
100
0 0.002 0.004 0.006
Tun
nelin
g S
plitt
ing
/ cm
-1
Coupling / a. u.
FMSQuantum
v=1
v=0
•Agreement of Tunneling Splitting is better than absolute frequencies
Single-Surface Spawning Wavefunction
Q
sQuantum FMS
Acknowledgments
Funded By:
CollaboratorsH.-J. Werner – FMS-MolProJ. J. P. Stewart – FMS-MOPACM. Persico – FMS-MOPACA. StolowC. Bardeen J. Moore K. Schulten
CHE, BESDMR, CISE
HanneliHudock
Joshua CoeAlexis
Thompson
ChutintornPunwong
Ben Levine
JiahaoChen
ChaehyukKo
HongliTao
AaronVirshup
MitchellOng
Krissie Lamothe