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