Photochemistry: Photochemistry: adiabatic and adiabatic and nonadiabaticnonadiabatic molecular dynamics molecular dynamics

withwith multireferencemultireference abab initioinitio methodsmethods

Mario BarbattiMario Barbatti

Institute for Theoretical ChemistryUniversity of Vienna

COLUMBUS in BANGKOK (3COLUMBUS in BANGKOK (3--TSTS22CC22))Apr. 2 Apr. 2 -- 5, 20065, 2006

BuraphaBurapha University, Bang University, Bang SaenSaen, Thailand, Thailand

OutlineOutline

First First LectureLecture: An : An introductionintroduction to to molecularmolecular dynamicsdynamics1. Dynamics, why?2. Overview of the available approaches

Second Second LectureLecture: : TowardsTowards an an implementationimplementation of of surfacesurface hoppinghopping dynamicsdynamics1. The NEWTON-X program2. Practical aspects to be adressed

Third Lecture: Some applications: theory and experiment• On the ambiguity of the experimental raw data• On how the initial surface can make difference• Intersection? Which of them?• Readressing the DNA/RNA bases problem

Part IIPart IITowardsTowards an an implementationimplementation of of

surfacesurface hoppinghopping dynamicsdynamics

Cândido Portinari, O lavrador de café, 1934

Quick overview of the methodQuick overview of the method

Surface hopping approach: adiabatic populationSurface hopping approach: adiabatic populationSurface hopping approach: adiabatic population

∑ −=k

kti

k tetct k ),;(),(),,( )( rRRRr φψ γ ψψeH

ti =

∂∂

h

*ikki cca =Population:

( )∑ ⋅−⋅= −

iki

iijij

ikikj

kjkj eaeaa hvhv γγ&

• Two electronic states are coupled only by the nonadiabatic coupling vector hij(adiabatic representation).

( ) )()( )()( tdetccki

kitti

ikik∑

≠

−−= γγ& where kiriRkrikki td hvv ⋅=∇⋅=

∂∂

= φφφφ

Time derivative Nonadiabatic coupling vector

Tully, JCP 93, 1061 (1990); Ferretti et al. JCP 104, 5517 (1996)

Surface hopping approach: fewest switchesSurface hopping approach: fewest switchesSurface hopping approach: fewest switches

( ) jki

kjjkjkeab hv ⋅−= γRe2

tab

gjj

jkjk ∆=→ Tully, JCP 93, 1061 (1990)

´´)(1 ttba

gtt

t jkjj

jk ∆= ∫∆+

→ Hammer-Schiffer and Tully, JCP 101, 4657 (1994)

15 16 17 180.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Pro

babi

lity

/ fs

Time (fs)

∆t = 0.1 fs, ms = 1, int = Butcher Tully Hammer-Schiffer and Tully

Surface hopping approach: semiclassical trajectoriesSurface hopping approach: Surface hopping approach: semiclassicalsemiclassical trajectoriestrajectories

2

2

dtdME I

IiI

RR =∇−• Nuclear motion is obtained by integrating the Newton eqs.

• At each time, the dynamics is performed on one unique adiabatic state, Ei = Hii.

• In the adiabatic representation, Ei(R), ∇Ei, and hji are obtained with traditional quantum chemistry methods.

• aji is obtained by integrating

( )∑ ⋅−⋅= −

iki

iijij

ikikj

kjkj eaeaa hvhv γγ&

• The transition probability gkj between two electronic states is calculated at each time step of the classical trajectory.

• A random event decides whether the system hops to other adiabatic state.

t

E

(Dis)advantages of the on-the-fly approach((Dis)advantagesDis)advantages of the onof the on--thethe--fly approachfly approach

Advantages:Advantages:• It is not need to get the complete surface. Only that regions spanned during the dynamics• It dispenses interpolation, extrapolation and fitting schemes

Disadvantages:Disadvantages:• Time-expensive dynamics• No non-local effects (tunneling)

Preparation of surface dynamicsConventionalConventional

dynamicsOnOn--thethe--flyfly

Total timeTotal time

PracticalPractical aspectsaspects to to bebeaddressedaddressed

How to prepare initial conditionsHow to prepare initial conditionsHow to prepare initial conditions

R,P

E

S0

Sn

Microcanonical distributions:• Classical harmonic distribution• R-P uncorrelated quantum harmonic distrib. (Wigner)• R-P correlated quantum harmonic ditrib.

Canonical distribution:• Boltzmann

A good one is the Velocity Verlet (Swope et al. JCP 76, 637 (1982)):

For each nucleus I

Classical dynamics: integrator Classical dynamics: integrator Classical dynamics: integrator

[ ]

ttttttt

ttEM

t

ttttt

ttttttt

III

IRI

I

III

IIII

∆∆++⎟⎠⎞

⎜⎝⎛ ∆+=∆+

∆+∇−=

∆+=⎟⎠⎞

⎜⎝⎛ ∆+

∆+∆+=∆+

)(21

2)(

)(1)(

)(21)(

2

)(21)()()( 2

avv

Ra

avv

avRR

Quantum chemistry calculation

Any standard method can be used in the integration of the Newton equations.

Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997).

Time-step for the classical equationsTimeTime--step for the step for the classicalclassical equationsequations

Time-step for the classical equationsTimeTime--step for the step for the classicalclassical equationsequations

Time step should not be larger than 1 fs (1/10v).

∆t = 0.5 fs assures a good level of conservation of energy most of time.

Exceptions: • Dynamics close to the conical intersection may require 0.25 fs• Dissociation processes may require even smaller time steps

TDSE: integratorTDSE: integratorTDSE: integrator

• Fourth-order Runge-Kutta (RK4)

• Bulirsh-Stoer

Adaptive works better than Adaptive works better than constant time stepconstant time step

Numerical Recipes in Fortran

Constant time step:Constant time step:

Adaptive time step:Adaptive time step:

TDSE: integratorsTDSE: integratorsTDSE: integrators

Some step-constant integrators available in NEWTON-X:

• Polynomial, 3rd order • Runge-Kutta, 4th order • Adams Moulton predictor-corrector, 5th order• Adams Moulton predictor-corrector, 6th order• Unitary propagator• Butcher, 5th order 14 16 18 20 22 24 26 28 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7∆t = 0.5 fs; ms = 1

Runge-Kutta Adams Moulton (6) Propagator Butcher

a 00

Time (fs)

h(t) h(t+∆t)

∆t/ms

)11( ))(-)((+)(=1 -..mn=ttt+mnt

mn-tt+ s

ss

hhhh ∆⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∆

. . .

0 10 20 30 40 50 60 70 80 90 100

1.00

1.01

1.02

1.03

1.04

2

3

ms = 20

ms = 10

Σi|ci|2

Time (fs)

ms = 1

∆t = 0.5 fs

Time-step for the quantum equationsTimeTime--step for the step for the quantumquantum equationsequations

)()( 2222 ttata ∆+→

NaN 222 =

( )0

)()()()(

21

2222

2212

=

∆+−=∆+−=

→

→

hop

hop

n

NttatattNtNn

Fewest switches: two statesFewest switches: two statesFewest switches: two states

Population in S2:

Trajectories in S2:

Minimum number of hoppings that keeps the correct number of trajectories:

⎪⎪⎩

⎪⎪⎨

⎧

≤

>⎟⎠⎞

⎜⎝⎛∆

≈= →→

0,0

0,)(

)( 11

1111

11

2

1212

dtdadt

dadt

datat

tNnP

hophop

Probability of hopping

0

1

P2→1

Fewest switches: several statesFewest switches: several statesFewest switches: several states

( ) ( )∑ ∑≠ ≠

⎥⎦⎤

⎢⎣⎡ −==

kl klklklklklklkk daHaba ** Re2Im2

h&

Tully, JCP 93, 1061 (1990)tabP

kk

klhoplk ∆=→

Example: Three states 313233 bba +=&

tabPhop ∆=→

33

3223

Only the fraction of derivativeconnected to the particular transition

0

1

P3→2

P3→2 +P3→1

´´)(1 ttba

Ptt

t lkkk

hoplk ∆= ∫

∆+

→ Hammer-Schiffer and Tully, JCP 101, 4657 (1994)

R

E

Total energy

Forbidden hop

Forbidden hop makes the classical statistical distributions deviate from thequantum populations.

How to treat them:• Reject all classically forbidden hop and keep the momentum.• Reject all classically forbidden hop and invert the momentum.• Use the time uncertainty to search for a point in which the hop is allowed(Jasper et al. 116 5424 (2002)).

Forbidden hopsForbidden hopsForbidden hops

Adjustment of momentum after hoppingAdjustment of momentum after hoppingAdjustment of momentum after hopping

R

E

KN(t)KN(t+∆t)

Total energy

After hop, what are the new nuclear velocities?

• Redistribute the energy excess equally among all degrees• Adjust velocities components in the direction of the nonadiabatic coupling vector h12• Adjust velocities components in the direction of the difference gradient vector g12• Adjust velocities in the direction

θθ cossin klkl ghe ))) +=

Phase controlPhase controlPhase control

0 2 4 6 8 10 12

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

h1x

Time (fs)

Component h1x Before the phase correction

CNH4+: MRCI/CAS(4,3)/6-31G*

Compare h(t) and h(t+ ∆t)

Phase controlPhase controlPhase control

0 2 4 6 8 10 12

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

h1x

Time (fs)

Component h1x Before the phase correction After the phase correction

CNH4+: MRCI/CAS(4,3)/6-31G*

Compare h(t) and h(t+ ∆t)

Abrupt changes controlAbrupt changes controlAbrupt changes control

11.75 fs

01h

12.00 fs

OrthogonalizationOrthogonalizationOrthogonalization

ββ

ββ

cossin~sincos~

ijijij

ijijij

hgh

hgg

+=

+−=

ijijijij

ijij

gghhhg

⋅−⋅⋅

=2

2tan β

• The routine also gives the linear parameters:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−

∆++±+=

2/12222 )(

2)(

21 yxyxyxdE gh

yxgh σσ

F

S1

S0

F

F

S1

S0

F

g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)]

When surface hopping failsWhen surface hopping failsWhen surface hopping fails

• SH is supposed to reproduce quantum distributions, in the sense thatfraction of trajectories (t) = adiabatic population (t) Eq. (1)

This statement should be true for:Number of forbidden hops → 0Number of trajectories → infinity

Granucci and Persico have shown that for some cases, even if these conditions are satisfied, Eq. (1) may be not true.

• SH, as any trajectory-independent semiclassical method, cannot account for quantum interference effects and quantization of vibrational and rotational motions.

• It is unclear how good the fewest switches approach in the proximity of conical intersections is.

NNEWTONEWTON--X: a package for Newtonian X: a package for Newtonian dynamics close to the crossing seamdynamics close to the crossing seam

M. M. BarbattiBarbatti, G. , G. GranucciGranucci, H. , H. LischkaLischkaand M. and M. RuckenbauerRuckenbauer (2005(2005--2006)2006)

NX aimsNX aimsNX aims

• Easy and practical of using: just make the inputs and start the simulations; monitor partial results on-the-fly; get relevant summary of results at the end;

• Robust: if the input is right, the job will run: in case of error, messages must guide the user to fix the problem;

• Flexible: some different case to study or new method to implement? It should be easy to change the code;

• Open source: in the future, NX should be opened to the community.

NX input facility: nxinpNX input facility:NX input facility: nxinpnxinp

------------------------------------------NEWTON-X

Newton dynamics close to the crossing seam------------------------------------------

MAIN MENU

1. GENERATE INITIAL CONDITIONS

2. SET BASIC INPUT

3. SET GENERAL OPTIONS

4. SET NONADIABATIC DYNAMICS

5. GENERATE TRAJECTORIES

6. SET STATISTICAL ANALYSIS

7. EXIT

Select one option (1-7):

NX input facility: nxinpNX input facility:NX input facility: nxinpnxinp

------------------------------------------NEWTON-X

Newton dynamics close to the crossing seam------------------------------------------

SET BASIC OPTIONS

nat: Number of atoms.There is no value attributed to natEnter the value of nat : 6Setting nat = 6

nstat: Number of states.The current value of nstat is: 2Enter the new value of nstat : 3Setting nstat = 3

nstatdyn: Initial state (1 - ground state).The current value of nstatdyn is: 2Enter the new value of nstatdyn : 2Setting nstatdyn = 2

prog: Quantum chemistry program and method

0 - ANALYTICAL MODEL1 - COLUMBUS2.0 - TURBOMOLE RI-CC22.1 - TURBOMOLE TD-DFT

The current value of prog is: 1Enter the new value of prog : 1

NX modular designNX modular designNX modular design

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

provide:provide:∇Ek(t+∆t), hkl(t+∆t)

v(t+∆t)

akk, Pk→l(t+∆t)

Initial condition generation

Statistical analysis

Tables and graphics

• Fortran 90 routines• Perl controller

Adiabatic dynamicsAdiabatic dynamicsAdiabatic dynamics

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

provide:provide:∇Ek(t+∆t)

v(t+∆t)

Methods availableMethods availableMethods available

Presently:• COLUMBUS [(non)adiabatic dynamics]

• MCSCF• MRCI

• TURBOMOLE [adiabadic dynamics]• TD-DFT• RI-CC2

• Analytical models [user provided]

Being implemented:• COLUMBUS + TINKER

• QM/MM [(non)adiabatic dynamics]

To be implemented:• ACES II

• EOM-CC [(non)adiabatic dynamics]

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

provide:provide:∇Ek(t+∆t), hkl(t+∆t)

v(t+∆t)

akk, Pk→l(t+∆t)

So many choicesSo many choices……What method should I use?What method should I use?

Method Single/Multi Reference

Analytical gradients

Coupling vectors

Computational effort

Typical implementation

MR-CISD MR √ √ Columbus EOM-CC SR √ √ Aces II RI-CC2 SR √ × Turbomole CASPT2 MR × × Molcas / Molpro MR-MP2 MR × × Gamess CISD/QCISD SR √ × Molpro / Gaussian CASSCF MR √ √ Columbus / Molpro TD-DFT SR √ × Turbomole FOMO/AM1 MR √ √ Mopac (Pisa)

Present situation of quantum chemistry methodsPresent situation of quantum chemistry methodsPresent situation of quantum chemistry methods

Comparison among methodsComparison among methodsComparison among methods

0 5 10 15 20 25 30

Ene

rgy

Time (fs)

RI-CC2

0 5 10 15 20 25 30

Time (fs)

TD-DFT

0 5 10 15 20 25 30

Time (fs)

CASSCF

Adiabatic dynamics can be used to find out the most relevant relaxation paths.

But be careful with the limitation of each method (CT states in TD-DFT for example).

CNH4+: MRCI/CAS(4,3)/6-31G*

A basic protocolA basic protocolA basic protocol

• Use TD-DFT for large systems (> 10 heavy atoms) with one single configuration dominating the region of the phase space spanned by the dynamics. Test against CASSCF and RI-CC2.

• Use RI-CC2 for medium systems (6-10 heavy atoms) under the same conditions as in the previous point.

• Use CASSCF for medium systems with strong multireference character in all phase space. Test against MRCI.

• Use MRCI for small systems (< 6 heavy atoms).

• In all cases, when the number of relevant internal coordinates is small (2-4) and they can easily be determined, test against wave-packet dynamics.

ConclusionsConclusionsConclusions

Basis set

Correlation Method

DZ TZ BS limit

HF

CASSCF

MRCI

Full-CI

…

…

ConclusionsConclusionsConclusions

Basis set

Correlation Method

DZ TZ BS limit

HF

CASSCF

MRCI

Full-CI

…

…

Dynamics method

Surf. Hopping and Mean Field

Multiple Spawning

Wavepacket dynamics

Static calculations

Adiabatic dynamics

< 6 heavy atoms

6-10 heavy atoms

Next lecture:Next lecture:• Adiabatic and nonadiabatic dynamics methods will be used in the investigation of some examples of photoexcited systems

This lecture:This lecture:• Surface hopping is one of the most popular methods available for nonadiabatic dynamics• Its implementation is direct and it can be used with any quantum chemistry method that can provide analytical excited-state gradients and analytical nonadiabatic coupling vectors• These requirements are fulfilled by only a few methods such as CASSCF, MRCI and (partially) EOM-CC