adiabatic and nonadiabatic molecular dynamics …...adiabatic and nonadiabatic molecular dynamics...

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 Rio: univie.ac.at/columbus/rio

Rio de Janeiro, Nov. 27- Dec. 02, 2005

OutlineOutline

First Lecture: An introduction to molecular dynamicsFirst Lecture: An introduction to molecular dynamics1. Dynamics, why?2. Overview of the available approaches

Second Lecture: Towards an implementation of surface hopping dynSecond Lecture: Towards an implementation of surface hopping dynamicsamics1. Practical aspects to be addressed2. The NEWTON-X program3. Some applications: theory and experiment

Part IIPart IITowards an implementation of Towards an implementation of

surface hopping dynamicssurface hopping dynamics

Practical aspects to be Practical aspects to be addressedaddressed

Initial conditionsInitial conditionsInitial conditions

R,P

E

E0 ± ∆E

Wigner distribution

S0

Sn

acceptdon’t accept

( )

2

2exp2exp1),(

22)0(

HO

eW

m

PRRPRP

ωα

αα

π

=

⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡ −−=

hhh

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 the time.

Exception: dynamics close to the conical intersection may require 0.25 fs.

TDSE: integratorTDSE: integratorTDSE: integrator

• Fourth-order Runge-Kutta (RK4)

• Bulirsh-Stoer

BS works better than RK4

Numerical Recipes in Fortran

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 derivative connected to the particular transition

0

1

P3→2

P3→2 +P3→1

R

E

Total energy

Forbidden hop

Forbidden hop makes the classical statistical distributions deviate from the quantum 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 incertainty to search for a point in which the hop is allowed (Jasper et al. 116 5424 (2002)).

Forbidden hopsForbidden hopsForbidden hops

After hop, what are the new nuclear velocities?

• Adjust the velocities components in the direction of the nonadiabatic coupling vector h12.

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

R

E

KN(t)KN(t+∆t)

Total energy

Phase controlPhase controlPhase control

0 2 4 6 8 10 12

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15h1

x

Time (fs)

Component h1x Before the phase correction

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

Phase controlPhase controlPhase control

0 2 4 6 8 10 12

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15h1

x

Time (fs)

Component h1x Before the phase correction After the phase correction

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

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)]

Abrupt changes controlAbrupt changes controlAbrupt changes control

01h 01~h

11.75 fs

12.00 fs

The NewtonThe Newton--X programX program

Adiabatic dynamicsAdiabatic dynamicsAdiabatic dynamics

Classical motion of the nuclei: Velocity Verlet [Swope et al. JCP 76, 637 (1982)]

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

COLUMBUS:COLUMBUS:E(t+∆t), ∇E(t+∆t)

v(t+∆t)

• Fortran 90 routines• Perl controler

Nonadiabatic dynamicsNonadiabaticNonadiabatic dynamicsdynamics

• Surface hopping: fewest switches

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

COLUMBUS:COLUMBUS:E(t+∆t), ∇E(t+∆t), hkl(t+∆t)

v(t+∆t)

ak, Pk→l(t+∆t)

∑ −=k

kti

kketct )()()( )( Rφψ γ

Integration:RK4 (with Granucci, Pisa)

Previous implementation:Bulirsh-Stoer (Pittner, Prague)

∑≠

− ⋅−=kl

klti

lkletctc hv)()()( γ&

)()( ttt

i el ψψ H=∂∂

h

)()()( * tctcta lkkl =

⎟⎟⎠

⎞⎜⎜⎝

⎛=→

kk

lklk a

taMaxtP δ&,0)(

Extensions to other methodsExtensions to other methodsExtensions to other methods

R(t), v(t)

t+∆t, R(t+∆t), v(t+∆t/2)

Any program:Any program:E(t+∆t), ∇E(t+∆t)

v(t+∆t)

Presently:

• COLUMBUS (nonadiabatic dynamics)• MCSCF• MRCI

• TURBOMOLE (adiabadic dynamics) • TD-DFT• RI-CC2

Features Options Dynamics On-the-fly adiabatic dynamics On-the-fly nonadiabatic dynamics with surface hopping (fewest switches) Adjustment of momentum along the nonadiabatic coupling vector Neglecting of forbidden hoppings Phase control of the CI vector (overlapping / inner-product) Yarkony orthogonalization of the nonadiabatic coupling vectors Topographic analysis of the conical intersections Damped dynamics (kinetic energy always null) TDSE integrated in substeps of the classical equations Portability COLUMBUS (CASSCF, MRCI)

TURBOMOLE (TD-DFT, RI-CC2) Analytical models

Initial conditions Wigner distribution / classical harmonic oscillator File management Friendly input via nxinp. Automatic management of files and directories in multiple

trajectories Documentation All options and routines are documented Output and analysis Statistical analysis of results (internal coordinates and forces, energies, wave function

properties) Recomputation of internal coordinates On-the-fly graphical outputs (MOLDEN, GNUPLOT)

Current capabilitiesCurrent capabilitiesCurrent capabilities

$NX/nxinp$NX/$NX/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):

$NX/nxinp$NX/$NX/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

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 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

TD-DFT MCSCF

Time0 tc tfinal

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

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 TD-DFT for large systems until a multireference region of the phase space be reached. Switch to CASSCF to continue the dynamics.

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

• 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.

Some applications:Some applications:theory and experimenttheory and experiment

On the ambiguity of the On the ambiguity of the experimental raw dataexperimental raw data

Bonačić-Koutecký and Mitrić, Chem. Rev. 105, 11 (2005).

Experimental: pump-probeExperimental: pumpExperimental: pump--probeprobe

Analysis of the experimental resultsAnalysis of the experimental resultsAnalysis of the experimental results

Fuß et al., Faraday Discuss. 127, 23 (2004).

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

Ave

rage

occ

upat

ion

Time (fs)

SS11

SS22

SS00

Barbatti, Granucci, Persico and Lischka, CPL 401, 276 (2005)

The lifetime of ethyleneThe lifetime of ethyleneThe lifetime of ethylene

Method Lifetime (fs) DTSH [1] S1→S2,S0 139±1a / 105±1b DTSH [1] S1,S2→S0 188±2a / 137±2b DTSH [2] S1→S2,S0 50 AIMS [3] > 250 AIMS [4] 180 MCDTH [5] >> 100 Experimental [6] 30±15 Experimental [7] 20±10 Experimental (τ1 + τ 2) [8] 25±5

[1] Barbatti, Granucci, Persico and Lischka, CPL 401, 276 (2005).[2] Granucci, Persico, and Toniolo, JCP 114 (2001) 10608.[3] Ben-Nun and Martínez, CPL 298 (1998) 57.[4] Ben-Nun, Quenneville, and Martínez, JPCA 104 (2000) 5161.[5] Viel, Krawczyk, Manthe, Domcke, JCP 120 (2004) 11000. [6] Farmanara, Stert, and Radloff, CPL 288 (1998) 518.[7] Mestagh, Visticot, Elhanine, Soep, JCP 113 (2000) 237.[8] Stert, Lippert, Ritze, and Radloff, CPL 388 (2004) 144.

a Evb = 6.2 ± 0.3 eV.

b Evc = 7.4 ± 0.15 eV.

Survey of theoretical and experimental predictionsSurvey of theoretical and experimental predictionsSurvey of theoretical and experimental predictions

Eion-Epot

S0

S1

C2H4+

MXS

0 20 40 60 80 100 120 140 160 180 2002

4

6

8

τbτc

172 nm (7.21 eV)

τnd

267 nm (4.6 eV)E ion-E

pot (

eV)

Time (fs)

Ev = 6.2 eV Ev = 7.4 eV

Radloff has a rather distinct explanation!CPL 388 (2004) 144.

The time-window questionThe timeThe time--window questionwindow question

On how the initial surface On how the initial surface can make a differencecan make a difference

0

2

4

6

8

10

12

14

1.2

1.3

1.4

1.5

1.6

015

3045

6075

90

Ener

gy (e

V)

CN Stre

tch (Å

)

Torsion (°)

Basic scenery expected:Torsion + decay at twisted MXS

CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44

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

This and other movies are available at:homepage.univie.ac.at/mario.barbatti

Barbatti, Aquino and Lischka, Mol. Phys. in press (2005)

0

2

4

6

8

10

12

14

1.2

1.3

1.4

1.5

1.6

015

3045

6075

90

Ener

gy (e

V)

CN Stre

tch (Å

)

Torsion (°)

Large momentum along CN stretch isacquired in the S2-S1 transition.

Stretching + pyramidalization dominate

CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44

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

0

2

4

6

8

10

12

14

1.2

1.3

1.4

1.5

1.6

015

3045

6075

90

Ener

gy (e

V)

CN Stre

tch (Å

)

Torsion (°)

If S1 (σπ*) is directly populated,the torsion should dominate thedynamics.

CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44

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

This and other movies are available at:homepage.univie.ac.at/mario.barbatti

Intersection? Intersection? Which of them?Which of them?

The S0/S1 crossing seamThe SThe S00/S/S11 crossing seamcrossing seam

4

5

6

7

8

-100

-50

0

50

100

4060

80100

120

Ene

rgy

(eV

)

β (degrees)

θ (degrees)

EthylideneEthylidene

PyramidalizedPyramidalized

HH--migrationmigration

Barbatti, Paier and Lischka, JCP 121, 11614 (2004).

θθββ

Ohmine 1985Freund and Klessinger 1998Ben-Nun and Martinez 1998Laino and Passerone 2004

CC3V3V

Evolution on the configuration spaceEvolution on the configuration spaceEvolution on the configuration space

20 40 60 80 100 120 1400

10

20

30

0

10

20

30

0

10

20

30

Hydrogen Migration (degrees)

Twisted

planarPy

ram

idal

izat

ion

(deg

rees

)

Ethylid. MXS

Pyramid. MXS

H-mig. c.i.

First S1→S0 hopping.

Torsion Torsion + Pyramid.+ Pyramid.

HH--migrationmigrationPyram. MXSPyram. MXS

Ethylidene MXSEthylidene MXS

~7.6 eV~7.6 eV

ττ ~ 100~ 100--140 fs140 fs

60%60%

11%11%23%23%

Barbatti, Ruckenbauer and Lischka, JCP 122, 174307 (2005)

Conical intersections in ethyleneConical intersections in ethyleneConical intersections in ethylene

Readressing the DNA bases problemReadressing the DNA bases problem

An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases

What has theory to say?

• The decay can be due to πσ*/S0 crossing Sobolewski and Domcke, Eur. Phys. J. D 20, 369 (2002).

• Or maybe due to the ππ*/S0 crossingMarian, JCP 122, 104314 (2005).

• Or nπ*/S0 crossing, who knows?Chen and Li, JPCA 109, 8443 (2005).

• Maybe, there’s no crossing at all!Langer and Doltsinis, PCCP 6, 2742 (2004)

• Our own dynamics simulations (TD-DFT(B3LYP)/SVP) do not show any crossing too…

• But the dynamics calculations are not reliable enough until now: or they miss the MR character, or the nonadiabatic character, or both!

Experimental lifetimes of the excited state of DNA/RNA basis: ~ 1 ps

An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases

Amino pyrimidine as a model for adenine

9H-adenine 4-amino pyrimidine

An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases

4-amino pyrimidineSA-3-CAS(5,6)/6-31G*

Time (fs)

The system moves toward the crossing seam with out-of-plane motions similar to that predicted by Marian for adenine (JCP 122, 104314 (2005))

An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases

4-amino pyrimidineSA-3-CAS(5,6)/6-31G*

Time (fs)

If the H-atoms that replace the ring have a big isotopic mass, the out-of-plane motion is inhibited and there is no crossing.

An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases

4-amino pyrimidineSA-3-CAS(5,6)/6-31G*

Average probability is 4×10-5

in each 0.05 fs.

In 1 ps, the probability is 0.8.

The system does not need a crossing to decay in 1 ps. The weak S1/S0 coupling is enough to induce the transfer.

0 100 200 300 400 500

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.008

Time (fs)

P(1→0) hopping probability in 0.05 fs

ConclusionsConclusions

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

ConclusionsConclusionsConclusions

Dynamics, Why?• Ab initio dynamics can be used to characterize lifetimes, and relaxations paths in the ground and excited state, occurring in the time scale of sub-picosecond.• Dynamics reveal features that is not easily found by static methods.• Dynamics can be used to locate the important regions of the crossing seam.

Excited-state dynamics around the world (a very incomplete and biased inventory…)• Semiclassical dynamics:Tully; Jasper and Thrular; Robb, Olivucci and Buss; Barbatti and Lischka; Marx and Doltsinis (Car-Parrinelo); Granucci and Persico (local diabatization); Martinez-Nuñez; Pittner and Bonačić-Koutecký; Neufeld.

•Multiple spawningBen-Nun and Martínez

• Wave packetGonzalez and Manz; Manthe and Cederbaum; Viel and Domcke; Jacubtz; de Vivie-Riedle and Hofmann; Schinke; Köppel.

Our contributionOur contributionOur contribution

Towards an implementation of surface hopping dynamics• NEWTON-X: new implementation of on-the-fly surface hopping dynamics.• Energies, gradients and nonadiabatic coupling vectors can be read from any quantum chemical package.• For MRCI and CASSCF dynamics, COLUMBUS has been used.• For TD-DFT and RI-CC2, TURBOMOLE.

Where are we going now?• QM/MM: investigation of the influence of the environment on the relaxations paths, lifetimes and efficiency of the crossing seam.

MRCI, CAS

Force field

CollaboratorsCollaboratorsCollaborators

Vienna:Hans LischkaAdélia AquinoMatthias Ruckenbauer (Master student)Gunther Zechmann (Master student, now at Warwick)Daniela Raab

Pisa:Giovanni Granucci

Prague:Jiri Pittner

Zagreb:Mario Vazdar

ConclusionsConclusionsConclusions

Columbus in Rio: univie.ac.at/columbus/rio

Rio de Janeiro, Nov. 27- Dec. 02, 2005

• References and additional information: homepage.univie.ac.at/mario.barbatti

• Support: CNPq (Brazil), Austrian Science Fund (Special project F16)