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

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

AdiabaticAdiabatic andand nonadiabaticnonadiabatic molecularmolecular dynamicsdynamics withwith multireferencemultireference abab initioinitio methodsmethods

Mario Barbatti

Institute for University of Vienna Outline

First Lecture: An introduction to 1. Dynamics, why? 2. Overview of the available approaches

Second Lecture: Towards an implementation of surface hopping dynamics 1. Practical aspects to be addressed 2. The NEWTON-X program 3. Some applications: theory and experiment PartPart IIII TowardsTowards anan implementationimplementation ofof surfacesurface hoppinghopping dynamicsdynamics PracticalPractical aspectsaspects toto bebe addressedaddressed InitialInitial conditionsconditions

2 2 (0) 1 ⎡− 2 (R − Re ) ⎤ ⎡− P ⎤ PW (R, P) = exp⎢ ⎥exp ⎢ 2 ⎥ E π h ⎣ α h ⎦ ⎣ h ⎦ α mω = HO 2 α E0 ± ∆E Sn

Wigner distribution

S0

accept R,P don’t accept ClassicalClassical dynamics:dynamics: integratorintegrator

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

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

For each nucleus I 1 R (t + ∆t) = R (t) + v (t)∆t + a (t)∆t 2 I I I 2 I ⎛ ∆t ⎞ 1 vI ⎜t + ⎟ = vI (t) + aI (t)∆t ⎝ 2 ⎠ 2 1 aI (t) = − ∇R E[]R I (t + ∆t) calculation M I ⎛ ∆t ⎞ 1 vI (t + ∆t) = vI ⎜t + ⎟ + aI (t + ∆t)∆t ⎝ 2 ⎠ 2 TimeTime-step-step forfor thethe classicalclassicalequationsequations

Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997). TimeTime-step-step forfor thethe classicalclassicalequationsequations

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:TDSE: integratorintegrator Numerical Recipes in Fortran

• Fourth-order Runge-Kutta (RK4)

• Bulirsh-Stoer

BS works better than RK4 TimeTime-step-step forfor thethe quantumquantumequationsequations

∆t/ms

. . . h(t) h(t+∆t) ⎛ ⎛ n ⎞⎞ n ⎜ ⎜ ⎟⎟ h⎜t+∆t⎜1- ⎟⎟ = h(t) + (h(t+∆t) - h(t)) (n=1..ms-1) ⎝ ⎝ ms ⎠⎠ ms

Σ|c |2 i i ∆t = 0.5 fs 3

m = 1 s 2

1.04

1.03 m = 10 s 1.02

1.01 m = 20 1.00 s

0 102030405060708090100 Time (fs) FewestFewest switches:switches: twotwo statesstates

Population in S2: a22 (t) → a22 (t + ∆t)

Trajectories in S2: N2 = a22 N

Minimum number of hoppings that keeps the correct number of trajectories: hop n2→1 = N2 (t) − N2 (t + ∆t)

= ()a22 (t) − a22 (t + ∆t) N nhop = 0 1→2 1

Probability of hopping

⎧ ∆t ⎛ da ⎞ da 11 , 11 0 hop ⎪ ⎜ ⎟ > hop n2→1 ⎪a11(t) ⎝ dt ⎠ dt P2→1 = ≈ ⎨ N (t) da 2 ⎪0 , 11 ≤ 0 ⎩⎪ dt P2→1

0 FewestFewest switches:switches: severalseveral statesstates

⎡2 * * ⎤ a = b = Im(a H )− 2Re(a d ) &kk ∑∑kl ⎢ kl kl kl kl ⎥ l≠k l≠k ⎣h ⎦

hop bkl Pk→l = ∆t Tully, JCP 93, 1061 (1990) akk

Example: Three states a&33 = b32 + b31

hop b32 1 P3→2 = ∆t a33 Only the fraction of derivative connected to the particular transition

P3→2 +P3→1

P3→2

0 ForbiddenForbidden hopshops

E Forbidden hop

Total energy

R

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)). AdjustmentAdjustment ofof momentummomentum afterafter hoppinghopping E

Total energy

KN(t) KN(t+∆t)

R

After hop, what are the new nuclear velocities? • Adjust the velocities components in the direction of the nonadiabatic

coupling vector h12. PhasePhase controlcontrol + CNH4 : MRCI/CAS(4,3)/6-31G*

Component h1x Before the phase correction 0.15

0.10

0.05

0.00 h1x

-0.05

-0.10

-0.15

024681012 Time (fs) PhasePhase controlcontrol + CNH4 : MRCI/CAS(4,3)/6-31G*

Component h1x Before the phase correction 0.15 After the phase correction

0.10

0.05

0.00 h1x

-0.05

-0.10

-0.15

024681012 Time (fs) AbruptAbrupt changeschanges controlcontrol

h01

11.75 fs

12.00 fs OrthogonalizationOrthogonalization

g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)] ~ gij = −gij cosββ + hij sin β ~ hij = gij sin + hij cos β

2g ⋅h tan 2β = ij ij hij ⋅hij − gij ⋅gij

S1 • The routine also gives the linear parameters: ⎡ 1/ 2 ⎤ σ ⎛ 1 ∆ gh ⎞ 2 2 2 2 S E = d gh ⎢ x x +σ y y ± ⎜ (x + y ) + (x − y )⎟ ⎥ 0 ⎣⎢ ⎝ 2 2 ⎠ ⎦⎥

F

F AbruptAbrupt changeschanges controlcontrol

~ h01 h01

11.75 fs

12.00 fs TheThe NewtonNewton--XX programprogram AdiabaticAdiabatic dynamicsdynamics

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

R(t), v(t) • Fortran 90 routines • Perl controler t+∆t, R(t+∆t), v(t+∆t/2)

COLUMBUS: E(t+∆t), ∇E(t+∆t)

v(t+∆t) NonadiabaticNonadiabaticdynamics dynamics ψ • Surface hopping: fewest switches

−iγ k (t ) (t) = ∑ck (t)e φk (R) ψ k ∂ i (t) = H ψ (t) R(t), v(t) h ∂t el

−iγ l (t ) c&k (t) = −∑cl (t)e v ⋅hkl t+∆t, R(t+∆t), v(t+∆t/2) l≠k * akl (t) = ck (t)cl (t)

⎛ a δt ⎞ COLUMBUS: ⎜ &lk ⎟ Pk→l (t) = Max⎜0, ⎟ E(t+∆t), ∇E(t+∆t), hkl(t+∆t) ⎝ akk ⎠

Integration: a , P (t+ t) k k→l ∆ RK4 (with Granucci, Pisa)

v(t+∆t) Previous implementation: Bulirsh-Stoer (Pittner, Prague) ExtensionsExtensions toto otherother methodsmethods

R(t), v(t) Presently:

t+∆t, R(t+∆t), v(t+∆t/2) • COLUMBUS (nonadiabatic dynamics) •MCSCF • MRCI Any program: E(t+∆t), ∇E(t+∆t) • TURBOMOLE (adiabadic dynamics) • TD-DFT • RI-CC2 v(t+∆t) CurrentCurrent capabilitiescapabilities

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 (, GNUPLOT)

$NX/$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): $NX/$NX/nxinpnxinp

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

SET BASIC OPTIONS

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

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

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

prog: Quantum chemistry program and method

0 - ANALYTICAL MODEL 1 - COLUMBUS 2.0 - TURBOMOLE RI-CC2 2.1 - TURBOMOLE TD-DFT The current value of prog is: 1 Enter the new value of prog : 1 SoSo manymany choiceschoices…… WhatWhat methodmethod shouldshould II use?use? PresentPresent situationsituation ofof quantumquantum chemistrychemistry methodsmethods

Method Single/Multi Analytical Coupling Computational Typical implementation Reference gradients vectors effort MR-CISD MR √ √ Columbus RI-CC2 SR √ × Turbomole CASPT2 MR × × Molcas / Molpro MR-MP2 MR × × Gamess CISD/QCISD SR √ × Molpro / CASSCF MR √ √ Columbus / Molpro TD-DFT SR √ × Turbomole FOMO/AM1 MR √ √ Mopac (Pisa)

ComparisonComparison amongamong methodsmethods

RI-CC2 TD-DFT CASSCF

Energy

0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time (fs) Time (fs) Time (fs)

TD-DFT MCSCF

0 tc tfinal Time AA basicbasic protocolprotocol

•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. SomeSome applications:applications: theorytheory andand experimentexperiment OnOn thethe ambiguityambiguity ofof thethe experimentalexperimental rawraw datadata Experimental:Experimental: pumppump-probe-probe

Bonačić-Koutecký and Mitrić, Chem. Rev. 105, 11 (2005). AnalysisAnalysis ofof thethe experimentalexperimental resultsresults

Fuß et al., Faraday Discuss. 127, 23 (2004). TheThe lifetimelifetime ofof ethyleneethylene

1.0

S1 0.8 S0

0.6

0.4 S2 Average occupation Average

0.2

0.0 0 100 200 300 400 Time (fs)

Barbatti, Granucci, Persico and Lischka, CPL 401, 276 (2005) SurveySurvey ofof theoreticaltheoretical andand experimentalexperimental predictionspredictions

Method Lifetime (fs) a b DTSH [1] S1→S2,S0 139±1 / 105±1 a b DTSH [1] S1,S2→S0 188±2 / 137±2 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 a b Ev = 6.2 ± 0.3 eV. b c Ev = 7.4 ± 0.15 eV.

[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. TheThe timetime-window-window questionquestion

+ C2H4

Eion-Epot S1 E = 6.2 eV v 8 E = 7.4 eV MXS v S0 172 nm (7.21 eV) 6 τc τb (eV) Radloff has a rather pot

-E distinct explanation! ion

E 267 nm (4.6 eV) CPL 388 (2004) 144. 4

τnd

2 0 20 40 60 80 100 120 140 160 180 200 Time (fs) OnOn howhow thethe initialinitial surfacesurface cancan makemake aa differencedifference CNH ++: MRCI/CAS(4,3)/6-31G* CNH44 : MRCI/CAS(4,3)/6-31G*

Basic scenery expected: Torsion + decay at twisted MXS

14

12

10 )

V 8

e

(

y

g 6

r

e n

E 4

2 1.6 ) Å 0 1.5 ( h tc e 90 1.4 tr 75 S 60 1.3 N 45 C This and other movies are available at: 30 T 1.2 homepage.univie.ac.at/mario.barbatti orsio 15 n (°) 0

Barbatti, Aquino and Lischka, Mol. Phys. in press (2005) CNH ++: MRCI/CAS(4,3)/6-31G* CNH44 : MRCI/CAS(4,3)/6-31G* Large momentum along CN stretch is acquired in the S2-S1 transition.

Stretching + pyramidalization dominate

14

12

10 )

V 8

e

(

y

g 6 r

e n

E 4

2 1.6 ) Å 0 1.5 ( h tc e 90 1.4 tr 75 S 60 1.3 N 45 C 30 T 1.2 orsio 15 n (°) 0 CNH ++: MRCI/CAS(4,3)/6-31G* CNH44 : MRCI/CAS(4,3)/6-31G*

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

14

12

10 )

V 8

e

(

y

g 6 r

e n

E 4

2 1.6 ) Å 0 1.5 ( h tc e 90 1.4 tr This and other movies are available at: 75 S 60 1.3 N 45 C homepage.univie.ac.at/mario.barbatti 30 T 1.2 orsio 15 n (°) 0 Intersection?Intersection? WhichWhich ofof them?them? The S /S crossing seam The S00/S11 crossing seam

8 θ β

) 7

V

e

(

y

g r

e C3V n 6

E Ohmine 1985 Freund and Klessinger 1998 5 H-migration H-migration Ben-Nun and Martinez 1998 Ethylidene 100 50 Laino and Passerone 2004 4 s) Pyramidalized 0 ee 40 gr 60 de -50 ( 80 β θ (d 100 egre -100 es) 120

Barbatti, Paier and Lischka, JCP 121, 11614 (2004). EvolutionEvolution onon thethe configurationconfiguration spacespace

30 Pyramid. MXS 20 Ethylid. MXS 10 H-mig. c.i. 0 30 planar 20 Twisted 10 0 30

Pyramidalization (degrees)Pyramidalization 20 10 0 20 40 60 80 100 120 140 Hydrogen Migration (degrees)

First S1→S0 hopping. ConicalConical intersectionsintersections inin ethyleneethylene

τ ~ 100-140 fs

Ethylidene MXS ~7.6 eV 11% 23%

60% Pyram. MXS H-migration Torsion + Pyramid.

Barbatti, Ruckenbauer and Lischka, JCP 122, 174307 (2005) ReadressingReadressing thethe DNADNA basesbases problemproblem AnAn example:example: photodynamicsphotodynamicsof of DNADNA basesbases

Experimental lifetimes of the excited state of DNA/RNA basis: ~ 1 ps 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 crossing Marian, 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! AnAn example:example: photodynamicsphotodynamicsof of DNADNA basesbases Amino pyrimidine as a model for adenine

9H-adenine 4-amino pyrimidine AnAn example:example: photodynamicsphotodynamicsof of DNADNA basesbases

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

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

Time (fs) AnAn example:example: photodynamicsphotodynamicsof of DNADNA basesbases

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

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.

Time (fs) AnAn example:example: photodynamicsphotodynamicsof of DNADNA basesbases

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

P(1→0) hopping probability in 0.05 fs 0.008

0.0014 Average probability is 4×10-5 0.0012 in each 0.05 fs.

0.0010 In 1 ps, the probability is 0.8. 0.0008

0.0006 The system does not need a

0.0004 crossing to decay in 1 ps. The weak S1/S0 coupling is enough 0.0002 to induce the transfer. 0.0000

0 100 200 300 400 500 Time (fs) ConclusionsConclusions ConclusionsConclusions

Correlation Method

Full-CI … MRCI

CASSCF

HF Basis set DZ TZ … BS limit ConclusionsConclusions

Correlation Method

Full-CI … MRCI < 6 heavy atoms CASSCF 6-10 heavy atoms HF Basis set DZ TZ … BS limit Sta tic A ca diab lcul at a Su i tions r c d f. H yn M op am ul p ic tipl ing s W e S an ave pa d M pa wni e cke n an t g Fi Dynamics method dyna eld m ics ConclusionsConclusions

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 spawning Ben-Nun and Martínez

• Wave packet Gonzalez and Manz; Manthe and Cederbaum; Viel and Domcke; Jacubtz; de Vivie- Riedle and Hofmann; Schinke; Köppel. OurOur contributioncontribution

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.

Force field

MRCI, CAS CollaboratorsCollaborators

Vienna: Hans Lischka Adélia Aquino Matthias Ruckenbauer (Master student) Gunther Zechmann (Master student, now at Warwick) Daniela Raab

Pisa: Giovanni Granucci

Prague: Jiri Pittner

Zagreb: Mario Vazdar ConclusionsConclusions

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

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

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

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