Photochemistry: adiabatic and nonadiabatic molecular dynamics with multireference ab initio methods Photochemistry: adiabatic and nonadiabatic molecular dynamics with multireference ab initio methods Mario Barbatti Institute for Theoretical Chemistry University of Vienna COLUMBUS in BANGKOK (3-TS 2 C 2 ) Apr , 2006 Burapha University, Bang Saen, Thailand

Outline First Lecture: An introduction to molecular dynamics 1.Dynamics, why? 2.Overview of the available approaches Second Lecture: Towards an implementation of surface hopping dynamics 1.The N EWTON -X program 2.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 II Towards an implementation of surface hopping dynamics Cândido Portinari, O lavrador de café, 1934

Quick overview of the method

Surface hopping approach: adiabatic population Population: Two electronic states are coupled only by the nonadiabatic coupling vector h ij (adiabatic representation). where Time derivativeNonadiabatic coupling vector Tully, JCP 93, 1061 (1990); Ferretti et al. JCP 104, 5517 (1996)

Surface hopping approach: fewest switches Tully, JCP 93, 1061 (1990) Hammes-Schiffer and Tully, JCP 101, 4657 (1994)

Surface hopping approach: semiclassical trajectories Nuclear motion is obtained by integrating the Newton eqs. At each time, the dynamics is performed on one unique adiabatic state, E i = H ii. In the adiabatic representation, E i (R),  E i, and h ji are obtained with traditional quantum chemistry methods. a ji is obtained by integrating The transition probability g kj 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 Advantages: It is not need to get the complete surface. Only that regions spanned during the dynamics It dispenses interpolation, extrapolation and fitting schemesDisadvantages: Time-expensive dynamics No non-local effects (tunneling) Preparation of surfacedynamicsConventional On-the-fly Total time

Practical aspects to be addressed

How to prepare initial conditions R,PR,P E S0S0 SnSn 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 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 equations

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: integrator Fourth-order Runge-Kutta (RK4) Bulirsh-Stoer Adaptive works better than constant time step Numerical Recipes in Fortran Constant time step: Adaptive time step:

TDSE: integrators Some step-constant integrators available in N EWTON -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

h(t)h(t) h(t+t)h(t+t) t/mst/ms...  t = 0.5 fs Time-step for the quantum equations

Fewest switches: two states Population in S 2 : Trajectories in S 2 : Minimum number of hoppings that keeps the correct number of trajectories: Probability of hopping 0 1 P2→1P2→1

Fewest switches: several states Tully, JCP 93, 1061 (1990) Example: Three states Only the fraction of derivative connected to the particular transition 0 1 P3→2P3→2 P 3→2 +P 3→1 Hammer-Schiffer and Tully, JCP 101, 4657 (1994)

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 uncertainty to search for a point in which the hop is allowed (Jasper et al (2002)). Forbidden hops

Adjustment of momentum after hopping R E KN(t)KN(t) KN(t+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 h 12 Adjust velocities components in the direction of the difference gradient vector g 12 Adjust velocities in the direction

Phase control CNH 4 + : MRCI/CAS(4,3)/6-31G* Compare h(t) and h(t+  t)

Phase control CNH 4 + : MRCI/CAS(4,3)/6-31G* Compare h(t) and h(t+  t)

Abrupt changes control fs fs

OrthogonalizationOrthogonalization The routine also gives the linear parameters: g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)]

When surface hopping fails SH is supposed to reproduce quantum distributions, in the sense that fraction of trajectories (t) = adiabatic population (t)Eq. (1) This statement should be true for: Number of forbidden hops → 0 Number 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.

N EWTON -X: a package for Newtonian dynamics close to the crossing seam M. Barbatti, G. Granucci, H. Lischka and M. Ruckenbauer ( )

NX 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: nxinp 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: nxinp 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 TURBOMOLE RI-CC TURBOMOLE TD-DFT The current value of prog is: 1 Enter the new value of prog : 1

NX modular design R(t), v(t) t+  t, R(t+  t), v(t+  t/2) provide:  E k (t+  t), h kl (t+  t) v(t+  t) a kk, P k  l (t+  t) Initial condition generation Statistical analysis Tables and graphics Fortran 90 routines Perl controller

Adiabatic dynamics R(t), v(t) t+  t, R(t+  t), v(t+  t/2) provide:  E k (t+  t) v(t+  t)

Methods 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:  E k (t+  t), h kl (t+  t) v(t+  t) a kk, P k  l (t+  t)

So many choices… What method should I use?

Present situation of quantum chemistry methods

Comparison among methods RI-CC2TD-DFTCASSCF 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). CNH 4 + : MRCI/CAS(4,3)/6-31G*

A 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.

ConclusionsConclusions Basis set Correlation Method DZTZBS limit HF CASSCF MRCI Full-CI … …

ConclusionsConclusions Basis set Correlation Method DZTZBS 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: Adiabatic and nonadiabatic dynamics methods will be used in the investigation of some examples of photoexcited systems 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