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Adiabatic and Nonadiabatic Molecular Dynamics with Multireference Ab 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 Theoretical Chemistry University of Vienna 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. 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 ⎡− 2α(R − Re ) ⎤ ⎡− P ⎤ PW (R, P) = exp⎢ ⎥exp⎢ ⎥ E πh ⎣ h ⎦ ⎣ 2αh ⎦ mω α = HO 2 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) Quantum chemistry 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 cos β + hij sin β ~ hij = gij sin β + hij cos β 2g ⋅h tan 2β = ij ij hij ⋅hij − gij ⋅gij S1 • The routine also gives the linear parameters: ⎡ 1/ 2 ⎤ ⎛ 1 ∆ gh ⎞ 2 2 2 2 S E = d gh ⎢σ 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 • Surface hopping: fewest switches −iγ k (t ) ψ (t) = ∑ck (t)e φk (R) k ∂ i ψ (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 (MOLDEN, 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 / Gaussian 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.
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