Ultrafast processes in molecules IX – Surface hopping implementation Mario Barbatti barbatti@kofo.mpg.de Energy Wave packet propagation Surface hopping propagation Reaction coordinate • Tully, Preston, JCP 55, 562 (1971) 2 the semi-classical propagation 3 r, R ,t c c j t Φ j r; R c t Φ | k Φk Φl r kl j i H (r , R , t ) 0 e t dc k i dt 2 dt Φ | H k Φ | H d k 2 kj kj c c c cj 0 Φk j c d Rm H kj i Fkj v c ,m F kj c Fm 0 Mm t c F kj v c Φj r Φk Rm Φ j A AR V k kj W kj , F kj 0 c c r t Adiabatic basis Diabatic basis 4 2 c d Rm dt 2 c Fm Fm R m H ll c 0 Mm c Any standard method can be used in the integration of the Newton’s equations. A good one is the Velocity Verlet For each nucleus m: R cm ( t t ) R cm ( t ) v cm ( t ) t 1 2 a m (t ) t c 2 t 1 c c v t v m (t ) a m (t ) t 2 2 c m a m (t ) c 1 Mm c R E R m ( t t ) t 1 c c c v m (t t ) v m t a m (t t ) t 2 2 • Swope et al. J. Chem. Phys. 76, 637 (1982) 5 • Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997) 6 Time step should not be larger than 1 fs (1/10v). Dt = 0.5 fs assures a good level of conservation of energy. Exceptions: • Dynamics close to the conical intersection may require 0.25 fs • Dissociation processes may require even smaller time steps 7 i dc k dt H c kj i Fkj v c c c j 0 SC-TDSE j The SC-TDSE is solved with standard methods (Unitary Propagator, Adams Moulton 6th-order, Butcher 5th-oder) 0.7 t = 0.5 fs; ms = 1 Runge-Kutta Adams Moulton U. Propagator Butcher 0.6 |c0| 2 0.5 0.4 0.3 0.2 0.1 0.0 14 16 18 20 22 24 Time (fs) 26 28 30 8 t/ms ... h(t) h(t+t) n n h t + t 1 = h ( t )+ ( h ( t + t )- h ( t )) ms ms |c | 2 i ( n = 1 ..m s - 1) t = 0.5 fs i 3 ms = 1 2 1.04 1.03 ms = 10 1.02 1.01 ms = 20 1.00 0 10 20 30 40 50 60 70 80 90 100 Time (fs) 9 1.0 Occupation Ad. pop. 0.8 0.6 0.4 0.2 0.0 0 20 40 60 80 100 Time (fs) Uncorrected H 2C C C Butatriene cation CH2 10 TDSE E Q SC-TDSE E Because in the SC-TDSE the “wave-packet” splitting among the several electronic surfaces is kept correlated by the coordinate Rc, the time propagation is fully coherent. Q • Schwartz, Bittner, Prezhdo, Rossky, J. Chem. Phys. 104, 5942 (1996) • Zhu, Jasper, Truhlar, J. Chem. Phys. 120, 5543 (2004) • Granucci, Persico, Zoccante, J. Chem. Phys. 133, 134111 (2010) 11 Decoherence is introduced ad hoc by correcting the time dependent coefficients: c k c k exp t / ki ' 2 ci ci i 1 ki ' 2 ck k i 1/ 2 C 1 Ek Ei E kin C 0.1 hartree • Granucci and Persico, J. Chem. Phys. 126, 134114 (2007) 12 1.0 1.0 Occupation Ad. pop. 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0 20 40 60 80 100 Time (fs) C C Butatriene cation 0.0 0 20 40 60 80 100 Time (fs) Uncorrected H 2C Occupation Ad. pop. Corrected CH2 13 F12 v H 12 exp 2 2 P * * 1 H 11 H 22 t 0.57 Energy (au) -224.65 * cs -224.70 * -224.75 0.43 -224.80 cs -224.85 0 2 4 6 8 10 12 14 Time (fs) 14 Pl k P opulation increm ent in k due to flux fro m l during t P opulation of l lk t c l c k * Pl k 2t m ax 0, ll 1 Im kl H • Tully, J Chem Phys 93, 1061 (1990) c lk R e kl F v c kl c 15 A hopping will take place if two conditions are satisfied: 1) A uniformly selected random number rt in the [0, 1] interval is such that k 1 k P t r P t l n n 1 l n t n 1 2) The energy gap between the final and initial states satisfies Vk R c t Vl R c t N at c c ,m v m F kl m N at 2 M 1 m 2 F c ,m kc 2 m 16 E Forbidden hop Total energy R How to treat such situations: Reject all classically forbidden hop and keep the momentum Reject all classically forbidden hop and invert the momentum Use the time uncertainty principle to search for a point where the hop is allowed • Jasper, Stechmann, Truhlar, J. Chem. Phys. 116, 5424 (2002) 17 E Total energy KN(t) KN(t+t) R 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 h12 Adjust velocities components in the direction of the difference gradient vector g12 • Pechukas, Phys. Rev. 181, 174 (1969) • Fabiano, Keal, Thiel, Chem. Phys. 349, 334 (2008) 18 1. Wigner Distribution for harmonic oscillator PW Q , P 3N 6 1 i 1 2. i2 i Hi O Q i 2 P exp i i R , P HO For each Rn in {R,P} H eΦ k r ; R n V k Φ k r ; R n V k R n , f1k R n 3. Compute cross section E 4. e 2 2 m c 0 N fs l2 1 N p Np n f 1 l R n g E V1 l R n , Compare to experiments 10 ln 10 3 NA • Barbatti, Aquino, Lischka, PCCP 12, 4959 (2010) • Barbatti, PCCP 13, 4686 (2011) 19 Wavelength (nm) 300 Ura 200 simulated {R,P} measured -1 Cross section (Å .molecule ) 0.4 250 2 0.3 0.2 *+nR3s 0.1 0.0 * n* * 4 5 6 7 8 Energy (eV) 20 surface hopping: panoramic view 21 1. Solve SE for Rc H eΦ k r ; R c V Φ r; R V c k k k , V k , Fkj Φ k Φ j 2. Solve Newton’s equation on one surface 2 c d Rm dt 2 mV l c 0 Step 1 is the computational bottleneck Mm 3. Integrate the SC-TDSE i dc k dt V c k kj i Fkj v c c c j 0 j 4. Compute transition probability Pl k 2t * c c m ax 0, R e c k c l F kl v 2 cl • If it hops, then adjust momentum • Repeat procedure until the end of the trajectory • Compute many trajectories 5. Decide surface for next time step k 1 k P t r P t , l n n 1 l n t rt random 0,1 n 1 22 Simple implementation • Propagation in Cartesian coordinates • Trivial connection to different quantum chemical methods, including QM/MM (regarding these methods can provide excited state energies, energy gradients and nonadiabatic couplings) • Independent trajectories: trivial parallelization Local approximation • No need of precomputing multidimensional potential energy surfaces • Straightforward on-the-fly implementation • All nuclear degrees of freedom are propagated 23 Simple implementation • High computational costs in the on-the fly approach Local approximation • Inconsistent treatment of zero point energy • No treatment of tunneling effects • Wrong coherence between states 24 Comparison to other methods • Cattaneo and Persico, J. Phys. Chem. A 101, 3454 (1997) • Worth, Hunt, Robb, J. Phys. Chem. A 107, 621 (2003) Comparison between hopping algorithms • Zhu, A. W. Jasper, and D. G. Truhlar, JCTC 1, 527 (2005) • Fabiano, Groenhof, Thiel, Chem. Phys. 351, 111 (2008) Conceptual background • • • • Herman, J. Chem. Phys. 103, 8081 (1995) Schwartz, Bittner, Prezhdo, Rossky, J. Chem. Phys. 104, 5942 (1996) Tully, Faraday Discuss. 110, 407 (1998) Schmidt, Parandekar, Tully, J. Chem. Phys. 129, 044104 (2008) Surface hopping reviews • Doltsinis, NIC series, 2002 • Barbatti, WIREs: Comp. Mol. Sci. 1, 620 (2011) 25 electrons • classical treatment • quantum treatment nuclei Atomic and molecular collision reactions Molecular photochemistry and photophysics Condensed phase dynamics See References in: • Barbatti, WIREs: Comp. Mol. Sci. 1, 620 (2011) 26 • classical treatment Slow degrees of freedom Fast degrees of freedom • quantum treatment • Tully, Faraday Discuss. 110, 407 (1998) • Herman, J. Chem. Phys. 103, 8081 (1995) Singlet-triplet transitions • Carbogno, Behler, Reuter, Gross, Phys. Rev. B 81, 035410 (2010) Electric field interactions • Mitric, Petersen, Bonacic-Koutecky, Phys. Rev. A 79, 053416 (2009) Solvent-induced vibrational relaxation • Hammes-Schiffer, Tully, J. Chem. Phys. 101, 4657 (1994) 27 the newton-x program 28 Barbatti, Granucci, Ruckenbauer, Plasser, Crespo-Otero, Pittner, Persico, Lischka NEWTON-X: A Package for Newtonian Dynamics Close to the Crossing Seam (2007-2013) Interface Method Spectrum Dynamics Adiabatic Nonadiabatic (TSH) NAC COLUMBUS TURBOMOLE GAUSSIAN GAMESS ANALYTICAL DFTB DFTB+ TINKER DFT-MRCI MRCI, MCSCF TDDFT CC2, ADC(2) CASSCF TDDFT MCSCF User defined TD-DFTB DFTB MM DFT-MRCI CIO +MM +MM +MM +MM +MM • Barbatti, Granucci, Persico, Ruckenbauer, Vazdar, Eckert-Maksic, Lischka, J Photochem Photobiol A 190, 228 (2007) LD 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: • NX is the first MQCD-oriented program freely available and open to the community 30 -----------------------------------------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): 31 -----------------------------------------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 32 FSSH needs coefficients c: Pl k 2t * m ax 0, R e c k c l Fkj v 2 cl 1) Through: dc k i dt V k c k Fkj v c j j A) First order nonadiabatic couplings (NAC) Explicit evaluation of Fkj Only few methods (MCSCF, MRCI) B) CI overlap (OVL) Numerical evaluation of Fkj v F jk v t S jk t 1 4t 3S t 3S t S t t S t t jk kj jk kj j t t k t Any method allowing to get a CI-like wavefunction (TDDFT, ADC(2), …) • Hammes-Schiffer and Tully, J Chem Phys 101, 4657 (1994) • Pittner, Lischka, MB, Chem Phys 356, 147 (2009) 34 FSSH needs coefficients c: 2t * Pl k m ax 0, R e c k c l Fkj v 2 cl 1 V t TV t t T 1 1 t c t 2) Through: c t t T exp i ħ 2 C) Local diabatization (LD) Löwdin orthogonalization of CI overlap matrix S T SO 1/ 2 O t S SO O t Any method allowing to get a CI-like wavefunction (TDDFT, ADC(2), …) • Granucci, Toniolo, Persico, J Chem Phys 114, 10608 (2001) • Plasser, Granucci, Pittner, MB, Persico, Lischka, J Chem Phys 137, 22A314 (2012) 35 Percent relative error 30 NAC OVL LD 20 10 Landau-Zener model system: Relative error for the asymptotic diabatic transition probability x t 0 -10 LD is especially adequate to weak nonadiabatic coupling regimes -20 -30 0.01 0.1 1 Timestep length (fs) • Plasser, Granucci, Pittner, MB, Persico, Lischka, J Chem Phys 137, 22A314 (2012) 36 The nuclear Ensemble Method OH 4 6 5 HN 3 2 N 4 9 67 7 hn 8 1 trans 6 5 HN 7 3 O 10 2 N 8 1 cis HO 9 • Review: Gibbs, Tye, Norval, Photochem Photobiol Sci 7, 655 (2008) O 10 OH 4 6 5 HN 2 N hn 8 67 7 3 4 9 6 O 2 10 cis 8 HO 9 O 10 UVA Ftrans→cis (289 nm) = 0.08 2 -1 Cross section (Å .molecule ) UVB N 1 trans UVC 7 3 1 1.0 5 HN 0.5 Ftrans→cis (302 nm) = 0.31 Ftrans→cis (313 nm) = 0.49 0.0 260 280 300 320 Wavelength (nm) 340 N NH OH HN OH N O O 2 Cross section (Å ) 1.0 Exp. 0.5 Theor. 0.0 280 320 Wavelength (nm) • Barbatti, PCCP 13, 4686 (2011) First order of the time-dependent perturbation theory E 3 c 0 n r E E 00 , nk R 00 R 0 r , R n nk R E 00 , nk R E d R 2 * r n ,k E 00 , nk E nk E 00 E 0 , n The problem can be recast in the time domain: E 1 6 2 c 0 n r E n R e E 0 , n R M 0 n R 00 R n R , t e M 0n R 0 e r; R n n t e iH n t / 2 r 2 * E 0 , n E 00 , nk i E E 00 t / dt d R k 00 • Sakurai (1994) Modern Quantum Mechanics • Tannor, Heller, J Chem Phys 77, 202 (1982) 41 Relation between cross section (cm2) and extinction coefficient (M-1cm-1) 10 ln 10 3 NA 42 The core of the method is to compute the overlap R , t 00 R n R , t * needed to integrate E 1 6 2 c 0 n r Re E R M R R R ,t e E 2 0 ,n 2 0n * 00 n i E E 00 t / n dt d R Tannor and Heller proposed a analytical solution based on harmonic oscillator 00 n t 2nj i j t i i j t a exp 1 e E 0 , n t 2 2 j 43 But we want to go beyond the Condon approximation. Starting from Tannor-Heller equation: 00 n t 2nj i j t i i j t a exp 1 e E 0 , n t 2 2 j The expansion to second order is 2 1 2 a 2 2 1 t nj j t nj j 0n 2 j 2 j 1 00 n t exp i This motivates to introduce the following functional: n R , t 00 R exp * 00 2 i E 0 ,n R t i nt i E 00 t 1 nt 2 8 • Crespo-Otero, Barbatti, Theor Chem Acc 131, 1237 (2012) 2 2 44 00 n R , t 00 R exp 2 * i E 0 ,n R t E E 1 6 2 c 0 n r e i nt i E 00 t 1 2 2 Re E R M R R R ,t e E 2 2 m c 0 n r E 2 0 ,n 2 0n * 00 n nt 2 8 i E E 00 t / n dt d R 00 R E 0 , n R f 0 n R g G auss E E 0 , n R n R , n d R 2 n g G auss E E 0 , n R n R , n 1 2 n / 2 2 1/ 2 E E R 2 0 ,n n exp 2 2 n / 2 45 E e 2 2 m c 0 n r E 00 R E 0 , n R f 0 n R g E E 0 , n R n R , n d R 2 n If we have a ground state distribution of points: 00 R l 2 We can integrate the cross section by Monte-Carlo and get: E e 2 2 m c 0 n r E N fs n 1 Np Np E R f R g E E R , 0 ,n l 0n l l n l 46 Using a Wigner distribution to a harmonic oscillator, we can sample the ground state: 00 q 2 3N 6 j 1 j j 1/ 2 exp j j q j / 2 Another way of sampling is to run a very long trajectory in the ground state and pick points from it. But be careful with the right temperature! 47 Pros: • Easy to use • Clear conceptual basis • Absolute heights • Absolute widths • Post Condon Approximation • Dark vibronic bands • Implemented in Newton-X Cons: • No vibrational resolution • No non-adiabatic info • No info on excited state wavefunction • One arbitrary parameter 48 0.3 3 0.010 0.2 2 0.005 0.1 1 Simulated Expt. Bolovinos et al. 1982 2 -1 Cross section (Å .molecule ) 0.015 0.000 4.5 5.0 Energy (eV) 5.5 0.0 5.6 5.8 6.0 6.2 Energy (eV) 6.4 0 4.5 5.0 5.5 6.0 6.5 7.0 Energy (eV) TD-CAM-B3LYP/TZVP 49 2 Cross section (Å ) NE LVC Expt. 0.002 H 3C N N CH3 0.001 azomethane 0.000 2.5 3.0 3.5 4.0 4.5 5.0 Energy (eV) • Szalay, Aquino, Barbatti, Lischka, Chem Phys 380, 9 (2011) tra n s -2 tra n s -1 7 2 1.0 5 -1 -1 (10 M cm ) 0.8 e trans-1 trans-2 cis-A1 Total Expt. d e' 0.6 d' N H 21 N 24 22 23 NH2 N N 9 H N H N NH2 N H N 12 17 c is -A 2 c is -A 1 b 0.4 c b' a c' 0.2 N H N H N N NH2 N N H N H N NH2 a' 0.0 300 400 500 600 700 800 900 c is -B 2 c is -B 1 Wavelength (nm) N N • Lan, Nonell, Barbatti, J Phys Chem A 116, 3366 (2012) N H NH2 N H H N H N Porphycene N N NH2 • Homem, Lopez-Castillo, Barbatti, Rosa, Iza, Cavasso-Filho, Farenzena, Lee, Iga, J Chem Phys 137, 184305 (2012) 52 Challenges in computational simulations of nucleobases dynamics E2 1H 6 MRCIS-2n 6H 1 2E 6 E2 1H 6 OM2/MRCI N1 6E 2E 2 E2 C2 H E6 Ab initio and semiempirical MRCI produce divergent results 54 TD-PBE0 TD-BHLYP E2 6 2 E H1 6 E 6 6 H1 2 E 2E E E2 E2 1H 6 E6 55 Ground state population % at 1 ps C2 puckering NH2 C6 puckering H elimination C Experimental S0 population at 1 ps (%) 85 8 80 68 2 60 57 12 59 8 N Right for different reasons C CH HC C N 40 5 8 10 16 0 wrong Ex pe r im en ta AD l C (2 M ) R O C I M 2/ S M R TD CI TD -P - BE B9 TD 7X -B D 3L TD TD YP -C -P AM BE -B 0 3 TD LY -B P TD HLY -M P 06 -H F 0 N H 25 16 20 21 20 15 12 8 20 N • MB, Lan, Crespo-Otero, Szymczak, Lischka, Thiel, J Chem Phys 137, 22A503 (2012) 56 Pump-probe signal depends on changes in the ionization cross section and in the ionization energy Variations in the ionization energy are not usually taken into account neither in simulations nor in experimental analysis • Barbatti and Ullrich, PCCP 13, 15492 (2011) 57 58 It is not only an excited-state problem. Ground state is also bad. 59 Ground state simulations: 1- 2 kcal/mol accuracy Excited state simulations: 5-10 kcal/mol accuracy Reason: • Excited states spectral region has high density of states • Small variation in geometry leads to a change of electronic character • No affordable method can describe all characters at the same level 60 • Merchan and Serrano-Andres, JACS 125, 8108 (2003) 61 Next lecture: • Case studies of dynamics simulations Contact barbatti@kofo.mpg.de