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Surface Hopping Dynamics with Direct Semiempirical Solution of the Electronic Problem. Maurizio Persico Universit`A Di Pisa Dipa

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Surface Hopping Dynamics with Direct Semiempirical Solution of the Electronic Problem. Maurizio Persico Universit`A Di Pisa Dipa Surface hopping dynamics with direct semiempirical solution of the electronic problem. Maurizio Persico Universit`adi Pisa Dipartimento di Chimica e Chimica Industriale Thanks to the work of Giovanni Cosimo Silvia Teodoro Alessandro Granucci Ciminelli Inglese Laino Toniolo Teresa Valentina Gloria Andi Luigi Cusati Cantatore Spighi Shehu Creatini Outline of this talk. Goals and problems • Trajectory based methods and the like • On the fly calculation of PES and couplings • An example from atmospheric chemistry: ClOOCl on ice • More examples from material science: the azobenzene chromophore • Trying to overcome size and time limitations: a proposal • Goals and problems. Simulate photophysical and photochemical processes, spin-changing reac- • tions, and electron transfer reactions, to interpret time-resolved experiments and propose new ones, and to help designing new materials and molecu- lar devices = Calculate excited electronic states, correctly represent the ⇒ nonadiabatic dynamics. Treat large systems: extended or multiple chromophores and reactive centers, • interacting with solvent, surfaces, proteins etc = Cannot apply (uniformly) ⇒ high level methods for electronic structure. Many nuclear degrees of freedom are important. Consider the interplay of processes with (sometimes slightly) different time • scales = Integration times from several picoseconds to several nanoseconds. ⇒ Trajectory surface hopping simulations. The nuclear dynamics is represented by a swarm of classical trajectories; each • trajectory runs on a given adiabatic PES, but it may jump to another PES at any time (surface hopping). The electronic wavefunction Ψ(t) evolves in time according to the TDSE: • i d Ψ(t) = ˆ (t) Ψ(t) dt | i Hel | i Ψ(t) is expanded in the basis of the N lowest adiabatic states ψ : • K Ψ(t) = K AK(t) ψK(t) | i P | i and P (t) = A 2 are the adiabatic probabilities. K | K| Switching from an adiabatic surface to another depends on the P (t) prob- • K abilities, according to a stochastic algorithm (Tully’s “fewest switches” with decoherence corrections). Initial coords. and momenta are sampled according to ground state quantal • or classical distributions; each trajectory starts with a vertical excitation. Observables are computed as averages over many trajectories. • The “on the fly” strategy. The cost of computing the potential energy surfaces (PES) grows exponen- • tially with the number of coordinates. The problem of representing analytically PES and couplings, in the presence • of surface crossings, is a hard one. The trajectory methods are ideally suited for the direct calculation of elec- • tronic energies and wavefunctions (one electronic calculation per time step). We have implemented three different options: Ab initio: NEWTON-X program, coupled to COLUMBUS, TURBOMOLE • etc (see Barbatti, Granucci, Persico, Ruckenbauer, Vazdar, Eckert-Maksic and Lischka, J. Photochem. Photobio. A 190, 228, 2007) Semiempirical, with CI wavefunctions and floating occupation SCF, to rep- • resent excited states and bond breaking (see Granucci, Persico and Toniolo, J. Chem. Phys. 114, 10608, 2001). The semiempirical parameters are opti- mized, to reproduce experimental and ab initio data for a specific compound. QM/MM with semiempirical wavefunctions (see Persico et al, THEOCHEM • 621, 119, 2003; Toniolo et al, Theoret. Chem. Acc. 93, 270, 2004) Semiempirical method Most semiempirical methods are based on: Single determinant SCF wavefunctions • Core electrons replaced by modified nuclear charges and core-core potentials • Minimal basis sets • One- and two-electron integrals replaced by analytic expressions dependent • on parameters, or altogether neglected. The parameters are optimized once for all, to yield good ground state prop- • erties of several classes of compounds. To represent the electronic wavefunctions for excited states and distorted geometries, we need: Configuration Interaction (CI) • 2 Fractional occupation SCF: ρ = i niφi • P Ad hoc reparameterization for every new compound/process to be studied. • The floating occupation SCF 2 εF 2 2 ni = Z exp[ (ε εi) /w ]dε √πw −∞ − − The occupation numbers depend on the MO energies, so: homolytic dissociation is correctly represented; • degenerate orbitals are equally occupied; • the lowest virtuals are partially optimized. • orbital orbital energies occupations 3 3 Fermi level 2 2 1 1 QM/MM strategy. The reactive portion of the system (“QM subsystem”) is treated quantum- • mechanically at semiempirical level. The “MM subsystem” is treated by a force-field: it may be a solvent, a solid • surface, a polymeric matrix... whatever takes part in the dynamics without undergoing bond breaking or getting electronically excited. The interaction between the two subsystems consists of Lennard-Jones and • electrostatic terms: Aαβ Bαβ ˆLJ = α β 12 6 H P P Rαβ − Rαβ qαqβ qβ ˆelec = α β i β H P P Rαβ − P P Riβ where α = QM nucleus, β = MM nucleus, i = QM electron. The electrostatic QM/MM interaction is added to the semiempirical hamil- • tonian (state-specific treatment of environmental effects). Electrostatic embedding and surface crossings. In PES crossing situations, while it makes sense to add solute-solvent inter- action terms to the electronic hamiltonian (or to the diabatic potentials), a direct modification of the adiabatic PES would lead to unphysical features. modified adiabatic potentials modified hamiltonian 20 20 in vacuo in vacuo 15 with solvent 15 with solvent 10 10 5 5 energy energy 0 0 -5 -5 -10 -10 2 3 4 5 6 7 8 2 3 4 5 6 7 8 internal coord. internal coord. Connection atom approach to covalent QM/MM interactions. The CA is part of the QM subsystem: since it owns one electron and one • basis function, of s type, it makes a single bond with the closest QM atom. The CA also participates of the MM force field. This ensures the correct de- • pendence of the potential on the bond lengths, angles and dihedrals involving the CA, the MM atoms and the closest QM atoms. Photodissociation of ClOOCl adsorbed on ice. Of interest for the ozone chemistry in the polar stratospheric clouds: What bonds are broken and what products are formed? • Are the products free or adsorbed on ice? • Is the photodissociation more or less efficient than in gas phase? • ... and all the details of the reaction mechanism to fuel the debate among • chemists. 5 5 5 ClOO + Cl . 2221 3 3 3 · 2 1 1 1 1 Occupation numbers of the fragments: 2ClO ( Π) . 22222 2 2 2 2 3 · 5 5 5 5 5 5 O2( Σ−) + 2Cl . 1 1 3 3 3 3 3 3 Azobenzene photochemistry. PP PP N=N torsion (rotamer) PPq 3 1 cis-azobenzene (CAB) - Q Q trans-azobenzene (TAB) Q N-inversion (invertomer) Q QQs -Internal Conversion to ground state symmetric NNC bending Questions we have answered: What is the reaction mechanism? • Why the photoisomerization quantum yields decrease when increasing the • excitation energy? Why a viscous solvent, that slows down the isomerization dynamics, does • increase the quantum yield? Why the fluorescence emitted by azobenzene is strogly polarized? • Azobenzene as a light powered engine: working against a pulling force. Experiment: Gaub and coworkers, Science 296, 1103 (2002); Macromolecules 36, 2015 (2003); ibid 39, 789 (2005). trans cis 13.59 A˚ 11.28 A˚ ←− −→ ←− −→ Simulations: Creatini, Cusati, Granucci, Persico, Chem. Phys. 347, 492 (2008) Azobenzene as a light powered engine: snapping eight hydrogen bonds. Experiment: Vollmer, Clark, Steinem, Reza Ghadiri, Angew. Chem. Int. Ed. 38, 1598 (1999); Steinem, Janshoff, Vollmer, Reza Ghadiri, Langmuir 15, 3956 (1999). cis trans Simulations: Ciminelli, Granucci, Persico, Chem. Phys. 349, 325 (2008) Good news. With the QM/MM strategy we can simulate fairly large systems (thousands • of atoms). The semiempirical PES can be as accurate as the best ab initio ones, de- • pending on the size of the QM system. The surface hopping method can yield results in very good agreement with • experiments (quantum yield, transient spectra, energy disposal in photodis- sociations, etc). Bad news. The reparameterization of the semiempirical hamiltonian is still a cumber- • some task, mainly trial-and-error (no gradients, many local minima, problems in converging the electronic structure calculations when trying bad parameter sets). We are still limited by the size/number of chromophores. • Trajectory methods with direct solution of the electronic problem scale lin- • early with the real simulation time: this is not good enough! In quantum dynamics the situation is even worse, but for multiscale problems we need to go orders of magnitude faster. A modest proposal to treat multiple chromophores and long integration times. Carry out a “model dynamics” with the usual methods (one chromophore, • simpler chemical environment, short time, few trajectories...) Extract from the results of the model dynamics an analytic expression of • the energy difference between excited and ground state, as a function of the (few) relevant internal coordinates, Uexc(Q). Assume the excited state decay obeys the rate equation • P˙ (t) = K(t)P (t) exc − exc whence t 0 K(t′)dt′ Pexc(t) = Pexc(0) e− R Choose a ground state force field U (Q). • 0 The effective potential is then: • Ueff (Q,t) = U0(Q) + Uexc(Q)Pexc(t) If we write the “rate constant” as • ˙ K = Qr fr X r the nuclear forces are given by ∂U0 ∂Uexc ˙ ˙ Fr = Pexc + UexcPexcfr Qr /Qr −∂Qr − ∂Qr The f factors will depend on the energy difference U and possibly also • r exc on some of the internal coordinates. They may vanish for most atoms, with the exception of the those directly involved in the excitation. They must be parameterized using the results of the model dynamics. The simulation is reduced to a Molecular Dynamics one, with a modified potential. It can be carried out on more complex systems (many interacting chromophores), for longer times, many trajectories... Still to be implemented! Collaborations are welcome!.
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