Surface hopping dynamics with directsemiempirical solution of the electronic problem.

Maurizio Persico

Universita di Pisa

Dipartimento di Chimica e Chimica Industriale

Thanks to the work of

Giovanni Cosimo Silvia Teodoro AlessandroGranucci Ciminelli Inglese Laino Toniolo

Teresa Valentina Gloria Andi LuigiCusati 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 ddt |Ψ(t)〉 = Hel(t) |Ψ(t)〉• Ψ(t) is expanded in the basis of the N lowest adiabatic states ψK :

|Ψ(t)〉 =∑

K AK(t) |ψK(t)〉and PK(t) = |AK |2 are the adiabatic probabilities.

• Switching from an adiabatic surface to another depends on the PK(t) prob-

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)

• Fractional occupation SCF: ρ =∑

i niφ2i

• Ad hoc reparameterization for every new compound/process to be studied.

The floating occupation SCF

ni =2√πw

∫ εF

−∞ exp[−(ε− εi)2/w2]dε

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.

3 2

1

3

2

1

occupationsorbital

energiesorbital

Fermi level

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:

HLJ =∑

α∑

βAαβ

R12αβ

− Bαβ

R6αβ

Helec =∑

α∑

βqαqβRαβ− ∑

i∑

βqβRiβ

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

with solvent

in vacuo

internal coord.

ener

gy

8765432

20

15

10

5

0

-5

-10

with solvent

in vacuo

internal coord.

ener

gy

8765432

20

15

10

5

0

-5

-10

Connection atom approach to covalent QM/MMinteractions.

• 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.

Occupation numbers of the fragments:

ClOO· + Cl . . . 2 2 2 1 53

53

53

2ClO·(2Π) . . . 2 2 2 2 2 12

12

12

12

O2(3Σ−) + 2Cl . . . 1 1 5

353

53

53

53

53

Azobenzene photochemistry.

trans-azobenzene (TAB)

��

��

��3

-

QQ

QQ

QQs

N=N torsion (rotamer)

N-inversion (invertomer)

symmetric NNC bending

PPPPPPq

������1

cis-azobenzene (CAB)

-Internal Conversionto ground state

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 multiplechromophores 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

Pexc(t) = −K(t)Pexc(t)

whence

Pexc(t) = Pexc(0) e−∫ t0 K(t′)dt′

• Choose a ground state force field U0(Q).

• The effective potential is then:

Ueff(Q, t) = U0(Q) + Uexc(Q)Pexc(t)

• If we write the “rate constant” as

K =∑

r

∣

∣

∣

∣

Qr

∣

∣

∣

∣

fr

the nuclear forces are given by

Fr = −∂U0

∂Qr− ∂Uexc

∂QrPexc + UexcPexcfr

∣

∣

∣

∣

Qr

∣

∣

∣

∣

/Qr

• The fr factors will depend on the energy difference Uexc and possibly also

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!