Intramolecular dynamics from statistical theories Pascal Parneix 1 Institut des Sciences Mol´ eculaires d’Orsay Universit´ e Paris-Sud, Orsay August 28, 2019 1 pascal.parneix@u-psud.fr ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 1 / 62
Introduction ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 2 / 62
Introduction D ynamics in the short time (sub-ps) is governed by non adiabatic couplings. Dynamics in the excited states. F ollowing this electronic relaxation, the molecule can be found in the fundamental electronic state. I n this course, we will focalize on the competition between different relaxation processes, may be sequential, of the system in this ground electronic state: • Dissociation • Isomerisation • IR Emission F ollowing dynamics of a molecular system over a long time is really a challenge both for experimentalists and theoreticians. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 3 / 62
Introduction ”Complex” molecular systems have some common properties: P otential Energy Surface (PES ) is characterized by a large number of local minima (isomers) and extrema (saddle points). A nharmonicity of the PES C haracteristics times of different processes on different orders of magnitude [coexistence of short time (ps-ns) and long time (ms-s) dynamics]. M olecular system with a large number of freedom. Difficult to follow the time evolution by solving Schr¨ odinger equation. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 4 / 62
Introduction Born-Oppenheimer adiabatic approximation is generally used to compute electronic states. T he molecular hamiltonian can be written as: H = T ( p ) + T ( P ) + U ( r , R ) (1) I n this expression, r is the set of the electronic coordinates, R is the set of the nuclear coordinates. p and P are the momenta linked to r and R , respectively. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 5 / 62
Introduction As the nuclei masses are larger of electrons, the electronic wavefunctions ϕ ( n ) e ( r ; R ) can be computed by fixing molecular geometry ( adiabatic approximation ). T hese electronic wave functions depend parametrically on the nuclear positions. For each value of R , the schr¨ odinger equation is solved: [ T ( p ) + U ( r , R )] ϕ ( n ) e ( r ; R ) = V n ( R ) ϕ ( n ) e ( r ; R ) (2) ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 6 / 62
Introduction T he function V n ( R ) corresponds to the electronic energy for the n ` eme adiabatic electronic state. F or each electronic state, we will have a function of whole of the nuclear coordinates called Potential Energy Surface (PES). B y the following, we will work on the ground PES V ( R ). A t the vicinity of a local minimum R e , the PES can be expressed as: V ( R ) = V ( R e ) + ( R − R e ) t H h ( R − R e ) + ... with H h the Hessian matrix. O n this PES, classical dynamics of the nuclei can be simulated. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 7 / 62
Introduction D ifferent methods can be used for the calculation of the PES V ( � R 1 , ...., � R n ): Atomistic model 1 ab-initio PES. Based on the calculation of the electronic wavefunction (or the electronic density). Semi-empirical PES (TB, DFTB, ...) Non reactive empirical PES (AMBER, CHARMM, ...) Reactive empirical PES (AIREBO, REAX) Coarse grained model 2 T his choice will be mainly governed by: The size and the nature of the molecular system The characteristics time of the microscopic phenomena The quality of the PES sampling ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 8 / 62
Introduction Figure: An example of Potential energy surface. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 9 / 62
Introduction W hy exploring this PES ? I mportant to find local minima and extrema which play a crucial role in the dynamics. F ollowing the time evolution of a given physical observable versus of E, T, ... U nderstanding thermodynamics of the system ... U nderstanding the reactional dynamics along a given path λ ( R ). H ow to properly explore this PES ? E xploration of the phase space. Dynamics in the (NVE), (NVT) statistical ensembles, ... Time average of physical observables. E xploration of the configuration space in different statistical ensembles. Ensemble average of physical observables. P roblem of ergodicity .... → Numerical strategies to follow. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 10 / 62
Introduction M olecular dynamics simulations in the (NVE) ensemble Propagation of Hamilton’s equations 1 ( R ( t ) , P ( t )) Perfectly adapted to follow the time evolution of physical properties as 2 a function of time N A ( t ) ≡ A ( R ( t )) et � A � t = 1 � A ( t i ) N i =1 Allow to compute rate constants for different processes (isomerisation, 3 dissociation, ...) from different initial conditions. But ... T he gradient of the PES has to be computed. D ifficult to extract information on rare events and/or for systems with N ≫ 1. The accessible characteristics times depend on the complexity of the PES. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 11 / 62
Competition between isomerization, fragmentation and IR emission IC Dissociation | 2 � fragment 2 IR Emission fragment 3 and UV-visible fragment 1 excitation Isomerization | 1 � | 0 � ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 12 / 62
Fragmentation Fragmentation ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 13 / 62
Introduction Following the non-adiabatic dynamics, the molecule can be found in the electronic ground state. A statistical approach could be used if T IVR ≪ T disso . T he characteristics time of dissociation T disso will depend on: I nternal energy (or temperature) D issociation energy T he number of degrees of freedom I n the framework of statistical theories, the density of states will naturally play an important role. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 14 / 62
RRK theory O ne simple model for a molecular dissociation of the parent X n ( → X n − 1 +X) is to consider X n as a set of harmonic oscillators, following the idea of Rice, Ramsperger et Kassel. The molecule will be considered as dissociated when the localized energy on a given mode will be larger than the dissociation energy. W e note g = 3 n − 6 the number of degrees of freedom of the parent molecule. Let us computing the probability P ( E ) for that E to be localized in a dissociative mode, will be larger than the dissociation energy D n . T he number of possibilities to distribute E over g oscillators is given by E g − 1 / ( g − 1)!. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 15 / 62
RRK theory T he probability P ( E ) is thus given by: � E − D n ǫ g − 2 d ǫ/ ( g − 2)! 0 P ( E ) = E g − 1 / ( g − 1)! � g − 1 � E − D n = (3) E T he dissociation constant k ( n ) d ( E ) is proportional to this probability. We thus obtain: � g − 1 � E − D n k d ( E ) = ν 0 (4) E T he ν 0 prefactor is generally fitted to reproduce experimental results. Only the reactant is taken into account in this approach. K assel has proposed a quantal version, much more adapted for small systems and/or at low energy. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 16 / 62
RRKM Theory N otion of transition state. S eparation between nuclear and electronic degrees of freedom. Nuclear dynamics on a PES. T here is a critical surface which separates reactant and produit. H ypothesis of ”non retour”. I n this approach, the dissociation rate is directly linked to the flux of trajectories through the critical surface. One of the major difficulty is to properly localize the transition state. A lso based on the quasi-equilibrium hypothesis: E nergy redistribution much more rapid than the dissociation reaction. S eparability at the transition state: 1 dissociation coordinate + spectator modes. E nergy equipartition in the spectator modes at the transition state. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 17 / 62
RRKM Theory W e note D n the energy of the transition state. We note v the derivative with respect time of the reaction coordinate at the TS. It thus simply corresponds to the velocity at the TS. W e note E † t the kinetic energy along the reaction coordinate at the TS. T he RRKM dissociation constant can be written: k d ( E ) ∝ v Ω † ( E † ) (5) Ω n ( E ) with E † = E − D n the energy available at the transition state. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 18 / 62
RRKM Theory T he density of states Ω † at the transition state is written as: � v ) ρ ( E † Ω † ( E † ) = N ( E † t ) (6) � − 1 / 2 and v ∝ A s ρ ( E † t ) ∝ E † E † t , we obtain: t k d ( E ) ∝ N † ( E ) (7) h Ω n ( E ) � with N † ( E ) = N ( E † v ) the number of vibrational states for the spectator modes which can be populated at the TS. ´ Pascal Parneix (ISMO) Ecole d’´ et´ e GDR EMIE et UP August 28, 2019 19 / 62
Recommend
More recommend
