Bifurcations in the evolution of molecular structure during photoisomerization

Abstract

Photoisomerization can, in most cases, be considered as a redistribution of atoms or molecular fragments among potential wells separated by barriers in the ground electronic state of the molecule initiated by the energy of a light wave. According to catastrophe theory, this process can be irreversible as a result of the jumpwise "choice" of one or another potential well (isomer) after the molecule has shifted to a position of unstable equilibrium, i.e., at the top of a potential barrier. In Prigogine's terminology, this is an "induced bifurcation" [1].In the present work, using mixed quantum-classical simulation of the photoisomerization dynamics, we demonstrated that such jumpwise changes in the character of the evolution of molecular structure are possible when the electronic subsystem of a molecule is excited by light and part of this energy is transferred to the nuclei. These jumps can be a result of fluctuations of the parameters that characterize the light and the molecule in the vicinity of some singularities (bifurcation values), for example, as will be shown below, when the frequency of the quasi-monochromatic irradiation is changed.The evolution of the electronic subsystem of a molecule and the radiation field in time was described using quantum theory, which allowed us to correctly consider the secondary radiation field ( σ photons, see below); the nuclear subsystem was described in the framework of classical theory. Using the expression for the force exerted by the electronic subsystem of the molecule on its nuclei, which is determined by the Ehrenfest theorem, and the Schrödinger equation for the electronic subsystem and the radiation field interacting with it, we obtained a system of self-consistent differential equations for the "reaction coordinate" Q , the amplitudes b λ 0 and b σ 0 , the populations of the initial and one of the final states of the total system, and the amplitude b 01 of the population of the resonance excited state of the molecule.In the resonance approximation, this system of equations takes the form (1) where χ = λ , σ is the set of the quantum numbers of the radiation photon and the secondary photon, respectively; E 0 ( Q [ t ]) and E 1 ( Q [ t ]) are the energies of, respectively, the ground and excited electronic states, which parametrically depend on Q ; m is a constant with the dimension of mass ( m = ប / Ω ); ω 10 (the resonance detuning, which changes in time in response to the change in the isomerization coordinate Q ;where e λ and e σ are the unit vectors of the λ photon and the σ photon, respectively; d is the dipole moment operator of the molecular system (independent of Q , the Condon approximation), and N λ = 1, 2, … is the number of radiation photons; P 0 ( t ) and P 1 ( t ) are the time-dependent populations of, respectively, the ground and excited electronic states of the molecule, which are determined as follows:

Simulation of Control of Photoisomerization by Light Pulses

“…As distinct from the works dealing with this line of studying the possibilities for control of chemical reactions, in [2,3], the intramolecular photoisomerization dynamics was simulated for the case where a molecule is irradiated with quasi-monochromatic light, which does not lead to considerable population of the excited state of its electronic subsystem. This irradiation does not give rise to packets of wave functions of the nuclear subsystem of the molecule while it transforms the light field.…”

“…The force exerted by the light-excited electronic subsystem of the molecule on the "isomerization oscillator" can induce the transition of this oscillator from the state corresponding to the initial isomer to the state corresponding to the other isomer (the displacement from one well of a double-well potential to the other). It was demonstrated in [2,3] that, at definite values of internal parameters that characterize the structure of a molecule (electronic transition frequencies, excited electronic state decay constants, the forms of the potentials of nuclear oscillators in the ground and excited electronic states of a molecule), there exists a range of controlling parameters (radiation frequency and intensity) at which the isomerization oscillator is irreversibly "photolocalized" in a certain potential well. Before the instant of choice of a particular well for location, the oscillator, generally speaking, passes several times through the position of unstable equilibrium (the top of the barrier of the double-well potential).…”

“…In the Newton equation, which determines the time dependence of the coordinate Q, the effective force exerted by the electrons on the nuclei of the molecule is described by an expression that follows from the Hellman-Feynman theorem, which was extended in [2,3] to the case of nonstationary electronic states of the molecule in the light field. For any normal mode of the nuclei characterized by the coordinate Q, this expression has the form…”

“…Next, we separated the normal coordinate of isomerization as in [2][3][4][5][6] and considered the one-dimensional problem on motion of the isomerization oscillator along this coordinate (designated as Q).…”

“…Let this position of the isomerization potential correspond to the cis isomer of the model system under consideration. It was assumed, as in [2,3], that ω C = 10 2 Ω = 10 3 γ. Here, ω C ≡ E 1 (Q = 0) -E 0 (Q = 0) is the frequency of the "vertical" 0 → 1 transition of the electronic subsystem of the molecule at Q = 0 (see Fig.…”