Born-Oppenheimer dynamics

One may consider the above equation as a generalization of Born-Oppenheimer dynamics in which electrons always stay on the Born-Oppenheimer surface. For a given conformation of nuclei, the numerical value of the fictitious mass associated with electronic degrees of freedom determines how far the electron density is allowed to deviate from the Born-Oppenheimer one. Each consecutive step along the trajectory, which involves electronic and nuclear degrees of freedom, can be obtained without determining the exact Born-Oppenheimer electron density. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

For further information on general concepts of non-Born-Oppenheimer dynamics, we refer to recent reviews [162, 163] and textbooks [169, 170]. 

Eq. (21), which directly reflects the non-Born-Oppenheimer dynamics of the system. Assuming that the system is initially prepared at x = 3 in the diabatic state /2), the corresponding initial distribution pg mainly overlaps with orbits A and C, since at x = 3 these orbits do occupy the state /2). Similarly, excitation of /i) mainly overlaps with orbits of type B, which at x = 3 occupy the state /i) (see Fig. 32). In a first approximation, the electronic population probability P t) may therefore be calculated by including these orbits in the... 

In this approach, the system is partitioned into a part described quantum mechanically (the ion plus hydration waters) and the other treated by molecular mechanics. A detailed description of the QM/MM method as implemented for condensed phase system is provided by Field et al. [231]. Interactions inside the hydration complex are calculated using ab initio Born-Oppenheimer dynamics [228], while all the other interactions are modeled by classical pair potentials. 

Our first attempt to generate Born-Oppenheimer dynamics by simulated annealing, shown in Figure 4, can be improved in several ways. Car and Parrinello enforced orthonormality of electronic orbitals as a holonomic constraint on the parameter dynamics. For N orbitals this gives N N - -1) holonomic constraints. For our single electron model problem (5) there is only one constraint. 

Figure 5. Simulated annealing nuclear dynamics under the holonomic constraint (6). The parameter masses are set to half the proton mass, chosen, for artistic reasons, so that some small deviation from exact Born-Oppenheimer dynamics would be visible to the reader. Further reduction of the parameter mass would lead to nuclear dynamics indistinguishable from the exact Born-Oppenheimer trajectory, also shown in the figure.

The basic elements of the description of non-Born-Oppenheimer dynamics in either the adiabatic or diabatic representations have been outlined in Chapters 1-4. The point of departure of the present discussion are the coupled equations in the diabatic representation... 

The goal of this review is to critically compare — from both a concep-tional and a practical point of view — various MQC strategies to describe non-Born-Oppenheimer dynamics. Owing to personal preferences, we will focus on the modeling of ultrafast bound-state processes following photoexcitation such as internal-conversion and nonadiabatic photoisomerization. To this end, Sec. 2 introduces three model problems Model I represents a three-mode description of the Si — S2 conical intersection in pyrazine. Model II accounts for the ultrafast C B X internal-conversion process in the benzene cation, and Model III represents a three-mode description of ultrafast photoisomerization triggered by a conical intersection. Allowing for exact quantum-mechanical reference calculations, all models have been used as benchmark problems to study approximate descriptions. 

In this review we have considered mixed quantum-classical (MQC) methods to describe non-Born-Oppenheimer dynamics at conical intersections. We have started with an attempt to classify existing MQC strategies in formulations resulting from a partial classical limit, from a connection ansatz, and from a mapping formalism. Focusing on methods that have been applied... 

It has been shown that in the limit of ultrashort laser pulses the stimulated-emission pump-probe signal is proportional to the population probability of the initially excited diabatic state [Tf)) Eq. (59) and Refs. 7, 99 and 141. As has been emphasized in Chapter 9, the electronic population probability P2 t) represents a key quantity in the discussion of internal-conversion processes, as it directly reflects the non-Born-Oppenheimer dynamics (in the absence of vibronic coupling, P2 t) = const ). It is therefore interesting to investigate to what extent this intramolecular quantity can be measured in a realistic pump-probe experiment with finite laser pulses. It is clear from Eq. (33) that the detection of P2(t) is facilitated if a probe pulse is employed that stimulates a major part of the excited-state vibrational levels into the electronic ground state, that is, the probe laser should be tuned to the maximum of the emission band. Figure 4(a) compares the diabatic population probability P2(t) with a cut of the stimulated-emission spectrum for uj2 3.4 eV, i.e. at the center of the red-shifted emission band. Apart from the first 20 fs, where the probe laser is not resonant with the emission [cf Fig. 2(b)], the pump-probe signal is seen to capture the overall time evolution of electronic population probability. Pump-probe experiments thus have the potential to directly monitor electronic populations and thus non-Born-Oppenheimer dynamics in real time. ... 

Similar experimental results were found for a variety of molecules. Furthermore, in these more complex systems, the coupling of the nuclear and the electronic degrees of freedom gives rise to new physical phenomena. As illustrative examples of such phenomena, we refer to the so-called ionization induced Coulomb explosion [71], and the production of even harmonics as a consequence of beyond-Born-Oppenheimer dynamics [72]. 

Molecular dynamics methods based on quantum chemical calculations to compute the energy of a cluster as a function of the nuclei positions avoid the difficult task of building an intermolecular potential. Tight-binding molecular dynamics, semiempirical and Hartree-Fock ab initio Born-Oppenheimer dynamics, Car-Parrinello DFT molecular dynamics, and ADMP molecular dynamics are becoming more and more popular. The more sophisticated techniques are still Hmited to the study of clusters over short simulation times (ps time scale). The tight-binding approach SCC-DFTB is cheap and appears to be quite accurate. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

In the Car-Parrinello method [6] (and see, e.g., [24, 25, 16, 4]), the adiabatic time-dependent Born-Oppenheimer model is approximated by a fictitious Newtonian dynamics in which the electrons, represented by a set of... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

With the characterized mechanism, the next key question is the origin of its catalytic power. A prerequisite for this investigation is to reliably compute free energy barriers for both enzyme and solution reactions. By employing on-the-fly Born-Oppenheimer molecular dynamics simulations with the ab initio QM/MM approach and the umbrella sampling method, we have determined free energy profiles for the methyl-transfer reaction catalyzed by the histone lysine methyltransferase SET7/9... 

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