Separability of motions

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

Calculations for H + H2 at energies below threshold for excitation of the first vibrational level were compared with two approximations which assumed separability of motions along the reaction path and transversal to it. One approximation assumed conservation of transversal vibrational energy, and the second one conservation of transversal vibrational quantum number, i.e. adiabatic motion. The second assumption was closer to the exact results both in magnitude and in the shape of the reaction threshold region. However, as collision energies were increased and other vibrational states of products became excitable, the behaviour became more nearly statistical, in that probabilities of reactions tended to decrease and become comparable. 

The above dynamic separation of motions reduces to the equations... 

In the above considerations, the collision problem is simplified by using either a rigorous dynamic or an approximate adiabatic procedure for a separation of the internal from the external (overall translation and rotation) motions. Both approaches are also applicable under certain conditions to the separation of some internal motions and,in particular, to the separation of motion along the reaction coordinate from the non-reactive modes of motion. 

The adiabatic and non-adiabatic separation of motions will be considered side by side throughout the following chapters of this book. 

If accurate tunneling corrections are required, the situation rapidly becomes more complicated. Issues related to separability of motion along the reaction coordinate, curvature of the reaction coordinate, and multidimensional tunneling arise and must be dealt with. Marcus and Coltrin [51] found that reaction path curvature forces the reaction to cut... 

In contrast to the corresponding coupled equations bijrij = 0 in mass-weighted coordinates, Eq. 6.49 shows that each normal coordinate Q,- oscillates independently with motion which is uncoupled to that in other normal coordinates Qj. This separation of motion into noninteracting normal coordinates is possible only if V contains no cubic or higher-order terms in Eq. 6.4. Anharmonicity will inevitably couple motion between different vibrational modes, and then the concept of normal modes will break down. In the normal mode approximation, no vibrational energy redistribution can take place in an isoFated molecule. 

The Other reactive resonance is called the trapped-state resonance or Feshbach resonance, shown in Fig. 4.1c. In this case, the ABC complex is dynamically trapped along the reaction coordinate, even the minimum energy path on the BO PES is totally repulsive. The trapping of the short-lived ABC complex is caused by the vibrationally adiabatic potential, which is based on the concept of vibrational adiabaticity [23, 75, 76, 120]. As the vibrational motions along the directions perpendicular to R are fast compared with the motion along R, the vibrational modes should approximately conserve the quantum number n, which is in the spirit of BO separation of motions with different time scale. A typical vibrationally adiabatic potential along the reaction coordinate R is shown in Fig. 4.2b (left), and it can be constructed as... 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Le Bihan D, Breton E, Lallemand D, Aubin M-L, Vignaud J and Laval-Jeantet M 1988 Separation of diffusion and perfusion in intravoxel inooherent motion MR imaging 1988 Radiology 168 497-505... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Consider a polyatomic system consisting of N nuclei (where > 3) and elecbons. In the absence of any external fields, we can rigorously separate the motion of the center of mass G of the whole system as its potential energy function V is independent of the position vector of G (rg) in a laboratory-fixed frame with origin O. This separation introduces, besides rg, the Jacobi vectors R = (R , , R , .. , Rxk -1) = (fi I "21 I fvji) fot nuclei and electrons,... 

In this representation, Newton s equations of motion separate to 3N — 6 equations... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The starting point for the theory of molecular dynamics, and indeed the basis for most of theoretical chemistry, is the separation of the nuclear and electionic motion. In the standard, adiabatic, picture this leads to the concept of nuclei moving over PES corresponding to the electronic states of a system. 

The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

---