Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

The time evolution of molecular processes in the END formalism employs a Cartesian laboratory frame of coordinates. This means that in addition to the internal dynamics overall translation and rotation of the molecular system are treated. The six extra degrees of freedom add work, but become less of a burden as the complexity of the system grows. The advantage is that the kinetic energy terms are simple and no mass polarization terms need to be discarded. Furthermore, the complications of having to choose different internal coordinates for product channels exhibiting different fragmentations are not present. One can treat all product channels on an equal footing in the same laboratory frame. Since the fundamental invariance laws with respect to overall translation and rotation are satisfied within END [4] it is straightforward to extract the internal dynamics at any time in the evolution.

Better END approximations are defined by the introduction of more general molecular wave functions leading to larger and more involved parameter spaces.

B.Free Electrons

In this context, it is interesting to explore the possibilities of the END theory to describe molecular processes that involve free electrons either as reagents or as products. Electron-molecule scattering or ionization processes in molecular collisions are commonly treated separately from general molecular reaction dynamics. The principal idea in extending END to include free electron capabilities is to center electronic basis functions on independent positions in space. This means that such basis centers, so-called free centers, move on their own and are not associated with nuclear positions, however, the positions and conjugate momenta of these free centers are dynamical variables, which evolve according to the appropriate Euler–Lagrange equations.

The electronic basis for the free centers is similar to that in Eq. (24) and more precisely can be written as

with x ¼ ðx; y; zÞ an electron coordinate, r the center coordinate, and p the average electronic momentum. We can add such electronic orbitals to the minimal END wave function considered in Section III.A. The electronic basis centered on the atomic nuclei are standard basis functions ui suitable for the particular element, while on the free centers the basis is the union wi ¼ ui [ vi, where vi is a set of diffuse functions. In order to create an initial state for an

ionizing atomic or molecular collision, one performs an self-consistent field

P

(SCF) calculation in the bound state basis ui to obtain the orbitals fi ¼ k ukcki. The initial state component on a free center is then constructed using the

In an electron scattering or recombination process, the free center of the incoming electron has the functions wi ¼ ui [ vi and the initial state of the free electron is some function vi the width of which is chosen on the basis of the electron momentum and the time it takes the electron to arrive at the target. Such choice is important in order to avoid nonphysical behavior due to the natural spreading of the wavepacket.

In a completely general and flexible application of END one may choose to include some number, say Nion, of nuclei described as in Eq. (20) completely void of electronic basis functions, and some number (NA) of nuclei with electronic basis functions, as well as some number (NF) of free centers.

C.General Electron Nuclear Dynamics

When constructing more general molecular wave functions there are several concepts that need to be defined. The concept of geometry is introduced to mean a (time-dependent) point in the generalized phase space for the total number of centers used to describe the END wave function. The notations R and P are used for the position and conjugate momenta vectors, such that

These notations are used for positions and momenta, when the nuclei are treated as classical particles and denote average positions and momenta when they are treated quantum mechanically.

Another concept is that of electronic structure, which is defined as an electronic wave function associated with a geometry. For the case that the electrons are described by a single determinantal wave function it would be meaningful to consider multiple different electronic structures associated with the same geometry. In general, it would also be meaningful to consider multiple geometries, each evolving with its own electronic structure. The reason for this particular definition of electronic structure is that it would not be meaningful to consider multiple geometries with a single electronic structure, since the BO approximation provides a very good description. In Table I, we list the three possible combinations of geometry and electronic structure.

The wave function for the electronic structure can in principle be any of the constructions employed in electronic structure theory. The preferred choice in this context is a wave functions that can be classified as single and multiconfigurational, and for the latter type only complete active space (CAS) wave

a single electronic structure (SES), multiple electronic structure (MES), single geometry (SG), and multiple geometry (MG).

The symbols X and x denote the quantum mechanical coordinates of the nuclei and electrons, respectively. The index m runs over electronic structures and g over geometries.

functions are really useful. The reason for this is that such constructions have a well-established coherent (or vector coherent) state description, so that the parameters define a well-behaved phase space for a dynamical Hamiltonian system. Because in the END formulation of molecular dynamics the wave function parameters are the dynamical variables it is essential that they are nonredundant and continuous.

The Thouless determinantal electronic wave function jzi ¼ det wiðxjÞ in Eq. (21) is an example of such proper parametrization. The dynamical spin orbitals are expressed in terms of atomic spin orbitals centered on the various nuclei

X

wi ¼ ui þ ujzji ð36Þ j

with time-dependent complex coefficients zji being the dynamical variables. This parameterization guarantees that all possible determinantal wave functions in terms of the atomic orbitals are accessible during imposed dynamical changes of the system. Numerical stability is ensured as long as the z coefficients are small in comparison to unity. This can be assured by the capability to switch from one local parameterization or chart that during the dynamics may have led to large parameter values and therefore numerically unstable equations, to another chart more suitable for that part of the dynamics. Such change of charts must be possible without introduction of any artificial discontinuities in trajectories and various calculated properties.

We consider the example of a particular trajectory of the Hþ þ H2ð0; 0Þ ! H2ðv; jÞ þ H at an energy of 1.2 eV in the center-of-mass frame. By using an atomic orbital basis and a representation of the electronic state of the system in terms of a Thouless determinant and the protons as classical particles, the leading term of the electronic state of the reactants is

which represents the state Eq. (38) when

za13 ¼ 1

za23 ¼ 0

ð43Þ

zb12 ¼ 0

zb13 ¼ 1

Although the leading term of the electronic wave function of the system is thus changed, the total wave function has not and the calculated trajectory and properties exhibit no discontinuous behavior.

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonorthogonal basis with electron translation factors.

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Furthermore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive.

IV. MOLECULAR PROCESSES

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield corresponding transition probability amplitudes. Details of the END dynamics can be depicted and crosssection cross-sections and rate coefficients calculated.

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonorthogonal spin

orbitals with electron translation factors centered on the dynamically changing nuclear positions, yield results for hyperthermal atomic and molecular reactive collisions that are usually in agreement with available experimental data. It is interesting to ask to what extent this level of treatment applies to low energy processes. The experience gained from several applications is that some quantities that are not too sensitive to the detailed dynamics, such as integral cross-sections, can be described quite well, while other properties, notably differential cross-sections, are not. This is understandable from the fact that at thermal energies the dynamics follows closely the ground-state potential energy surface, which for minimal END is the ground-state SCF surface.

In order to make END better suited to the application of low energy events it is important to include an explicitly correlated description of the electron dynamics. Therefore multiconfigurational [25] augmentations of the minimal END are under development.

However, for molecular events involving more than one electronic state, even when they take place at low energies, minimal END direct dynamics appear to do well. Electron transfer is an example of such processes. Ion–atom collisions have been studied at a great variety of energies [27–29], ranging from a few tens of an electron volt to hundreds of kiloelectron volts, usually achieving agreement with available experimental data. Minimal END for Hþ2 þ H2 at 0.5– 4.0 eV [30] yields integral cross-sections for formation of Hþ3 and for electron transfer in good agreement with experiment.

A.Reactive Collisions

Bimolecular reactive encounters, atom–molecule, ion–molecule, and ion–atom collisions at a great variety of energies and initial states can be studied with the END theory. If we use classical nuclei this means that in addition to the initial electronic state of the system the nuclear geometries or internal states of the participating molecular species must be chosen. Several END trajectories have to be calculated, which means that for, say, gas-phase processes a sufficient number of relative orientations of the reactants must be considered so that directional averages can be obtained. Also, a range of impact parameters must be employed ranging from zero for head on collisions to such values that produce nonreactive trajectories. This simply corresponds to studying the processes for a range of total angular momenta.

The general problem of molecular reactive scattering can be studied with the machinery of formal time-dependent (or time-independent) scattering theory. However, for the implementation of END theory with classical nuclei it is useful to remind ourselves of some of the concepts of classical potential scattering. The consideration of the scattering of two structureless particles interacting via a potential energy UðRÞ can suffice for reminding the reader of some of the features of classical scattering. The collision energy is E ¼ mv2=2 with m the

with J0 a Bessel function of order zero, and where q ¼ jki kf j is the momentum transferred during the collision, y is the angle between the initial wave vector of the projectile ki and final wave vector in the direction of the detector kf . The semiclassical phase shift dðbÞ is related to the deflection function through (see [31])

The END trajectories for the simultaneous dynamics of classical nuclei and quantum electrons will yield deflection functions. For collision processes with nonspherical targets and projectiles, one obtains one deflection function per orientation, which in turn yields the semiclassical phase shift and thus the scattering amplitude and the semiclassical differential cross-section

For a particular process, this expression should be multiplied with the probability for that process as determined by projection of the END evolved state cðtÞ for the system on the appropriate final state cf described within the same basis set and at the same level of approximation as the evolved state, that is, the amplitude hcf jcðtÞi at a sufficiently large time t.

It is interesting to note the similarity of the expression in Eq. (48) with the result obtained through a WKB or eikonal type of argument [37,38]. The eikonal approximation resorts to straight-line trajectories, while the END application of the Schiff approximation uses fully dynamical trajectories. Schiff [36] demonstrates that the scattering wave function obtained through his procedure of summing the Born series contains an additional term, which is essential for the correct treatment of the scattering and is not present in the eikonal or WKB approaches to the problem. This formula of the scattering amplitude [Eq. (48)] is also considered to be in principle valid for all scattering angles (see [38], p. 604).

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi’s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography.

We have found that display of nuclear trajectories and the simultaneous evolution of charge distributions to yield insightful details of complicated processes. Such descriptions also map more readily to the actual experimental conditions than do the more conventional time-independent scattering matrix descriptions.

As an illustration of how results from time-dependent treatments of reactive molecular collisions can be represented, we present some recent results [61] on the D2 þ NHþ3 reaction at energies from 6 to 16 eV in the center-of-mass frame. Recent molecular beam experiments have been carried out on this system in the group of Zare [39–41] at energies from 1 to 10 eV in the center of mass. These studies seek to gain insight into the mechanisms of the reaction by considering several different initial conditions with varying amounts of energy in translational and vibrational degrees of freedom of the reactants. At these energies the two main mechanisms are the abstraction

and the competing exchange reaction

In applying minimal END to processes such as these, one finds that different initial conditions lead to different product channels. In Figure 1, we show a somewhat truncated time lapse picture of a typical trajectory that leads to abstraction. In this rendering, one of the hydrogens of NH3Dþ is hidden. As an example of properties whose evolution can be depicted we display interatomic distances and atomic electronic charges. Obviously, one can similarly study the time dependence of various other properties during the reactive encounter.

At low energies the abstraction process dominates and at higher energies the exchange mechanism becomes more important. The cross-sections for the two processes crossing at 10 eV. The END calculations yield absolute crosssections that show the same trend as the experimentally determined relative cross-sections for the two processes. The theory predicts that a substantial fraction of the abstraction product NH3Dþ, which are excited above the dissociation threshold for an N H bond actually dissociates to NH2Dþ þ H or NHþ3 during the almost 50-ms travel from the collision chamber to the detector, and thus affects the measured relative cross-sections of the two processes.

One can note some interesting features from these trajectories. For example, the Mulliken population on the participating atoms in Figure 1 show that the departing deuterium carries a full electron. Also, the deuterium transferred to the NHþ3 undergoes an initial substantial bond stretch with the up spin and down spin populations separating so that the system temporarily looks like a biradical before it settles into a normal closed-shell behavior.