Reactive Scattering Dynamics on Conically Intersecting Potential Energy Surfaces: The H + H2 Exchange Reaction

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

Electronic nonadiabatic effects in the state-to-state dynamics of the H + H2 → H2 + H exchange reaction with a vibrationally excited reagent

Out of the many major breakthroughs that the hydrogen-exchange reaction has led to, electronic nonadiabatic effects that are mainly due to the geometric phase has intrigued many. In this work we investigate such effects in the state-to-state dynamics of the H + H2 (v = 3, 4, j = 0) → H2 (v', j') + H reaction with a vibrationally excited reagent at energies corresponding to thermal conditions. The dynamical calculations are performed by a time-dependent quantum mechanical method both on the lower adiabatic potential energy surface (PES) and also using a two-states coupled diabatic theoretical model to explicitly include all the nonadiabatic couplings present in the 1E' ground electronic manifold of the H3 system. The nonadiabatic couplings are considered here up to the quadratic term; however, the effect of the latter on the reaction dynamics is found to be very small. Adiabatic population analysis showed a minimal participation of the upper adiabatic surface even for the vibrationally excited reagent. A strong nonadiabatic effect appears in the state-to-state reaction probabilities and differential cross sections (DCSs). This effect is manifested as "out-of-phase" oscillations in the DCSs between the results of the uncoupled and coupled surface situations. The oscillations persist as a function of both scattering angle and collision energy in both the backward and forward scattering regions. The origins of these oscillations are examined in detail. The oscillations that appear in the forward direction are found to be different from those due to glory scattering, where the latter showed a negligibly small nonadiabatic effect. The nonadiabatic effects are reduced to a large extent when summed over all product quantum states, in addition to the cancellation due to integration over the scattering angle and partial wave summation.

Time-dependent quantum mechanical wave packet dynamics

Starting from a model study of the collinear (H, H₂) exchange reaction in 1959, the time-dependent quantum mechanical wave packet (TDQMWP) method has come a long way in dealing with systems as large as Cl + CH₄. The fast Fourier transform method for evaluating the second order spatial derivative of the wave function and split-operator method or Chebyshev polynomial expansion for determining the time evolution of the wave function for the system have made the approach highly accurate from a practical point of view. The TDQMWP methodology has been able to predict state-to-state differential and integral reaction cross sections accurately, in agreement with available experimental results for three dimensional (H, H₂) collisions, and identify reactive scattering resonances too. It has become a practical computational tool in predicting the observables for many A + BC exchange reactions in three dimensions and a number of larger systems. It is equally amenable to determining the bound and quasi-bound states for a variety of molecular systems. Just as it is able to deal with dissociative processes (without involving basis set expansion), it is able to deal with multi-mode nonadiabatic dynamics in multiple electronic states with equal ease. We present an overview of the method and its strength and limitations, citing examples largely from our own research groups.

Effect of Reagent Vibration and Rotation on the State-to-State Dynamics of the Hydrogen Exchange Reaction, H + H2 → H2 + H

State-to-state dynamics of the benchmark hydrogen exchange reaction H + H₂ (v = 0-4, j = 0-3) → H₂ (v', j') + H is investigated with the aid of the real wave packet approach of Gray and Balint-Kurti (J. Chem. Phys. 1998, 108, 950-962) and electronic ground BKMP2 potential energy surface of Boothroyd et al. (J. Chem. Phys. 1996, 104, 7139-7152). Initial state-selected and product state-resolved reaction probabilities, integral cross section, and product diatom vibrational and rotational level populations at a few collision energies are reported to elucidate the energy disposal mechanism. State-specific thermal rate constants are also calculated and compared with the available literature results. Coriolis coupling terms of the nuclear Hamiltonian are included, and calculations are parallelized over the helicity quantum number, Ω'. Attempts are made, in particular, to study the effect of reagent vibrational and rotational excitations on the dynamical attributes. It is found that the calculations become computationally expensive with reagent vibrational and rotational excitation. Reagent vibrational excitation is found to enhance the reactivity and has significant impact on the energy disposal to the vibrational and rotational degrees of freedom of the product. The interplay of reagent translational and vibrational energy on the product vibrational distribution unfolds an important aspect of the energy disposal mechanism. The effect of reagent rotation on the state-to-state dynamics is found not to be very significant, and the weak effect turns out to be specific to v'.

Observation of the geometric phase effect in the H + HD → H2+ D reaction

Theory has established the importance of geometric phase (GP) effects in the adiabatic dynamics of molecular systems with a conical intersection connecting the ground- and excited-state potential energy surfaces, but direct observation of their manifestation in chemical reactions remains a major challenge. Here, we report a high-resolution crossed molecular beams study of the H + HD → H2+ D reaction at a collision energy slightly above the conical intersection. Velocity map ion imaging revealed fast angular oscillations in product quantum state–resolved differential cross sections in the forward scattering direction for H2products at specific rovibrational levels. The experimental results agree with adiabatic quantum dynamical calculations only when the GP effect is included.

Assessment of Real-Time Time-Dependent Density Functional Theory (RT-TDDFT) in Radiation Chemistry: Ionized Water Dimer

Ionization in the condensed phase and molecular clusters leads to a complicated chain of processes with coupled electron-nuclear dynamics. It is difficult to describe such dynamics with conventional nonadiabatic molecular dynamics schemes since the number of states swiftly increases as the molecular system grows. It is therefore attractive to use a direct electron and nuclear propagation such as the real-time time-dependent density functional theory (RT-TDDFT). Here we report a RT-TDDFT benchmark study on simulations of singly and doubly ionized states of a water monomer and dimer as a prototype for more complex processes in a condensed phase. We employed the RT-TDDFT based Ehrenfest molecular dynamics with a generalized gradient approximate (GGA) functional and compared it with wave-function-based surface hopping (SH) simulations. We found that the initial dynamics of a singly HOMO ionized water dimer is similar for both the RT-TDDFT/GGA and the SH simulations but leads to completely different reaction channels on a longer time scale. This failure is attributed to the self-interaction error in the GGA functionals and it can be avoided by using hybrid functionals with large fraction of exact exchange (represented here by the BHandHLYP functional). The simulations of doubly ionized states are reasonably described already at the GGA level. This suggests that the RT-TDDFT/GGA method could describe processes following the autoionization processes such as Auger emission, while its applicability to more complex processes such as intermolecular Coulombic decay remains limited.

Single and double ionization of the camphor molecule excited around the C 1s edge

RATIONALE

An interesting class of volatile compounds, the monoterpenes, is present in some plants although their functions are not yet fully understood. We have studied the interaction of the camphor molecule with monochromatic high-energy photons (synchrotron radiation) using time-of-flight mass spectrometry and coincidence techniques.

METHODS

A commercial sample of S-camphor was admitted into the vacuum chamber, without purification, through an inlet system. Monochromatic light with energy around the C 1s edge was generated by the TGM beamline at the Brazilian Synchrotron Facility. A Wiley-McLaren mass spectrometer was used to characterize and detect the ions formed by the camphor photoionization. The data analysis was supported by energy calculations.

RESULTS

Although the fragmentation patterns were basically the same at 270 eV and 330 eV, it was observed that above the C 1s edge the contribution to the spectrum from lower mass/charge fragment ions increased, pointing to a higher degree of dissociation of the molecule. Projections of the PEPIPICO spectra demonstrated the existence of unstable doubly charged species. The Gibbs free energy was calculated using the Møller-Plesset perturbation theory (MP2) for the neutral, singly and doubly excited camphor molecule.

CONCLUSIONS

Our PEPIPICO spectrum clearly demonstrated the formation of doubly ionic dissociative species. From a slope analysis, we propose a secondary decay after a deferred charge separation mechanism in which, after a few steps, the camphor dication dissociates into C2 H3 (+) and C3 H5 (+) . This is the main relaxation route observed at 270 eV and 330 eV. The large energy difference between the mono and the dication (of the order of 258.2 kcal/mol) may explain the experimentally observed absence of stable dications in the spectra, because their formation is disadvantaged energetically.

The diabatic picture of electron transfer, reaction barriers, and molecular dynamics

Diabatic states have a long history in chemistry, beginning with early valence bond pictures of molecular bonding and extending through the construction of model potential energy surfaces to the modern proliferation of methods for computing these elusive states. In this review, we summarize the basic principles that define the diabatic basis and demonstrate how they can be applied in the specific context of constrained density functional theory. Using illustrative examples from electron transfer and chemical reactions, we show how the diabatic picture can be used to extract qualitative insight and quantitative predictions about energy landscapes. The review closes with a brief summary of the challenges and prospects for the further application of diabatic states in chemistry.

Excited electronic states and nonadiabatic effects in contemporary chemical dynamics

The question of how to describe the crossing of molecular electronic states is one of the most challenging issues in contemporary chemical dynamics. In recent years, the fundamental concept of conical intersections (CIs) of electronic potential energy surfaces (PESs) has emerged, which allows extremely fast and efficient switching of a molecule between its excited and ground electronic states. CIs are ubiquitous in polyatomic molecules. Because they generically allow the crossings of the Born-Oppenheimer (BO) adiabatic PESs, they have become the crucial mechanistic elements of the rapidly growing area of nonadiabatic chemistry. The most critical consequence of CIs is a complete breakdown of the adiabatic BO approximation. That means that the reorganization of fast-moving electrons and nuclear vibrations must be treated concurrently. Ideally, the theoretical description should be quantum mechanical in this situation. However, because of the complexity, the necessary approximations often make it difficult to conclusively predict dynamic behavior of large polyatomic molecules. In addition, a nonunique diabatic electronic representation (describing coupling between states in the electronic Hamiltonian) is essential to avoid the singular nature of the nuclear kinetic coupling terms of the unique adiabatic electronic representation. This Account describes both the challenges and some recent advances in quantum mechanical studies of nonadiabatic molecular processes, highlighting results from our recent work examining the static aspects of CIs and their dynamical consequences. The spectroscopic implications of the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) intersections in complex molecular systems are discussed. Our work probes the underlying details of complex vibronic spectra of systems of growing sizes in terms of both electronic and nuclear degrees of freedom. The necessity of extension of the theoretical treatment beyond a linear vibronic coupling approach is addressed. Our results establish highly overlapping band structures due to JT and PJT CIs, a bimodal distribution of spectral intensity that originates from strong JT coupling, and the role of intermode bilinear coupling in the progressions of vibronic bands. Investigations of the quantum dynamics of the prototypical naphthalene radical cation were aimed at understanding its photostability, lack of fluorescence emissions, and diffuse interstellar bands. This work established extremely fast relaxation of this radical cation through CIs. Simulations of the interplay of electronic and relativistic spin-orbit coupling in the photodetachment spectroscopy of ClH(2)(-), in conjunction with experimental data, support the existence of a shallow van der Waals well in the reactive Cl + H(2) PES. These results also reveal a quenching of electronic coupling by the relatively strong spin-orbit coupling. In addition, we studied the dynamics of the prototypical H + H(2) reaction from a new perspective by explicitly including the coupling between the two energy surfaces of its JT split degenerate ground electronic state. Although individual reaction probabilities show partial sensitivity to nonadiabatic effects, the theoretical results reveal that they are not important for the dynamical outputs such as integral reaction cross sections and thermal rate constants.

Multidimensional Quantum Dynamical Study of β-Hydrogen Transfer in a Cationic Rhodium Complex

The dynamics of migratory insertion and beta-hydrogen elimination in the cationic complex [CpRh(PH3)H(C2H4)]+ is studied from a quantal point of view. On the basis of DFT results for the relevant stationary points of the potential energy surface, three coordinates are identified that vary strongly during the reaction. A suitable three-dimensional grid, along with an appropriate kinetic energy operator, are constructed that are employed in the subsequent wave packet propagations. The latter are performed in the spirit of transition state spectroscopy and start from the various saddle points of the potential energy surface. Vibrational periods and lifetimes for these elementary processes, relevant to homogeneous catalysis, are obtained in this way for the first time. This work is considered to provide the basis for a subsequent treatment of equilibrium rate constants and to shed new light on the electronic factors governing these prototypical reaction steps.

Effect of the geometric phase on the dynamics of the hydrogen-exchange reaction

A recent puzzle in nonadiabatic quantum dynamics is that geometric phase (GP) effects are present in the state-to-state opacity functions of the hydrogen-exchange reaction, but cancel out in the state-to-state integral cross sections (ICSs). Here the authors explain this result by using topology to separate the scattering amplitudes into contributions from Feynman paths that loop in opposite senses around the conical intersection. The clockwise-looping paths pass over one transition state (1-TS) and scatter into positive deflection angles; the counterclockwise-looping paths pass over two transition states (2-TS) and scatter into negative deflection angles. The interference between the 1-TS and 2-TS paths thus integrates to a very small value, which cancels the GP effects in the ICS. Quasiclassical trajectory (QCT) calculations reproduce the scattering of the 1-TS and 2-TS paths into positive and negative deflection angles and show that the 2-TS paths describe a direct insertion mechanism. The inserting atom follows a highly constrained "S-bend" path, which allows it to avoid both the other atoms and the conical intersection and forces the product diatom to scatter into high rotational states. By contrast, the quantum 2-TS paths scatter into a mainly statistical distribution of rotational states, so that the quantum 2-TS total ICS is roughly twice the QCT ICS at 2.3 eV total energy. This suggests that the S-bend constraint is relaxed by tunneling in the quantum system. These findings on H+H(2) suggest that similar cancellations or reductions in GP effects are likely in many other reactions.

General explanation of geometric phase effects in reactive systems: Unwinding the nuclear wave function using simple topology

We describe a simple topological approach which was used recently to explain geometric phase (GP) effects in the hydrogen-exchange reaction [Juanes-Marcos et al., Science 309, 1227 (2005)]. The approach is general and applies to any reactive system in which the nuclear wave function encircles a conical intersection (CI) and is confined to one adiabatic surface. The only numerical work required is to add and subtract nuclear wave functions computed with normal and GP boundary conditions. This is equivalent to unwinding the nuclear wave function onto a double cover space, which separates out two components whose relative sign is changed by the GP. By referring to earlier work on the Aharanov-Bohm effect, we show that these two components contain all the Feynman paths that follow, respectively, an even and an odd number of loops around the CI. These two classes of path are essentially decoupled in the Feynman sum, because they belong to different homotopy classes (meaning that they cannot be continuously deformed into one another). Care must be taken in classifying the two types of path when the system can enter the encirclement region from several different start points. This applies to bimolecular reactions with identical reagents and products, for which our approach allows a symmetry argument developed by Mead [J. Chem. Phys. 72, 3839 (1980)] to be generalized from nonencircling to encircling systems. The approach can be extended in order to unwind the wave function completely onto a higher cover space, thus separating contributions from individual winding numbers. The scattering boundary conditions are ultimately what allow the wave function to be unwound from the CI, and hence a bound state wave function cannot be unwound. The GP therefore has a much stronger effect on the latter than on the wave function of a reactive system.

Interpolation of diabatic potential-energy surfaces: Quantum dynamics on ab initio surfaces

A method for constructing diabatic potential-energy matrices from ab initio quantum chemistry data is described and tested for use in exact quantum reactive scattering. The method is a refinement of that presented in a previous paper, in that it accounts for the presence of the nonremovable derivative coupling. The accuracy of quantum dynamics on this type of diabatic potential is tested by comparison with an analytic model and for an ab initio description of the two lowest-energy states of H3.

On the (E⊗e)-Jahn-Teller conical intersections in the 3p(E′) and 3d(E″) Rydberg electronic states of triatomic hydrogen

The static and dynamic aspects of the Jahn-Teller (JT) interactions in the 3p(E') and 3d(E") Rydberg electronic states of H3 are analyzed theoretically. The static aspects are discussed based on recent ab initio quantum chemistry results, and the dynamic aspects are examined in terms of the vibronic spectra and nonradiative decay behavior of these states. The adiabatic potential-energy surfaces of these degenerate electronic states are derived from extensive ab initio calculations. The calculated adiabatic potential-energy surfaces are diabatized following our earlier study on this system in its 2p(E') ground electronic state. The nuclear dynamics on the resulting conically intersecting manifold of electronic states is studied by a time-dependent wave-packet approach. Calculations are performed both for the uncoupled and coupled state situations in order to understand the importance of nonadiabatic interactions due to the JT conical intersections in these excited Rydberg electronic states.

Theoretical study of geometric phase effects in the hydrogen-exchange reaction

The crossing of two electronic potential surfaces (a conical intersection) should result in geometric phase effects even for molecular processes confined to the lower surface. However, recent quantum simulations of the hydrogen exchange reaction (H + H2 --> H2 + H) have predicted a cancellation in such effects when product distributions are integrated over all scattering angles. We used a simple topological argument to extract reaction paths with different senses from a nuclear wave function that encircles a conical intersection. In the hydrogen-exchange reaction, these senses correspond to paths that cross one or two transition states. These two sets of paths scatter their products into different regions of space, which causes the cancellation in geometric phase effects. The analysis should generalize to other direct reactions.

Geometric phase effects in the H+H2 reaction: Quantum wave-packet calculations of integral and differential cross sections

We report quantum wave-packet calculations on the H+H(2) reaction, aimed at resolving the controversy over whether geometric phase (GP) effects can be observed in this reaction. Two sets of calculations are reported of the state-to-state reaction probabilities, and integral and differential cross sections (ICSs and DCSs). One set includes the GP using the vector potential approach of Mead and Truhlar; the other set neglects the phase. We obtain unequivocal agreement with recent results of Kendrick [J. Phys. Chem. A 107, 6739 (2003)], predicting GP effects in the state-to-state reaction probabilities, which cancel exactly on summing the partial waves to yield the ICS. Our results therefore contradict those of Kuppermann and Wu [Chem. Phys. Lett. 349 537 (2001)], which predicted pronounced GP effects in the cross sections. We also agree with Kendrick in predicting that there are no significant GP effects in the full DCS at energies below 1.8 eV, and in the partial (0<or=J<or=10) DCS at energies above this. However, we find that in the full DCS above 1.8 eV (which was not reported by Kendrick), there are GP effects, which may be experimentally measurable.

Conical intersections and semiclassical trajectories: Comparison to accurate quantum dynamics and analyses of the trajectories

Semiclassical trajectory methods are tested for electronically nonadiabatic systems with conical intersections. Five triatomic model systems are presented, and each system features two electronic states that intersect via a seam of conical intersections (CIs). Fully converged, full-dimensional quantum mechanical scattering calculations are carried out for all five systems at energies that allow for electronic de-excitation via the seam of CIs. Several semiclassical trajectory methods are tested against the accurate quantum mechanical results. For four of the five model systems, the diabatic representation is the preferred (most accurate) representation for semiclassical trajectories, as correctly predicted by the Calaveras County criterion. Four surface hopping methods are tested and have overall relative errors of 40%-60%. The semiclassical Ehrenfest method has an overall error of 66%, and the self-consistent decay of mixing (SCDM) and coherent switches with decay of mixing (CSDM) methods are the most accurate methods overall with relative errors of approximately 32%. Furthermore, the CSDM method is less representation dependent than both the SCDM and the surface hopping methods, making it the preferred semiclassical trajectory method. Finally, the behavior of semiclassical trajectories near conical intersections is discussed.