Diabatic Representation Research Articles

Theoretical studies of intersystem crossing effects in the O(3P, 1D)+H2 reaction

We have studied the influence of intersystem crossing on the reaction dynamics of the O+H2 reaction by performing trajectory surface hopping (TSH) calculations with accurate potential-energy surfaces and global spin–orbit coupling surfaces that we have generated using a four state model proposed by Hoffmann and Schatz. In the TSH calculations, we develop a new mixed representation that treats the reactant and product asymptotes in the adiabatic representation, and the singlet–triplet crossing region in the diabatic representation. This representation thus correctly describes O and OH fine structure-resolved cross sections, and it also treats intersystem crossing effects arising from the singlet–triplet crossing. Our calculations are based on the 1 3A′ and 1 3A″ states of Walch and Kuppermann, and the 1 1A′ state of Dobbyn and Knowles. The globally determined spin–orbit coupling matrix is derived from complete active space self-consistent field calculations using the two-electron Breit–Pauli Hamiltonian. Our dynamics calculations show that the triplet O+H2 cross section is modestly increased (up to 20% at collision energies >10 kcal/mol above the reactive threshold) by intersystem crossing, and product rotational excitation is also increased. In addition, we find that the OH spin–orbit distributions favor the Π3/22 state by a 2:1 ratio over Π1/22. This result is consistent with observations for O atom reactions with alkanes.

Transition state spectroscopy of the excited electronic states of Li–HF

In this work the LiHF(A,B,B′←X) electronic spectrum is simulated and compared with the experimental one obtained by Hudson et al. [J. Chem. Phys. 113, 9897 (2000)]. High level ab initio calculations of three A′2 and one A″2 electronic states have been performed using a new atomic basis set and for a large number of nuclear configurations (about 6000). Four analytic global potential energy surfaces have been fitted. The spectrum involved very excited rovibrational states, close to the first dissociation limit, at high total angular momentum. Two different methods have been used, one based on bound state and the second one on wave packet calculations. Different alternatives have been used to simulate the relatively high temperatures involved. The agreement obtained with the experimental spectrum is very good allowing a very simple assignment of the peaks. They are due to bending progressions on the three excited electronic states. A simple model is used in which only rotational degrees of freedom are included, which simulates the spectrum in excellent agreement with the experimental one, providing a nice physical interpretation. Moreover, the remaining theoretical/experimental discrepancies have been attributed to nonadiabatic effects through the extension of this model to a diabatic representation of excited coupled electronic states.

Split diabatic representation

A split diabatic representation is proposed as a technique for generating diabatic potential curves with the maximal physical content and favorable computational characteristics. This method is a mixed adiabatic-diabatic representation, in which smoothly varying couplings appear in both the kinetic- and potential-energy matrices. It requires the solution of the first-order differential equation for the transformation matrix of the standard strict diabatic representation for which an efficient numerical scheme is also presented. A transformation propagator, akin to the Cayley-Hamiltonian time evolution operator, is employed to obtain the diabatic states while preserving unitarity. Several examples illustrate the advantages of the proposed split diabatic representation.

Multidimensional Hydrogen Bond Dynamics in Salicylaldimine: Coherent Nuclear Wave Packet Motion versus Intramolecular Vibrational Energy Redistribution

The multidimensional wave packet dynamics of salicylaldimine following ultrafast infrared laser excitation of the OH-bending and -stretching vibrations is discussed. A seven-dimensional Cartesian reaction surface Hamiltonian in the diabatic representation is employed comprising the five skeleton normal modes which have the greatest influence on the in-plane hydrogen motion. For the numerical solution of the time-dependent Schrodinger equation the multiconfiguration time-dependent Hartree method is used. Coherent wave packet dynamics is observed after excitation of the OH-bending fundamental transition, whereas stiong anharmonic mode couplings to the stretching transition cause intramolecular vibrational energy redistribution on a time scale of about 700 fs.

Photoinduced dynamics of the valence states of ethene: A six-dimensional potential-energy surface of three electronic states with several conical intersections

A six-dimensional analytic potential-energy surface of the three valence states (N, V, Z) of ethene has been constructed on the basis of complete-active-space ab initio calculations and ab initio calculations with perturbation theory of second order based on a complete active reference space. The nuclear coordinate space is spanned by the torsion, the C–C stretch coordinate, the left and right pyramidalization and the symmetric and antisymmetric scissor coordinates. The C–H stretch coordinates and the CH2 rocking angles are kept frozen at their ground-state equilibrium value. A diabatic representation of the valence states of ethene has been constructed within the framework of a Hückel-type model. The diabatic potential-energy elements are represented as analytic functions of the relevant coordinates. The parameters of the analytic functions have been determined by a least-squares fit of the eigenvalues of the diabatic potential-energy matrix to the ab initio data for one-dimensional and two-dimensional cuts of the six-dimensional surface. As a function of the torsion, the analytic potential-energy surface describes the intersections of the V and Z states for torsional angles near 90°, which are converted into conical intersections by the antisymmetric scissor mode. As a function of pyramidalization of perpendicular ethene, it describes the intersections of the diabatic N and Z states, which are converted into conical intersections by displacements in the torsional mode. The analytic potential-energy surfaces can provide the basis for a quantum wave packet description of the internal conversion of photoexcited ethene to the electronic ground state via conical intersections.

Unified Semiclassical Theory for the Nonadiabatic Transition

AbstractUnified semiclassical theory for general two‐state nonadiabatic transition is discussed by demonstrating an important quantity that represents a type of nonadiabatic transition. This quantity is essentially defined by the rotation angle which determines transformation between diabatic and adiabatic representation, and it runs from unity to infinity continuously. The unified semiclassical formula is numerically demonstrated, especially in the high energy range where the transition probability is sensitive to the type of nonadiabatic transition. The present semiclassical theory can be incorporated with the framework of the Tully's fewest‐switches trajectory surface hopping method to deal with nonadiabatic transitions in multi‐dimensional systems.

Construction of electronic diabatic states within a molecular orbital scheme

A new procedure is proposed to construct a diabatic representation that is readily implemented in the molecular orbital-self-consistent field-configuration interaction scheme. It is based on the calculation of adiabatic wave functions at a reference geometry Q0 and of the appropriate modifications to be made to molecular orbitals for Q≠Q0 in order to force the derivative couplings for all electronic states to be exactly zero in the space around Q0. This approach is applied to the construction of the diabatic basis and to the calculation of the associated (adiabatic) vibronic coupling for a number of well-characterized systems. The properties and the limitations of this diabatic basis are discussed.

Direct calculation of complex resonance energies for Al2 electronic predissociation

Abstract More than one hundred electronic predissociative resonances of the excited Al 2 vibrations on 1,2,3 3 Π g -coupled surfaces are accurately calculated by an algorithm which directly produces complex resonance energies for many resonances efficiently and accurately. We show that it is possible to differentiate two families of resonances originated from the 2,3 3 Π g adiabatic surfaces using the electronic state components of the diabatic representation of the pseudo-resonance wavefunctions corresponding to resonance states. The resonances on 2 3 Π g adiabatic surface show a simple oscillatory structure of lifetimes, which may be caused by the proximity between dissociative 1 3 Π g and bound 2 3 Π g surfaces, consequently, by slow oscillation of overlaps of dissociating and bound wavefunctions as energy increases.

Dissipative wave packet dynamics of the intramolecular hydrogen bond in o-phthalic acid monomethylester

We investigate the infrared laser-driven ultrafast dynamics of the carboxy-deuterated title compound in the condensed phase using a system–bath approach. The three-dimensional relevant system comprises the OD stretching and bending motion as well as a low-frequency out-of-plane twisting vibration modulating the hydrogen-bond strength. The dominant relaxation and dephasing channels are identified giving rise to three terms which contribute to the system–bath interaction: (i) A linear system-solvent coupling leading to the relaxation of the low-frequency mode. Here, a classical molecular dynamics simulation is performed to obtain the spectral density and thus the relevant relaxation time scale for this mode. (ii) A coupling which is quadratic in the system coordinates and responsible for pure dephasing. (iii) A fourth-order coupling which involves the release of vibrational energy into the environment by means of the simultaneous excitation of two intramolecular bath vibrations and a solvent mode. Using a quantum master equation approach it is demonstrated that this model is in accord with the results of recent infrared pump-probe and four-wave mixing experiments. The dynamics is discussed in terms of a diabatic representation defined with respect to the high-frequency stretching and bending modes. The behaviour of the low-frequency mode can be characterized as a dissipative wave packet motion in the diabatic ground state which contains contributions from bath-induced coherence transfer out of the initially excited diabatic state.

Semiclassical simulation of photochemical reactions in condensed phase

We compare two strategies for the semiclassical simulation of photochemical reactions in condensed phase (solutes in liquids, impurities in solid matrices, adsorbates, or photoreactive units in biological environments). Both strategies are based on classical nuclear trajectories, with surface hopping to simulate nonadiabatic transitions. In the first approach we calculate electronic energies and couplings by ab initio methods for many nuclear geometries of the reactive system (the ‘solute’), and we fit the results by analytic functions of the internal coordinates. In the case of surface crossings it is mandatory to resort to a (quasi-)diabatic representation of the electronic states. A condensed state environment (the ‘solvent’) is described by ordinary molecular mechanics (MM), and the solute–solvent interactions can be state-specific. The other strategy is direct, i.e. it involves the calculation of electronic energies and wavefunctions at each integration step of a nuclear trajectory. The method we have implemented is based on semiempirical configuration interaction calculations with floating occupation SCF orbitals; within this approach we have developed a hybrid quantum mechanics/molecular mechanics (QM/MM) procedure to represent the solvent, which is here briefly described for the first time. While the intra- and intermolecular potentials for the solvent molecules are of MM type, the QM/MM interaction is introduced in the semiempirical electronic hamiltonian, so that it influences in a state-specific way electronic energies and wavefunctions. Applications of both approaches to photoreaction dynamics are briefly described.

Direct diabatization of electronic states by the fourfold way. II. Dynamical correlation and rearrangement processes

Diabatic representation of coupled potential energy surfaces and their scalar couplings provides a compact and convenient starting point for dynamics calculations carried out in either the adiabatic or diabatic representation. In a previous paper we presented a general, path-independent scheme, called the fourfold way, for calculating diabatic surfaces and their scalar couplings from adiabatic surfaces and electronic density matrices such that the manifold of diabatic states spans the variationally optimized space of a finite number of adiabatic states. In the present paper we extend that scheme in these ways: (1) We show how to include dynamical electronic correlation energy by multireference perturbation theory or configuration interaction based on a complete active reference space. (2) We present a more general strategy for treating rearrangements. (3) We present consistency criteria for testing the validity of the assumptions for a particular choice of reference geometries, diabatic molecular orbital (DMO) ordering, dominant configuration-state-function lists, and choice(s) for reference DMO(s) for systems involving rearrangements. The first extension is illustrated by multiconfiguration quasidegenerate perturbation theory (MC-QDPT) calculations on LiF, and all three extensions are illustrated by MC-QDPT calculations on the reaction Li(2 2S,2 2P)+HF→LiF+H.

Spin−Orbit Coupling and Conical Intersections. IV. A Perturbative Determination of the Electronic Energies, Derivative Couplings, and a Rigorous Diabatic Representation near a Conical Intersection. The General Case†

Conical intersections play an essential role in electronically nonadiabatic processes. For molecules with an odd number of electrons the spin−orbit interaction produces essential changes in the topography and connectivity of points of conical intersection. In the nonrelativistic case, or when the molecule has an even number of electrons, η, the dimension of the branching space, the space in which the conical topography is evinced, is 2. By contrast, for molecules with an odd number of electrons the branching space is 5 dimensional (η = 5) in general, or 3-dimensional (η = 3) when Cs symmetry is present. Recently, we have used degenerate perturbation theory to obtain analytic representations of the energy, and of the derivative couplings, and a “rigorous” diabatic basis in the vicinity of a conical intersection for the η = 3 case. Here, we extend this analysis to the general, no symmetry, η = 5, case. The perturbative results provide valuable insights into the nature of this singular point.

Hopping and Jumping between Potential Energy Surfaces

The crossing of potential energy surfaces (and the flow of nuclear wave function amplitudes between them) becomes important in many different contexts in chemical physics. Photochemical processes often involve dynamics on surfaces that cross (or can be made to cross in an appropriate diabatic representation). Photophysical processes involving radiationless transitions between different Born-Oppenheimer potential energy surfaces are extremely common. Finally, once we “dress” an initial potential energy surface with the energy of a photon, radiative processes involving absorption or emission of a photon are also seen to involve transitions between crossing or closelying potential energy surfaces. Some years ago, Tully and Preston 1 introduced an approach for approximate treatment of dynamics at regions of close-lying potential energy surfaces. In many subsequent trials and refinements, it has proved a worthy computational tool, simple to implement and also very intuitive. The “surface hopping” method has its roots in the Landau-Zener theory 2 but goes beyond it in dealing with multiple crossings and with many degrees of freedom. Related theories include the exponential energy gap law of radiationless transitions 3 and the exponential momentum gap law of Ewing 4 for vibrational predissociation (in which a highfrequency vibrational mode plays the role of the electronic state, relative to a low-frequency van der Waals mode). In these studies, considerable effort has been devoted to developing new intuition for nonclassical Franck-Condon factors. The emphasis has been on the mode competition problem and the overall dependence of rates on the energy or quantum number gaps. Our emphasis is instead directly on the features of the potential energy surfaces, which control the Franck-Condon factors, which in turn control the rates. The fundamental idea that permits a classical treatment of surface hopping is contained in the Landau-Zener-Stuckelberg model, 2 which shows that the hops from one surface to another are localized to the region where the surfaces cross or almost cross. For weak coupling, the probability for hopping is given in terms of the overall coupling, W, between the surfaces, the difference in slopes, j¢F┴j, normal to the intersection of the two potentials at the crossing, and the speed, p┴/m, with which the trajectory goes through the crossing region:

Ab initioadiabatic and diabatic energies and dipole moments of the RbH molecule

For nearly all the states dissociating below the ionic limit [i.e., Rb $(5s,$ $5p,$ $4d,$ $6s,$ $6p,$ $5d,$ $7s,$ and $4f)+\mathrm{H}$ $(1s)]$ in ${}^{1}{\ensuremath{\Sigma}}^{+}$ and ${}^{3}{\ensuremath{\Sigma}}^{+}$ symmetries, we present an adiabatic and diabatic study. Adiabatic results are also reported for ${}^{1}\ensuremath{\Pi},$ ${}^{3}\ensuremath{\Pi},$ ${}^{1}\ensuremath{\Delta},$ and ${}^{3}\ensuremath{\Delta}$ symmetries. The ab initio calculations rely on pseudopotential, operatorial core-valence correlation, and full valence configuration-interaction approaches, combined to an efficient diabatization procedure. For the low-lying states, our vibrational level spacings and spectroscopic constants are in good agreement with the available experimental data. Diabatic potentials and dipole moments are analyzed, revealing the strong imprint of ionic state in the ${}^{1}{\ensuremath{\Sigma}}^{+}$ adiabatic states while improving the results. As for LiH, vibrational spacing of the A state is bracketed by our results, with and without the improvement, taking into account the diabatic representation. Experimental suggestions are also given.

Ab initio adiabatic and diabatic energies and dipole moments of the KH molecule

An ab initio adiabatic and diabatic study of the KH molecule is performed for all states below the ionic limit [i.e., K (4s, 4p, 5s, 3d, 5p, 4d, 6s, and 4f)+H(1s)] in 1Σ+ and 3Σ+ symmetries. Adiabatic results are also reported for 1Π, 3Π, 1Δ, and 3Δ symmetries. The ab initio calculations rely on pseudopotential, operatorial core valence correlation, and full valence CI approaches, combined to an efficient diabatization procedure. For the low-lying states, our vibrational level spacings and spectroscopic constants are in very good agreement with the available experimental data. Diabatic potentials and dipoles moments are analyzed, revealing the strong imprint of the ionic state in the 1Σ+ adiabatic states while improving the results. The undulations of the diabatic curves and of the triplet–singlet diabatic energy difference which we found positive, as in Hund’s rule, are related to the Rydberg functions. As for LiH, the vibrational spacing of the A state is bracketed by our results with and without the improvement taking into account the diabatic representation. Experimental suggestions are also given.

Spin-orbit coupling and conical intersections in molecules with an odd number of electrons. III. A perturbative determination of the electronic energies, derivative couplings and a rigorous diabatic representation near a conical intersection

When the spin–orbit interaction is included, the character of a conical intersection in a molecule with an odd number of electrons differs dramatically from that of its nonrelativistic counterpart. In contrast to the two-dimensional branching space (η=2) in the nonrelativistic case, for these conical intersections the branching space is five-dimensional (η=5) in general, or three-dimensional (η=3) when Cs symmetry is present. Recently we have introduced an algorithm, based on analytic gradient techniques, to locate such conical intersections and used related techniques to efficiently construct and study the properties of the vectors defining the branching space. Here we extend this analysis. A perturbative description of the η=3 case is reported and used to determine the energy, derivative couplings, and a “rigorous” diabatic basis in the vicinity of a conical intersection. The perturbative results are compared with those of exact numerical calculations employing model Hamiltonians. The implications for the nuclear motion problem are discussed.

A local diabatic representation of non-Born–Oppenheimer dynamics

We propose a new canonical procedure for investigating non-Born–Oppenheimer dynamics. This procedure is based on Van Vleck's theory and expressed in the normal coordinates of one of the diabatically coupled electronic surfaces. Up to intermediate energies, the successive unitary transformations convert the smooth, non-resonant part of the energy shifts due to the diabatic coupling into anharmonicities and off-diagonal couplings inside each surface. As a result, in this energy range, the two transformed electronic surfaces remain fairly decoupled. We also show that, because of the reduced coupling range, the size of the matrix needed to calculate the exact energy levels is substantially reduced, which is promising for future practical applications.

Dipole-allowed excited states of N2: Potential energy curves, vibrational analysis, and absorption intensities

The three lowest adiabatic potential energy curves for each of the two dipole-allowed symmetries, Σu+1 and Πu1, are calculated in the multireference configuration–interaction framework. Diabatic potentials and corresponding coupling elements are obtained by diagonalizing the electronic operator r2 which serves to discriminate Rydberg and valence type states. A large basis set and judiciously chosen active orbital and configuration spaces furnish smooth and reliable potential curves. However, a vibrational analysis of the coupled systems in diabatic representation still shows some disappointing deviations from the experimental interference patterns of overlapping absorption bands that are highly sensitive to potential energy differences. Starting from the calculated curves, a fitting procedure accounting also for empirical information yields potential energy curves and diabatic coupling elements that reproduce all details of the experiment very well. These recommended results also serve to identify residual defects in the ab initio curves mainly as vertical shifts. The performance of other commonly used ab initio methods for the calculation of excited states is briefly discussed.

Semiclassical treatment of charge transfer in molecule-surface scattering

We have treated the ionization probability of iodine molecules scattered from diamond by a semiclassical surface hopping scheme, namely Tully’s fewest-switches algorithm [J. Chem. Phys. 93, 1061 (1990)]. The interaction is described by a model potential that has been adjusted to empirical data. We start with a one-dimensional two-state model in which just the molecular distance from the surface and the neutral and negatively charged state of I2 are considered. We determine the ionization probability within the adiabatic and diabatic representation and compare it with exact quantum calculations. For this particular problem we find that the diabatic picture shows too little coherence, while the adiabatic representation yields satisfactory results. In the second part we have successively increased the complexity of the simulation by additionally taking a surface oscillator coordinate, the molecular rotation and vibration into account. Including more degrees of freedom damps out the Stückelberg oscillations present in the one-dimensional model. Our results qualitatively reproduce the observed dependence of the ionization probability on the incident energy of the molecules. This dependence is not given by the electronic coupling per se, but rather due to energy transfer to substrate and internal degrees of freedom during the scattering event. Finally, we are also able to reproduce the measured dissociation probability which can be explained in a centrifugal model.

Infinite-order non-Born-Oppenheimer perturbation theory for systems with intersecting potentials

We analyze models for two intersecting electronic states coupled by one nuclear coordinate. The corresponding model-Hamiltonians depend on a perturbation parameter \ensuremath{\alpha}. Its variation allows us to switch on the nonadiabatic coupling continuously. We derive analytical expressions for the eigenstates and energies as function of \ensuremath{\alpha} in the diabatic representation. We rederive the expression for the energy eigenvalues by complete resummation of the Brillouin-Wigner perturbation series in the Born-Oppenheimer basis. We find that the divergences of the nonadiabatic coupling cancel strictly order by order. The admixture of high energy components is shown explicitly.