Diabatic Representation Research Articles

Avoided resonance overlapping beyond the energy independent formalism. II. Electronic predissociation

The transition between the diabatic and the adiabatic zero order representation when the electronic potential coupling increases, is studied in the partitioning method (PM) and in the complex coordinate method (CCM) implemented in a discrete variable representation (DVR). The model is a C+ type predissociation in a diatomic system with a crossing between a Morse potential energy curve and an exponentially repulsive curve. In the weak coupling regime (isolated diabatic metastable states), both methods confirm the linear variation of the resonance widths with the strength of the electronic interaction, as expected in a perturbative treatment. When the coupling is large in the diabatic representation (strong overlap regime), the formation of narrow resonances supported by the upper adiabatic potential can be related, in PM, to the process of avoided resonance overlapping among interfering states. The complete change of representation can be described in the initial diabatic basis set in PM. However, the full energy dependence of the discrete-continuous matrix elements must then be taken into account. The deformation and the final splitting of a diffuse spectral line with increasing coupling is re-examined in terms of these energy dependent eigenvalues of the effective Hamiltonian. The use of the appropriate representation, either diabatic or adiabatic, according to the strength of the electronic coupling, has been found decisive in CCM so as to observe the correct migration of the resonance positions towards the zero order adiabatic states, and the decrease of their widths. No relevant results have been obtained for the intermediate strength of the coupling in CCM. Analytical expressions for the derivative coupling matrix elements (∂/∂R) in the fixed node DVR (corresponding to the particle-in-a-box wave functions) have been established.

Electronically diabatic quantum dynamics of molecular desorption

The photodesorption of a diatomic from a metal surface, following absorption of visible or UV light, involves electronic transitions of the desorbing species coupled to the lattice vibrations and electron–hole excitations of the substrate. We present a general treatment of these phenomena, based on the Liouville–Von Neuman equation for the density operator, and a stochastic theory of localized perturbations in an extended system. The Hamiltonian of the extended molecular system is divided into a term for the localized primary degrees of freedom (DFs) affected by the desorption, coupled to secondary DFs that acts as a time-evolving bath. A self-consistent field treatment gives an effective (non-Hermitian) Hamiltonian for the primary DFs that accounts for energy fluctuation and dissipation in terms of the properties of adsorbate and substrate. A diabatic electronic representation is used to eliminate momentum couplings between adsorbate electronic states. The bath dynamics is studied for lattice vibrations and for electronic excitations. Electron–hole excitations of the substrate are considered for intraband and interband transitions. The assumption of Brownian motion leads to expressions for the dissipative potentials in terms of the time-correlation functions of lattice displacements and of electron density fluctuations. The dissipation depends on time, allowing for time-dependent substrate temperatures and generalizing the Langevin description. Dissipation contributes to the time evolution of both ground and excited electronic states of the desorbing species. The model is discussed for the special case of Ni(001)(ads)CO.

Doubly periodical in time and energy exactly soluble system with two interacting systems of states

The time-dependent matrix Schrodinger equation 1/ic( delta Psi / delta t)=H(t) Psi describing two bands of an infinite number of equidistant states with different energy spacings omega +or- in each band is studied. Both bands are linearly dependent on time t. The interaction upsilon =( square root ( omega - omega +)/ pi )tan pi s between the bands is considered to be equal for any pair of states from each band. Using the Fourier series transformation the instant eigenvalues E(t, s) are calculated which reveal the double periodicity in the energy-time plane. The corresponding eigenvalue surface in the (E, t, s)-space, apart from the triple periodicity, shows quite unexpected symmetry properties relative to the exchange of t and s, and relative to some inversions in the (E, t) plane. The latter one leads to a new equivalence between weak and strong coupling, a new kind of pseudocrossing and a new concept of antidiabatic states. The Fourier transformation reduces the problem to a 2*2 first-order differential operator. The diagonalization of H(r) for fixed t produces explicit expressions for the eigenvalues (adiabatic potential curves) and eigenstates (adiabatic basis). The time evolution operator is calculated both in the diabatic and adiabatic representations. The results are simplified for the special value of the interaction parameter.

Theoretical study of single and double charge transfer in He2+-He collisions at kilo-electron-volt energies in a diabatic molecular representation.

Theoretical study of single and double charge transfer in He2+-He collisions at kilo-electron-volt energies in a diabatic molecular representation.

From torsional spectra to Hamiltonians and dynamics: effects of coupled bright and dark states of 9-(N-carbazolyl)-anthracene

The torsional dynamics of the 9-(N-carbazolyl)-anthracene (C9A) molecule is investigated by means of time-independent (1) and time-dependent (2) quantum-mechanical simulations in a diabatic representation. The study includes effects of surface crossing of the brightS 1 state with a dark state. (1) The intensity pattern of theS 0 →S 1 fluorescence excitation spectrum is used to fit an effective one-dimensional Hamiltonian with a single-minimum potential for the dark state together with diabatic couplings to the double well potential of the bright state. (2) Based on this Hamiltonian, first predictions for a pump-probe scheme are made. In the pump process the molecules are excited to theS 1 state followed by competing torsions in the bright state and diabatic curve crossings to the dark state, depending on the pump frequency. Assuming the probe process to be an ionization from the bright state, the interfering effects of the dark state on the dynamics in the bright state can be monitored in a directly time-dependent way on a fs-ps time scale.

Avoided crossings in potential curves of BF2+: A study of models for bonding in diatomic dications

An ab initio study of the potential energy curves for BF2+ is reported. The 1Σ+ state is metastable and is characterized by an unusually strong bond with a well depth of 6.06 eV. The origin of this bond is clearly traced to the interaction between the strongly bonding configuration B2+ + F and the repulsive configurations B++F+. Bound metastable states are also present for 1Π and 3Π manifolds. Dipole moments are used to assign types of bonding in each region of internuclear distance for the most important electronic states. An approximate diabatic representation is given for the two low-lying states derived from the 1Σ+ adiabatic curves. This study confirms the utility of an avoided crossing-diabatic coupling plus polarization model for systems of intermediate polarity.

Application of generalized quantum defect theory to van der Waals complex bound state calculations

Bound states of atom–diatom van der Waals complexes are calculated in the generalized quantum defect theory framework (GMQDT). Due to very strong interchannel couplings, the diabatic GMQDT formalism is extended in order to account for the strongly closed channels (associated with potential curves lying above the total energy E) in the asymptotic analysis of the wave function. An alternative GMQDT treatment combining diabatic and adiabatic representations is also presented. The influence of these strongly closed channels on the level positions and oscillator strengths is analyzed.

Emission spectroscopy of dissociating H2S: Influence of nonadiabatic coupling

The emission spectroscopy of H2S excited in the first absorption band around 195 nm is investigated theoretically using ab initio potential energy surfaces (PES) and transition dipole moment functions. As shown in our previous studies, the photodissociation involves two excited electronic states, one which is binding and another one which is dissociative. The nonadiabatic coupling between these two states is very strong and described in a diabatic representation in which only the binding state is optically excited while the dissociative state is dark. As in the case of H2O excited in the 165 nm band, the emission spectrum shows a long progression of stretching states up to seven HS vibrational quanta. In contrast to water, however, some weak activity in the bending mode is also observed. Most remarkable is a prominent wavelength dependence which is attributed to the strong nonadiabatic coupling between the two excited electronic states. The agreement with experimental data is only fair; the essential features of the measurements are qualitatively reproduced, finer details such as the wavelength dependence are, however, not well described. It is concluded that more accurate ab initio input data are required in order to reproduce all details of the measured emission spectra.

Optical potential coupled to discrete variable representation for calculations of quasibound states: Application to the CO(B 1Σ+–D′1 Σ+) predissociating interaction

The optical potential method initially proposed by Jolicard and Austin in the context of the stabilization method is reviewed here and used with the accurate and the efficient discrete variable representation method to obtain the energies and the widths (respectively, the real and the imaginary part of the resonance energies) of rovibrational predissociated states of diatomic molecules. In this method the resonances for an n coupled states problem are obtained by a direct diagonalization of the Hamiltonian matrix in the diabatic representation. This Hamiltonian matrix is directly evaluated in the discrete variable representation using the Fourier grid Hamiltonian method proposed by Marston and Balint-Kurti. In this approach, two optical potentials are tested and used here to impose the asymptotic behaviors of the boundary conditions which are compatible with the resonance states. The method is exemplified for the B 1Σ+–D′1 Σ+ Rydberg–valence predissociating interaction in the CO molecule.

Dissipative dynamics in a curve-crossing system. The N 2 molecule

Density matrix theory is applied to the study of environmental effects on the dynamics of vibrational degrees of freedom in coupled potential energy surfaces. For this purpose we formulate the theory in a diabatic state representation and incorporate the environment to second order of perturbation theory. As an example we study the dynamics on the coupled b′ 1Σ u +-c′ 1Σ u + potential energy surfaces of the N 2 molecule which is supposed to be immobilized in a solid state matrix. The resulting dependence of the vibrational dynamics on the coupling strength to the matrix degrees of freedom is discussed in the context of a pump probe experimental configuration. It is found that with increasing coupling strength oscillatory structures in the probe spectrum are strongly suppressed.

Ab initiostudy of NO2

Large-scale variational calculations of high-lying vibrational levels of the ground electronic state of NO2 are reported in both the diabatic and adiabatic representations of the potential. We have used and modified the MRD-CI analytical surfaces by Hirsch et al. and the Carter-Handy method. Vibrational bands of the diabatic potential have been computed and converged up to 18000 cm−1, and the vibrational levels of the adiabatic surface have been compared with the experimental results up to 5000 cm−1.

Quantum rearrangement scattering calculations using the invariant imbedding method

We derive an algorithm to calculate quantum transmission and reflection amplitudes (S-matrix elements) for rearrangement reactions using the invariant imbedding method. Our approach can incorporate closed channels, treat complex potentials (to simulate decay), and can be used within an adiabatic representation (in which derivative coupling terms are present) or within a diabatic representation. The method can also be used to determine bound-state energies when only closed channels are present. The method is demonstrated on a rearrangement problem with two coupled potential surfaces (four channels) as a function of energy in the energy region from the lowest channel threshold through the highest channel threshold.

From concepts to algorithms for the characterization of reaction mechanisms. H2CS as a case study

All the stationary points on the S0 surface of H2CS and their quadratic force fields have been determined by correlated methods using large multiply polarized basis sets. Complete quartic force fields of all stationary points and reaction paths connecting each pair of minima through a first-order saddle point have been obtained at the MP2 level using smaller polarized basis sets. These data allow the computation of reliable isomerization and fragmentation rates by semiclassical expressions which take into account tunneling as well as curvature and mode–mode couplings. Further insight into the characteristics of the different reaction channels has been gained from the parameters of the so-called reaction path Hamiltonian (RPH) both in adiabatic and diabatic representations. The bookkeeping and processing of the large body of data involved in this study has been possible through the development of a package which is able to perform anharmonic and reaction path computations. The results are discussed with special reference to the H2CO system and to the interplay of potential, kinetic, and statistic effects in determining reaction mechanisms and rates.

Characterization of the S1–S2 conical intersection in pyrazine using ab initio multiconfiguration self-consistent-field and multireference configuration-interaction methods

Potential-energy surfaces of the three lowest singlet states of pyrazine have been calculated as a function of ab initio determined ground-state normal coordinates, using complete-active-space self-consistent-field (CASSCF) and multireference configuration interaction (MRCI) techniques. The conical intersection of the S1 and S2 adiabatic potential-energy surfaces has been mapped out in selected subspaces spanned by the most relevant vibrational coordinates. A unitary transformation from the adiabatic to a quasidiabatic electronic representation is performed, which eliminates the rapid variations of the wave functions responsible for the singularity of the nonadiabatic coupling element. Transition-dipole-moment functions have been obtained in the adiabatic and in the diabatic representation. The leading coefficients of the Taylor expansion of the diabatic potential-energy and transition-dipole-moment surfaces in terms of ground-state normal coordinates at the reference geometry have been obtained at the CASSCF/MRCI level. Using a vibronic-coupling model Hamiltonian based on this Taylor expansion, the absorption spectrum of the interacting S1–S2 manifold has been calculated, taking account of the four spectroscopically most relevant modes.

Invariant imbedding method for multichannel Schrödinger equations with first derivative coupling

We derive and demonstrate an invariant imbedding method for calculating the S matrix for scattering problems involving a first derivative term in the Schrödinger equation for the relative motion. This method can be used directly on scattering problems derived using molecular structure calculations that determine adiabatic potential energies and their derivative coupling matrix elements. A critical discussion of the procedures that transform from an adiabatic to a diabatic representation is presented.

Laser-induced-resonance calculations for the photodissociation of H2+ in an adiabatic electronic-field representation using the radiation-field gauge.

The properties of the laser-induced first resonance state in the photodissociation of ${\mathrm{H}}_{2}^{+}$ are obtained from calculations in the radiation-field (RF) gauge, also called the velocity gauge, as a function of intensity and wavelength of the field within an adiabatic electronic-field representation, and the results are compared to the corresponding ones obtained using a diabatic RF gauge representation and an adiabatic electric-field gauge representation. Gauge transformation within the coupled equations used for these calculations is also discussed.

Adiabatic and approximate diabatic potential energy surfaces for the B...H2 van der Waals molecule

We report multireference configuration-interaction calculations for the lowest potential energy surfaces of the B(2s22p 2P)...H2 van der Waals molecule. The degeneracy of the 2p orbital implies that there exist three adiabatic potential energy surfaces (two of A′ symmetry and one of A″ symmetry in Cs geometry) which become degenerate at large B–H2 separation. By assuming that the two adiabatic states of A′ symmetry correspond primarily to an orthogonal transformation of the in-plane B 2p orbitals, one can use calculated matrix elements of the electronic orbital angular momentum to transform to an approximate diabatic representation, which involves four potential energy functions. The proper angular expansion of these functions in terms of reduced rotation matrix elements is discussed and an analytic representation of the calculated points is obtained. The minimum energy of the B(2s22p 2P)...H2 van der Waals molecule is predicted to occur in C2v geometry with an electronic symmetry of 2B2, at a B–H2 distance of 3.11 Å, and a dissociation energy De of 121 cm−1. For the interaction of B(2P) with p-H2, assumed spherical in j=0, the zero-point corrected dissociation energy is D0=25 cm−1.

Theoretical study of line broadening in the (3sσ) 3Π g vibrational levels of the O 2 molecule employing a diabatic CI representation

Ab initio CI calculations have been carried out for the lowest three 3Π g electronic states of the O 2 molecule and a diabatic representation has been obtained via a diagonalization procedure involving the electric quadrupole operator. A variation of the complex scaling procedure is applied to the coupled vibrational Hamiltonian for these three states and linewidths are computed for the v′ = 0–4 levels of the (3sσ) 3Π g state for a series of four isotopomers for which experimental data have recently been reported by Sur, Nguyen and Nikoi. Trends in v′−0 isotopomeric shifts for ground-state transitions are discussed and some changes in previous experimental assignments are suggested on this basis.

Nonadiabatic effects in the photodissociation of H2S in the first absorption band: An ab initio study

The photodissociation of H2S through excitation in the first absorption band (λ≊195 nm) is investigated by means of extensive ab initio calculations. Employing the MRD-CI method we calculate the potential energy surfaces for the lowest two electronic states of 1A″ symmetry varying both HS bond distances as well as the HSH bending angle. (In the C2v point group these states have electronic symmetry 1B1 and 1A2, respectively.) The lower adiabatic potential energy surface is dissociative when one H atom is pulled away whereas the upper one is binding. For the equilibrium angle of 92° in the electronic ground state they have two conical intersections, one occurring near the Franck–Condon point. Because of the very small energy separation between these two states nonadiabatic coupling induced by the kinetic energy operator in the nuclear degrees of freedom are substantial and must be incorporated in order to describe the absorption and subsequent dissociation process in a realistic way. In the present work we treat the coupling between the two electronic states in a diabatic representation extracting the coordinate-dependent mixing angle from the CI coefficients of the electronic wave functions. The nuclear motion is treated in three dimensions in an exact quantum mechanical approach by propagation of a two-component time-dependent wave packet. The calculated absorption spectra for H2S and D2S satisfactorily agree with the measured spectra. In particular, the calculations reproduce the diffuse structures with energy spacing of about 1200 and 850 cm−1 for H2S and D2S, respectively. Furthermore, the calculated rotational- and vibrational-state distributions of the HS and DS fragments reproduce recent measurements in a convincing way. The photodissociation of H2S is a prototype for very fast electronic predissociation. The photon preferentially excites the binding (diabatic) state. This state, however, is quickly depleted by strong coupling to the dissociative (diabatic) state with the complex finally breaking up into products H and HS. The electronic quenching takes place on the time scale of one internal vibrational period only. Our calculations unambiguously confirm that the diffuse structures superimposed to the broad background are caused by symmetric stretch motion—in the binding state—and not by activity in the bending mode as originally assumed.

Internal conversion funnel in benzene and pyrazine: adiabatic and diabatic representation

The two lowest singlet surfaces of benzene and pyrazine exhibit a low-lying conical intersection on the reaction path to a biradical form called prefulvene. The potential-energy surfaces have been calculated with the complete-active-space self-consistent-field method. The conical intersection has been mapped out in two dimensions, the reaction coordinate to prefulvene and the coordinate of maximal interstate coupling. Quasi-diabatic potential-energy surfaces have been obtained via a block-diagonalization of the CI matrix. The ab initio diabatic potentials reveal that the conical intersection is locally well described by a simple linear model in both benzene and pyrazine.