Nonadiabatic Coupling Matrix Elements Research Articles

Charge transfer in cold Yb++Rb collisions

Charge-transfer cold Yb$^+$ + Rb collision dynamics is investigated theoretically using high-level {\it ab initio} potential energy curves, dipole moment functions and nonadiabatic coupling matrix elements. Within the scalar-relativistic approximation, the radiative transitions from the entrance $A^1\Sigma^+$ to the ground $X^1\Sigma^+$ state are found to be the only efficient charge-transfer pathway. The spin-orbit coupling does not open other efficient pathways, but alters the potential energy curves and the transition dipole moment for the $A-X$ pair of states. The radiative, as well as the nonradiative, charge-transfer cross sections calculated within the $10^{-3}-10$ cm$^{-1}$ collision energy range exhibit all features of the Langevin ion-atom collision regime, including a rich structure associated with centrifugal barrier tunneling (orbiting) resonances. Theoretical rate coefficients for two Yb isotopes agree well with those measured by immersing Yb$^+$ ion in an ultracold Rb ensemble in a hybrid trap. Possible origins of discrepancy in the product distributions and relations to previously studied similar processes are discussed.

Experimental and Theoretical Study of Multi-Quantum Vibrational Excitation: NO(v = 0→1,2,3) in Collisions with Au(111)

We measured absolute probabilities for vibrational excitation of NO(v = 0) molecules in collisions with a Au(111) surface at an incidence energy of translation of 0.4 eV and surface temperatures between 300 and 1100 K. In addition to previously reported excitation to v = 1 and v = 2, we observed excitation to v = 3. The excitation probabilities exhibit an Arrhenius dependence on surface temperature, indicating that the dominant excitation mechanism is nonadiabatic coupling to electron-hole pairs. The experimental data are analyzed in terms of a recently introduced kinetic model, which was extended to include four vibrational states. We describe a subpopulation decomposition of the kinetic model, which allows us to examine vibrational population transfer pathways. The analysis indicates that sequential pathways (v = 0 → 1 → 2 and v = 0 → 1 → 2 → 3) alone cannot adequately describe production of v = 2 or 3. In addition, we performed first-principles molecular dynamics calculations that incorporate electronically nonadiabatic dynamics via an independent electron surface hopping (IESH) algorithm, which requires as input an ab initio potential energy hypersurface (PES) and nonadiabatic coupling matrix elements, both obtained from density functional theory (DFT). While the IESH-based simulations reproduce the v = 1 data well, they slightly underestimate the excitation probabilities for v = 2, and they significantly underestimate those for v = 3. Furthermore, this implementation of IESH appears to overestimate the importance of sequential energy transfer pathways. We make several suggestions concerning ways to improve this IESH-based model.

Adjustment of Born-Oppenheimer electronic wave functions to simplify close coupling calculations

Technical problems connected with use of the Born-Oppenheimer clamped-nuclei approximation to generate electronic wave functions, potential energy surfaces (PES), and associated properties are discussed. A computational procedure for adjusting the phases of the wave functions, as well as their order when potential crossings occur, is presented which is based on the calculation of overlaps between sets of molecular orbitals and configuration interaction eigenfunctions obtained at neighboring nuclear conformations. This approach has significant advantages for theoretical treatments describing atomic collisions and photo-dissociation processes by means of ab initio PES, electronic transition moments, and nonadiabatic radial and rotational coupling matrix elements. It ensures that the electronic wave functions are continuous over the entire range of nuclear conformations considered, thereby greatly simplifying the process of obtaining the above quantities from the results of single-point Born-Oppenheimer calculations. The overlap results are also used to define a diabatic transformation of the wave functions obtained for conical intersections that greatly simplifies the computation of off-diagonal matrix elements by eliminating the need for complex phase factors.

Explorative computational study of the singlet fission process

Different ab initio methods, namely multi-reference and nonorthogonal configuration interaction techniques, are explored for their applicability in studying the singlet fission problem. It has been shown for 2-methyl-1,5-hexadiene that the 1TT state can be identified using multi-reference techniques. The geometrical and vibrational properties of the 1TT state are such that they can be approximated with those of the 5TT state. A proof of principle is given for the calculation of the singlet fission pathway driven by nuclear motion: efficient singlet fission can take place if the 1TT and S1 states are close in energy with a large non-adiabatic coupling matrix element at the S1 geometry, and the energy of the S0 state is well below that of the 1TT state at the 1TT geometry. The nonorthogonal configuration interaction method was used to treat a tetracene trimer. It has been shown that the first excited states can be interpreted as delocalised states; interaction with charge-transfer base states plays an important role. The 1TT states are localised on one pair of molecules. The electronic coupling between the diabatic S[n] and 1TT[m] states is in the meV range, confirming previous estimates. The charge-transfer base states enhance the coupling between the S[1]/S[2] and 1TT[2] excited states.

Rovibrational dynamics of the strontium molecule in the ${\rm A}\sideset{^1}{_{u}^{+}}{\Sigma}$AΣu+1, c3Πu, and ${\rm a}\sideset{^3}{_{u}^{+}}{\Sigma}$aΣu+3 manifold from state-of-the-art ab initio calculations

State-of-the-art ab initio techniques have been applied to compute the potential energy curves for the electronic states in the ${\rm A}\sideset{^1}{_{u}^{+}}{\Sigma}$AΣu+1, c3Πu, and ${\rm a}\sideset{^3}{_{u}^{+}}{\Sigma}$aΣu+3 manifold of the strontium dimer, the spin-orbit and nonadiabatic coupling matrix elements between the states in the manifold, and the electric transition dipole moment from the ground ${\rm X}\sideset{^1}{_{g}^{+}}{\Sigma}$XΣg+1 to the nonrelativistic and relativistic states in the A+c+a manifold. The potential energy curves and transition moments were obtained with the linear response (equation of motion) coupled cluster method limited to single, double, and linear triple excitations for the potentials and limited to single and double excitations for the transition moments. The spin-orbit and nonadiabatic coupling matrix elements were computed with the multireference configuration interaction method limited to single and double excitations. Our results for the nonrelativistic and relativistic (spin-orbit coupled) potentials deviate substantially from recent ab initio calculations. The potential energy curve for the spectroscopically active (1)$0_u^+$0u+ state is in quantitative agreement with the empirical potential fitted to high-resolution Fourier transform spectra [A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 64, 227 (2011)]10.1140/epjd/e2011-20229-6. The computed ab initio points were fitted to physically sound analytical expressions, and used in converged coupled channel calculations of the rovibrational energy levels in the A+c+a manifold and line strengths for the ${\rm A}\sideset{^1}{_{u}^{+}}{\Sigma}\leftarrow {\rm X}\sideset{^1}{_{g}^{+}}{\Sigma}$AΣu+1←XΣg+1 transitions. Positions and lifetimes of quasi-bound Feshbach resonances lying above the 1S0 + 3P1 dissociation limit were also obtained. Our results reproduce (semi)quantitatively the experimental data observed thus far. Predictions for on-going and future experiments are also reported.

Ultrafast coherent dynamics of nonadiabatically coupled quasi-degenerate excited states in molecules: Population and vibrational coherence transfers

Results of a theoretical study of ultrafast coherent dynamics of nonadiabatically coupled quasi-degenerate π-electronic excited states of molecules were presented. Analytical expressions for temporal behaviors of population and vibrational coherence were derived using a simplified model to clarify the quantum mechanical interferences between the two coherently excited electronic states, which appeared in the nuclear wavepacket simulations [M. Kanno, H. Kono, Y. Fujimura, S.H. Lin, Phys. Rev. Lett 104 (2010) 108302]. The photon-polarization direction of the linearly polarized laser, which controls the populations of the two quasi-degenerate electronic states, determines constructive or destructive interference. Features of the vibrational coherence transfer between the two coupled quasi-electronic states through nonadiabatic couplings are also presented. Information on both the transition frequency and nonadiabatic coupling matrix element between the two states can be obtained by analyzing signals of two kinds of quantum beats before and after transfer through nonadiabatic coupling.

Ab initiostudy of charge-transfer dynamics in collisions of C2+ions with hydrogen chloride

Ab initio quantum chemistry molecular calculations followed by a semiclassical dynamical treatment in the keV collision energy range have been developed for the study of the charge-transfer process in collisions of ${\mathrm{C}}^{2+}$ ions with hydrogen chloride. The mechanism has been investigated in detail in connection with avoided crossings between states involved in the reaction. A simple mechanism driven by a strong nonadiabatic coupling matrix element has been pointed out for this process. A comparative analysis with the halogen fluoride target corresponding to a similar electronic configuration shows a quite different charge-transfer mechanism leading to a very different behavior of the cross sections. Such behavior may be correlated to specific nonadiabatic interactions observed in these collision systems.

Sympathetic cooling of the Ba+ion by collisions with ultracold Rb atoms: Theoretical prospects

State-of-the-art {\em ab initio} techniques have been applied to compute the potential energy curves of the (BaRb)$^+$ molecular ion in the Born-Oppenheimer approximation. The long-range coefficients describing the electrostatic, induction, and dispersion interactions at large interatomic distances are also reported. The electric transition dipole moments governing the X$^1\Sigma \to ^1\Sigma, ^1\Pi$ have been obtained as the first residue of the polarization propagator computed with the linear response coupled-cluster method restricted to single and double excitations. Nonadiabatic radial and angular coupling matrix elements, as well as the spin-orbit coupling matrix elements have been evaluated using the multireference configuration interaction method restricted to single and double excitations with a large active space. With these couplings, the spin-orbit coupled (relativistic) potential energy curves for the 0$^+$ and 1 states relevant for the running experiments have been obtained. Finally, relativistic transition moments and nonadiabatic coupling matrix elements were obtained from the nonrelativistic results and spin-orbit eigenvectors. The electronic structure input has been employed in the single channel scattering calculations of the collisional cross sections between the Ba$^+$ ion and Rb atom. Both nonrelativistic and relativistic potentials were used in these calculations. Our results show that the inelastic cross section corresponding to the charge transfer from the Rb atom to the Ba$^+$ ion is much smaller than the elastic one over a wide range of energies up to 1 mK. This suggests that sympathetic cooling of the Ba$^+$ ion by collisions with ultracold Rb atoms should be possible.

Quantum dynamics of the charge transfer in C+ + S at low collision energies

Following a recent semiclassical investigation by Bacchus-Montabonel and Talbi (2008 Chem. Phys. Lett. 467 28), the C+(2s22p)2P + S(3s23p4)3P charge transfer process involved in the modellization of the interstellar medium chemistry and its reverse reaction are revisited by combining a wave packet approach and semiclassical dynamics in a quasimolecular approach for doublet and quartet states. New radial non-adiabatic coupling matrix elements have been calculated and the mixed treatment gives access to new precise values of the rate coefficients for the direct and reverse charge transfer processes. For this system, quantum and semiclassical results are in good agreement even at low collision kinetic energies. The dominance of the quartet states in the process is confirmed. In the quantum treatment, the collision matrix elements are extracted from wave packets by the flux method with an absorbing potential. The formation of resonances due to a centrifugal barrier is illustrated.

Ab initio adiabatic and quasidiabatic potential energy surfaces of lowest four electronic states of the H++O2 system

Ab initio global adiabatic and quasidiabatic potential energy surfaces of lowest four electronic (1-4 (3)A(")) states of the H(+)+O(2) system have been computed in the Jacobi coordinates (R,r,γ) using Dunning's cc-pVTZ basis set at the internally contracted multireference (single and double) configuration interaction level of accuracy, which are relevant to the dynamics studies of inelastic vibrational and charge transfer processes observed in the scattering experiments. The computed equilibrium geometry parameters of the bound [HO(2)](+) ion in the ground electronic state and other parameters for the transition state for the isomerization process, HOO(+)⇌OOH(+) are in good quantitative agreement with those available from the high level ab initio calculations, thus lending credence to the accuracy of the potential energy surfaces. The nonadiabatic couplings between the electronic states have been analyzed in both the adiabatic and quasidiabatic frameworks by computing the nonadiabatic coupling matrix elements and the coupling potentials, respectively. It is inferred that the dynamics of energy transfer processes in the scattering experiments carried out in the range of 9.5-23 eV would involve all the four electronic states.

Nonadiabatic Dynamics on the Two Coupled Electronic PESs: The H+ + O2 System

Multistate adiabatic and diabatic PESs were computed for the H+ + O2 collision system in Jacobi coordinates, (R,r,γ) using the cc-pVTZ basis set and the ic-MRCI level of theory. In addition, all possible interaction potentials and nonadiabatic coupling matrix elements among those different electronic states were also computed. Comparisons with earlier computed interaction potentials were made wherever possible, and the differences between them is attributed to the multistate diabatization and the chosen level of theory and basis set. Focusing our attention on the ground-state (GS) and the first excited-state (ES) PES, quantum dynamics were performed using the 2 × 2 diabatic potential submatrix obtained from the multistate (four) diabatic potential matrix within the VCC-RIOSA scheme at two experimentally reported collision energies, E(cm) = 9.5 and 23 eV. The scattering quantities were computed for two experimentally observed collision processes, namely, the inelastic vibrational excitation (IVE), H+ + O2 (X3Σg(−),v = 0) → H+ + O2 (X3Σg(−),v′), and the vibrational charge transfer (VCT), H+ + O2 (X3Σg(−),v = 0) → H (2S) + O (X2Πg,v′′). Comparisons were made with experimental results and found an overall improvement relative to the earlier computed results, and the discrepancies, if any, could be brought down to minimum by further modification in employed ab initio PESs and the interaction potential.

Ab initiocalculation ofH+He+charge-transfer cross sections for plasma physics

The charge-transfer in low-energy ($0.25$ to $150$ eV/amu) $\mathrm{H}(\mathit{nl})+{\mathrm{He}}^{+}(1s)$ collisions is investigated using a quasimolecular approach for the $n=2,3$ as well as the first two $n=4$ singlet states. The diabatic potential energy curves of the HeH${}^{+}$ molecular ion are obtained from the adiabatic potential energy curves and the nonadiabatic radial coupling matrix elements using a two-by-two diabatization method, and a time-dependent wave-packet approach is used to calculate the state-to-state cross sections. We find a strong dependence of the charge-transfer cross section on the principal and orbital quantum numbers $n$ and $l$ of the initial or final state. We estimate the effect of the nonadiabatic rotational couplings, which is found to be important even at energies below $1$ eV/amu. However, the effect is small on the total cross sections at energies below $10$ eV/amu. We observe that to calculate charge-transfer cross sections in an $n$ manifold, it is only necessary to include states with ${n}^{\ensuremath{'}}\ensuremath{\leqslant}n$, and we discuss the limitations of our approach as the number of states increases.

Ab initio ground and the first excited adiabatic and quasidiabatic potential energy surfaces of H + + CO system

Ab initio global adiabatic as well as quasidiabatic potential energy surfaces for the ground and the first excited electronic states of the H + + CO system have been computed as a function of the Jacobi coordinates ( R, r, γ) using Dunning’s cc-pVTZ basis set at the internally contracted multi-reference (single and double) configuration interaction level of accuracy. In addition, nonadiabatic coupling matrix elements arising from radial motion, mixing angle and coupling potential have been computed using the ab initio procedure [Simah et al. (1999) [66]] for the purpose of dynamics study. The geometrical properties corresponding to the minimum energy of the bound HCO + and HOC + isomers have been obtained and compared with those predicted by previous theoretical and experimental results. The HCO + has been found to be more stable than the HOC +. The minimum energy pathway in the ground electronic state for the isomerization process, HCO + ⇌ HOC + has also been obtained as a function of γ.

Ab initio calculation of the 66 low-lying electronic states of HeH+: adiabatic and diabatic representations

Using the quantum chemistry package MOLPRO and an adapted basis set, we have calculated the adiabatic potential energy curves of the first 20 1Σ+, 19 3Σ+, 12 1Π, 9 3Π, 4 1Δ and 2 3Δ electronic states of the HeH+ molecular ion in CASSCF and CI approaches. The results are compared with previous works. The radial and rotational non-adiabatic coupling matrix elements as well as the dipole moments are also calculated. The asymptotic behaviour of the potential energy curves and of the various couplings between the states is also studied. Using the radial couplings, the diabatic representation is defined and we present an example of our diabatization procedure on the 1Σ+ states.

Photodissociation of ozone in the Hartley band: Potential energy surfaces, nonadiabatic couplings, and singlet/triplet branching ratio

The lowest five (1)A(') states of ozone, involved in the photodissociation with UV light, are analyzed on the basis of multireference configuration interaction electronic structure calculations with emphasis on the various avoided crossings in different regions of coordinate space. Global diabatic potential energy surfaces are constructed for the lowest four states termed X, A, B, and R. In addition, the off-diagonal potentials that couple the initially excited state B with states R and A are constructed to reflect results from additional electronic structure calculations, including the calculation of nonadiabatic coupling matrix elements. The A/X and A/R couplings are also considered, although in a less ambitious manner. The photodissociation dynamics are studied by means of trajectory surface hopping (TSH) calculations with the branching ratio between the singlet, O((1)D)+O(2)((1)Delta(g)), and triplet, O((3)P)+O(2)((3)Sigma(g) (-)), channels being the main focus. The semiclassical branching ratio agrees well with quantum mechanical results except for wavelengths close to the threshold of the singlet channel. The calculated O((1)D) quantum yield is approximately 0.90-0.95 across the main part of the Hartley band, in good agreement with experimental data. TSH calculations including all four states show that transitions B-->A are relatively unimportant and subsequent transitions A-->X/R to the triplet channel are negligible.

First-order nonadiabatic couplings from time-dependent hybrid density functional response theory: Consistent formalism, implementation, and performance

First-order nonadiabatic coupling matrix elements (NACMEs) are key for phenomena such as nonradiative transitions and excited-state decay, yet a consistent and practical first principles treatment has been elusive for molecules with more than a few heavy atoms. Here we present theory, implementation using Gaussian basis sets, and benchmarks of first-order NACMEs between ground and excited states in the framework of time-dependent hybrid density functional theory (TDDFT). A time-dependent response approach to NACMEs which avoids explicit computation of excited-state wave functions is outlined. In contrast to previous approaches, the present treatment produces exact analytical derivative couplings between time-dependent Kohn-Sham (TDKS) determinants in a finite atom-centered basis set. As in analytical gradient theory, derivative molecular orbital coefficients can be eliminated, making the computational cost independent of the number of nuclear degrees of freedom. Our expression reduces to the exact Chernyak-Mukamel formula for first-order NACMEs in the complete basis-set limit, but greatly improves basis-set convergence in finite atom-centered basis sets due to additional Pulay type terms. The Chernyak-Mukamel formula is shown to be equivalent to the Hellmann-Feynman contribution in analytical gradient theory. Our formalism may be implemented in TDDFT analytical excited-state gradient codes with minor modifications. Tests for systems with up to 147 atoms show that evaluation of first-order NACMEs causes total computation times to increase by an insignificant 10% on average. The resolution-of-the-identity approximation for the Coulomb energy (RI-J) reduces the computational cost by an order of magnitude for nonhybrid functionals, while errors are insignificant with standard auxiliary basis sets. We compare the computed NACMEs to full configuration interaction (FCI) in benchmark results for diatomic molecules; hybrid TDDFT and FCI are found to be in agreement for regions of the potential energy curve where the Kohn-Sham ground-state reference is stable and the character of the excitation is properly captured by the present functionals. With these developments, nonadiabatic molecular dynamics simulations of molecular systems in the 100 atoms regime are within reach.

Symmetry forbidden vibronic spectra and internal conversion in benzene

The spectra of symmetry-forbidden transitions and internal conversion were investigated in the present work. Temperature dependence was taken into account for the spectra simulation. The vibronic coupling, essential in the two processes, was calculated based on the Herzberg-Teller theory within the Born-Oppenheimer approximation. The approach was employed for the symmetry-forbidden absorption/fluorescence, and internal conversion between 1(1)A(1g) and 1(1)B(2u) states in benzene. Vibrational frequencies, normal coordinates, electronic transition dipole moments, and non-adiabatic coupling matrix elements were obtained by ab initio quantum chemical methods. The main peaks, along with the weak peaks, were in good agreement with the observed ones. The rate constant of the 1(1)A(1g)← 1(1)B(2u) internal conversion was estimated within the order of 10(3) s(-1). This could be regarded as the lower limit (about 4.8 × 10(3) s(-1)) of the internal conversion. It is stressed that the distortion effect was taken into account both in the symmetry-forbidden absorption/fluorescence, and the rate constants of internal conversion in the present work. The distortion effects complicate the spectra and increase the rate constants of internal conversion.

Effectiveness of Perturbation Theory Approaches for Computing Non-Condon Electron Transfer Dynamics in Condensed Phases

A description of electron transfer in condensed-phase media requires models that adequately describe the coupling of the electronic degrees of freedom to the surrounding nuclear coordinates. The spin-boson model has been the canonical model used to understand quantum dynamic processes in condensed-phase media over the last 25 years. Inherent in the standard model of a two-state quantum system coupled to a bosonic bath is the assumption that the Condon approximation is valid. In this context, the Condon approximation assumes that the bath configurations (coordinates) have no effect on the nonadiabatic coupling matrix element. While this is a useful model for electron transfer in small molecular systems, the validity of this approximation is less likely when large-scale motions of solvent molecules are strongly coupled to the electron transfer event, e.g., in molecular clamps and long-range electron transfer in biopolymers. In the present paper a general model for two-state electron transfer which allows for system-bath coupling in both the diagonal and off-diagonal (nonadiabatic) terms is studied. Time-dependent perturbation theory for this Hamiltonian is developed using a small polaron transformation. As noted in several recent studies, in a certain regime of parameter space, the relevant Hamiltonian admits an exact solution, termed the exactly solvable non-Condon Hamiltonian (or NCE). This limit, for which exact solutions are available, is used to benchmark the short- and long-time accuracy of various perturbative approaches. The validated perturbation equations are subsequently used to explore the role of non-Condon effects on electron transfer by systematically increasing the strength of the non-Condon coupling term from zero (i.e., the canonical spin-boson model) to the value that pertains to the exactly solvable non-Condon model (where non-Condon effects are significant).

Potential curves and nonadiabatic coupling matrix elements for the O+−Ne system

An ensemble-average one-electron potential is used to define a molecular orbital basis suitable for the simultaneous description of a set of low-lying electronic states. The common basis simplifies the construction of the coupling between different Born-Oppenheimer states, and the required matrix elements are calculated. A new excitation mechanism is developed for inelastic scattering of O+ on Ne.

Efficient parabolic evaluation of coupling terms in hybrid quantum/classical simulations

A parabolic interpolation function of time is proposed to evaluate the non-adiabatic coupling matrix elements and the adiabatic energies at intermediate times within the classical time integration interval in hybrid quantum/classical simulations. The accuracy and the computational efficiency of this parabolic approximation are illustrated by carrying out a numerical application to the well-studied vibrational relaxation of I 2 in liquid xenon.