A semiclassical hybrid approach to many particle quantum dynamics

We analytically derive a correlated approach for a mixed semiclassical many particle dynamics, treating a fraction of the degrees of freedom by the multitrajectory semiclassical initial value method of Herman and Kluk [Chem. Phys. 91, 27 (1984)] while approximately treating the dynamics of the remaining degrees of freedom with fixed initial phase space variables, analogously to the thawed Gaussian wave packet dynamics of Heller [J. Chem. Phys. 62, 1544 (1975)]. A first application of this hybrid approach to the well studied Secrest-Johnson [J. Chem. Phys. 45, 4556 (1966)] model of atom-diatomic collisions is promising. Results close to the quantum ones for correlation functions as well as scattering probabilities could be gained with considerably reduced numerical effort as compared to the full semiclassical Herman-Kluk approach. Furthermore, the harmonic nature of the different degrees of freedom can be determined a posteriori by comparing results with and without the additional approximation.     

Herman-Kluk semiclassical dynamics in action-angle representation: new approaches to mapping quantum degrees of freedom

A general approach to mapping a discrete quantum mechanical problem by a continuous Hamiltonian is presented. The method is based on the representation of the quantum number by a continuous action variable that extends from -infinity to infinity. The projection of this Hilbert space onto the set of integer quantum numbers reduces the Hamiltonian to a discrete matrix of interest. The theory allows the application of the semiclassical methods to discrete quantum mechanical problems and, in particular, to problems where quantum Hamiltonians are coupled to continuous degrees of freedom. The Herman Kluk semiclassical propagation is used to calculate the nonadiabatic dynamics for a model avoided crossing system. The results demonstrate several advantages of the new theory compared to the existing mapping approaches.     

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schro?dinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.     

Conical intersections involving the dissociative 1pisigma* state in 9H-adenine: a quantum chemical ab initio study

The conical intersections of the dissociative 1pisigma* excited state with the lowest 1pipi* excited state and the electronic ground state of 9H-adenine have been investigated with multireference electronic structure calculations. Adiabatic and quasidiabatic potential energy surfaces and coupling elements were calculated as a function of the NH stretch coordinate of the azine group and the out-of-plane angle of the hydrogen atom, employing MultiReference Configuration-Interaction (MRCI) as well as Complete-Active-Space Self-Consistent-Field (CASSCF) methods. Characteristic properties of the 1pipi*-1pisigma* and 1pisigma*-S0 conical intersections, such as the diabatic-to-adiabatic mixing angle, the geometric phase of the adiabatic electronic wavefunctions, the derivative coupling, as well as adiabatic and diabatic transition dipole moment surfaces were investigated in detail. These data are a prerequisite for future quantum wavepacket simulations of the photodissociation and internal-conversion dynamics of adenine.     

Conical intersections in solution: formulation, algorithm, and implementation with combined quantum mechanics/molecular mechanics method

The significance of conical intersections in photophysics, photochemistry, and photodissociation of polyatomic molecules in gas phase has been demonstrated by numerous experimental and theoretical studies. Optimization of conical intersections of small- and medium-size molecules in gas phase has currently become a routine optimization process, as it has been implemented in many electronic structure packages. However, optimization of conical intersections of small- and medium-size molecules in solution or macromolecules remains inefficient, even poorly defined, due to large number of degrees of freedom and costly evaluations of gradient difference and nonadiabatic coupling vectors. In this work, based on the sequential quantum mechanics and molecular mechanics (QM/MM) and QM/MM-minimum free energy path methods, we have designed two conical intersection optimization methods for small- and medium-size molecules in solution or macromolecules. The first one is sequential QM conical intersection optimization and MM minimization for potential energy surfaces; the second one is sequential QM conical intersection optimization and MM sampling for potential of mean force surfaces, i.e., free energy surfaces. In such methods, the region where electronic structures change remarkably is placed into the QM subsystem, while the rest of the system is placed into the MM subsystem; thus, dimensionalities of gradient difference and nonadiabatic coupling vectors are decreased due to the relatively small QM subsystem. Furthermore, in comparison with the concurrent optimization scheme, sequential QM conical intersection optimization and MM minimization or sampling reduce the number of evaluations of gradient difference and nonadiabatic coupling vectors because these vectors need to be calculated only when the QM subsystem moves, independent of the MM minimization or sampling. Taken together, costly evaluations of gradient difference and nonadiabatic coupling vectors in solution or macromolecules can be reduced significantly. Test optimizations of conical intersections of cyclopropanone and acetaldehyde in aqueous solution have been carried out successfully.     

Conical and glancing Jahn-Teller intersections in the cyclic trinitrogen cation

The ground and electronically excited states of cyclic N(3) (+) are characterized at the equilibrium D(3h) geometry and along the Jahn-Teller distortions. Lowest excited states are derived from single excitations from the doubly degenerate highest occupied molecular orbitals (HOMOs) to the doubly degenerate lowest unoccupied molecular orbitals (LUMOs), which give rise to two exactly and two nearly degenerate states. The interaction of two degenerate states with two other states eliminates linear terms and results in a glancing rather than conical Jahn-Teller intersection. HOMO-2-->LUMOs excitations give rise to two regular Jahn-Teller states. Optimized structures, vertical and adiabatic excitation energies, frequencies, and ionization potential (IP) are presented. IP is estimated to be 10.595 eV, in agreement with recent experiments.     

Multidimensional reactive scattering with quantum trajectories: dynamics with Morse vibrational modes

The reactive scattering of a wave packet is studied by the quantum trajectory method for a model system with up to 25 Morse vibrational modes. The equations of motion are formulated in curvilinear reaction path coordinates with the restriction to a planar reaction path. Spatial derivatives are evaluated by the least squares method using contracted basis sets. Dynamical results, including trajectory evolution and time-dependent reaction probabilities, are presented and analyzed. For the case of one Morse vibrational mode, the results are in good agreement with those derived through direct numerical integration of the time-dependent Schrodinger equation.     

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.     

Next-generation quantum theory of atoms in molecules for the S1/S0 conical intersections in dynamics trajectories of a light-driven rotary molecular motor

Next-generation quantum theory of atoms in molecules was applied to analyze, along an entire bond path, intramolecular interactions known to influence the photoisomerization dynamics of a light-driven rotary molecular motor. The 3D bond-path framework set B_0,1 constructed from the least and most preferred directions of electronic motion, provided new insights into the bonding leading to different S₁ state lifetimes including the first quantification of covalent character of a closed-shell intramolecular bond path. We undertook the first use of the stress tensor trajectory T_σ(s) analysis on selected nonadiabatic molecular dynamics trajectories with the electron densities obtained using the ensemble density functional theory method. The stress tensor T_σ(s) analysis was found to be well suited to follow the dynamics trajectories that included the S₀ and S₁ electronic states through the conical intersection and also provided to a new measure to assess the degree of purity of the axial bond rotation for the design of rotary molecular motors.     

Detailed dynamics of the nonradiative deactivation of adenine: a semiclassical dynamics study

A realistic dynamics simulation study is reported for the ultrafast radiationless deactivation of 9H-adenine. The simulation follows two different excitations induced by two 80 fs (fwhm) laser pulses that are different in energy: one has a photon energy of 5.0 eV, and the other has a photon energy of 4.8 eV. The simulation shows that the excited molecule decays to the electronic ground state from the (1)pipi* state in both excitations but through two different radiationless pathways: in the 5.0 eV excitation, the decay channel involves the out-of-plane vibration of the amino group, whereas in the 4.8 eV excitation, the decay strongly associates with the deformation of the pyrimidine at the C 2 atom. The lifetime of the (1) npi* state determined in the simulation study is 630 fs for the 5.0 eV excitation and 1120 fs for the 4.8 eV excitation. These are consistent with the experimental values of 750 and 1000 fs. We conclude that the experimentally observed difference in the lifetime of the (1) npi* state at various excitations results from the different radiationless deactivation pathways of the excited molecule to the electronic ground state.     

Multidimensional reactive scattering with quantum trajectories: dynamics with 50-200 vibrational modes

The dynamics of ensembles containing thousands of quantum trajectories are studied for multidimensional systems undergoing reactive scattering. The Hamiltonian and equations of motion are formulated in curvilinear reaction path coordinates, for the case of a planar (zero-torsion) reaction path. In order to enhance the computational efficiency, an improved least squares fitting procedure is introduced. This scheme involves contracted basis sets and the use of inner and outer stencils around points where fitting is performed. This method is applied to reactive systems with 50-200 harmonic vibrational modes which are coupled to motion along the reaction coordinate. Dynamical results, including trajectory evolution and time-dependent reaction probabilities, are presented and power law scaling of computation time with the number of vibrational modes is described.     

The ground state tunneling splitting of malonaldehyde: accurate full dimensional quantum dynamics calculations

Benchmark calculations of the tunneling splitting in malonaldehyde using the full dimensional potential proposed by Yagi et al. are reported. Two exact quantum dynamics methods are used: the multiconfigurational time-dependent Hartree (MCTDH) approach and the diffusion Monte Carlo based projection operator imaginary time spectral evolution (POITSE) method. A ground state tunneling splitting of 25.7+/-0.3 cm(-1) is calculated using POITSE. The MCTDH computation yields 25 cm(-1) converged to about 10% accuracy. These rigorous results are used to evaluate the accuracy of approximate dynamical approaches, e.g., the instanton theory.     

Independent trajectory implementation of the semiclassical Liouville method: application to multidimensional reaction dynamics

We describe an independent trajectory implementation of semiclassical Liouville method for simulating quantum processes using classical trajectories. In this approach, a single ensemble of trajectories describes all semiclassical density matrix elements of a coupled electronic state problem, with the ensemble evolving classically under a single reference Hamiltonian chosen on the basis of physical grounds. In this paper, we introduce an additional uncoupled trajectory approximation, allowing the members of the ensemble to evolve independently of one another and eliminating the major computational costs of our previous coupled trajectory implementation. The accuracy of the method is demonstrated for model one-dimensional problems. In addition, the approach is applied to the chemical reaction dynamics of a collinear triatomic system, yielding excellent agreement with exact calculations. This method allows molecular dynamics involving coupled electronic surfaces to be modeled with essentially the same effort as classical molecular dynamics and ensemble averaging.     

Reconciling semiclassical and Bohmian mechanics: IV. Multisurface dynamics

In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schr?dinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.     

Photochemistry of pyrrole: time-dependent quantum wave-packet description of the dynamics at the 1pi sigma*-S0 conical intersections

The photoinduced hydrogen-elimination reaction in pyrrole via the conical intersections of the two (1)pi sigma(*) excited states with the electronic ground states [(1)B(1)(pi sigma(*))-S(0) and (1)A(2)(pi sigma(*))-S(0)] have been investigated by time-dependent quantum wave-packet calculations. Model potential-energy surfaces of reduced dimensionality have been constructed on the basis of accurate multireference ab initio electronic-structure calculations. For the (1)B(1)-S(0) conical intersection, the model includes the NH stretching coordinate as the tuning mode and the hydrogen out-of-plane bending coordinate as the coupling mode. For the (1)A(2)-S(0) conical intersection, the NH stretching coordinate and the screwing coordinate of the ring hydrogens are taken into account. The latter is the dominant coupling mode of this conical intersection. The electronic population-transfer processes at the conical intersections, the branching ratio between the dissociation channels, and their dependence on the initial preparation of the system have been investigated for pyrrole and deuterated pyrrole. It is shown that the excitation of the NH stretching mode strongly enhances the reaction rate, while the excitation of the coupling mode influences the branching ratio of different dissociation channels. The results suggest that laser control of the photodissociation of pyrrole via mode-specific vibrational excitation should be possible. The calculations provide insight into the microscopic details of ultrafast internal-conversion processes in pyrrole via hydrogen-detachment processes, which are aborted at the (1)pi sigma(*)-S(0) conical intersections. These mechanisms are of relevance for the photostability of the building blocks of life (e.g., the DNA bases).     

Reduced density matrix hybrid approach: an efficient and accurate method for adiabatic and non-adiabatic quantum dynamics

We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into "core" and "reservoir" modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems.     

Multidimensional quantum trajectories: applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.