Time-dependent Discrete Variable Representation Research Articles

Construction of Beyond Born-Oppenheimer Based Diabatic Surfaces and Generation of Photoabsorption Spectra: The Touchstone Pyrazine (C4 N2 H4 ).

We construct theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) based diabatic potential energy surfaces (PESs) of pyrazine (C4 N2 H4 ) molecule involving lowest four excited adiabatic PESs (S1 to S4 ) and nonadiabatic coupling terms (NACTs) among those surfaces as functions of nonadiabatically active normal modes (Q1 , Q6a , Q9a and Q10a ) to compute its photoabsorption (PA) spectra. Those adiabatic PESs are calculated using CASSCF as well as MRCI based methodologies, where NACTs are obtained from CP-MCSCF approach. Employing ab initio quantities (adiabatic PESs and NACTs), it is possible to depict the conical intersections (CIs) and develop matrices of diabatic PESs over six normal mode planes. Once single-valued, smooth, symmetric and continuous 2×2 and 4×4 diabatic surface matrices are in hand for the first time, such matrices are used to perform multi-state multi-mode nuclear dynamics with the aid of Time-Dependent Discrete Variable Representation (TDDVR) methodology initializing the product type wavefunction on (S1 ) and (S2 ) states to obtain the corresponding PA spectra. TDDVR calculated spectra for those states (S1 and S2 ) obtained from BBO based 2×2 and 4×4 diabatic surface matrices show good and better agreement with the experimental results, respectively. Both of these calculated results depict better peak progression over the existing profiles of Multi-Configuration Time-Dependent Hartree (MCTDH) dynamics over 2×2 Vibronic Coupling Model (VCM) Hamiltonian.

Mixed-quantum-classical or fully-quantized dynamics? A unified code to compare methods.

Three methods for non-adiabatic dynamics are compared to highlight their capabilities. Multi-configurational time-dependent Hartree is a full grid-based solution to the time-dependent Schrödinger equation, variational multi-configurational Gaussian (vMCG) uses a less flexible but unrestricted Gaussian wavepacket basis, and trajectory surface hopping (TSH) replaces the nuclear wavepacket with a swarm of classical trajectories. Calculations with all methods using a model Hamiltonian were performed. The vMCG and TSH were also then run in a direct dynamics mode, with the potential energy surfaces calculated on-the-fly using quantum chemistry calculations. All dynamics calculations used the Quantics package, with the TSH calculations using a new interface to a surface hopping code. A novel approach to calculate adiabatic populations from grid-based quantum dynamics using a time-dependent discrete variable representation is presented, allowing a proper comparison of methods. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

A beyond Born-Oppenheimer treatment of C6H6 + radical cation for diabatic surfaces: Photoelectron spectra of its neutral analog using time-dependent discrete variable representation.

We employ theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C6H6 +) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn-Teller interactions and accidental conical intersections/seams to pseudo Jahn-Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X̃2E1g and B̃2E2g-C̃2A2u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches.

A comparative multi-state multi-dimensional quantum-classical dynamics on compact polycyclic aromatic hydrocarbons (CPAHs) by parallel TDDVR method

The newly implemented parallelized quantum–classical dynamical approach, namely, Time Dependent Discrete Variable Representation (TDDVR) method is applied to the spectroscopically important phenanthrene, acenaphthene and pyrene, where several conical intersections exist in their four lowest singlet excited electronic states (X, A, B and C), to illustrate their various dynamical aspects. This parallel version shows almost linear scalability with increasing number of computing processors. First principles dynamics is performed to the adopted Hamiltonians to obtain photoelectron (PE) spectra, zero electron kinetic energy (ZEKE) spectra, population dynamics and many other dynamical observables. The reduced densities of the wave packet (WP) in coupled electronic surfaces are used to elucidate the intrinsic dynamical features. The TDDVR calculated PE and assigned ZEKE spectra of all CPAHs are found to be good agreement with the previous experimental data and other available theoretical results.

The time-dependent discrete variable representation (TDDVR) method to the quantum dynamics on ethylene (C2H4+)

The time-dependent discrete variable representation (TDDVR) method to the quantum dynamics on ethylene (C2H4+)

Classical blended quantum formulation of the parallel TDDVR method to the dynamics on furan

A major portion (∼98%) of the quantum-classical molecular dynamics approach, namely, Time-Dependent Discrete Variable Representation (TDDVR) method, is recently parallelized using shared-memory parallelization scheme with the aim of performing dynamics on relatively large molecular systems. Therefore, we have chosen the furan as a model system for dynamical simulation. Four lowest singlet excited electronic states 1A2(3s), 1B2(V), 1A1(V*), and 1B1(3p) of furan are vibronically coupled through several conical intersections in the vicinity of Frank-Condon region. The major focus of the present paper is to explore the efficiency of our newly implemented parallelized TDDVR algorithm to perform dynamics on large dimensional quantum systems in vibronically coupled electronic manifold. The present version of parallelized TDDVR algorithm show closely linear speed up with increasing number of computing processors. As a significant speed up is achieved by cycling in the correct way over arrays, all dynamical simulations are performed within a reasonable wall clock time. The photoelectron spectra calculated by the TDDVR method show peak by peak correspondence with the experimental spectra. The TDDVR calculated quantum-classical dynamical outcomes, viz., population and photoelectron spectra, etc. show good agreement with the findings of the well-established quantum dynamical method, i.e. Multi Configuration Time-Dependent Hartree (MCTDH) approach.

Multisurface multimode molecular dynamical simulation of naphthalene and anthracene radical cations by using nearly linear scalable time-dependent discrete variable representation method.

The major portion of the algorithm of the time-dependent discrete variable representation (TDDVR) method is recently parallelized using the shared-memory parallelization scheme with the aim of performing dynamics on relatively large molecular systems. Because of the astronomical importance of naphthalene and anthracene, we have investigated their radical cations as models for theoretical simulation of complex photoelectron spectra and nonradiative decay process using the newly implemented parallel TDDVR code. The strong vibronic coupling among the six lowest doublet electronic states makes these polynuclear hydrocarbons dynamically important. The aim of the present investigation is to show the efficiency of our current TDDVR algorithm to perform dynamics on large dimensional quantum systems in vibronically coupled electronic manifold. Both the sequential and the parallelized TDDVR algorithms are almost linear scalable for an increase in number of processors. Because a significant speed-up is achieved by cycling in the correct way over arrays, all of the simulations are performed within a reasonable wall clock time. Our theoretical spectra well reproduce the features of the corresponding experimental analog. The dynamical outcomes, for example, population, photoelectron spectra, and diffused interstellar bands, etc., of our quantum-classical approach show good agreement with the findings of the well-established quantum dynamical method, that is, multi configuration time-dependent Hartree (MCTDH) approach.

NEARLY LINEAR SCALABILITY OF TIME-DEPENDENT DISCRETE VARIABLE REPRESENTATION (TDDVR) METHOD FOR THE DYNAMICS OF MULTI-SURFACE MULTI-MODE HAMILTONIAN

Time-Dependent Discrete Variable Representation (TDDVR) method was implemented by involving "classical" trajectories on each degrees of freedom (DOF) for the dynamics of multi-surface multi-mode Hamiltonian. The major focus of this article is to explore the efficiency of the serial and parallelized TDDVR algorithm for relatively large dimensional quantum dynamics in presence of non-adiabaticity among the electronic states. As a model system, the complex photoelectron spectra and non-radiative decay dynamics of trifluoroacetonitrile radical cation ( CF3CN+ ) are theoretically simulated with the aid of such parallelized algorithm, where the five lowest electronic states (X2E, A2A1, B2A2, C2A1, and D2E) of the Hamiltonian are interconnected through several conical intersections in the vicinity of Frank–Condon region with twelve (12) active vibrational modes. The Jahn–Teller splitting of the X2E and D2E states makes the coupled five-surface system to a more challenging quantum dynamical seven-surface twelve-mode model. The results obtained from the TDDVR approach show very good agreement with the profiles of both Multi Configuration Time-Dependent Hartree (MCTDH) methodology and experimental technique, where its' sequencial and parallelized algorithm depict closely linear scalability with the increasing number of basis set vis-a-vis DOFs.

A quantum-classical simulation of a multi-surface multi-mode nuclear dynamics on $ {\rm C_{6}H}_{6}^{+} $ incorporating degeneracy among electronic states

We have performed a nuclear dynamics simulation to calculate photoelectron spectra and population profiles of benzene radical cation (C6H\(_6^+\)) employing the parallelized Time Dependent Discrete Variable Representation (TDDVR) approach. For this purpose, we have considered two multi-state multi-mode model Hamiltonians of C6H\(_6^+\) with degeneracy among the eectronic states: (a) One consists of three states and eight modes, which in turn leads to a five state thirteen mode Hamiltonian (X 2E1g -B 2E2g -C 2A2u ) due to the degeneracy; and (b) The other is constituted of three states and thirteen modes which is basically a five state twenty mode Hamiltonian (B 2E2g -D 2E1u -E 2B2u ) for the same reason. Since these electronic states are interconnected by several conical intersections in the vicinity of the Franck Condon region, it will be challenging to pursue such large dynamical calculation in the presence of nonadiabaticity among the electronic states. The spectral as well as population profiles calculated with the advent of TDDVR approach show reasonably good agreement with the results obtained by the Multi Configuration Time Dependent Hartree (MCTDH) methodology. The photoelectron spectra of benzene is quantum - classically simulated using parallelized time dependent discrete variable representation (TDDVR) method, where the degeneracy among different states is considered in the realistic model Hamiltonian of corresponding system.

The effect of phonon modes on the D2(v=0, j=0)–Cu(111) scattering processes

We include the phonon modes originating from the three layers of Cu(111) surface atoms on the dynamics of incoming molecular [D2(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering the Bose–Einstein probability factor for the initial state distribution of the surface modes calculated within the harmonic approximation. Such a time- and temperature-dependent effective Hamiltonian is further subdivided assuming a weak coupling between two sets of molecular DOFs, namely (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost, and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of the time-dependent discrete variable representation (TDDVR) approach. We demonstrate the applicability of the TDDVR method to investigate the collision of H2(v, j) on the Cu(100) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results for the D2(v=0, j=0)–Cu(111) system show that the phonon modes affect the state-to-state transition probabilities of the scattered D2 molecule substantially and chemisorption–physisorption processes noticeably.

Multi-state multi-mode nuclear dynamics on three isomers of C6H4F+2 using parallelized TDDVR approach

We have performed molecular dynamics on the three isomers of the difluorobenzene radical cation (C(6)H(4)F(2)(+)) after excitation from the ground state to a specific higher electronically excited state by using our recently implemented parallelized Time-Dependent Discrete Variable Representation (TDDVR) methodology. A five-state eleven-mode realistic model Hamiltonian for o-C(6)H(4)F(2)(+) and two separate five-state ten-mode Hamiltonians for m- and p-isomer of the same radical cation are considered, where those five electronic states are interconnected through several conical intersections in the vicinity of the Franck-Condon (FC) region and thus the dynamics for each case become complex. The photoelectron, mass analyzed threshold ionization spectra and population profiles obtained by using our TDDVR approach show reasonably good agreement with the results obtained by multiconfiguration time dependent Hartree (MCTDH) method. It is worthwhile to mention that the parallelized TDDVR algorithm reduces the computation time by more than an order of magnitude compared to its serial analog and, therefore, such approach appears to be a good compromise between accuracy and speed for a large molecular system.

The effect of phonon modes on the H2(v, j)/D2(v, j)–Cu(1nn) scattering processes

We include the effect of the phonon modes originating from the three layers of Cu(1nn) surface atoms on the dynamics of incoming molecular [H(2)(v, j)/D(2)(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes calculated within harmonic approximation. Such time and temperature dependent effective Hamiltonian is further subdivided assuming a weak coupling between the two sets of molecular DOFs, namely, (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of Time Dependent Discrete Variable Representation (TDDVR) approach. We demonstrate the workability of TDDVR method to investigate the scattering of H(2)(v, j) on Cu(1nn) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results show that the phonon modes affect (a) the state-to-state transition probabilities of the scattered H(2) molecule substantially but chemisorption and physisorption processes negligibly and (b) the reaction probability of the incoming D(2) molecule noticeably.

The multistate multimode vibronically coupled nuclear dynamics of monofluorobenzene radical cation using a parallelized TDDVR approach

We have investigated the molecular dynamics of C 6 H 5 F + after excitation from the ground to different electronically excited states by using our parallelized Time Dependent Discrete Variable Representation method. Since the system consists of five electronic states with several conical intersections, a complex dynamical phenomena arises. TDDVR calculated population dynamics and various spectra, namely photoelectron, mass analyzed threshold ionization and photoinduced Rydberg ionization, etc. show good agreement with the results obtained by multiconfiguration time-dependent Hartree method as well as experimental findings. The parallelized TDDVR algorithm reduces the computation time drastically and appears to be a good compromise between accuracy and speed for such large molecular system.

A quantum-classical simulation of the nuclear dynamics in NO 2 − and C6H 6 + with realistic model Hamiltonian

We perform the nuclear dynamics simulation to calculate the broad band as well as better resolved (a) photodetachment spectra of nitrogen dioxide anion (NO −2 ), and (b) photoelectron spectra of benzene radical cation (C6H +6 ) with degeneracy among the electronic states using our parallelized time dependent discrete variable representation approach. For this purpose, we first consider a theoretical model that describes the photodetachment spectrum originating due to the Franck-Condon (FC) transition from the anionic ground state to the conically intersecting \(\tilde X^2 A_1 - \tilde A^2 B_2\) electronic manifold of the neutral NO2, whereas two multi-state, multi-mode model Hamiltonians of C6H +6 with degeneracy among the electronic states are explored for its’ photoelectron spectra. The dynamics of C6H +6 has been carried out with two Hamiltonians: (a) one consists with four states and nine modes, and (b) the other is constituted with three states and seven modes. Since the electronic states of both the systems are interconnected by several conical intersections in the vicinity of the FC region, it would be challenging to persue the dynamical calculation due to the impact of non-adiabaticity among the electronic states. The spectral profiles obtained with the advent of TDDVR approach show reasonably good agreement with the results obtained by other theoretical methodologies and experimental measurements.

A “classical” trajectory driven nuclear dynamics by a parallelized quantum-classical approach to a realistic model Hamiltonian of benzene radical cation

We explore the workability of a parallelized algorithm of time-dependent discrete variable representation (TDDVR) methodology formulated by involving “classical” trajectories on each DOF of a multi-mode multi-state Hamiltonian to reproduce the population dynamics, photoabsorption spectra and nuclear dynamics of the benzene radical cation. To perform such dynamics, we have used a realistic model Hamiltonian consists of five lowest electronic states (X2E1g, B2E2g, C2A2u, D2E1u, and E2B2u) which are interconnected through several conical intersections with nine vibrational modes. The calculated nuclear dynamics and photoabsorption spectra with the advent of our parallelized TDDVR approach show excellent agreement with the results obtained by multiconfiguration time-dependent Hartree method and experimental findings, respectively. The major focus of this article is to demonstrate how the “classical” trajectories for the different modes and the “classical” energy functional for those modes on each surface can enlight the time-dependent feature of nuclear density and its' nodal structure. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Tunneling dynamics and spawning with adaptive semiclassical wave packets

Tunneling through a one-dimensional Eckart barrier is investigated using a recently developed propagation scheme based on semiclassical wave packets. This version of the time-dependent discrete variable representation method yields linear equations for the parameters, is fully adaptive, and does not require a frozen ansatz in order to approximate the exact solution of the Schrödinger equation accurately. We rely on an analytical result to derive a new algorithm to spawn a second family of semiclassical wave packets after the tunneling has occurred. Numerical results for a benchmark problem demonstrate the accuracy of the new method.

A parallelised quantum-classical approach to the molecular dynamics of allene () radical cation

We are investigating the molecular dynamics of the allene system using a parallelised Time Dependent Discrete Variable Representation (TDDVR) methodology by employing a corresponding cation () where its two electronic surfaces (A 2 E and B 2 B 2) are vibronically coupled with each other. In fact the allene radical cation exhibits a three-surface system due to the presence of degeneracy in the A 2 E state. Our initial investigation is carried out on a linear vibronically coupled model Hamiltonian consisting of 11 vibrationally active modes with two potential energy surfaces. We included both the bilinear and quadratic coupled terms in the effective three-surface Hamiltonian of the radical cation. The spectral profiles obtained from the higher order Hamiltonian show better agreement with the experimental spectrum than the result corresponding to the linearly coupled one. Along with the spectral calculation, we also analyse the nuclear cum population dynamics of the same system. The TDDVR calculated spectral profile as well as the population dynamics show reasonably good agreement with the results calculated by an exact quantum mechanical (multiconfiguration time-dependent Hartree, MCTDH) approach as well as experimental measurement. It appears that the TDDVR approach for those large systems where the quantum mechanical description is needed in a restricted region, is a good compromise between accuracy and speed.

The multistate multimode vibronic dynamics of benzene radical cation with a realistic model Hamiltonian using a parallelized algorithm of the quantumclassical approach

We demonstrate the workability of a parallelized algorithm of the time-dependent discrete variable representation (TDDVR) method to explore the detailed dynamical aspects of vibronic interaction in two three-state model Hamiltonians (X (2)E(1g), B (2)E(2g), C (2)A(2u) and B (2)E(2g), D (2)E(1u), E (2)B(2u)) of benzene radical cation along with a preliminary investigation on its five electronic states (X (2)E(1g), B (2)E(2g), C (2)A(2u), D (2)E(1u), and E(2)B(2u)). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, respectively, in order to perform the quantum dynamics on such system. The population profiles calculated by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mechanical (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calculated) photoabsorption spectra originating from various electronic states agree well with the experimental ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large molecular system, where quantum mechanical description is needed in a restricted region.

Layered discrete variable representations and their application within the multiconfigurational time-dependent Hartree approach

The multiconfigurational time-dependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by employing a layered representation: in the upper layer of the representation the multidimensional wave function is expanded in a basis of time-dependent single-particle functions, in the lower layer then these single-particle functions are represented in a larger time-independent basis. This two layer approach can be generalized to a multilayer MCTDH representation to further increase the numerical efficiency. In recent work [U. Manthe, J. Chem. Phys. 128, 164116 (2008)], a quadrature scheme has been introduced which facilitates multilayer MCTDH calculations for Hamiltonians with general potential energy functions. Based on time-dependent discrete variable representation (DVR) grids corresponding to the single-particle function bases the scheme employs the correlation DVR (CDVR) approach. The present work shows that serious problems arise when this original multilayer CDVR approach is applied for accurate calculations of vibrational states. A solution to these problems is presented which uses a revised scheme to define the time-dependent DVR grids entering into the CDVR scheme. Applications studying the vibrational states of CH(3) demonstrate that the resulting revised multilayer MCTDH/CDVR approach yields smoothly convergent and accurate results.

A quantum-classical approach to the molecular dynamics of butatriene cation with a realistic model Hamiltonian

We are investigating the molecular dynamics of the butatriene cation after excitation from the ground state (X(2)B(2g)) to the first excited electronic state (A(2)B(2u)) by using the time-dependent discrete variable representation (TDDVR) method. The investigation is being carried out with a realistic 18-mode model Hamiltonian consisting of all the vibrational degrees of freedom of the butatriene molecule. First, we perform the simulation on a basic five mode model, and then by including additional thirteen modes as bath on the basic model. This sequential inclusion of bath modes demonstrates the effect of so called weak modes on the subsystem, where the calculations of energy and population transfer from the basic model to the bath quantify the same effect. The spectral profile obtained by using the TDDVR approach shows reasonably good agreement with the results calculated by the quantum mechanical approach/experimental measurement. It appears that the TDDVR approach for those large systems where quantum mechanical description is needed in a restricted region, is a good compromise between accuracy and speed.