Fourier Grid Hamiltonian Research Articles

The double minimum E(3)[formula omitted] state in Cs[formula omitted]

The double minimum E1Σu+ state in caesium dimer was investigated by analysing spectra of the E1Σu+← X1Σg+ band system, simplified by polarisation labelling. A total of 6727 rotationally resolved transitions to levels situated in the inner well and above the internal barrier were identified and a potential energy curve allowing to reproduce their energies was constructed using the Fourier grid Hamiltonian and inverted perturbation approach methods. The unambiguity of the potential curve in view of lack of data related to levels located in the outer well is discussed.

Tunnelling Dynamics of Kicked Systems: A Route to Tunnelling Control.

The effects of discontinuously time-varying perturbations on the dynamics of a particle moving in harmonic, symmetric double well and symmetric triple well potentials, are investigated both classically and quantum mechanically. The quantum dynamics is followed using the time-dependent Fourier grid Hamiltonian (TDFGH) method while the classical dynamics is analyzed within the framework of classical Hamiltonian mechanics. Depending on the spatial symmetry of the perturbation and the characteristic features of the reversal time , different types of 'phase space' structures are observed in each of the potentials. For symmetric double and triple well potentials, quantum dynamics reveals that complete destruction of tunnelling (CDT) can be achieved in the presence of a time-dependent spatially asymmetric perturbing field that is continuous in time. Any discontinuity in time-variation of the perturbation may induce over the barrier transition. The relevance of these results in the context of (i) tunnelling control and (ii) quantum computing with 3-state or 2-state quantum registers is briefly discussed.

Design of laser pulse to enhance the photo-dissociation of a tri-atomic molecular system using optimal control theory based adaptive simulated annealing technique

In this communication, we design and optimize laser pulses to enhance the dissociation of a tri-atomic molecule using optimal control theory based adaptive simulated annealing technique. ClCN is chosen as tri-atomic molecular system as molecules containing fluorine and chlorine atoms are involved in the depletion of ozone layer and also dissociative electron attachment is possible. Fourier Grid Hamiltonian based 2D mean field methodology has been used for numerical calculations. We find an effective laser pulse by combining polychromatic Gaussian pulses and polychromatic linear pulses with different frequencies and different field intensities.

Multicomponent heat-bath configuration interaction with the perturbative correction for the calculation of protonic excited states.

In this study, we extend the multicomponent heat-bath configuration interaction (HCI) method to excited states. Previous multicomponent HCI studies have been performed using only the variational stage of the HCI algorithm as they have largely focused on the calculation of protonic densities. Because this study focuses on energetic quantities, a second-order perturbative correction after the variational stage is essential. Therefore, this study implements the second-order Epstein-Nesbet correction to the variational stage of multicomponent HCI for the first time. Additionally, this study introduces a new procedure for calculating reference excitation energies for multicomponent methods using the Fourier-grid Hamiltonian (FGH) method, which should allow the one-particle electronic basis set errors to be better isolated from errors arising from an incomplete description of electron-proton correlation. The excited-state multicomponent HCI method is benchmarked by computing protonic excitations of the HCN and FHF- molecules and is shown to be of similar accuracy to previous excited-state multicomponent methods such as the multicomponent time-dependent density-functional theory and equation-of-motion coupled-cluster theory relative to the new FGH reference values.

Theoretical study of Zeno and anti-Zeno effects on photodissociation dynamics: A model approach

Zeno and anti-Zeno effects in the evolution of the multi-photonic dissociation dynamics of the diatomic molecule HBr[Formula: see text] owing to repeated measurements demand if the system in the initial state have been studied. The effects have been calculated numerically for the case of vibrational population transfer and dissociation dynamics of HBr[Formula: see text] taking it as a model. We use time-dependent Fourier grid Hamiltonian (TDFGH) method as a mathematical tool in presence of intense radiation field as perturbation. The effects have been explored through a probable mechanism of population transfer from the ground vibrational state to the different upper vibrational states which ultimately go to the dissociation continuum. The results show significant differences in the mechanism of population transfer and the significant role of time interval of measurement ([Formula: see text] in Zeno and anti-Zeno effects. In case of survival probability of ground vibrational states, there is Zeno effect when the frequency of the laser to which the molecule is submitted is near the vibrational [Formula: see text] to [Formula: see text] resonance, while there is anti-Zeno effect if it is far from this resonance.

Fourier Grid Hamiltonian Method for calculating the Einstein coefficients, Franck–Condon factors, r-Centroids, average internuclear separations and radiative lifetimes for N2 and CO molecules

The Einstein coefficients and radiative lifetimes for diatomic molecules such as N2 and CO have been computed using the Fourier Grid Hamiltonian (FGH) method and a comparison with other methods has been made. The Hamiltonian matrix is constructed using the Hulburt and Hirschfelder (HH) potential model and the Rydberg-Klein-Rees (RKR) potential model for diatomic molecules. The FGH method works well for any model of the potential including the tabulated forms of experimental potential energy data. The eigenvalues and vibrational wave-functions for the ground and the exited state of N2 and CO molecules have been computed by diagonalizing the Hamiltonian matrix. The vibrational wave-functions and eigenvalues for the ground and the excited states are used to calculate the radiative transition parameters such as Einstein coefficients (Av″v′), radiative lifetimes, electronic transition dipole moments (Re(x)), Franck–Condon factors (FCFs), r-Centroids and average internuclear separations for both the models. FGH method work well for both the models. There are good agreements between our computed values of radiative transition parameters for both the models and the values estimated by other methods.

Role of polar medium on laser induced dissociation dynamics of ClCN: A theoretical study

Photo dissociation dynamics of linear triatomic molecule, cyanogen chloride (ClCN) in polar medium have been explored using Fourier grid Hamiltonian (FGH) method based two-dimensional mean field methodology. Ground electronic state potential of the molecule is calibrated with coupled cluster singles and doubles (CCSD) level ab initio theory using Gaussian 03 package taking 6-311++G(3df,2pd) as the basis set and fitted with two-dimensional coupled Morse potential and this potential is used for numerical calculation. Time-integrated probability current density is used as the measure of dissociation probability. Monochromatic and bichromatic laser fields are used as the external field and dissociation probability is calculated as the function of field frequency, dipolar position and pulse shape.

Correlation between Franck-condon Factors and Average Internulcear Separations for Diatomics Using the Fourier Grid Hamiltonian Method

The Fourier Grid Hamiltonian (FGH) method is used to compute the vibrational eigenvalues and eigenfunctions of bound states of diatomic molecules. For these computations, the Hulburt and Hirschfelder (HH) potential model for diatomics is used. These potential energy functions are used for constructing and diagonalizing the molecular Hamiltonians. The vibrational wave functions for the ground and the excited states are used to calculate the Franck-Condon factors (FCFs), r-Centroids and average internuclear separations which play a significant role in determining the intensity of the bands in electronic transitions. The results of FCFs and r-Centroids for diatomic molecules such as H2, N2, CO, I2 and HF using the FGH method are compared with other methods. The Franck-Condon profiles indicate a strong correlation between the values of Franck-Condon factors and the mean internuclear separations for the corresponding transitions.

Franck–Condon Factors for Diatomics: Insights and Analysis Using the Fourier Grid Hamiltonian Method

Franck–Condon factors (FCFs) play a crucial role in determining the intensities of the vibrational bands in electronic transitions. In this article, a relatively simple method to calculate the FCFs is illustrated. An algorithm for the Fourier Grid Hamiltonian (FGH) method for computing the vibrational wave functions and the corresponding energy values for different electronic states of diatomics is outlined. For these computations, the Morse potential energy forms are used for constructing and diagonalizing the molecular Hamiltonians. Once the vibrational wave functions for the ground and the excited states are known, the vibrational overlap integrals are calculated by using the formula ∫ψ*v″ψv′ dτn, where dτn is the volume element for nuclear coordinate, v′ and v″ are the vibrational quantum numbers in the ground and exited electronic states, and ψv′ and ψv″ denote the vibrational wave functions. The effects of the changes in the location, height, and the nature of the excited state electronic energy curve (relative to the ground state) on the FCFs have been evaluated. The method is also illustrated for N2 and O2. This method can be effectively used for introducing the FCFs to the students of undergraduate molecular spectroscopy course and also for post-graduate students.

Program package for numerical solving time-independent Schrödinger equation for vibrational problems: Inclusion of coordinate dependent reduced masses

In this article an improved program package for variational solving of the time-independent Schrödinger equation (SE) is presented. The program is coded in FORTRAN-95 and is aimed to solve the vibrational SE on a generalized potential energy hypersurface (PES), typically constructed pointwise by quantum chemical calculations. The procedure consists of fitting of the hypersurface to either cubic splines or displaced Gaussians. The central part is the discrete representation of the vibrational Hamiltonian on a uniform grid according to the Fourier Grid Hamiltonian (FGH) scheme. The FGH is tuned for expression of the vibrational problem in terms of generalized internal coordinates (bond distances, angles, torsions, etc.), which is the most convenient and hence widely used representation for chemical systems. In the formalism of generalized internal coordinates the reduced masses assume quite a sophisticated coordinate-dependent form. The FGH matrix is diagonalized and its eigenvalues and eigenfunctions are analyzed giving rise to observables that can be compared with the experimental data (vibrational spectroscopy in the first place). We tested the package for three distinct chemical problems; namely, the umbrella inversion of ammonia, OH stretching motion of sodium hydrogen bissulfate, and hydrogen dynamics in soybean lipoxygenaze-1 (SLO-1), yielding good agreement with the available experimental and previously published computational data. The program package is available from the authors on request.

Laser control of photodissociation process in diatomic molecule

AbstractIn this work, we first aim to realize the complete laser‐induced photodissociation of the OH molecule, and then intend to control the wavepacket generated on the continuum state, i.e., to achieve the laser control of the above‐threshold dissociation (ATD) spectrum. To numerically solve the Schrödinger equation, we adopt the split operator method (SOM), which conserves the norm of the state vector, and can treat both discrete and continuum states simultaneously and correctly. This photodissociation process induced by the multiphoton absorption involves the ATD spectrum due to the continuum‐continuum transition by the intense electric field. First, we investigate the detailed mechanism of the complete photodissociation with the one‐color laser pulse by changing the laser parameters. Then, we investigate the control of the ATD spectrum by using the two‐color laser field, where we focus on the role of the relative phase and position of two laser pulses. To analyze the population of both discrete and continuum states involved in the resultant wavepacket, we show the effective method by means of the quasicontinuum state on the Morse potential obtained by numerically diagonalizing the Fourier grid Hamiltonian (FGH). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Modeling of thermal effects on photo-dissociation dynamics of diatomic molecular ion HCl +

The dynamics of photo-dissociation of a diatomic molecular ion (HCl +) interacting with ultra short infrared laser pulses have been explored at both zero and non-zero temperatures. The potential that we have used for the systems was calibrated with the help of the ab initio theoretical calculation at the CCSD/aug-cc-pVTZ level and then fitted with an appropriate Morse potential. The vibrational eigenvalues and eigenfunctions were obtained by the time-independent Fourier grid Hamiltonian (FGH) method. Then, the time dependent Schrödinger equation is solved by invoking the time dependent FGH method. The changes of photo-dissociation probability of HCl + have been explored with the variation of the light intensity, and temperature. Our model has been introduced to mimic the medium induced perturbation.

Fourier Grid Hamiltonian Method for Solving the Vibrational Schrödinger Equation in Internal Coordinates: Theory and Test Applications

A derivation of the vibrational Hamiltonian in generalized (internal) coordinates within a grid representation using the Fourier Grid Hamiltonian (FGH) method is presented. The objective of the the...

Fourier grid hamiltonian method and lagrange-mesh calculations

Bound state eigenvalues and eigenfunctions of a Schrodinger equation or a spinless Salpeter equation can be simply and accurately computed by the Fourier grid Hamiltonian (FGH) method. It requires only the evaluation of the potential at equally spaced grid points, and yields the eigenfunctions at the same grid points. The Lagrange-mesh (LM) method is another simple procedure to solve a Schrodinger equation on a mesh. It is shown that the FGH method is a special case of a LM calculation in which the kinetic energy operator is treated by a discrete Fourier transformation. This gives a firm basis for the FGH method and makes possible the evaluation of the eigenfunctions obtained with this method at any arbitrary values.

Fourier grid Hamiltonian multiconfigurational self-consistent-field: A method to calculate multidimensional hydrogen vibrational wavefunctions

The Fourier Grid Hamiltonian Multiconfigurational Self-Consistent-Field (FGH-MCSCF) method for calculating vibrational wavefunctions is presented. This method is designed to calculate multidimensional hydrogen nuclear wavefunctions for use in mixed quantum/classical molecular dynamics simulations of hydrogen transfer reactions. The FGH-MCSCF approach combines a MCSCF variational method, which describes the vibrational wavefunctions as linear combinations of configurations that are products of one-dimensional wavefunctions, with a Fourier grid method that represents the one-dimensional wavefunctions directly on a grid. In this method a full configuration interaction calculation is carried out in a truncated one-dimensional wavefunction space [analogous to complete active space self-consistent-field (CASSCF) in electronic structure theory]. A state-averaged approach is implemented to obtain a set of orthogonal multidimensional vibrational wavefunctions. The advantages of the FGH-MCSCF method are that it eliminates the costly calculation of multidimensional integrals, treats the entire range of the hydrogen coordinates without bias, avoids the expensive diagonalization of large matrices, and accurately describes ground and excited state hydrogen vibrational wavefunctions. This paper presents the derivation of the FGH-MCSCF method, as well as a series of test calculations on systems comparing its performance with exact diagonalization schemes.

Adiabatic evolution of eigenspectra of model 1-D and 2-D Hamiltonians: Quantum adiabatic switching algorithm in a time-independent Fourier grid Hamiltonian framework

We explore the viability of a time-independent quantum adiabatic switching algorithm in the Fourier grid Hamiltonian (FGH) framework in the presence of degeneracy, avoided crossing, and chaos. The algorithm is simple and cost effective and provides information about the full eigenspectrum of the evolving Hamiltonian. It is shown to be capable of capturing accurately the change in the pattern of level spacing distribution statistics as one switches from a nonchaotic region of parameter values into the chaotic region. The Transition turns out to be less sharp than anticipated. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 133–141, 1998

Quasibound states in long-range alkali dimers: Grid method calculations

A local minimum is found in the 0g+ long range potential curves of the K2 and Rb2 alkali dimers. This well-of magnitude 42 cm−1 for K2 and 93 cm−1 for Rb2—is located above the first ns+n2P3/2 dissociation limit and metastable states could be populated using laser light blue detuned compared to the resonance line. To compute the previously unknown energies and lifetimes of these quasibound states, two grid methods are employed. One method is based on diagonalizing a Fourier grid Hamiltonian, the other uses a propagation technique in imaginary time to filter out vibrational eigenfunctions. Equivalent results are given by both methods. Then the lifetimes are extracted from the correlation function obtained by propagation in real time of these numerical vibrational wave functions. The methods are employed both in adiabatic representation with one electronic potential curve and in diabatic representation with two potential curves. Two quasibound states are found for K2, and three for Rb2 above seven stable bound states. Their lifetimes vary from 20 ps to 3 ns.

Energetic and Dynamic Aspects of Intramolecular Proton Transfer in 4-Methyl-2,6-diformylphenol: A Detailed Analysis with AM1 Potential Energy Surfaces

The energetics of ground and excited state proton transfer in an isolated 4-methyl-2,6-diformylphenol molecule has been systematically studied by the configuration interaction method at the AM1 level of approximation. The ground singlet and the excited triplet are predicted to have rather large activation barriers on the respective proton transfer paths while the barrier height is much lower on the corresponding singlet surface. The process is predicted to be endothermic in the ground state and exothermic in the excited singlet and more so in the triplet state. From an analysis of the under-barrier vibrational levels supported by the asymmetric double-well potential characterizing two forms, it appears that proton transfer may occur by a vibrationally assisted over-barrier process as well as by a tunneling mechanism following the S0 → S1 excitation. Complex coordinate rotation calculation in the Fourier grid Hamiltonian (FGH) framework shows that the tunneling rate constant from the ν = 0 vibrational level in the S1 state is of the same order of magnitude as the experimentally obtained rate constant.

The Fourier Grid Hamiltonian Method Based on Even Grid Numbers

A new grid based method to solve the Schr\"{o}dinger equation is presented. The method based on the discrete Fourier transform algorithm is used to calculate bound state eigenvalues and eigenfunctions for diatomic molecules. The Hamiltonian matrix elements are calculated analytically which requires only the evaluation of the potential at certain grid points.

Dynamics of vibrationally assisted tunnelling in model multidimensional systems. Three routes involving the Fourier grid Hamiltonian method

The multidimensional complex-scaled Fourier grid Hamiltonian (FGH) and time-dependent FGH method are invoked to study the dynamics of vibrationally assisted or subdued tunnelling in model two-dimensional systems that can mimick the behaviour of many real proton transfer systems. In the time-dependant description a Hartree-type meanfield method and a propagator method in a grid basis set are explored. The coupling between tunnelling and anharmonic vibrational modes are seen to affect the tunnelling rates in different ways depending upon the types and magnitudes of the coupling. Implications of our results with respect to H-atom or proton transfer reactions are considered.