Log-derivative Method Research Articles

ABC+D: A time-independent coupled-channel quantum dynamics program for elastic and ro-vibrational inelastic scattering between atoms and triatomic molecules in full dimensionality.

We discuss the details of a time-independent quantum mechanical method and its implementation for full-dimensional non-reactive scattering between a closed-shell triatomic molecule and a closed-shell atom. By solving the time-independent Schrödinger equation within the coupled-channel framework using a log-derivative method, the state-to-state scattering matrix (S-matrix) can be determined for inelastic scattering involving both the rotational and vibrational modes of the molecule. Various approximations are also implemented. The ABC+D code provides an important platform for understanding an array of physical phenomena involving collisions between atoms and molecules.

Ultracold atom-atom collisions by renormalized Numerov method

The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in <sup>39</sup>K-<sup>133</sup>Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.

A Time-Independent Quantum Approach to Ro-vibrationally Inelastic Scattering between Atoms and Triatomic Molecules

A full-dimensional time-independent quantum mechanical theory for ro-vibrationally inelastic scattering of triatomic molecules with atoms is formulated. The Jacobi-Radau coordinate system used in the calculation allows not only a near perfect description of the vibrational problem but also the adaptation of the exchange symmetry for A2B type triatoms. The S-matrix elements are obtained by solving the close-coupling equations with contracted basis using the log-derivative method. This method is applied to the inelastic scattering of the water molecule by a chlorine atom, which sheds light on the energy gap law in energy transfer in atom-triatom collisions.

A new proof of the septuple product identity

By means of Liouville’s theorem and the log-derivative method, we give a new proof of the septuple product identity.

Riccati differential equation for quantum mechanical bound states: Comparison of numerical integrators

AbstractComputational comparison is presented of several integrators for the Riccati differential equation for the one‐dimensional quantum mechanical bound state problem. The computational method for the quantum Hamilton‐Jacobi equation for stationary states is briefly reviewed. An interpolation formula for the quantum momentum function is proposed for the evaluation of the phase integral. The Möbius integrator, the R‐matrix propagation method, the log derivative method, and the symplectic integrator can accurately pass through singularities in the solution. These four integrators are actually Möbius transformations with different coefficients. Analytical analysis and numerical results demonstrate that the classical momentum is a good choice for the initial condition to obtain the bound state solution of the Riccati differential equation as long as the starting point of the propagation is far enough from the origin. The computational comparison is demonstrated for the harmonic oscillator. The numerical results indicate that the symplectic integrators achieve highest accuracy, and the simplicity of these algorithms shows an advantage over the other methods. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

A five-dimensional quantum scattering model for the type AB+XCD3⇌ABX+CD3 reaction in hyperspherical coordinates: Application to OH+CH4⇌H2O+CH3

A reduced dimensionality (RD) reactive quantum scattering model has been presented for the general type AB+XCD3⇌ABX+CD3 reaction, where the CD3 fragment has C3V symmetry. The model Hamiltonian and quantum scattering formulas are derived in hyperspherical coordinates, together with a single hyperradius hyperspherical projection method for boundary conditions. The time-independent coupled channel equations are propagated using a log-derivative method. The surface functions are produced by a guided spectral transform (GST) Lanczos algorithm. We also describe a GST Lanczos method to solve the generalized eigenvalue problem in a nonorthogonal basis. The RD model has been applied to the OH+CH4⇌H2O+CH3 reaction. A comparison of calculated results with previous theoretical and experimental ones is made.

Quantum dynamics of the O(3P)+CH4→OH+CH3 reaction: An application of the rotating bond umbrella model and spectral transform subspace iteration

We have applied the rotating bond umbrella (RBU) model to perform time-independent quantum scattering calculations of the O(3P)+CH4→OH+CH3 reaction based on a realistic analytic potential energy surface. The calculations are carried out in hypercylindrical coordinates with a log-derivative method incorporating a guided spectral transform (GST) subspace iteration technique. A single sector hyperspherical projection method is used for applying the boundary conditions. The results show that ground-state CH4 gives CH3 that is rotationally cold. For CH4 initially vibrationally excited in the C–H stretch or the H–CH3 bending mode, a bimodal CH3 rotational distribution has been observed. The product OH is a little vibrationally excited, while the umbrella mode of CH3 is moderately excited. Vibrational excitation enhances the reactivity substantially. The calculated rate constants are in good agreement with experimental measurements.

Four-dimensional quantum scattering calculations on the H+CH4→H2+CH3 reaction

Time-independent quantum scattering calculations have been performed to study the H+CH4→H2+CH3 reaction, using the analytic potential-energy surface developed by Jordan and Gilbert. A rotating bond umbrella (RBU) approximation with the implementation of a guided spectral transform subspace iteration technique has been applied together with a log-derivative method in hyperspherical coordinates. A single sector hyperspherical projection method was used to apply the boundary conditions to extract the S matrix at a large hyperradius. The results show that the H+CH4→H2+CH3 reaction occurs via a direct mechanism. The tunneling effect is pronounced, while there is little recrossing. Vibrational excitation of the C–H stretch and/or the H–CH3 bending modes of CH4 significantly enhance the reactivity. Exciting the umbrella mode of CH4 also enhance the reactivity, although less efficiently. The calculated thermal rate constants are larger than the experimental ones. However, good agreement has been obtained by including a barrier height correction of the potential function to make it agree with ab initio results. Finally, vibrational and rotational distributions of the reaction products are discussed in detail.

Error estimation in molecular vibrational energies calculated by Johnson's log-derivative method

Estimates are made of the truncation errors in molecular vibrational energies when calculated by Johnson's log-derivative method. Numerical comparisons are made for some potentials whose eigenenergies are known analytically.

A four dimensional quantum scattering study of the Cl+CH4⇌HCl+CH3 reaction via spectral transform iteration

We present a quantum dynamics study of the Cl+CH4⇌HCl+CH3 reaction using a four-dimensional rotating bond umbrella (RBU) model. A semiempirical potential energy surface is employed, where the zero point energy of the modes not explicitly treated in the RBU calculations is approximately included. The potential gives a vibrationally adiabatic ground state barrier height of 3.48 kcal/mol. The calculations have been performed in hypercylindrical coordinates using a log-derivative method. A single sector hyperspherical projection method has been developed for applying boundary conditions. A guided spectral transform (GST) Krylov subspace method has been constructed to find the eigenstates of the coupling matrix appearing in the coupled channel equations. The results show that the product methyl is rotationally cold for the forward reaction. A pronounced tunneling effect on the rate constants was obtained. The calculated thermal rate constants are 12%–45% smaller than the experimental results over the temperature range 200–800 K for the Cl+CH4⇌HCl+CH3 reaction. State-to-state differential cross sections have been studied at a kinetic energy of 0.159 eV. A detailed comparison with experimental measurements is made.

Three-dimensional quantum scattering calculations on the Cl+CH4HCl+CH3 reaction

Reduced dimensionality quantum scattering calculations are performed on the Cl+CH4HCl+CH3 reaction using a rotating line umbrella (RLU) model. The breaking and forming bonds are treated explicitly as is the umbrella motion (ν2) in CH3 and the correlating motion (ν4) in CH4. A potential energy surface, where the zero-point energy of the modes not explicitly treated in the scattering calculations are approximately included, has been constructed from a surface for CH4, due to Hase and collaborators, and an LEPS function, treating the reactive bonds. The calculations are carried out using a log-derivative propagation method in hypercylindrical coordinates. A single-sector hyperspherical projection method has been developed for applying boundary conditions. The forward reaction rate is enhanced by placing one quantum either in the ν3b mode of CH4, corresponding to the breaking bond, or the ν4 mode. The reverse reaction is enhanced by placing one quantum in the HCl vibration but is impeded by placing a quantum in the CH3 umbrella mode. For Cl+CH4(ν3b=1)→HCl(ν=1)+CH3 and a translational energy of 0.159 eV, calculated differential cross-sections give predominantly forward and sideways scattering without excitation of the umbrella motion of CH3. For Cl+CH4(ν3b=1)→HCl(ν=0)+CH3 there is predominantly backward scattering with excitation of the umbrella mode. Comparison with experimental measurements is made.

Reduced dimensionality quantum scattering calculations on the Cl+CH4→HCl+CH3 reaction

Reduced dimensionality quantum scattering calculations using the Rotating Line Approximation, RLA, are performed on the Cl+CH4→HCl+CH3 reaction, treating CH4 as a pseudo-diatom QH. A LEPS potential energy surface is used, where the zero-point energy of the modes not explicitly treated in the RLA calculations are included. The calculations are performed using hyperspherical coordinates and the improved log-derivative method of Manolopoulos. Boundary conditions have been applied using a hyperspherical projection method and an approximate method where the boundary conditions are applied directly in the hyperspherical coordinates. It is explicitly shown that the agreement between the methods is good. Scattering resonances are observed and related to the shape of the hyperspherical adiabats. Cumulative and state-to-state reaction probabilities are also presented. Vibrational adiabaticity is found to dominate without being exclusive. For Cl+CH4(v=1)→HCl(n=1)+CH3 and a translational energy of 0.159 eV, measured differential cross sections give predominantly forward scattering, while for Cl+CH4(v=1)→HCl(n=0)+CH3 there is predominantly backward and sideways scattering. Our calculated differential cross sections are in agreement with these general observations. Thermal rate constants have been calculated for the temperature range 200–800 K. The calculated rate constants are smaller than the experimental ones by less than 40% over the entire temperature range.

An improved classical approach quantum encounter treatment of collision-induced vibrational energy transfer. Application to He + CO (ni = 1, 2)

Abstract The vibrational energy transfer in a three-dimensional molecular collision is assumed, in the recently published Classical Approach Quantum Encounter (CAQE) theory [J. Chem. Phys. 99 (1993) 2567], to occur in a brief and sharply defined encounter which can be formulated in terms of an effectively one-dimensional atom-diatomic collision, with the parameters for the encounter reflecting the local features of the potential in the neighborhood of the collision,geometry. Here we describe improvements to the original CAQE theory. A mathematical recasting renders the theory more conducive to use with accurate quantum encounter methods. The first-order distorted wave Born approximation that was used in the original CAQE theory is here replaced by a numerically exact one-dimensional encounter method. We illustrate the need for the new method using collinear calculations for Ar + HCl and He + CO, and then apply it using a log-derivative method in a three-dimensional calculation of single and multiple quantum vibrational cross sections for He + CO. The results are judged by comparison to the results of coupled states and semiclassical calculations, with good agreement being obtained for all conditions over a wide range of energies.

An adaptation of the log-derivative method to massively parallel computers

We present a strategy for parallelization of the log-derivative quantum dynamics method suitable for smaller basis set calculations where many different energies must be examined. Parallelization of work specific to both the scattering coordinate grid and the energy grid is accomplished but all matrix manipulations are carried out on single processors. We execute this strategy for a realistic triatomic scattering problem on three parallel computers: one modestly parallel, shared-memory computer and two massively parallel, distributed-memory computers. Results show successful parallelization on the two massively parallel computers.

Symplectic integrators for the multichannel Schrödinger equation

The multichannel radial Schrödinger equation that arises in time-independent inelastic scattering theory and certain bound state problems has a classical Hamiltonian structure in which the radial coordinate plays the role of time. One consequence of this Hamiltonian structure is that the Schrödinger equation has symplectic symmetries, which lead in the context of inelastic scattering to the unitarity and symmetry of the S matrix. Another consequence is that so-called symplectic integrators can be used to solve the radial Schrödinger equation, both for bound state and scattering problems. This idea is used here to derive a new family of symplectic integrator-based log derivative methods for solving the multichannel radial Schrödinger equation. In addition to being simpler to write down and program, these methods are shown to be highly competitive with Johnson’s original log derivative method for several inelastic scattering and bound state test problems. An equivalent solution following version of the symplectic integrator family is also introduced and shown to have similar advantages over the DeVogelaere method. A number of more formal consequences of the classical Hamiltonian structure of the radial Schrödinger equation are also noted.

On Johnson's log-derivative algorithm for a single equation

Johnson's log-derivative method is presented as a propagator for the log-derivative itself; this propagator is derived in a simple way from the Riccati equation and it is given in a more efficient form. Two numerical examples are given.

A linear reference potential algorithm for the study of molecular photodissociation

This paper describes the extension of a linear-reference potential, log-derivative method to the solution of the inhomogeneous equations which arise in a particularly useful time-independent formulation of molecular photodissociation. Explicit expressions are given here for the inhomogeneous propagators, which complement previously published expressions for the corresponding homogeneous propagators [M.H. Alexander and D.E. Manolopoulos, J. Chem. Phys. 86 (1987) 2044].

The coupled equations problem of the quantum theory of atom-diatom reactive scattering

Abstract Various formulations of the coupled equations in the quantum theory of atom-diatom reactive scattering—based on the ordinary Schrodinger equation and on the generalized equations of the arrangement channel quantum mechanics—are summarized with emphasis on symmetry properties of these formulations. New simplified and/or symmetrized versions are presented of the coupled ordinary differential equations derived previously (using the mixed Jacobi/hyperspherical coordinates) from the two distinct generalized Schrodinger equations which incorporate the arrangement channel permuting and the Fock arrangement coupling scheme, respectively. Inhomogeneous coupled equations arising from the close-coupling description of some related half-collision processes are also included in the considerations. Numerical approaches to solution of the coupled equations problems are recapitulated. The log-derivative method is presented in various versions. The material included in the presentation supplements previous review...

Lidar measurement of atmospheric aerosol extinction profiles: a comparison between two techniques—Klett inversion and pure rotational Raman scattering methods

Two lidar methods of determining an atmospheric extinction coefficient profile are compared. The methods are the Klett inversion method for elastic lidar return and the log-derivative method for rotational Raman backscattered signal processing. The comparison includes numerical modeling and processing of lidar measurements when both the elastic and the rotational Raman backscattered signals are measured simultaneously. The suggested idea is that such a comparison can be used as a criterion for the reliability of the results of lidar measurements, similar to the comparison between the results of lidar and contact measurements.

Complex log derivative method for nonreactive coupled-channel scattering calculations

A new version of Johnson's log derivative method for integrating the (nonreactive) Schrodinger equation is described. The new feature is that the log derivative equation is integrated inward (from large r to small r) with initial conditions that correspond to asymptotically incoming (or outgoing) radial waves. The result is that the log derivative function Q(r) (which is now a complex function of r) is a very smooth, nonsingular function (compared to the oscillatory, singular behavior of Q(r) with the usual real boundary conditions), meaning that many fewer grid points are required for its numerical integration. The number of grid points necessary to achieve a given level of accuracy is also only a weak function of the scattering energy E or the mass of the particles. Test calculations for a one-dimensional and a multichannel example illustrate this behavior.