Kohn Variational Principle Research Articles

Predicting observables on different potential energy surfaces using feature sensitivity analysis: Application to the collinear H+H2 exchange reaction

Two sensitivity-analysis techniques are used to show how one can predict observables on new or perturbed potential energy surfaces (PES) without doing any additional dynamics calculations on the new PESs. Both techniques require the computation of the observable (O) and its sensitivity to variations in the potential (δO/δV) on just one surface. The first approach uses a simple first order expansion of the observable with respect to δV. The second approach uses a nonlinear least-squares fit of particular features in the observable, and then uses the same functional form to predict the observable on a different PES but with a modified set of fitting parameters. The new fitting parameters are related to the old via a functional Taylor expansion. Both approaches work well when variations in the potential are small. When the potential difference is large, the second approach gives reasonable results even in cases where the first approach is giving spurious predictions. These ideas are tested for the collinear H+H2 reaction within the framework of quantum reactive scattering. The log-derivative version of the Kohn variational principle is used for the scattering calculations.

The complex coordinate scattering theory and the Kohn variational method: A general formulation and application to long range potentials

The complex coordinate scattering theory for the calculation of T-matrix elements, as was introduced by Engdahl, Moiseyev, and Maniv [J. Chem. Phys. 94, 1636 (1991)] and by Peskin and Moiseyev [J. Chem. Phys. 96, 2347 (1992)], is shown to satisfy the complex version of the Kohn variational principle introduced by Nuttall and Cohen [Phys. Rev. 188, 1542 (1969)]. This theory and the related S-matrix version of the Kohn variational principle, developed by Zhang, Chu, and Miller [J. Chem. Phys. 88, 6233, (1988)] are combined to formulate a generalized variational basis set approach for quantum scattering calculations. In this approach the Kohn variational procedure to optimize the linear parameters in the T matrix is followed by an optimization of the complex nonlinear parameters. This enables the application of the complex coordinate analytical continuation of the T matrix to the calculation of scattering probability amplitudes for long range potentials. Illustrating numerical applications to short and long range potentials are given.

Quantum functional sensitivity analysis within the log-derivative Kohn variational method for reactive scattering

A new approach to calculating quantum functional sensitivity maps of transition probabilities is described in this paper. This approach is based on the log-derivative version of the Kohn variational principle and is applied here to the collinear H+H2 hydrogen exchange reaction. The sensitivity maps provide detailed quantitative information about how variations in the potential energy surface affect the state-to-state transition probabilities. The key issues investigated are (i) the evolution of sensitivity structure in the 0.30–1.50 eV range of total energy; (ii) the comparison of sensitivity structure on the Porter–Karplus, the Liu–Siegbahn–Truhlar–Horowitz, and the double-many-body-expansion potential energy surfaces; and (iii) the range of linearity for first order sensitivity predictions.

Quantum functional sensitivity analysis for the collinear H+H2 reaction rate coefficient

The effects of features in the potential energy surface on the collinear H+H2 reaction rate coefficient are investigated by the method of quantum functional sensitivity analysis (QFSA). The calculations use QFSA to connect features in the microscopic realm, with their response upon macroscopic quantities of chemical interest, via the intermediary sensitivities of the reactive transition probabilities. While the sensitivities of the individual transition probabilities show considerable structure, there is an attendant loss of structure in the rate coefficient sensitivities because of the thermal averaging. For the range of temperatures used in our study (200–2400 K), the most important region of the potential energy surface is found to be not at the top of the barrier, but rather at the lower energy shoulders of the barrier. There are also regions near the barrier where an increase in the potential surface actually increases the reaction rate! The effects of using different underlying potentials [the Porter–Karplus (PK2), Liu–Siegbahn–Truhlar–Horowitz (LSTH), and double many-body expansion (DMBE) surfaces] on the nature of the results were also compared. The absolute sensitivity magnitudes on the PK2 surface vary considerably from the other two, but the relative change in the rate coefficient is about the same on all three surfaces. Furthermore, the identified regions of importance on the potential surfaces remain essentially the same. The reactive scattering calculations were performed with the log-derivative version of the Kohn variational principle.

Representation of electronic wavefunctions by Lobatto shape functions: application to the photoionization cross section of H +2

It is shown that a convenient and flexible representation of molecular electronic continuum wavefunctions may be constructed from Lobatto shape functions, especially in combination with conventional Gaussian-type basis functions. When used in the logarithmic derivative version of the Kohn variational principle, an efficient procedure for calculating molecular photoionization cross sections is obtained. As a first example, the method is applied to the benchmark system H + 2. Excellent results are found with rather moderate effort for the full investigated range of photoelectron energies (up to 200 eV).

Finite-volume variational method for the Dirac equation.

A finite-volume variational principle based on the Dirac Hamiltonian is formulated in a way similar to the Kohn variational principle for the logarithmic derivative used in nonrelativistic electron-atom scattering. As in {ital R}-matrix methods, the essential idea is to solve the Dirac equation variationally within a finite reaction volume for a given total energy of the system. In contrast to previous methods, we do not need to impose any unphysical boundary conditions on the trial function. The continuation of the variational solution into the outer region is represented by a standard close-coupling expansion. The {ital K} matrix is obtained by matching the variational solutions with the solutions in the outer region on the surface enclosing the reaction volume.

Quantum dynamics of the H+D2→D+HD reaction: Comparison with experiment

The Kohn variational principle for the log-derivative matrix is used to calculate integral cross sections for H+D2 (v=0, j=0) to D+HD (v′=0,1,2, all j′) at the experimentally accessible collision energies of 0.55 and 1.3 eV. Comparison is made with experimental and theoretical studies in the literature. Product state relative rotational distributions, vibrational branching ratios, and energy partitioning fractions are all in good agreement with the recent experimental results of Rinnen, Kliner, and Zare. Absolute cross sections are compared with the experimental work of Levene et al. and Johnson et al. Our results agree very well with their experiments. It is found that the quasiclassical results of Blais and Truhlar compare well with the present exact quantum mechanical predictions in many respects, however, the product rotational distributions are ‘‘hotter’’ than the quantal ones.

Schwinger and anomaly-free Kohn variational principles and a generalized Lanczos algorithm for nonsymmetric operators.

The Schwinger variational principle is combined with the Lanczos algorithm. It is shown that the straightforward approach where the Lanczos basis is generated by V-${\mathit{VG}}_{0}$V is not very useful. The difficulties arise because of the unboundedness of the Schwinger operator V-${\mathit{VG}}_{0}$V for unbound potentials (such as, e.g., the Yukawa potential). A modification of the Schwinger operator is presented that resolves the problem. This approach leads to a new Lanczos recursion for nonsymmetric (and non-Hermitian) operators yielding nonorthogonal basis functions. The resulting approach is shown to be equivalent to the continued-fraction method of Hor\'a\ifmmode \check{c}\else \v{c}\fi{}ek and Sasakawa. To illuminate the particular properties of the Schwinger-Lanczos basis we have applied it as well to other variational principles like the C\ifmmode \tilde{}\else \~{}\fi{} functional, the Newton variational principle, and the Kohn variational principle. To treat the multichannel problem we have adopted a modified band-Lanczos algorithm. This allows for a more efficient computation of the off-diagonal T-matrix elements than previous approaches.

Approximations for rough surface scattering: Beyond plane waves

Most theoretical work on rough surface scattering concerns monochromatic plane waves incident on infinite irregular interfaces in a constant sound-speed environment. Experiments, on the other hand, most often use pulsed point sources or beams in a ducted environment. Here, various approximations used for plane-wave scattering are extended to point sources in the time domain. The Kirchhoff and small-slope approximations prove to have simple forms for both the near and far field. Dashen's formula for plane waves can be generalized to a limited extent for point sources in the time domain in variable environments. Since, for plane-wave scattering from sinusoidal surfaces, the Schwinger variational principle gives scattering amplitudes whose phases are closer to exact phases than any of the standard approximations, the Kohn and Schwinger variational principles will be examined as starting points for approximations of reverberation in ducted environments. [Work supported by NRL.]

Quasi-adiabatic basis functions for the S-matrix Kohn variational approach to quantum reactive scattering

A basis set contraction scheme is described for carrying out reactive scattering calculations within the S-matrix version of the Kohn variational principle. The new basis functions are distributed Gaussians for the radial (translational) degrees of freedom, as used before, but with quasi-adiabatic internal eigenfunctions for the internal degrees of freedom; i.e., each translational Gaussian has a different internal function, the adiabatic internal eigenfunction for that translational coordinate at which the Gaussian is centered. A very efficient and easy-to-use criterion is given for selecting which of these L{sup 2} functions to include in the basis. Application to the three-dimensional H + H{sub 2} reaction for various values of total angular momentum shows that accurate results can be obtained with half (or fewer) the number of basis functions that are necessary when asymptotic channel eigenfunctions are used for the internal degrees of freedom. The basis function selection criterion allows one to obtain moderately accurate results with very few basis functions and then to improve the accuracy systematically by increasing the basis in a determined fashion.

Translational basis set contraction in variational reactive scattering

A new translational basis set is introduced for quantum reactive scattering calculations that use the log derivative version of the Kohn variational principle. This basis set, which is similar in many respects to that used in electron–atom scattering calculations by Burke and Robb, is obtained by contracting a primitive basis of Lobatto shape functions to the box eigenfunctions of a one-dimensional reference Hamiltonian H0. In addition, a single energy-dependent scattering function is included in the variational expansion to ensure completeness at the boundary of the box. One fairly obvious choice for the reference Hamiltonian in an atom–diatom reaction is suggested, and all of the equations which are actually needed to implement the method in this context are carefully described. Example applications to the three-dimensional F+H2 reaction are then chosen to illustrate the practical potential of the approach.

Converged variational quantum scattering results for the three-dimensional F+HD reaction

The log derivative version of the Kohn variational principle is used to calculate J=0 reaction probabilities and state-to-state time delays for the F+HD reaction, on the T5A potential energy surface, in the total energy range E tot=0.236 to 0.350 eV. While the computed reaction probabilities to H+DF are comparatively bland, those to D+HF show considerable resonant structure as a function of the scattering energy. For example, the computed state-to-state time delays for the predominant HF( v′=2) resonance et E tot=0.2575 eV imply the existence of a metastable collision complex with a half-life of about 0.1 ps. These results are discussed in the light of the high-resolution molecular beam experiments of Lee and coworkers and the bending corrected rotating linear model calculations of Hayes and Walker.

A comment on quantum reactive scattering via the S matrix version of the Kohn variational principle: Differential and integral cross sections for D+H2→HD+H

A comment on quantum reactive scattering via the <i>S</i> matrix version of the Kohn variational principle: Differential and integral cross sections for D+H2→HD+H

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory.

We propose new basis sets for linear algebraic variational calculations of transition amplitudes for quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained even for basis sets with the majority of the members independent of energy.

Converged variational quantum scattering results for the three dimensional F+D 2 reaction

Converged J=0 quantum reaction probabilities over the translational range 0.04 to 4.84 kcal/mol are computed for the 3D F+D 2 reaction, using the log derivative version of the Kohn variational principle. Rotational distributions for the DF ν′=3 and ν′=4 product states formed from the ν=0, j=0 initial state are plotted. The results are considered in the light of available experimental and theoretical work on the F+H 2, F+D 2 and F+HD reactions.

Variational determination of optimal orbitals for electron scattering

The Kohn variational principle for the (negative) logarithmic derivative can be used to solve the Schrödinger equation variationally for given energies in the continuum. The MCSCF formulation of the problem permits the optimization of the configuration expansion coefficients as well as the orbitals, out of which the configuration state functions are constructed. We elucidate the correspondence between the variational determination of the orbitals and the Configuration-Interaction approach.

Iterative solution in quantum scattering theory. The log derivative Kohn approach

The log derivative version of the Kohn variational principle is reviewed in the context of a general inelastic molecular collision. Particular emphasis is placed on the possibility of solving the resulting linear equations iteratively for a single initial state column of the scattering matrix, S, and several important practical observations are made in this regard. In particular, it is argued that the discrete variable representation proposed by Light and coworkers leads to an extremely sparse coefficient matrix, and so has distinct advantages over the more obvious variational basis approach. The resulting methodology is applied to the diffractive scattering of a beam of helium atoms from the (001) face of LiF. Here test calculations with up to 2601 coupled channels and 216 translational grid points clearly demonstrate the practical potential of the iterative technique. The implications of these tests for more general scattering problems are also briefly discussed.

H+H2(0,0)→H2(v′, j′)+H integral cross sections on the double many body expansion potential energy surface

The log derivative version of the Kohn variational principle is used to calculate integral cross sections for H+H2(0,0)→H2(v′, j′)+H, in the energy range Etot =0.95−1.35 eV, on the double many body expansion (DMBE) potential energy surface. The resulting DMBE cross sections are qualitatively very similar to, but somewhat larger than, the Liu–Siegbahn–Truhlar–Horowitz cross sections reported by Zhang and Miller. In particular, they show none of the interesting resonant structure seen in the recent experiments of Nieh and Valentini. While this does not prove that these experiments are wrong, it does at least indicate the extent to which errors in the best available H3 potential energy surfaces are likely to affect the calculated dynamical results.

Comparison of quantum scattering calculations for the H + H2 reaction using the LSTH and DMBE potentials

The results of a comparison of quantum scattering calculations based on the S-matrix version of the Kohn variational principle, using the Liu–Siegbahn–Truhlar–Horowitz (LSTH) and double many-body expansion (DMBE) potential-energy surfaces for the H + H2 reaction, are reported. We focus on the particular state-to-state cross-sections [(v= 0, j= 0)→(v′= 1, j′= 1, 3)] for which there is an unresolved discrepancy between experimental and theoretical results, in order to estimate the sensitivity of the theoretical results to the accuracy of the potential. The potentials differ most significantly in their bending dependence near the H3 transition state. The difference between the potentials at the saddle point varies between 1 and 2% for various fixed angles. Partial cross-sections are presented for total J= 0 and 10 over a wide range of energy (0.9–1.3 eV total energy). The energy-dependent state-to-state cross-sections for the above transitions are quite insensitive to the difference between the potentials for all values of total J considered. The product state distributions for the energies at which scattering resonances have been experimentally reported are similarly insensitive to the difference in the H3 bending potential. We conclude from this brute-force comparison that the accuracy of the quantum scattering calculations is not subject to anomalous sensitivity.

A collocation approach for quantum scattering based on the S-matrix version of the Kohn variational principle

A collocation approach to quantum scattering is presented. The method is based on the S-matrix version of the Kohn variational principle with a different linear expansion used for the two wave functions—one is a linear combination of basis functions and the other is a pointwise representation with proper asymptotic conditions imposed. The resulting equations are similar in structure to the usual version of the Kohn variational principle, however, in the present approach there are no integrals between the square integrable (L2) basis functions. In addition, the method does not require the knowledge of quadrature weights associated with the collocation points as was the case in a previous pointwise method for quantum scattering. This property means that the method is readily applicable to reactive scattering problems which use different sets of coordinates for reactants and products. Appliction to a simple inelastic test problem (collinear He–H2 vibrationally inelastic scattering) shows the accuracy of the approach to be comparable to that of the usual variatinal form of the S-matrix Kohn method.