A novel discrete variable representation (DVR) is introduced for use as the L2 basis of the S‐matrix version of the Kohn variational method [Zhang, Chu, and Miller, J. Chem. Phys. 88, 6233 (1988)] for quantum reactive scattering. (It can also be readily used for quantum eigenvalue problems.) The primary novel feature is that this DVR gives an extremely simple kinetic energy matrix (the potential energy matrix is diagonal, as in all DVRs) which is in a sense ‘‘universal,’’ i.e., independent of any explicit reference to an underlying set of basis functions; it can, in fact, be derived as an infinite limit using different basis functions. An energy truncation procedure allows the DVR grid points to be adapted naturally to the shape of any given potential energy surface. Application to the benchmark collinear H+H2→H2+H reaction shows that convergence in the reaction probabilities is achieved with only about 15% more DVR grid points than the number of conventional basis functions used in previous S‐matrix Kohn...

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A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method

A three dimensional potential energy surface for the F+H2→HF+H reaction has been computed using the internally contracted multireference configuration interaction (MRCI) method with complete active space self‐consistent field (CASSCF) reference functions and a very large basis set. Calibration calculations have been performed using the triple‐zeta plus polarization basis set employed in previous nine‐electron full CI (FCI) calculations of Knowles, Stark, and Werner [Chem. Phys. Lett. 185, 555 (1991)]. While all variational MRCI wave functions yield considerably larger barrier heights than the FCI, excellent agreement with the FCI barrier height and the exothermicity was obtained when the Davidson correction was applied (MRCI+Q). The convergence of the barrier height and exothermicity, spectroscopic constants of the HF and H2 fragments, and the electron affinity of the fluorine atom with respect to the basis set has been carefully tested. Using the largest basis sets, which included 5d, 4f, 3g, and 2h func...

An accurate multireference configuration interaction calculation of the potential energy surface for the F+H2→HF+H reaction

A new method is suggested for the calculation of the microcanonical cumulative reaction probability via flux autocorrelation relations. The Hamiltonian and the flux operators are computed in a discrete variable representation (DVR) and a well‐behaved representation for the Green’s operator, G(E+), is obtained by imposing absorbing boundary conditions (ABC). Applications to a one‐dimensional‐model problem and to the collinear H+H2 reaction show that the DVR‐ABC scheme provides a very efficient method for the direct calculation of the microcanonical probability, circumventing the need to compute the state‐to‐state dynamics. Our results indicate that the cumulative reaction probability can be calculated to a high accuracy using a rather small number of DVR points, confined to the vicinity of the transition state. Only limited information regarding the potential‐energy surface is therefore required, suggesting that this method would be applicable also to higher dimensionality problems, for which the complete potential surface is often unknown.

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Calculation of the cumulative reaction probability via a discrete variable representation with absorbing boundary conditions

The new recently introduced [J. Chem. Phys 102, 7390 (1995)] empirical recursion formula for the scattering solution is here proved to yield an exact polynomial expansion of the operator [E−(H+Γ)]−1, Γ being a simple complex optical potential. The expansion is energy separable and converges uniformly in the real energy domain. The scaling of the Hamiltonian is trivial and does not involve complex analysis. Formal use of the energy‐to‐time Fourier transform of the ABC (absorbing boundary conditions) Green’s function leads to a recursion polynomial expansion of the ABC time evolution operator that is global in time. Results at any energy and any time can be accumulated simultaneously from a single iterative procedure; no actual Fourier transform is needed since the expansion coefficients are known analytically. The approach can be also used to obtain a perturbation series for the Green’s function. The new iterative methods should be of a great use in the area of the reactive scattering calculations and o...

A simple recursion polynomial expansion of the Green’s function with absorbing boundary conditions. Application to the reactive scattering

A new method of implementing scattering calculations is presented. For the S‐matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal L2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−H)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.

Spectral projection approach to the quantum scattering calculations

A new basis set approach for quantum scattering calculations is described and tested on model problems of elastic and inelastic collisions. The approach is essentially the Kohn variational method, but applied to the S or T matrix directly rather than to the K matrix as is normally done; it is seen that the result of the present approach is not equivalent to the usual Kohn method (i.e., for the K matrix) and is indeed preferable to it. The present approach is seen to have the same structure as the complex scaling/coordinate rotation expressions for the T matrix (but with some added features). Its potential advantage over the Schwinger variational method, another useful basis set technique, is that matrix elements of the Green’s function for some reference Hamiltonian are not required; the present method requires only matrix elements of the Hamiltonian itself between the basis functions. The essential reason for all of these desirable features is that the basis set which is used incorporates the correct sca...

Bernadette M. D. D. Jansen op de Haar

Papers

A new basis set method for quantum scattering calculations