Volume-Bloch Operator Approach to Quantum Reactive Scatteringon Non-orthogonal Scattering Coordinates

Abstract

The most detailed observables in theoretical studies of chemical reactions are the quantum state-to-state reaction probabilities determined as the modulus squared of the scattering matrix (S matrix) elements. Quantum mechanical methodology for S matrix calculation is well developed, especially in the level of Born-Oppenheimer approximation used for separating the nuclear dynamics from electronic motions. One of such numerical methods adopts boundary value Bloch operator in setting up the basis set representation of Schrodinger equation for the nuclear dynamical solution in the continuum region of energy (scattering energy) such as in the log derivative Kohn variational principle (Y-KVP) method of Manolopoulos and Wyatt. It has virtues of “totally energy-independent and real-valued basis functions/intermediate integrals” up to the point just before the final complex-valued S matrix evaluation. Since one must investigate the scattering solutions for many energies in the general practice of scattering problem, these properties substantially increase the computational efficiency than otherwise. However, when applied for general reactive multi-arrangement scattering on non-orthogonal coordinate system (e.g., the well-known collinear H2+H reactive scattering), it turns out to be inevitable to adopt wasteful redundant basis sets, each of which suitable for entrance and exit channels (at least, in the primitive direct product basis set description), and the subsequent cumbersome exchange-type integrations between the functions defined on different coordinate systems are unavoidable (about half of the integrals are of this type for the above example). Such formal inefficiency in the YKVP method can be avoided in the S-matrix version of Kohn variational principle (S-KVP) method by replacing the real-valued and energy-independent continuum-type basis functions with complex-valued and energy-dependent ones. However, it accompanies adverse effects, i.e., making a small rectangular part of Hamiltonian matrix to be complexvalued and energy-dependent. This work was motivated by noting that a delta function is included in the definition of conventional Bloch operator, consequently, it becomes a surface operator in multi-dimensional problem. By changing the effective region of the Bloch operator from surface to volume, we might hope to remove such unfavorable features of the Y-KVP method in a different way, and we intend to pursue this idea further. In the long run, no redundant basis sets are needed and all basis functions are real-valued and energy-independent, subsequently most of the intermediate integrals are real-valued and energy-independent except Nopen (the number of open channels) integrals which are complex-valued and energydependent in the present approach. All such features add up to suggest better performance over both of the Y-KVP and S-KVP methods, at least formally. Unfortunately, a small rectangular Bloch-operator-related matrix becomes complexvalued (though it can be evaluated from energy-independent intermediate integrals) just like in the S-KVP method. Perhaps, more importantly, it is a pity that this method is not benefited from the variational property, thus the calculated numerical S matrix is not guaranteed to be symmetric, unlike in the Y-KVP and S-KVP methods, as a result, it might be suffered from unstable features. Now we detail the derivation of the present approach. Scattering wave function ψ is chosen to satisfy the usual Smatrix boundary condition in the asymptotic region as a linear combination of incoming (I) and outgoing (O) waves with S matrix as the expansion coefficient,