disagreement between the vibrational levels of it and those of the Bowman potential and for remaining uncertainties in the assignment of the observed levels. We have suggested in the above analysis a corrected well depth of >1075 cm'1 11for the Bowman potential. This is to be compared with the ab initio well depth of -1109 cm'1. There is clearly reasonable agreement among these two values and the 1062-cm"1 well depth of the Bowman potential. There is, however, a large discrepancy between the zero-point levels of the two surfaces that arises primarily from what appears to be an overly large contribution (~440 cm"1) from the bending vibration on the ab initio surface. This leads not only to a systematic shift in the computed vdW stretching levels compared to experiment, but to the prediction of fewer excited bending states.The fact that experimental estimates of the dissociation energy, which range from 718 to 742 cm"1, are so closely in accord with the present calculation (724 cm"1) and estimation (~739 cm"1) on the Bowman surface strongly suggests that the bending po-tential is better described by that surface. This inference is independent of the manner in which the non-vdW stretching levels are assigned. The well depth of the ab initio surface would have to be in error by enough to allow a 126-cm'1 increase in the dissociation energy. While Chakravarty and Clary10 are inclined to accept a 15% error in the ab initio well depth, one must expect such an error also to affect the bending potential.

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A general, uniformly convergent series representation of operator-valued functions in terms of Faber polynomials is presented. The method can be used to evaluate the action of any operator-valued function which is analytic in a simply connected region enclosed by a curve, Lγ. The three most important examples include the time-independent Green’s operator, G+(E)=1/[E−(H−iε)], where H may be Hermitian or may also contain a negative imaginary absorbing potential, the time-dependent Green’s or evolution operator, exp(−iHt/ℏ), and the generalized collision operator from nonequilibrium statistical mechanics, 1/[E−(ℒ−iε)], where ℒ is the Liouvillian operator for the Hamiltonian. The particular uniformly convergent Faber polynomial expansion employed is determined by the conformal mapping between the simply connected region external to the curve Lγ, which encloses the spectrum of H−iε (or ℒ−iε), and the region external to a disk of radius γ. A locally smoothed conformal mapping is introduced containing a finite number of Laurent series terms. This results in an equal number of terms in the recursion of the Faber polynomials and avoids a serious memory problem in a calculation for a large system. In addition, this conformal mapping uniquely determines a scaled Hamiltonian, which when combined with the radius γ, ensures a completely stable recursion relation for calculating the Faber polynomials of the operator of interest (i.e., the Hamiltonian or Liouvillian). We earlier showed that for Lγ chosen to be an ellipse, the Faber polynomial expansion provides the generalization to non-Hermitian H of the Chebychev polynomial expansion of G+(E) [Chem. Phys. Lett. 225, 37 (1994); 206, 96 (1993)]; the present results provide a similar generalization for the Chebychev expansion of e−iHt/ℏ [Tal-Ezer and Kosloff, J. Chem. Phys. 81, 3967 (1984)]. Nonelliptic Lγ lead to other, new polynomial representations having superior convergence properties.

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The reactant-product decoupling (RPD) equations are a rigorous formulation of state-to-state reactive scattering recently introduced by Peng and Zhang. For an N-arrangement reaction there are a total of N RPD equations, each of which describes the dynamics in just one region of coordinate space. One of the regions (the r-region) encloses the reactant channel and the strong interaction region; each of the other N−1 regions encloses one of the product channels. In this paper we develop a suggestion later made by Kouri and co-workers: that the original RPD equations can be further partitioned into a set of new RPD equations, in which the original r-region is now partitioned into three regions—two enclosing the reactant channel, and one enclosing the strong interaction region. After introducing the new RPD equations, we derive the time-independent wave-packet (TIW) form of the equations, and show how to solve them using an extended version of the Chebyshev propagator. We test the new RPD equations (and the method) by calculating state-to-state reaction probabilities and inelastic probabilities for the three-dimensional (J = 50) H+H2 reaction.

A practical method based on distributed approximating functionals ͑DAFs͒ is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo ͓Phys. Rev. E 54, 931 ͑1996͔͒ to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ''flat'' region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schrödinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem.

Some time ago we published our first article on the Renner-Teller ͑RT͒ model to treat the electronic interaction for a triatomic molecule ͓J. Chem. Phys. 124, 081106 ͑2006͔͒. The main purpose of that Communication was to suggest considering the RT phenomenon as a topological effect, just like the Jahn-Teller phenomenon. However, whereas in the first publication we just summarized a few basic features to support that idea, here in the present article, we extend the topological approach and show that all the expected features that characterize a three ͑multi͒ state RT-type'3 system of a triatomic molecule can be studied and analyzed within the framework of that approach. This, among other things, enables us to employ the topological D matrix ͓Phys. Rev. A 62, 032506 ͑2000͔͒ to determine, a priori, under what conditions a three-state system can be diabatized. The theoretical presentation is accompanied by a detailed numerical study as carried out for the HNH system. The D-matrix analysis shows that the two original electronic states 2 A 1 and 2 B 1 ͑evolving from the collinear degenerate ⌸ doublet͒, frequently used to study this Renner-Teller-type system, are insufficient for diabatization. This is true, in particular, for the stable ground-state configurations of the HNH molecule. However, by including just one additional electronic state-a B state ͑originating from a collinear ⌺ state͒-it is found that a rigorous, meaningful three-state diabatization can be carried out for large regions of configuration space, particularly for those, near the stable configuration of NH 2 . This opens the way for an accurate study of this important molecule even where the electronic angular momentum deviates significantly from an integer value.