Classical Saddle Point Research Articles

The effect of higher than double excitations on the F+H2→FH+H barrier

The classical barrier height and saddle point geometry are computed using the averaged coupled-pair functional (ACPF) method, correlating both seven and nine electrons. The size-consistent ACPF method indicates that 2s correlation substantially reduces the barrier. The excellent agreement between the ACPF and MRCI+Q results at both the seven and nine electron level provides additional support for the +Q correction. Thus the ACPF treatment supports a low (1.65 kcal/mol) barrier. The seven-electron CISDTQ results are shown to be consistent with the MRCI+Q and ACPF results.

Lifetime measurements of interference-narrowed sodium Stark resonances.

We report measurements of the temporal decay of sodium Stark resonances over regions of interference narrowing and compare with calculations using Wentzel-Kramers-Brillouin quantum-defect (WKB-QD) Stark theory. The measurements are performed with a beam of sodium atoms in a uniform electric field of \ensuremath{\sim}3.2 kV/cm excited via the 3 $^{2}P$ state to \ensuremath{\sim}285 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ (${n}^{\mathrm{*}}$\ensuremath{\sim}20) below the zero-field ionization threshold, and hence substantially above the classical saddle-point energy. For mixed states at level crossings, the discrete-continuum couplings interfere, yielding resonances that are stabilized against ionization over a small interval of the field. Measurements of the maximum lifetimes of \ensuremath{\Vert}m\ensuremath{\Vert}=1 levels here are not limited by laser bandwidth, as were our previous spectral observations of m=0 resonances. For three narrowing regions studied here, the observed maximum lifetime values of 127\char21{}191 nsec are 10\char21{}20 % less than values obtained from WKB-QD calculations, while experimental error limits are \ensuremath{\sim}5%. After considering estimates for competing processes, namely radiative decay and transfer by blackbody radiation, a slight discrepancy remains.

Saddlepoints in group and semigroup minimization

The group or semigroup minimization problem, derived from integer programming, is discussed. A dual form of this problem is stated and weak and strong duality theorems, together with complementarity conditions, are shown. Moreover, a Lagrangean function is introduced and it is shown that the classical saddlepoint theorems still hold good. The objective function of the minimization problem is formed using elements drawn from an ordered d-monoid, thereby treating sum, bottleneck and lexicographic objectives from a unified point of view.

Variational transition state theory calculations of the reaction rates of F with H2, D2, and HD and the intermolecular and intramolecular kinetic isotope effects

We use variational transition state theory to calculate rate constants and kinetic isotope effects for the reactions F+H2→HF+H (with rate constant k1), F+D2→DF+D(k2), and two other isotopic analogs as functions of temperature. The calculations are performed using a recently proposed partly empirical, partly ab initio potential energy surface, called surface No. 5, and also using a new surface, called surface No. 5A, introduced here to test the effect of a higher classical saddle point on the reaction rates, kinetic isotope effects, and reaction thresholds. The various theoretical results are compared to the available experiments to test the validity of these potential energy surfaces. For those rate constants and kinetic isotope effects for which there is more than one experimental value at a given temperature, the theoretical results for reactions on surface No. 5 agree with experiment about as well as the individual experiments agree with each other. At T>373 K where there is only one experimental measurement for k1 and k2, the theoretical rate constants for surface No. 5 are up to 44% lower than experiment, and at T<190 K where there is only one experimental measurement of k1/k2, the theoretical rate constant ratio is as much as 43% low. These discrepancies could be due to experimental error or they could indicate that the theoretical activation energy is low, but by no more than 0.5 kcal/mol, which is considerably less than the 2.5 kcal/mol discrepancy from the most recently published large-scale calculation. If, however, the potential energy surface were adjusted to raise the activation energy by even 0.5 kcal/mol above the value for surface No. 5, the threshold energies would no longer be in such good agreement with experiment, and several other aspects of the results presented here would be in much worse agreement with experiment. Taken as a whole then, we interpret the present calculations as providing further confirmation of the methods used to obtain surface No. 5. With this interpretation, the most important implication about the experiments is that Persky’s low-temperature measurements of k1/k2, especially those for T≤200 K, are systematically high.

Uniform Asymptotic Expansions for Saddle Point Integrals-Application to a Probability Distribution Occurring in Noise Theory

The noncentral chi-square distribution occurs in noise interference problems. When the number of degrees of freedom becomes large, the middle portion of the distribution is given by the central limit theorem, and the tails by a classical saddle point expansion. Here recent work by N. Bleistein and F. Ursell on “uniform” asymptotic expansions is combined and extended to obtain an asymptotic series which apparently holds over the entire range of the distribution. General methods for expanding saddle point integrals in uniform asymptotic series are discussed. Recurrence relations are given for the coefficients in two typical cases, (i) when there are two saddle points and (ii) when there is only one saddle point but it lies near a pole or a branch point.