On simple saddle points of a potential surface, the conservation of nuclear symmetry along paths of steepest descent, and the symmetry of transition states

1976 ◽  
Vol 64 (4) ◽  
pp. 1516-1521 ◽  
Author(s):  
Philip Pechukas
Keyword(s):  
Potential Surface ◽  
Saddle Points ◽  
Transition States ◽  
Steepest Descent

Instantons in quantum mechanics (QM)

2021 ◽  
pp. 899-918
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Mechanics ◽  
Saddle Points ◽  
Descent Method ◽  
Classical Field ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Gradient Term ◽  
Leading Order ◽  
Field Equations ◽  
Barrier Penetration

Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.

Ab initio analysis of the transition states on the lowest triplet H2O2 potential surface

1999 ◽  
Vol 110 (24) ◽  
pp. 11918-11927 ◽  
Author(s):  
Sergei P. Karkach ◽  
Vladimir I. Osherov
Keyword(s):  
Ab Initio ◽  
Potential Surface ◽  
Transition States

Integrals with a large parameter: a double complex integral with four nearly coincident saddle-points

1980 ◽  
Vol 87 (2) ◽  
pp. 249-273 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Critical Points ◽  
Saddle Points ◽  
Steepest Descent ◽  
Large Parameter ◽  
Double Complex ◽  
Contour Integrals ◽  
Complex Integral ◽  
Uniform Expansions

AbstractThe method of steepest descents for finding the asymptotic expansion of contour integrals of the form ∫ g(z) exp (Nf(z)) dz where N is a real parameter tending to + ∞ is familiar. As is well known, the principal contributions to the asymptotic expansion come from certain critical points; the most important are saddle-points where df/dz = 0. The original contour is deformed into an equivalent contour consisting of paths of steepest descent through certain saddle-points, the relevant saddle-points. The determination of these is a global problem which can be solved explicitly only in simple cases. The function f (z) may also depend on parameters. The position of the saddle-points depends on the parameters and at a certain set of values of the parameters it may happen that two or more saddle-points coincide. The ordinary expansion is then non-uniform, but appropriate uniform expansions have been shown to exist in earlier work.

Integrals with a large parameter: paths of descent and conformal mapping

1970 ◽  
Vol 67 (2) ◽  
pp. 371-381 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Points ◽  
Steepest Descent ◽  
Large Parameter ◽  
Local Behaviour ◽  
Complete Asymptotic Expansion ◽  
Contour Integrals ◽  
Global Problems ◽  
Non Local ◽  
Image Position

AbstractThe method of steepest descents is widely used to find the asymptotic expansion of contour integrals of the formwhere N is a large positive parameter, and f(z) and g(z) are analytic functions of z. (For the sake of simplicity it will here be supposed that f(z) and g(z) are regular except at ∞, but our arguments can be extended to include singularities at other points.) In the method of steepest descents the path of integration is deformed into a new equivalent path consisting of the paths of steepest descent through a and b, together with paths of steepest descent through certain saddle points, the relevant saddle points. (Paths of descent, etc., are defined in §1 below.) Watson's Lemma then shows that the complete asymptotic expansion depends on the local behaviour of f(z) and g(z) at the end-points and at the relevant saddle-points. On the other hand to determine which saddle-points are relevant is a global (non-local) problem and requires either the tracing of paths of steepest descent all the way from saddle-points and endpoints to ∞, or some other global process. It has often been noted that in applications of Watson's Lemma the paths of steepest descent can be replaced by a wider class of paths of descent. In this way we are led to global problems like the following: Given 2points z = α and z = β where Nf(a) > Nf(β), can apath of descent be drawn from α to β? If also Ff(α) = Ff(β), can a path of steepest descent be drawn from α to β?

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