Wentzel–Kramers–Brillouin theory of multidimensional tunneling: General theory for energy splitting

A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens‐type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.

[1]  J. Bertrán,et al.  Bidimensional tunneling dynamics of malonaldehyde and hydrogenoxalate anion. A comparative study , 1990 .

[2]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[3]  J. Crank.  Free and moving boundary problems , 1984 .

[4]  H. Sekiya,et al.  Electronic spectra of jet-cooled tropolone. Effect of the vibrational excitation on the proton tunneling dynamics , 1990 .

[5]  B. C. Garrett,et al.  Variational transition state theory and tunneling for a heavy–light–heavy reaction using an ab initio potential energy surface. 37Cl+H(D) 35Cl→H(D) 37Cl+35Cl , 1983 .

[6]  A. Leggett,et al.  Quantum tunnelling in a dissipative system , 1983 .

[7]  Hiroki Nakamura,et al.  Dynamics of collinear asymmetric light atom transfer reactions , 1985 .

[8]  W. Miller.  Semiclassical limit of quantum mechanical transition state theory for nonseparable systems , 1975 .

[9]  M. Wilkinson.  Tunnelling between tori in phase space , 1986 .

[10]  A. Schmid.  Quasiclassical wave function in multidimensional quantum decay problems , 1986 .

[11]  Tucker Carrington,et al.  Reaction surface description of intramolecular hydrogen atom transfer in malonaldehyde , 1986 .

[12]  C. Herring.  CRITIQUE OF THE HEITLER-LONDON METHOD OF CALCULATING SPIN COUPLINGS AT LARGE DISTANCES , 1962 .

[13]  D. Makarov,et al.  Two-dimensional trajectories of tunneling in the symmetric double-well potential , 1992 .

[14]  W. Miller,et al.  Complex‐Valued Classical Trajectories for Linear Reactive Collisions of H + H2 below the Classical Threshold , 1972 .

[15]  V. Aquilanti,et al.  The two‐state linear curve crossing problems revisited. I. Analysis of Stokes phenomenon and expressions for scattering matrices , 1992 .

[16]  A. Zichichi.  The Whys of Subnuclear Physics , 1979 .

[17]  D. Makarov,et al.  Quantum chemical dynamics in two dimensions , 1993 .

[18]  Hiroki Nakamura.  What are the basic mechanisms of electronic transitions in molecular dynamic processes , 1991 .

[19]  A. Messiah.  Quantum Mechanics , 1961 .

[20]  Joseph B. Keller,et al.  Corrected bohr-sommerfeld quantum conditions for nonseparable systems , 1958 .

[21]  M. Ovchinnikova.  The tunneling dynamics of the low temperature H exchange reactions , 1979 .

[22]  Michael E. Coltrin,et al.  A new tunneling path for reactions such as H+H2→H2+H , 1977 .

[23]  B. Chance.  Tunneling in biological systems , 1979 .

[24]  E. Pollak,et al.  Quantum variational transition state theory revisited , 1993 .

[25]  D. W. Noid,et al.  Bound state semiclassical wave functions , 1986 .

[26]  A. Voros.  Semi-classical approximations , 1976 .

[27]  J. Heading,et al.  An introduction to phase-integral methods , 1962 .

[28]  Cutler,et al.  Wentzel-Kramers-Brillouin method in multidimensional tunneling. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[29]  Curtis G. Callan,et al.  Fate of the false vacuum. II. First quantum corrections , 1977 .

[30]  William H. Miller,et al.  Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants , 1974 .

[31]  Victor Pavlovich Maslov,et al.  Semi-classical approximation in quantum mechanics , 1981 .

[32]  J. Light,et al.  Efficient pointwise representations for vibrational wave functions: Eigenfunctions of H+3 , 1989 .

[33]  Paul F. Barbara,et al.  A theoretical study of multidimensional nuclear tunneling in malonaldehyde , 1989 .

[34]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[35]  Hänggi,et al.  Numerical study of tunneling in a dissipative system. , 1990, Physical Review B (Condensed Matter).

[36]  S. Kivelson,et al.  THE PATH DECOMPOSITION EXPANSION AND MULTIDIMENSIONAL TUNNELING , 1985 .

[37]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[38]  D. Makarov,et al.  Low-temperature chemical reactions. Effect of symmetrically coupled vibrations in collinear exchange reactions , 1991 .

[39]  L. Trakhtenberg,et al.  Tunneling Phenomena in Chemical Physics , 1988 .

[40]  Richard W. Duerst,et al.  Microwave spectroscopic study of malonaldehyde. 3. Vibration-rotation interaction and one-dimensional model for proton tunneling , 1984 .

[41]  R. L. Redington,et al.  Laser fluorescence excitation spectrum of jet‐cooled tropolone: The à 1B2–X̃ 1A1 system , 1988 .

[42]  W. Miller,et al.  Basis set methods for describing the quantum mechanics of a ‘‘system’’ interacting with a harmonic bath , 1987 .

[43]  Michael Baer,et al.  Theory of chemical reaction dynamics , 1985 .

[44]  R. T. Skodje,et al.  The semiclassical quantization of nonseparable systems using the method of adiabatic switching , 1985 .

[45]  Gregory,et al.  Multidimensional tunneling and complex momentum. , 1991, Physical review. D, Particles and fields.