### Abstract

We derive a simple, analytic expression for the energy splitting ΔE between the lowest pair of H^{+}
_{2} states (1sσ_{g} and 2pσ_{u}) that results from electron exchange between two protons. The calculation employs the semiclassical instanton method, with two unorthodox features which markedly simplify the treatment: (1) The double-minimum potential and corresponding wavefunctions that govern the electronic tunneling are evaluated in the large-dimension limit. (2) The time variable is rescaled to cure divergent behavior of fluctuations about the instanton path that otherwise appears because of the potential develops sharp cusps as the internuclear distance increases. By virtue of exact interdimensional degeneracies, the large-D limit yields valid results for a 3D molecule. Indeed, a simple dimensional scaling law gives ΔE for all pairs of g, u states that stem from separated atom states with m=l=n-1, for n=1→∞. For a wide range of internuclear distances, our analytic expression for ΔE, which pertains to the leading order in ł>, gives for such states good agreement with comparable semiclassical methods as well as with exact numerical calculations. It is remarkable that use of the effective potential for the large-dimension limit, which is exactly calculable from classical electrostatics, yields quantitatively results for electronic tunneling, an intrinsically quantal phenomenon.

Original language | English |
---|---|

Pages (from-to) | 393-402 |

Number of pages | 10 |

Journal | Chemical Physics |

Volume | 161 |

Issue number | 3 |

Publication status | Published - 15 Apr 1992 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Spectroscopy
- Physical and Theoretical Chemistry

### Cite this

^{+}

_{2}evaluated from the large-dimension limit.

*Chemical Physics*,

*161*(3), 393-402.

**Electronic tunneling in H ^{+}
_{2} evaluated from the large-dimension limit.** / Kais, Sabre; Frantz, Donald D.; Herschbach, Dudley R.

Research output: Contribution to journal › Article

^{+}

_{2}evaluated from the large-dimension limit',

*Chemical Physics*, vol. 161, no. 3, pp. 393-402.

^{+}

_{2}evaluated from the large-dimension limit. Chemical Physics. 1992 Apr 15;161(3):393-402.

}

TY - JOUR

T1 - Electronic tunneling in H+ 2 evaluated from the large-dimension limit

AU - Kais, Sabre

AU - Frantz, Donald D.

AU - Herschbach, Dudley R.

PY - 1992/4/15

Y1 - 1992/4/15

N2 - We derive a simple, analytic expression for the energy splitting ΔE between the lowest pair of H+ 2 states (1sσg and 2pσu) that results from electron exchange between two protons. The calculation employs the semiclassical instanton method, with two unorthodox features which markedly simplify the treatment: (1) The double-minimum potential and corresponding wavefunctions that govern the electronic tunneling are evaluated in the large-dimension limit. (2) The time variable is rescaled to cure divergent behavior of fluctuations about the instanton path that otherwise appears because of the potential develops sharp cusps as the internuclear distance increases. By virtue of exact interdimensional degeneracies, the large-D limit yields valid results for a 3D molecule. Indeed, a simple dimensional scaling law gives ΔE for all pairs of g, u states that stem from separated atom states with m=l=n-1, for n=1→∞. For a wide range of internuclear distances, our analytic expression for ΔE, which pertains to the leading order in ł>, gives for such states good agreement with comparable semiclassical methods as well as with exact numerical calculations. It is remarkable that use of the effective potential for the large-dimension limit, which is exactly calculable from classical electrostatics, yields quantitatively results for electronic tunneling, an intrinsically quantal phenomenon.

AB - We derive a simple, analytic expression for the energy splitting ΔE between the lowest pair of H+ 2 states (1sσg and 2pσu) that results from electron exchange between two protons. The calculation employs the semiclassical instanton method, with two unorthodox features which markedly simplify the treatment: (1) The double-minimum potential and corresponding wavefunctions that govern the electronic tunneling are evaluated in the large-dimension limit. (2) The time variable is rescaled to cure divergent behavior of fluctuations about the instanton path that otherwise appears because of the potential develops sharp cusps as the internuclear distance increases. By virtue of exact interdimensional degeneracies, the large-D limit yields valid results for a 3D molecule. Indeed, a simple dimensional scaling law gives ΔE for all pairs of g, u states that stem from separated atom states with m=l=n-1, for n=1→∞. For a wide range of internuclear distances, our analytic expression for ΔE, which pertains to the leading order in ł>, gives for such states good agreement with comparable semiclassical methods as well as with exact numerical calculations. It is remarkable that use of the effective potential for the large-dimension limit, which is exactly calculable from classical electrostatics, yields quantitatively results for electronic tunneling, an intrinsically quantal phenomenon.

UR - http://www.scopus.com/inward/record.url?scp=0011503574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011503574&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011503574

VL - 161

SP - 393

EP - 402

JO - Chemical Physics

JF - Chemical Physics

SN - 0301-0104

IS - 3

ER -