LeRoy radius

The LeRoy radius, as proposed by Robert J. LeRoy, defines the internuclear distance between two atoms at which LeRoy-Bernstein theory (sometimes called near-dissociation theory) becomes valid.

LeRoy-Bernstein theory is a semiclassical (WKB) approach for describing vibrational energy levels near the molecular dissociation limit. [cite journal|last=Leroy|first=Robert J.|coauthors=Richard B. Bernstein|date=1970|title=Dissociation energies of diatomic molecules from vibrational spacings of higher levels: application to the halogens|journal=Chemical Physics Letters|volume=5|issue=1|pages=42-44|doi=10.1016/0009-2614(70)80125-7 ] In this limit, the interaction potential between two atoms can be approximated as $V(r)\; =\; mathfrak\; D\; -\; C\_\{n\}/r^\{n\}$, which gives rise to a simple analytical approximation for the vibrational energy levels:

:$G(v)\; =\; mathfrak\{D\}-X\_n(C\_n)\; [v\_\{mathfrak\{D\; -\; v]\; ^\{frac\{2n\}\{n-2.$

In this expression, $X\_n(C\_n)$ is a simple function depending only upon "n" and "C"_"n", and $v\_\{mathfrak\; D\}$ can be identified as an effective vibrational quantum number at dissociation.

LeRoy later defined an expression for the radius that approximates a boundary between the region where electron exchange (quantum-mechanical) terms are prominent, and the region where atoms and molecules approximately interact through the laws of classical physics and, thus, LeRoy-Bernstein theory (as independent charge distributions and van der Waals interactions expressible as a power series in the internuclear separation).

This radius is defined as [cite journal|last=LeRoy|first=Robert J.|date=1974|title=Long-Range Potential Coefficients From RKR Turning Points: C6 and C8 for B(3ΠOu+)-State Cl2, Br2, and I2|journal=Canadian Journal of Physics|volume=52|issue=3|pages=246–256|doi=10.1139/p74-035]

:$R\_\{mathrm\{LR\; =\; 2\; [langle\; r\_A^2\; angle\; ^\{1/2\}\; +\; langle\; r\_B^2\; angle\; ^\{1/2\}]$,

where "r_A" and "r_B" denote the atomic radii of the two atoms.

For $r>R\_\{LR\}$ , the internuclear potential can be reasonably approximated by charge independent atomic distributions, and the vibrational levels can be well described by LeRoy-Bernstein theory.

For

A derivation of a more general expression, called the "m"-dependent LeRoy radius, which depends on the magnetic quantum number ("m"), was derived in 1995. [cite journal|last=Ji|first=Bing|coauthors=Tsai, Chin-Chun; Stwalley, William C.|date=1995|title=Proposed modification of the criterion for the region of validity of the inverse-power expansion in diatomic long-range potentials|journal=Chemical Physics Letters|volume=236|issue=3|pages=242-246|doi=10.1016/0009-2614(95)00216-Q ] This expression yields the traditional LeRoy Radius in the special case of a spherical, S-state, atom.

The LeRoy radius is described in high school level chemistry textbooks in Ontario (particularly, in Nelson Chemistry 12, which is the standard required textbook for grade 12 chemistry education in Ontario).

References