Origin of a Theory or

Where did “near-dissociation theory” & the “Le Roy Radius”come from?

In the beginning ... I was doing experiments studying rates of reaction and trying tounderstand the reaction mechanism.

Question: How did we measure the amount of I2 present at each instant as thereaction proceeded?

Ans. Using spectroscopy: I2 vapour absorbs light in the visible, and changesin the fraction of the incident light absorbed tell us about changes in the I2

concentration.

Question: How do we quantitatively relate the fraction of light absorbed to theamount of I2 present?

Ans. (a) By experiment, when possible.

Ans. (b) Using quantum mechanical theory to calculate“absorption coefficients”, and how they vary with temperature !

However, this requires a knowledge of the forces (or interatomic potential energyfunctions) between the atoms in the molecule.

Aside: What is Spectroscopy?

Ans. It is the study of the patterns of energies and intensities of the particular“colours” (i.e., frequencies or wavelengths) of light absorbed or emitted by molecules.

What does it tell us?The discrete energies (Eν = h ν ) associated with the particular colours (frequencies)of light absorbed or emitted by molecules tell us:

• molecules can only have energies with a discrete particular values

• the energies associated with these discrete colours (frequencies) of light tell usthe spacings between the energy levels in molecules

• the pattern of level spacings associated with radial or vibrational motion tells usabout the forces between the particles

• recall your discussion of the H atom and of atomic orbital energies:

... and the level spacing pattern depends on the potential

energy function V (x) , or inter-particle forces ~F = − d V (x)

dx

What About the Spectroscopy of I2 ?

Our experimental study of the kinetics of the I + I + M −→ I2 + M recombinationreaction had left me interested in the spectroscopy of the I2 molecule, and led to thefollowing paper.

This work used a conventional technique for determining the molecular dissociationenergy from a plot of the vibrational level spacings ∆Gv+1/2 vs. the vibrationalquantum number v .

It turned out later that my result was wrong – but working on this led me to wonder:

What is the characteristic functional behaviour of vibrationalspacings for levels near a dissociation limit ?

or more generally

What is a better way of determining molecular bond dissociationenergy from the observed vibrational spacings ?

What Forces Hold Matter Together?

1. Covalent bonding: in “network solids” such as diamond & graphite

2. Polymeric solids: very long chain hydrocarbon molecules which are

all tangled up with one another, and sometimes also joined by hydrogen

bonding; e.g., plastics

3. Metals: a “sea” of electrons loosely distributed around a structured cage

of + ve charged ion cores

4. Ionic solids: closely packed regular arrays of + ve & − ve ions; e.g.,

NaCl, MgCl2 , MgO, ...

• strength of binding ∝ (+ ve charge)×(− ve charge)(ion separation distance) ∝ Z1 Z2 e

2

r

5. Hydrogen bonding ... a special case ...

6. Van der Waals or “Physical” forces

• dispersion forces, “induction” & dipole-dipole forces

• the longest-range interactions: of a sum of inverse-power terms

V (R) = D−∑m≥n

Cm/Rm ≈

very large rD− Cn/Rn

How do molecules behave in levels very near dissociation ?

Because of the anharmonicity of the intermolecular potential, they spend

most of their time in the long range region where

V (r) ≈ D− Cn/Rn

Using this function in a quantum mechanical expression for the energy levels

led to the fundamental near-dissociation theory expression for the pattern

of vibrational level energies near a molecular dissociation limit

G(v) = D−Xn(Cn) [vD − v]2nn−2

Treating observed level spacings using this expression:

• allow us to obtain much more accurate bond dissociation energies

• allowed us to make accurate predictions of the numbers and energies of

unobserved higher levels

• provided the first ever quantitative experimental way of determining the

limiting long-range potential function coefficient Cn from experimental

data

... but ...

... but for the new theory to be used reliably, we needed acriterion for determining when it was valid !

This led to the development of what came to be known as the “Le Roy

radius”

RLR ≈ 2[〈rA

2〉1/2 + 〈rB2〉1/2

]

• For distances R > RLR the intermolecular forces are dominantly inverse-

power “physical” forces (i.e., based on the classical physics of charge

independent atomic distributions).

Our “near-dissociation theory” expression

G(v) = D−Xn(Cn) [vD − v]2nn−2

realistically describes the the spacings of vibrational levels whose outer

“turning points” lie in this region.

• For distances R < RLR the intermolecular forces are dominantly “chem-

ical” interactions which depend on the sharing and interchanging of elec-

trons between the two atom.

Away from the potential minimum there is no general analytic form

for the intermolecular potential in this region, and hence no analogous

simple expression for the level energies.

This criterion for estimating the distance RLR defining the boundary be-

tween the regions where chemical vs. physical forces are dominant seems to

have been quite generally useful, and it is referred to fairly frequently in the

literature. This led to its appearance as a margin note in your textbook.