Morse Potential Energy - Study Guide | Wiki

Core idea

Overview

The Morse potential is a convenient interatomic interaction model that describes the potential energy of a diatomic molecule as a function of atomic separation. It provides a more realistic approximation for molecular vibrations than the harmonic oscillator by accounting for bond dissociation and the anharmonicity of real chemical bonds.

When to use: Use the Morse potential when modeling diatomic systems where large atomic displacements occur, such as near the bond dissociation limit. It is the preferred choice for interpreting vibrational spectra where the spacing between energy levels decreases as energy increases. It assumes the potential energy is zero at the equilibrium bond length and reaches the dissociation energy as the distance approaches infinity.

Why it matters: This equation allows chemists to calculate dissociation energies and predict vibrational frequencies that align with experimental data better than simple harmonic models. It is fundamental in physical chemistry for understanding the stability of molecules and the kinetics of reaction pathways involving bond breaking.

Remember it

Memory Aid

Phrase: Very Deep (1 minus e against r minus re) Squared.

Visual Analogy: A marble in a bowl where one side flattens into a table. If you push the marble hard enough (reaching De), it rolls off the table and never returns, representing the bond breaking (dissociation).

Exam Tip: Always distinguish between De (well depth) and D0 (dissociation energy). De is measured from the curve's minimum, while D0 starts from the v=0 ground state.

Why it makes sense

Intuition

The Morse potential is visualized as an asymmetric potential well: it starts at zero at the equilibrium bond length, dips to a minimum representing the stable bond, rises steeply for shorter distances (repulsion), and

Symbols

Variables

V(r) = Potential Energy, D_e = Well Depth, a = Width Parameter, r = Bond Length, r_e = Equilibrium Bond Length

V (r)

Potential Energy

e V

D_{e}

Well Depth

e V

a

Width Parameter

\overset{˚}{A}^{- 1}

r

Bond Length

\overset{˚}{A}

r_{e}

Equilibrium Bond Length

\overset{˚}{A}

Walkthrough

Derivation

Understanding the Morse Potential

A more realistic potential energy function for a diatomic molecule that includes bond dissociation and anharmonicity.

The molecule is diatomic.
The potential is spherically symmetric (depends only on internuclear distance r).

1

State the Morse potential:

$D_{e}$ is the well depth (dissociation energy from the bottom of the well), $r_{e}$ is the equilibrium bond length, and a controls the width of the well.

V (r) = D_{e} (1 - e^{- a (r - r_{e})})^{2}

2

Compare with the harmonic oscillator:

Near the bottom of the well, the Morse potential is approximately harmonic with force constant k = 2 $D_{e}$ a².

V_{harm} (r) = \frac{1}{2} k (r - r_{e})^{2} \approx D_{e} a^{2} (r - r_{e})^{2}

3

Energy levels of the Morse oscillator:

Unlike the harmonic oscillator, the energy levels are not equally spaced — they converge as v increases, eventually reaching the dissociation limit.

E_{v} = h ν_{e} (v + \frac{1}{2}) - h ν_{e} x_{e} (v + \frac{1}{2})^{2}

Note: The anharmonicity constant $x_{e}$ = hν_e/(4 $D_{e}$ ) determines how quickly the levels converge. This explains overtone absorptions in IR spectroscopy.

Result

E_{v} = h ν_{e} (v + \frac{1}{2}) - h ν_{e} x_{e} (v + \frac{1}{2})^{2}

Source: Atkins & de Paula — Physical Chemistry

Where it shows up

Real-World Context

Analyzing the IR spectrum of HCl gas to determine its bond strength and anharmonicity constant.

Avoid these traps

Common Mistakes

Confusing De (well depth) with the experimental dissociation energy D0 which includes zero-point energy.
Incorrectly calculating the exponential term by forgetting the negative sign.

Study smarter

Tips

The variable 'De' represents the depth of the potential well, equivalent to the bond's dissociation energy.
The parameter 'a' is a constant related to the force constant of the bond; higher values indicate a narrower potential well.
Ensure units for 'a' and '(r - re)' are inverse of each other so the exponent remains dimensionless.
When 'r' equals 're', the potential energy 'V(r)' is exactly zero.

Common questions

Frequently Asked Questions

A more realistic potential energy function for a diatomic molecule that includes bond dissociation and anharmonicity.

Use the Morse potential when modeling diatomic systems where large atomic displacements occur, such as near the bond dissociation limit. It is the preferred choice for interpreting vibrational spectra where the spacing between energy levels decreases as energy increases. It assumes the potential energy is zero at the equilibrium bond length and reaches the dissociation energy as the distance approaches infinity.

This equation allows chemists to calculate dissociation energies and predict vibrational frequencies that align with experimental data better than simple harmonic models. It is fundamental in physical chemistry for understanding the stability of molecules and the kinetics of reaction pathways involving bond breaking.

Confusing De (well depth) with the experimental dissociation energy D0 which includes zero-point energy. Incorrectly calculating the exponential term by forgetting the negative sign.

Analyzing the IR spectrum of HCl gas to determine its bond strength and anharmonicity constant.

The variable 'De' represents the depth of the potential well, equivalent to the bond's dissociation energy. The parameter 'a' is a constant related to the force constant of the bond; higher values indicate a narrower potential well. Ensure units for 'a' and '(r - re)' are inverse of each other so the exponent remains dimensionless. When 'r' equals 're', the potential energy 'V(r)' is exactly zero.