Chemistry · Lesson

Model systems and the Born-Oppenheimer approximation

Solve two stipulated model Hamiltonians before using them as molecular approximations. Connect clamped-nuclei electronic energies to nuclear motion and identify what a Born–Oppenheimer separation leaves out.

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By the end, you can

Use infinite-well boundary conditions and energy scaling.
Convert a harmonic force constant and reduced mass into a frequency and zero-point energy.
Explain electronic surfaces, nuclear motion and limits of uncoupled-surface approximations.

Boundary conditions define the infinite-well model

For one nonrelativistic particle of mass m in a one-dimensional infinite well, the potential is zero for 0 < x < L and infinite outside. The normalized stationary states are ψₙ = √(2/L) sin(nπx/L), with n = 1,2,3,…, and Eₙ = n²h²/(8mL²). The boundary zeros determine the allowed wavelengths; n = 0 does not give a nonzero normalized state.

The internal node count is n − 1, excluding the two boundaries. At fixed n and mass, doubling L divides the energy by 4. At fixed n and L, doubling mass divides it by 2. These are exact statements within this Hamiltonian, not a calibrated prediction of a real molecule’s size, colour or electron confinement.

A finite-region probability comes from integration

Set u = x/L. The dimensionless density L|ψₙ|² = 2 sin²(nπu) integrates to one over 0 ≤ u ≤ 1. The plotted dashed curve is the ground-state density; n = 2 is shown only at selected markers. Negative wavefunction amplitudes would still give nonnegative density.

For the ground state, integrating between u = 1/4 and 3/4 gives 1/2 + 1/π ≈ 0.8183. The central half therefore has about 81.8% probability rather than the 50% of a uniform distribution. An exact point has zero probability for this regular continuous density, while a finite detector interval can have nonzero probability.

A normalized box density assigns probability to regions

L |ψₙ|²

Position ratio u = x / L

Ground state n = 1
Excited state n = 2
— Ground-state density 2 sin²(πu)

Underlying data

Position ratio u = x / L	L \|ψₙ\|²
0	0 · Ground state n = 1
0.25	0.9999999999999998 · Ground state n = 1
0.5	2 · Ground state n = 1
0.75	1.0000000000000002 · Ground state n = 1
1	2.999519565323715e-32 · Ground state n = 1
0	0 · Excited state n = 2
0.25	2 · Excited state n = 2
0.5	2.999519565323715e-32 · Excited state n = 2
0.75	2 · Excited state n = 2
1	1.199807826129486e-31 · Excited state n = 2

Original exact one-dimensional infinite-well model. u=x/L and L|ψₙ|² are dimensionless. The dashed curve draws n=1; the n=2 density is represented only by its markers. Boundary zeros are excluded from internal-node counts. No real molecular potential or measured electron-density profile is asserted.

A harmonic displacement uses a reduced mass

For a quadratic displacement potential V(q) = k_force q²/2, the diatomic reduced mass is m_red = m₁m₂/(m₁ + m₂). The angular frequency is ω = √(k_force/m_red), and ordinary frequency is ν = ω/(2π). The force constant k_force is in N m⁻¹ and differs from the Boltzmann constant.

The ideal oscillator levels are Eᵥ = ħω(v + 1/2), v = 0,1,2,…, measured relative to the potential minimum. Their spacing is ħω = hν and the ground zero-point energy is ħω/2. A real molecular potential is only approximately quadratic near a stable minimum; anharmonicity, dissociation and mode coupling require more information.

Separate fast electronic structure from nuclear motion carefully

In a Born–Oppenheimer treatment, solve the clamped-nuclei electronic problem at specified nuclear positions. Including the corresponding nuclear-repulsion term provides potential-energy surfaces for subsequent nuclear motion. Fixing the nuclei while solving that electronic problem does not mean actual nuclei never move; treating their later motion classically is an additional approximation.

Nuclear kinetic terms can couple electronic surfaces. Neglecting those couplings can fail near closely spaced states or during rapid passages through coupled regions. A supplied electronic gap alone does not determine a transition probability: nuclear motion and coupling information also matter. A one-dimensional avoided crossing is not the general geometry of a polyatomic conical intersection.

Worked examples

Convert a model gap into a matching photon wavelength

Take a stipulated electron infinite-well model with L = 1.00 nm. Using current constants and electron mass, E₁ ≈ 0.376030 eV. Find E₂ − E₁ and its matching vacuum photon wavelength.

E₂ = 4E₁, so the gap is 3E₁ ≈ 1.128090 eV.
One electronvolt is exactly 1.602176634 × 10⁻¹⁹ J.
For energy matching, use λ_vac = hc/ΔE with h = 6.62607015 × 10⁻³⁴ J s and c = 299792458 m s⁻¹.
The unrounded wavelength is approximately 1099.06 nm.

The gap is approximately 1.13 eV and λ_vac ≈ 1.10 × 10³ nm at the supplied length precision. Energy matching alone does not determine transition strength.

Keep angular frequency and zero-point energy distinct

Use invented harmonic inputs k_force = 5.00 × 10² N m⁻¹, m₁ = 2.00 × 10⁻²⁶ kg and m₂ = 3.00 × 10⁻²⁶ kg. Find frequency and zero-point energy relative to the potential minimum.

m_red = m₁m₂/(m₁ + m₂) = 1.20 × 10⁻²⁶ kg.
ω = √(k_force/m_red) ≈ 2.04124 × 10¹⁴ rad s⁻¹.
Divide by 2π: ν ≈ 3.24874 × 10¹³ Hz.
The level spacing hν ≈ 2.15264 × 10⁻²⁰ J; half gives the ground energy.

ν ≈ 3.25 × 10¹³ Hz and E₀ ≈ 1.08 × 10⁻²⁰ J per oscillator. These are supplied harmonic-model results, not values assigned to a named bond.

Try it yourself

At the same n and mass, what happens to an infinite-well energy when L doubles?
Show the answer and reasoning
It becomes one quarter as large.
The dependence is L⁻².
How many internal nodes does the one-dimensional well state n = 3 have?
Show the answer and reasoning
Two.
The internal count is n − 1; boundary zeros are separate.
Must ω be divided by 2π before reporting ordinary frequency in Hz?
Show the answer and reasoning
Yes.
Angular and ordinary frequency describe the same oscillation with different conventions.
Does a clamped-nuclei electronic calculation assert that real nuclei remain motionless?
Show the answer and reasoning
No.
Their motion is treated in a later nuclear problem, with additional approximations stated.

What to carry forward

Boundary conditions are part of the exact model.
Keep wavefunction sign, density and integrated probability distinct.
State the energy zero and frequency convention.
Electronic surfaces and nuclear motion require separate modeling decisions.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.