Morse Potential Calculator

Inputs

Enter parameters, pick units, then compute the potential and related quantities.

Well depth D_e

Positive energy scale of the bond.

Range parameter a

Controls stiffness and anharmonicity.

Equilibrium distance r_e

Bond length at minimum energy.

Bond distance r

Evaluate V(r) and F(r) here.

Energy unit eVJkJ/mol

Used for D_e, V(r), and energy levels.

Length unit Ånmpmm

Used for r, r_e, and a.

Potential reference Shifted (minimum at −Dₑ) Unshifted (minimum at 0)

Affects absolute energy reporting.

Mass m₁ (amu)

Atomic mass for atom 1.

Mass m₂ (amu)

Atomic mass for atom 2.

Vibrational index v

Used in E_v approximation.

Compute Reset

Default values represent a typical diatomic example for demonstration.

Example Data Table

Formula Used

The Morse potential models an anharmonic diatomic bond using:

V(r) = D_e[1 − e^{−a(r−r_e)}]²

If you choose the shifted form, the minimum energy is set to −D_e:

V_shift(r) = D_e[1 − e^{−a(r−r_e)}]² − D_e

The force follows from the negative derivative:

F(r) = −dV/dr = −2D_ea e^{−a(r−r_e)}[1 − e^{−a(r−r_e)}]

Near equilibrium, the curvature gives an effective spring constant:

k = (d²V/dr²)_r=re = 2D_ea²

With reduced mass μ = m₁m₂/(m₁+m₂), the small-amplitude frequency is:

ω = √(k/μ), f = ω/(2π)

How to Use This Calculator

1. Enter D_e, a, r_e, and the evaluation distance r.
2. Pick consistent energy and length units for all parameters.
3. Choose shifted or unshifted potential, depending on your convention.
4. Provide masses m₁ and m₂ to compute μ and vibrational quantities.
5. Optionally set v to estimate an anharmonic energy level.
6. Press Compute to see results above the form.

Professional Notes on the Morse Potential

1) Why the Morse model is used

The Morse potential is a compact, practical description of a diatomic bond. Unlike a purely harmonic spring, it captures bond softening during stretching and predicts a finite dissociation energy. This makes it useful in spectroscopy, molecular dynamics, and teaching workflows where an analytic, interpretable curve is preferred.

2) What D_e, a, and r_e represent

D_e sets the depth of the well (bond strength). The range parameter a controls how quickly the curve rises away from r_e and therefore influences stiffness and anharmonicity. r_e is the equilibrium distance at the minimum of the potential.

3) Typical parameter scales

For many diatomic molecules, D_e often falls between about 1–10 eV (roughly 100–1000 kJ/mol). Equilibrium distances are commonly 0.7–2.5 Å, while a is frequently near 1–3 Å⁻¹. These ranges vary with bonding type, and literature or fitted force fields should be used for precision.

4) Energy reference: shifted versus unshifted

The shifted form places the minimum at −D_e, aligning the dissociation limit with 0. This is convenient for thermodynamic intuition and absolute energy reporting. The unshifted form sets the minimum at 0, which can be handy when comparing relative energies without an offset.

5) Force and curvature in practice

The calculator reports force from −dV/dr, so the sign indicates direction relative to increasing r. At r = r_e, the force is zero. The second derivative at r_e gives k = 2D_ea², a useful stiffness proxy for small oscillations and for sanity-checking fitted parameters.

6) Frequency depends on reduced mass

Using μ = m₁m₂/(m₁+m₂), the small-amplitude angular frequency is ω = √(k/μ). Lighter atoms raise ω and f, while heavier isotopes lower them. For example, changing H to D approximately increases μ and reduces the vibrational frequency, consistent with isotope shifts seen in infrared spectra.

7) Anharmonic level estimate

The energy-level approximation E_v ≈ (v+1/2)ħω − [(v+1/2)ħω]²/(4D_e) captures first-order anharmonicity. As v increases, spacing decreases until levels approach dissociation. This is a controlled approximation; high-v accuracy requires solving the quantum problem more directly.

8) Practical modeling tips

Keep units consistent: if r and r_e are in Å, then a must be in Å⁻¹. Use trusted parameters from spectroscopy or validated force fields, then verify that V(r_e) matches your chosen reference and that the computed k and frequencies are physically plausible. Export CSV/PDF for lab notes and reports. When comparing datasets, compute V(r) at several r values around rₑ to visualize asymmetry and record results with the export buttons for traceable documentation.

FAQs

1) What does the Morse potential describe?

It describes the potential energy of a diatomic bond as a function of bond distance, including anharmonic behavior and a finite dissociation energy, unlike a simple harmonic spring.

2) Why might my V(r) be negative?

If you select the shifted form, the energy minimum is −D_e. Energies near r_e are therefore negative relative to the dissociation limit set to zero.

3) How should I choose units for a?

a must be the inverse of your chosen length unit. If r is in Å, use a in Å⁻¹; if r is in nm, use a in nm⁻¹. Mixed units produce incorrect results.

4) What do m₁ and m₂ affect?

The masses determine the reduced mass μ, which directly controls ω and f through ω = √(k/μ). They do not change V(r) itself, only vibrational quantities derived from k and μ.

5) Is the reported energy level exact?

No. E_v is an anharmonic approximation suitable for quick estimates. For high vibrational states or precision spectroscopy, use a dedicated quantum solver or fitted spectroscopic constants.

6) Why is the force sign important?

Force is computed as −dV/dr. A negative value indicates attraction toward smaller r (restoring pull), while a positive value indicates repulsion toward larger r, given the chosen coordinate direction.

7) Can I use this for polyatomic molecules?

It is designed for a single bond coordinate. For polyatomic systems, you typically need multiple bond terms plus angle and torsion potentials, or a full force field parameterization.