Questions: Intermolecular Forces and Lennard-Jones Potential

Question 1

In the Lennard-Jones potential V(r) = 4ε[(σ/r)¹² − (σ/r)⁶], what does σ represent?

• A. The equilibrium separation between two molecules — where the potential energy is at its minimum
• B. The depth of the potential well, measuring the maximum attraction between two molecules
• C. The distance at which the potential energy equals zero — interpreted as the effective molecular diameter
• D. The distance at which the repulsive term first exceeds the attractive term

Answer: (C)

σ is the distance at which the potential crosses zero on the way from positive (repulsive) to negative (attractive) values. It represents the effective 'size' of a molecule — the hard-sphere diameter below which overlap becomes extremely costly. The equilibrium separation (minimum energy) is at r = 2^(1/6)·σ ≈ 1.12σ, which is slightly larger than σ. ε (epsilon) is the depth of the potential well, measuring the binding energy at equilibrium. These two parameters fully characterize a Lennard-Jones interaction and are fitted to experimental data.

Question 2

Why is the repulsive term in the Lennard-Jones potential written as r⁻¹² rather than a form derived from quantum mechanics?

• A. Because the r⁻¹² form is derived rigorously from the Pauli exclusion principle and matches experimental repulsion exactly
• B. Because r⁻¹² produces a repulsive wall steep enough to mimic hard-sphere behavior, and r¹² = (r⁶)² is computationally convenient — requiring no extra power calculation
• C. Because r⁻¹² ensures the potential minimum occurs exactly at r = σ
• D. Because the repulsive exponent must always be twice the attractive exponent for mathematical consistency

Answer: (B)

The r⁻¹² exponent is not derived from first principles — it is a pragmatic choice. The true quantum mechanical repulsion (from Pauli exclusion and electron-electron overlap) does not have a simple power-law form. The r⁻¹² term was chosen because: (1) it rises steeply enough at short range to mimic hard-sphere repulsion, and (2) since the attractive term goes as r⁻⁶, computing r⁻¹² simply means squaring the already-computed r⁻⁶ term — a significant computational saving in molecular dynamics simulations. The convenience is computational, not physical.

Question 3

The equilibrium separation between two Lennard-Jones molecules (where the potential energy is minimum) occurs at the distance σ.

• T. True
• F. False

Answer: False

σ is where the potential energy equals zero, not where it is minimum. The minimum occurs at r = 2^(1/6)·σ ≈ 1.12σ — slightly larger than σ. At r = σ, the repulsive and attractive terms are equal in magnitude, so the net potential is zero, but the curve is still descending toward the minimum. This distinction matters: at σ, molecules would still attract each other and move closer; only at 2^(1/6)σ are they in equilibrium.

Question 4

Doubling the intermolecular separation reduces the London dispersion attraction by a factor of 64.

• T. True
• F. False

Answer: True

The attractive term in the Lennard-Jones potential scales as r⁻⁶. Doubling r means the attraction scales as (2r)⁻⁶ = r⁻⁶/64 — a reduction by a factor of 64. This steep distance dependence is why London dispersion forces are short-range and why closely packed molecules in a liquid experience far stronger cohesion than the same molecules in a gas. It also explains why large polarizable molecules (with stronger dispersion) have much higher boiling points than small ones.

Question 5

What are the physical origins of the two terms in the Lennard-Jones potential, and why is the repulsive term written as r⁻¹² rather than a physically derived form?

Model answer: The attractive r⁻⁶ term models London dispersion forces — quantum mechanical correlated fluctuations in electron density that create instantaneous dipole-induced dipole attractions. Its r⁻⁶ dependence comes from perturbation theory. The repulsive r⁻¹² term models Pauli exclusion repulsion when electron clouds overlap, but it is not derived from quantum mechanics — it was chosen because r¹² = (r⁶)² is computationally efficient and produces a sufficiently steep repulsive wall.

Understanding why r⁻¹² is a convenience rather than a derived result matters for knowing the model's limitations. Real repulsion falls off somewhat differently, and the LJ potential is known to be imperfect at very short and very long ranges. More accurate models exist, but LJ remains the workhorse of molecular simulation because its two-parameter simplicity (ε, σ) captures the essential physics at a low computational cost.