Question

The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, $$U(r)=A\left[\left(e^{(R-r) / S}-1\right)^{2}-1\right]$$ where \(r\) is the distance between the two atoms and \(A, R,\) and \(S\) are positive constants with \(S \ll R .\) Sketch this function for \(0 < r < \infty\). Find the equilibrium separation \(r_{\mathrm{o}}\), at which \(U(r)\) is minimum. Now write \(r=r_{\mathrm{o}}+x\) so that \(x\) is the displacement from equilibrium, and show that, for small displacements, \(U\) has the approximate form \(U=\) const \(+\frac{1}{2} k x^{2} .\) That is, Hooke's law applies. What is the force constant \(k ?\)

Short answer

The equilibrium separation is \( r_o = R \), and the force constant is \( k = \frac{2A}{S^2} \).

Step-by-step solution

Understanding the Morse Potential Function

The given function is the Morse potential: \( U(r) = A\left[\left(e^{(R-r) / S}-1\right)^{2}-1\right] \). This function describes how the potential energy \( U \) varies with the distance \( r \) between two atoms.

Sketching the Morse Potential

For \( 0 < r < \infty \), sketch \( U(r) \). As \( r \to 0 \), \( U(r) \to \infty \) because \( (R-r)/S \to \infty \) leads to a large exponent. As \( r \to \, \) \( \infty \), \( U(r) \to 0 \). For a minimum at some \( r_o \), \( U(r) \) starts high, drops to a minimum, then approaches zero as \( r \to \infty \).

Finding the Equilibrium Separation \( r_o \)

To find the minimum (equilibrium), set the derivative \( \frac{dU}{dr} = 0 \). Calculate: \[ \frac{dU}{dr} = A \cdot 2 \cdot e^{(R-r) / S}\left(e^{(R-r) / S} - 1\right) \cdot \frac{-1}{S} = 0. \] This simplifies to: \[ e^{(R-r_o)/S} = 1, \] hence, at the equilibrium separation \( r_o \), \( R = r_o \).

Applying Displacement from Equilibrium

Set \( r = r_o + x \), substitute into the expression: \[ U(r_o + x) = A \left( \left(e^{(R-(r_o + x)) / S} - 1\right)^2 - 1 \right). \]

Approximating for Small Displacements

For small \( x \), use \( e^{-(x/S)} \approx 1 - x/S + (x/S)^2/2 \) to approximate: \[ U(r_o + x) \approx A \left( \left(1 - 1 + x/S - (x/S)^2/2 \right)^2 - 1 \right). \] Simplifying gives: \[ U \approx A \left( (x/S)^2 - 1 \right). \]

Expressing in Hooke's Law Form

Expand the term \((x/S)^2\) and simplify: \[ U \approx -A + \frac{A}{S^2}x^2. \] This is in the form Hooke's law: \( U = \text{const} + \frac{1}{2}kx^2 \) where \( k = \frac{2A}{S^2} \).

Determining the Force Constant \( k \)

Thus, for Hooke's law approximation, the spring constant \( k \) is \( \frac{2A}{S^2} \).

Key concepts

Equilibrium Separation

In the context of the Morse potential, the equilibrium separation, denoted as \( r_o \), is the point at which the potential energy \( U(r) \) is at a minimum between two atoms. Think of it as the most comfortable distance where the atoms feel least "stressed" by their interaction forces. Mathematically, this equilibrium point is found by setting the derivative of the potential energy function with respect to distance \( r \) to zero, i.e., \( \frac{dU}{dr} = 0 \). This condition reveals where the system is in a stable state with no net force acting on the atoms. In simpler terms, it's the distance where the atoms are naturally balanced without external forces pulling them closer or pushing them apart.
The equilibrium separation \( r_o \) is crucial when discussing molecular vibrations, as it serves as the reference point from which deviations are measured during vibrational motions.

Hooke's Law Approximation

When atoms are slightly displaced from their equilibrium separation \( r_o \), their interaction can often be described by Hooke's Law. This approximation simplifies the complex Morse potential to a more manageable form for small deviations. Here's how it works:
For tiny displacements \( x \) from \( r_o \), denoted as \( r = r_o + x \), we can approximate \( U(r) \) using a quadratic function similar to a spring in Hooke's Law. This results in \( U = \text{const} + \frac{1}{2} k x^2 \), where \( k \) is the "spring constant". Such an approximation holds well for many molecular vibrations, allowing physicists to analyze these movements effectively using simple harmonic motion.

• The linear restoring force is straightforward and intuitive.
• This approach facilitates easier mathematical handling of vibrational analyses.

Force Constant

In the context of the Morse potential, the force constant, denoted \( k \), is a measure of the stiffness of the potential energy curve around the equilibrium separation \( r_o \). It is analogous to the spring constant in classical mechanics, representing how much "force" is required to displace an atom from its equilibrium position. This is essential for understanding how strongly bonded atoms resist being stretched or compressed.
Mathematically, the force constant is derived by examining the second derivative of the potential energy function at equilibrium, which results in \( k = \frac{2A}{S^2} \). It forms a key parameter in modeling molecular vibrations, providing insights into the strength of the interaction between atoms. The higher the force constant, the more energy is required to change the distance between the atoms, indicating a stronger bond.

Potential Energy Function

The Morse potential energy function describes the energy interaction between two atoms as they move relative to each other. It is given by the equation:\[ U(r) = A \left[ \left(e^{(R-r) / S}-1\right)^{2}-1\right]\]where \( r \) is the separation between the atoms, and \( A, R, \) and \( S \) are constants with \( S \ll R \). This potential gives a more realistic approximation compared to simpler models, as it accounts for the anharmonic behavior of diatomic molecules.
The function features a well-defined minimum at \( r_o \), representing the equilibrium separation.

• It rises steeply as \( r \) approaches zero, indicating strong repulsion when atoms are very close.
• As \( r \) becomes very large, the potential approaches zero, suggesting that the atoms exert negligible forces on each other when far apart.

The Morse potential effectively models how molecular bonds stretch and compress during vibrational motions, capturing phenomena that simpler, idealized models like the harmonic oscillator cannot fully describe. This sensitivity to real-world molecular interactions makes it a vital tool in chemical physics.