In order to describe the vibrational motion of a diatomic molecule (essentially variations in the bond length \(r\)), it is convenient to define a variable \(x\) which measure the magnitude of displacement from the equilibrium bond length \(r_e\). \(x\) is defined as follows:
\[ x = r - r_e. \]
When the bond is at equilibrium, \(r = r_e\) and \(x = 0\). Positive values of \(x\) correspond to bond stretching, while negative values correspond to bond compression.
Writing the potential energy as \(U(x)=\tfrac{1}{2}kx^2\) emphasizes that the energy depends only on how far the bond is displaced from equilibrium, not on its absolute position.
Big idea: the harmonic oscillator treats molecular vibrations as motion about the equilibrium bond length, with \(x = r - r_e\) measuring the deviation from equilibrium.
To describe molecular vibrations quantum mechanically, we begin by defining the potential energy surface (PES) associated with a vibrating bond. The potential energy surface specifies how the potential energy of a system changes as the positions of the atoms change.
For a diatomic molecule (or a single vibrational coordinate in a polyatomic molecule), the vibration can be described by the displacement \(x\) from the equilibrium bond length. The equilibrium position corresponds to the minimum of the potential energy.
The harmonic approximation
Near the equilibrium position, the potential energy can be approximated by a quadratic function of the displacement:
\[ U(x) = \frac{1}{2} k x^2. \]
This form is known as the harmonic oscillator potential. The constant \(k\) is the force constant, which measures the stiffness of the bond. A larger value of \(k\) corresponds to a steeper potential well and a stronger bond.
The harmonic potential is symmetric about \(x=0\), meaning that stretching and compressing the bond by the same amount costs the same amount of energy.
Connection to real molecular potentials
Real molecular potential energy curves are not exactly quadratic. At large displacements, bonds eventually break, and the potential energy flattens rather than increasing indefinitely.
Taylor series origin of the harmonic potential
The harmonic oscillator potential arises naturally from a Taylor series expansion of a general potential energy surface about the equilibrium position.
Let \(U(x)\) be the potential energy as a function of displacement \(x\) from equilibrium. Expanding \(U(x)\) about \(x=0\) gives
\[ U(x) = U(0) + \left.\frac{dU}{dx}\right|_{0} x + \frac{1}{2}\left.\frac{d^2U}{dx^2}\right|_{0} x^2 + \frac{1}{6}\left.\frac{d^3U}{dx^3}\right|_{0} x^3 + \cdots \]
Why the first two terms vanish
At the equilibrium position, the potential energy is at a minimum. Therefore the first derivative of the potential with respect to displacement is zero:
\[ \left.\frac{dU}{dx}\right|_{0} = 0. \]
The constant term \(U(0)\) represents the absolute reference energy and can be set to zero without loss of generality, since only energy differences are physically meaningful.
The first non-zero term: the quadratic contribution
The first non-zero term in the expansion is therefore the second-derivative term:
\[ U(x) \approx \frac{1}{2} \left.\frac{d^2U}{dx^2}\right|_{0} x^2. \]
The second derivative of the potential at equilibrium is identified as the force constant:
\[ k = \left.\frac{d^2U}{dx^2}\right|_{0}. \]
Substituting this definition gives the harmonic oscillator potential:
\[ U(x) = \frac{1}{2} k x^2. \]
Higher-order terms (\(x^3, x^4, \dots\)) describe anharmonicity and become important only for large displacements. For small vibrations near equilibrium, these terms are negligible.
Big idea: the harmonic oscillator potential is not an arbitrary choice — it is the simplest, physically justified approximation to any smooth potential energy surface near its minimum.
However, for small vibrations near equilibrium, the true potential energy curve is well approximated by the harmonic form. This approximation captures the essential physics of molecular vibrations while remaining mathematically simple.
In fact, the harmonic potential can be viewed as the first nonzero term in a Taylor expansion of the true potential energy about the equilibrium position.
Why the harmonic oscillator is important
The harmonic oscillator model provides:
- a simple and physically reasonable potential energy surface,
- an exactly solvable Schrödinger equation,
- a natural starting point for understanding molecular vibrations.
Although real molecules exhibit anharmonic behavior, the harmonic oscillator serves as the foundation upon which more accurate models are built.
Big idea: near equilibrium, molecular vibrations experience a restoring force that leads to a quadratic (harmonic) potential energy surface, making the harmonic oscillator a powerful and widely applicable model.
To set up the quantum-mechanical description of a vibrating bond, we now combine the kinetic energy and potential energy terms into the Hamiltonian operator.
For motion along a single vibrational coordinate \(x = r - r_e\), the kinetic energy operator is
\[ \hat{T} = -\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}, \]
where \( \mu \) is the reduced mass of the two atoms.
The reduced mass
A vibrating bond involves the motion of two atoms, not just one. In principle, this is a two-body problem. However, the motion can be simplified by transforming the system into an equivalent one-body problem.
This is accomplished by introducing the reduced mass, \( \mu \), defined as
\[ \mu = \frac{m_1 m_2}{m_1 + m_2}. \]
The reduced mass accounts for the fact that both atoms move as the bond stretches and compresses. If one atom is much heavier than the other, the reduced mass approaches the mass of the lighter atom.
Using the reduced mass allows the vibrational motion of the two atoms to be treated as if a single particle of mass \( \mu \) were moving in the harmonic potential.
Big idea: the reduced mass converts a two-body vibrational problem into an effective one-body problem, making the quantum harmonic oscillator model mathematically tractable.
The potential energy for a harmonic oscillator is
\[ U(x) = \frac{1}{2}kx^2. \]
Adding these together gives the Hamiltonian:
\[ \hat{H} = -\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2} + \frac{1}{2}kx^2. \]
This operator fully defines the quantum harmonic oscillator problem. Solving the Schrödinger equation with this Hamiltonian yields the allowed vibrational energy levels and wavefunctions.
Big idea: the harmonic oscillator Hamiltonian is the simplest quantum model that captures the essential physics of molecular vibrations.