Normal Vibrational Modes

A standard result from classical mechanics is that the vibrational energy of a $N$-body harmonic oscillator (4) can be written in terms of $3N-6$ mass-weighted linear combinations of the $u_j$ which are called vibrational normal coordinates $Q_r$:

$$E_{vib}^0 = \frac{1}{2}\sum_{r=1}^{3N-6} [\dot{Q}_r^2 + \lambda_rQ_r^2],$$ (5)

where

$$m_i^{1/2}u_i = \sum_{r=1}^{3N}l_{ui,r}Q_r.$$ (6)

The quantum mechanical Hamiltonian of a vibrating polyatomic molecule can be obtained from eq.(5) by replacing the classical variables $Q_r$ and $\dot{Q_r}$ by their quantum mechanical analogues. Great advantage of the vibrational energy expression in eq.(5) is that there is no cross terms in the potential energy. Therefore, the solution (wavefunction) of the corresponding Schrödinger equation is greatly simplified as can be presented as a product of the normal mode wavefunctions which are known solution of the harmonic oscillator problem:

$$\Phi_{vib} = \Phi_{v1}(Q_1)\Phi_{v2}(Q_2)\ldots \Phi_{v3N-6}(Q_{3N-6}).$$ (7)

The corresponding vibrational energy $E_{vib}$ is a sum of the each normal mode energy

$$E_{vib} = E_{v1} + E_{v2} + \cdots + E_{v3N-6},$$ (8)

where $E_{vk} = \omega_{ek}(v_k + 1/2)$.

Each of the $3N-6$ ($3N-5$) vibrational normal coordinate $Q_r$ describes a collective normal mode of vibration.

In general, any vibrational of the molecular system may be represented as a superposition of normal vibrations with suitable amplitudes. Within each of the normal mode $k$ all nuclei move with the same frequency $\nu_k$ according to simple harmonic motion. Two, or more normal modes are degenerate if they all have the same frequency.

As an example, consider vibration of a mass suspended by an elastic bar of rectangular cross section. If mass is displaced slightly from its equilibrium position in the $x$ direction and then left, it will carry our simple harmonic in this direction with a frequency

$$\nu_x = \frac{1}{2\pi}\sqrt{\frac{k_x}{m}},$$ (9)

where $k_x$ is a force constant in the $x$ direction.

If mass is displaced slightly from its equilibrium position in the $y$ direction and then left, it will carry our simple harmonic in this direction with a frequency

$$\nu_y = \frac{1}{2\pi}\sqrt{\frac{k_y}{m}},$$ (10)

where $k_y$ is a force constant in the $y$ direction.

If mass is displaced in a direction different from $x$ and $y$, it will not carry out a simple harmonic oscillation, but more complicated type of motion, so named Lissajous motion. This is because the restoring force $F$ whose components are $F_x = - k_xx$ and $F_y = - k_yy$ is not directed toward the origin since $k_x\ne k_y$. However, this motion can be always presented as linear superposition of two simple harmonic motions of different frequency:

$$x = x_0 \cos2\pi\nu_xt, \:\:\:\:\:\:\:\: y = y_0 \cos2\pi\nu_yt,$$ (11)

where $x_0$, $y_0$ are coordinates of initial position of the mass (point A).

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