The Vibrational Hamiltonian

The classical expression for the vibrational energy, using molecular-fixed $x, y, z$ coordinates ( $u_1, u_2, ... u_{3N}$) = ( $\Delta x_i, \Delta y_i, ... \Delta z_N$) is

$$E_{vib} = \frac{1}{2}\sum_{i=1}^{3N}m_i \dot{u}_i^2 + V_N(u_i),$$ (2)

where $V_N(u_i)$ is the potential energy which is zero at the equilibrium together with its first derivative. The Taylor's series expansion about the equilibrium is

$$V_N = \frac{1}{2}\sum_{i,j=1}^{3N} k_{ij}u_iu_j +\frac{1}{6} \sum... ...u_iu_ju_k + \frac{1}{24} \sum_{i,j,k,l=1}^{3N}k_{i,j,k,l}u_iu_ju_ku_l + \cdots,$$ (3)

where $k_{ij}, k_{i,j,k}$ and $k_{i,j,k,l}$ are force constants.

The lowest order terms in the expansion are quadratic and for small displacement only these terms can be preserved in eq. (3), while all other terms can be neglected. Corresponding expression for the potential $V_N^0$ is called the harmonic-oscillator approximation. In the harmonic-oscillator approximation the vibrational energy can be written as

$$E_{vib}^0 = \frac{1}{2}\sum_{i=1}^{3N} m_{i}\dot{u}_i^2 +\frac{1}{2} \sum_{i,j=1}^{3N}k_{i,j}u_iu_j,$$ (4)

where $k_{i,j}$ are harmonic force constants.

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