# Energy Levels¶

## Energy Quantisation¶

Energy is quantised with Planck’s constant (\(h\)) as the proportionality factor

## Energy Levels¶

### Electronic¶

These the quantum electronic states of the molecule

### Translational (3 DOF)¶

The molecule as a whole can move in space (as a single unit)

### Rotational (linear = 2 DOF nonlinear = 3 DOF)¶

The molecule can rotate in space

Where \(I=\) The moment of inertia of the system (\(I=\sum_{j=1}^nm_j(x_j-x_{cm})^2\) for linear molecules) and the degeneracy of a given level is \(\mathrm{g}_J=2J+1\)

Momentum for nonlinear molecules

For nonlinear molecules, we need to consider the symmetry, as it will have multiple moments of inertia, based on how it rotates

So this looks like:

### Vibrational (linear = 3n-5 DOF nonlinear = 3n-6 DOF)¶

Note

The subtraction from the degrees of freedom is removing the translational and rotational degrees of freedom. All molecules have a total of \(3n\) degrees of freedom

The regions bonds can vibrate with energy

This is a harmonic oscillator model and the degeneracy is \(\mathrm{g}_v=1\)

The space between each of the energy levels is \(=h\nu\), and the first energy level is separated from the depth of the well by the ZPVE \(\bigg(\frac{h\nu}{2}\bigg)\)

For the Morse potential, we can use the equation

The individual vibrations have different *normal modes* (types of vibration), and so the total vibrational energy is the sum of all the normal modes

### Total energy¶

The total energy of a molecule can be described as the sum of all of these energies

## Spacing of Energy Levels¶

These energy levels build upon each other, so for every electronic level is a series of vibrational levels and for every vibrational levels there are a series of rotational levels and for every rotational level there are a series of translational levels