Rotational, Vibrational, and Vibrational-Rotational Spectroscopy

Flashcards generated from lecture notes on rotational spectroscopy, vibrational spectroscopy, harmonic and anharmonic oscillators, selection rules, overtones, hot bands, and vibrational-rotational fine structure.

What is the gross selection rule for a molecule to be rotationally active?

The molecule must have a permanent dipole moment ($\mu \neq 0$).

What is the specific selection rule for rotational spectroscopy?

$$\Delta J = \pm 1$$

What is the formula for the rotational energy levels $E_J$ of a rigid rotor?

$E_J = B J(J+1)$, where $J = 0, 1, 2, 3 \dots$

What is the energy gap $\Delta E$ between two successive rotational energy levels $J$ and $J+1$?

$$\Delta E = 2B(J+1)$$

What is the spacing between consecutive rotational spectral lines?

The spacing between rotational spectral lines is constant and equal to $2B$.

What is the formula for $J_{max}$, the rotational energy level with the maximum population?

$$J_{max} = \sqrt{\frac{k_B T}{2 B h c}} - \frac{1}{2}$$

What is the rotational energy level expression $E_J$ for a non-rigid rotor?

$E_J = B J(J+1) - D J^2 (J+1)^2$, where $D = \frac{4 B^3}{\bar{\nu}^2}$

According to the law of equipartition of energy, what are the degrees of freedom for linear and non-linear molecules containing $N$ atoms?

Total degrees of freedom = $3N$. Linear: Translational = $3$, Rotational = $2$, Vibrational = $3N - 5$. Non-linear: Translational = $3$, Rotational = $3$, Vibrational = $3N - 6$.

What is the gross selection rule for vibrational spectroscopy?

There must be a change in dipole moment after vibration.

What is the vibrational energy level formula $E_v$ for a Simple Harmonic Oscillator (SHO)?

$E_v = \left(v + \frac{1}{2}\right) h \nu$, where $v = 0, 1, 2, 3 \dots$ is the vibrational quantum number.

What is the zero-point energy ($v = 0$) of a Simple Harmonic Oscillator?

$$E_0 = \frac{1}{2} h \nu$$

What is the specific selection rule and expected number of IR spectral peaks for a Simple Harmonic Oscillator?

The specific selection rule is $\Delta v = \pm 1$, which results in only 1 peak at $\nu = \nu_0$ in the IR spectrum.

What is the vibrational energy level expression $E_v$ for an Anharmonic Oscillator?

$E_v = \left(v + \frac{1}{2}\right) \bar{\omega}_e - \left(v + \frac{1}{2}\right)^2 x_e \bar{\omega}_e\,cm^{-1}$, where $x_e$ is the anharmonicity constant ($x_e \ll 1$).

What is the specific selection rule for an Anharmonic Oscillator?

$$\Delta v = \pm 1, \pm 2, \pm 3 \dots$$

What is the wavenumber formula for the fundamental band ($v'' = 0 \rightarrow v' = 1$) in an anharmonic oscillator?

$$\bar{\nu} = \bar{\omega}_e (1 - 2 x_e)\,cm^{-1}$$

What are the wavenumber expressions for the 1st, 2nd, and 3rd overtones in an anharmonic oscillator?

1st Overtone ($0 \rightarrow 2$): $\bar{\nu}_{0 \rightarrow 2} = 2 \bar{\omega}_e (1 - 3 x_e)$; 2nd Overtone ($0 \rightarrow 3$): $\bar{\nu}_{0 \rightarrow 3} = 3 \bar{\omega}_e (1 - 4 x_e)$; 3rd Overtone ($0 \rightarrow 4$): $\bar{\nu}_{0 \rightarrow 4} = 4 \bar{\omega}_e (1 - 5 x_e)$

What are the wavenumber expressions for the 1st, 2nd, and 3rd hot bands in an anharmonic oscillator?

1st Hot Band ($1 \rightarrow 2$): $\bar{\nu} = \bar{\omega}_e (1 - 4 x_e)\,cm^{-1}$; 2nd Hot Band ($2 \rightarrow 3$): $\bar{\nu} = \bar{\omega}_e (1 - 6 x_e)\,cm^{-1}$; 3rd Hot Band ($3 \rightarrow 4$): $\bar{\nu} = \bar{\omega}_e (1 - 8 x_e)\,cm^{-1}$

What are the formulas for dissociation energy $D_e$ and maximum vibrational quantum number $v_{max}$?

$D_e = \frac{\bar{\omega}_e}{4 x_e}$ and $v_{max} = \frac{1}{2 x_e} - 1$

What are the characteristic energy separations between rotational levels compared to vibrational levels?

The energy separation between rotational levels is $\approx 10\,cm^{-1}$, whereas the energy separation between vibrational levels is $\approx 3000\,cm^{-1}$.

What is the total energy equation $E_{TOTAL}$ for vibrational-rotational spectroscopy?

$$E_{TOTAL} = \left(v + \frac{1}{2}\right) \bar{\omega}_e - \left(v + \frac{1}{2}\right)^2 x_e \bar{\omega}_e + B J(J+1)$$

In vibrational-rotational spectroscopy, how are the P-branch and R-branch distinguished?

The P-branch corresponds to $\Delta J = -1$ (lower energy transition), while the R-branch corresponds to $\Delta J = +1$ (higher energy transition).