Advanced Chemistry - Mono Mole

August 1, 2025

Microwave spectroscopy (Instrumentation)

Microwave spectroscopy is a powerful analytical technique that analyses the rotational transitions of molecules. Operating in the microwave region of the electromagnetic spectrum, this method provides highly precise information about molecular structure, bond lengths, bond angles, and even the distribution of electrons within a molecule.

The diagram above outlines a conventional microwave spectrometer. A stable and tunable microwave source is essential for probing molecular energy transitions. This is fulfilled by a device known as a Klystron amplifier, which is a vacuum tube engineered to generate and amplify narrow-band microwave signals that can be precisely tuned to the desired frequency. It was invented in 1937 by American electrical engineers Russell and Sigurd Varian.

A Klystron amplifier includes an electron gun, two resonant cavities (buncher and catcher), a drift tube and a collector (see diagram above). The electron gun operates on the principle of thermionic emission, where electrons are emitted from a heated coil (cathode) and accelerated towards an anode. The buncher cavity is connected to a weak external AC supply (with frequency in the microwave range), which creates an oscillating flow of free electrons within the cavity walls. This, in turn, generates an alternating electric field across the perforated section (grid) of the cavity.

As electrons pass through the grid, they are either accelerated or decelerated depending on the polarity of the electric field. When the entrance grid is negative and the exit grid is positive, electrons are accelerated. When the polarity reverses, they are decelerated. This periodic acceleration and deceleration causes the electrons to bunch together as they travel through the drift tube, forming dense clusters that correspond to the frequency of the external AC source.

When these bunched electrons enter the catcher cavity, they induce a stronger alternating current (due to their higher density) at the same frequency as the AC input. Through an inductor, this energy is converted into strong monochromatic microwave radiation, which is then propagated along a waveguide.

Question

Is the frequency of the generated microwave radiation tunable? How does induction occur in the catcher cavity?

Answer

The resonant cavities are precisely engineered in both shape and size to resonate at a narrow band of microwave frequencies. Each Klystron amplifier module has a narrow operational bandwidth that can be tuned by mechanically adjusting the cavity dimensions using tuning screws. To cover a wider frequency range, multiple modules with different cavity dimensions must be employed.

Electromagnetic theory states that a flowing current generates a magnetic field, and that a changing magnetic field induces an alternating current in a conductor. The varying density of electrons flowing through the catcher cavity constitutes a varying current, which in turn creates a time-varying magnetic field. This results in the induction of an alternating current within the cavity walls.

The waveguide, constructed using copper of stainless steel, is a hollow evacuated tube designed to confine and direct the microwaves to the sample cell, which is often a section of the waveguide. Gases to be analysed are introduced via a gas inlet system, and the cell is often equipped with pumps and valves to control pressure and flush previous samples.

If the monochromatic microwave radiation matches the energy of a specific rotational transition of the molecules, a portion of it will be absorbed by the sample. The remaining (transmitted) microwave signal then reaches the detector, which typically consists of an inductive pickup (antenna), a diode rectifier, capacitors, and an amplifier. The inductor acts as an antenna that couples the microwave radiation into an alternating current (AC), which is then rectified by the diode to produce a pulsating direct current (DC) signal. A low-pass filter, often just a capacitor, smooths the rectified signal into a more stable DC voltage. This process is repeated as the microwave frequency is swept or incremented across a range. At each frequency step, the resulting DC voltage, which is proportional to the transmitted microwave intensity, is amplified and sent to a computer for data acquisition and analysis. Finally, the amplitudes of the DC voltages are plotted as a function of frequency, producing a microwave absorption spectrum that reveals the rotational transitions of the sample.

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August 1, 2025September 29, 2025

Rotational selection rules

Rotational selection rules for molecules determine the probabilities of rotational state transitions observed in spectroscopy.

According to the time-dependent perturbation theory, the transition probability $\vert\mu_{fi}\vert^2$ between the orthogonal rotational states $\psi(\theta,\phi)_{l,m_l}$ and $\psi(\theta,\phi)_{l',m'_l}$ within a given vibrational state of a molecule, as observed by microwave spectroscopy, is proportional to $\vert\langle\psi(\theta,\phi)_{l',m'_l}\vert\boldsymbol{\mathit{\hat{\mu}}}\vert\psi(\theta,\phi)_{l,m_l}\rangle\vert^2$ , where $\boldsymbol{\mathit{\hat{\mu}}}$ is the operator for the molecule’s electric dipole moment. In other words, a molecule must possess a permanent electric dipole moment ( $\boldsymbol{\mathit{\hat{\mu}}}\neq 0$ ) to exhibit a rotational spectrum. Homonuclear diatomic molecules and spherical rotors with no net permanent dipole moment (such as H₂ and CH₄ in their ground states) are generally rotationally inactive.

Since $\boldsymbol{\mathit{\hat{\mu}}}=\boldsymbol{\mathit{i}}\hat{\mu}_x+\boldsymbol{\mathit{j}}\hat{\mu}_y+\boldsymbol{\mathit{k}}\hat{\mu}_z$ , the components of the dipole moment in polar coordinates are:

$\hat{\mu}_x=\mu sin\theta cos\phi\;\;\;\;\;\;\;\;60$

$\hat{\mu}_y=\mu sin\theta sin\phi\;\;\;\;\;\;\;\;61$

$\hat{\mu}_z=\mu cos\theta\;\;\;\;\;\;\;\;62$

Suppose the perturbation on the molecule is caused by a plane-polarised electromagnetic wave with an electric component $\varepsilon_z$ oscillating in the $z$ -direction. We have $\vert\mu_{z,fi}\vert^2\propto\vert\langle\psi(\theta,\phi)_{l',m'_l}\vert\hat{\mu_z}\vert\psi(\theta,\phi)_{l,m_l}\rangle\vert^2$ , with

$\mu_{z,fi}=\int_0^{\pi}\int_0^{2\pi}\psi^*(\theta,\phi)_{l',m'_l}\mu cos\theta\psi(\theta,\phi)_{l,m_l}sin\theta d\theta d\phi\;\;\;\;\;\;\;\;63$

Substituting the explicit expression of the spherical harmonics wavefunction $\psi(\theta,\phi)$ into eq63 gives:

$\mu_{z,fi}=\mu N'N\int_0^{\pi}cos\theta P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}sin\theta d\theta\int_0^{2\pi}e^{i(m_l-m'_l)\phi}d\phi\;\;\;\;\;\;\;\;64$

where $N'N=\frac{1}{2\pi}\biggr\[\biggr$\frac{2l'+1}{2}\biggr$\frac{(l'-\vert m'_l\vert)!}{(l'+\vert m'_l\vert)!}\biggr\]^{\frac{1}{2}}\biggr\[\biggr$\frac{2l+1}{2}\biggr$\frac{(l-\vert m_l\vert)!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}}$ , and $P_{l'}^{\vert m'_l\vert}$ and $P_{l}^{\vert m_l\vert}$ are functions of $cos\theta$ .

When $m_l\neq m'_l$ , the integral with respect to $\phi$ is $\int_0^{2\pi}e^{i(m_l-m'_l)\phi}d\phi =0$ . When $m_l=m'_l$ , it evaluates to $\int_0^{2\pi}e^{i(m_l-m'_l)\phi}d\phi =2\pi$ . Since eq63 must be non-zero for a transition to be probable in the $z$ -direction, $m_l=m'_l$ or

$\Delta m_l=0\;\;\;\;\;\;\;\;65$

Substituting $\int_0^{2\pi}e^{i(m_l-m'_l)\phi}d\phi =2\pi$ back into eq64 and noting that $d(cos\theta)=-sin\theta d\theta$ yields:

$\mu_{z,fi}=\mu N''\int_{-1}^{1}cos\theta P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}d(cos\theta)\;\;\;\;\;\;\;\;66$

where $N''=\frac{1}{2}\biggr\[\frac{(2l'+1)(2l+1)(l'-\vert m'_l\vert)!(l-\vert m_l\vert)!}{(l'+\vert m'_l\vert)!(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}}$ .

Substituting the polar coordinate form $x=cos\theta$ into eq370 (a recurrence relation of the associated Legendre polynomials) results in:

$cos\theta P_{l}^{\vert m_l\vert}=\frac{l+\vert m_l\vert}{2l+1}P_{l-1}^{\vert m_l\vert}+\frac{l-\vert m_l\vert+1}{2l+1}P_{l+1}^{\vert m_l\vert}\;\;\;\;\;\;\;\;67$

Substituting eq67 into eq66 gives:

$\mu_{z,fi}=\mu N''\biggr\[\frac{l+\vert m_l\vert}{2l+1}\int_{-1}^{1}P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}d(cos\theta)+\frac{l-\vert m_l\vert+1}{2l+1}\int_{-1}^{1}P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}d(cos\theta)\biggr\]\;\;\;\;\;\;\;\;68$

For $\mu_{z,fi}\neq 0$ , either integral in eq68 must be non-zero. Each integral is non-zero when the corresponding pair of spherical harmonics is not orthogonal. This occurs if $l'=l-1$ and $m'_l=m_l$ for the first integral and $l'=l+1$ and $m'_l=m_l$ for the second integral. In other words, $\mu_{z,fi}\neq 0$ if

$\Delta l=\pm 1\;\;\;\;and\;\;\;\;\Delta m_l=0$

or using rotational spectroscopy notations:

$\Delta J=\pm 1\;\;\;\;and\;\;\;\;\Delta M_J=0\;\;\;\;\;\;\;\;69$

For an electric component $\varepsilon_x$ oscillating in the $x$ -direction, eq63 becomes

$\mu_{x,fi}=\int_0^{\pi}\int_0^{2\pi}\psi^*(\theta,\phi)_{l',m'_l}\mu sin\theta cos\phi\psi(\theta,\phi)_{l,m_l}sin\theta d\theta d\phi\;\;\;\;\;\;\;\;70$

Substituting the explicit expression of the spherical harmonics wavefunction $\psi(\theta,\phi)$ and $cos\phi=\frac{e^{i\phi}+e^{-i\phi}}{2}$ into eq70 gives:

$\mu_{x,fi}=\frac{\mu}{2}N'N\int_0^{\pi}sin\theta P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}sin\theta d\theta\int_0^{2\pi}[e^{i(m_l-m'_l+1)\phi}+e^{i(m_l-m'_l-1)\phi}] d\phi\;\;\;\;\;\;\;\;71$

When $m'_l=m_l$ , the integral with respect to $\phi$ equals zero. When $m_l\pm 1=m'_l$ , it evaluates to $2\pi$ . Since eq71 must be non-zero for a transition to be probable in the $x$ -direction, it must satisfy:

$\Delta m_l=\pm 1\;\;\;\;\;\;\;\;72$

Substituting $\int_0^{2\pi}[e^{i(m_l-m'_l+1)\phi}+e^{i(m_l-m'_l-1)\phi}] d\phi=2\pi$ back into eq71, and noting that $d(cos\theta)=-sin\theta d\theta$ , yields:

$\mu_{x,fi}=\frac{\mu}{2}N''\int_{-1}^{1}sin\theta P_{l'}^{\vert m_l\pm 1\vert}P_l^{\vert m_l\vert} d(cos\theta)\;\;\;\;\;\;\;\;73$

Substituting the polar coordinate form $x=cos\theta$ and into eq371 (another recurrence relation of the associated Legendre polynomials) results in:

$sin\theta P_{l}^{\vert m_l\vert}=\frac{1}{2l+1}P_{l+1}^{\vert m_l\vert+1}-\frac{1}{2l+1}P_{l-1}^{\vert m_l\vert+1}\;\;\;\;\;\;\;\;74$

Substituting eq74 into eq73 gives:

$\mu_{x,fi}=\frac{\mu N''}{2(2l+1)}\biggr\[\int_{-1}^{1}P_{l'}^{\vert m_l\pm 1\vert}P_{l+1}^{\vert m_l\vert+1} d(cos\theta)-\int_{-1}^{1}P_{l'}^{\vert m_l\pm 1\vert}P_{l-1}^{\vert m_l\vert+1} d(cos\theta)\biggr\]\;\;\;\;\;\;\;\;75$

For $\mu_{x,fi}\neq 0$ , either integral in eq75 must be non-zero. Each integral is non-zero when the corresponding pair of spherical harmonics is not orthogonal. This occurs if

Integrals	Condition 1	Cases	Condition 2	Results
1^st	$l'=l+1$	$m_l+1=m'_l$	$\vert m_l+1\vert-\vert m_l\vert=+1$	$\Delta m_l=+1$ if $m_l\geq 0$
		$m_l+1=m'_l$	$\vert m_l\vert-\vert m_l+1\vert=-1$	$\Delta m_l=-1$ if $m_l\geq 0$
		$m_l-1=m'_l$	$\vert m_l-1\vert-\vert m_l\vert=+1$	$\Delta m_l=+1$ if $m_l\leq 0$
		$m_l-1=m'_l$	$\vert m_l\vert-\vert m_l-1\vert=-1$	$\Delta m_l=-1$ if $m_l\leq 0$
2^nd	$l'=l-1$	$m_l+1=m'_l$	$\vert m_l+1\vert-\vert m_l\vert=+1$	$\Delta m_l=+1$ if $m_l\geq 0$
		$m_l+1=m'_l$	$\vert m_l\vert-\vert m_l+1\vert=-1$	$\Delta m_l=-1$ if $m_l\geq 0$
		$m_l-1=m'_l$	$\vert m_l-1\vert-\vert m_l\vert=+1$	$\Delta m_l=+1$ if $m_l\leq 0$
		$m_l-1=m'_l$	$\vert m_l\vert-\vert m_l-1\vert=-1$	$\Delta m_l=-1$ if $m_l\leq 0$

Combining the results, $\mu_{x,fi}\neq 0$ when $\Delta l=\pm 1$ and $\Delta m_l=\pm 1$ , or in rotational spectroscopy notations:

$\Delta J=\pm 1\;\;\;\;and\;\;\;\;\Delta M_J=\pm 1\;\;\;\;\;\;\;\;76$

Repeating the derivation for $\mu_{y,fi}$ , we arrive at the same selection rules expressed by eq76. Therefore, the rotational selection rules for polar molecules (linear rotors and spherical rotors) subjected to isotropic radiation are:

$\Delta J=\pm 1\;\;\;\;and\;\;\;\;\Delta M_J=0,\pm 1\;\;\;\;\;\;\;\;77$

Question

Is eq77 applicable both to transitions between rotational levels within a single vibrational state (e.g. $v=0$ ) and to transitions between rotational levels of two different vibrational states (e.g. $v=0$ and $v=1$ )?

Answer

Yes, the selection rule applies to both pure rotational transitions within a vibrational state and to rotational transitions accompanying vibrational transitions (rovibrational transitions).

For symmetric rotors, the electric dipole moment lies along the principal molecular axis ( $z$ -axis). However, there is a third quantum number $K$ to consider. If $\hat{\mu}_{z}$ also lies along the principal axis, the wavefunction $\psi(J,M_J,K)$ can be approximated as the product of three functions $f_1f_2f_3$ , where $f_3=e^{-iK\gamma}$ depends solely on the quantum number $K$ and $\gamma$ is one of three Euler angles (see diagram above). The matrix element $\langle J'M'K'\vert\hat{\mu}_{z}\vert JMK\rangle$ can then be expressed as a product of three integrals over the Euler angles, with one of the integrals being $\int f_3'^*(K')\hat{\mu}_{z}f_3(K)d\gamma$ . Since $\hat{\mu}_{z}$ is independent of $\gamma$ ,

$\int f_3'^*(K')\hat{\mu}_{z}f_3(K)d\gamma=\hat{\mu}_z\int_0^{2\pi}e^{i(K'-K)\gamma}d\gamma$

For this integral to be non-zero, we must have $K'=K$ . In other words, the rotational transition selection rules for symmetric rotors are:

$\Delta J=\pm 1,\;\;\;\;\Delta M_J=0,\pm 1\;\;\;\;and\;\;\;\;\Delta K=0\;\;\;\;\;\;\;\;78$

Question

Is the effect of nuclear statistics on rotational states different from rotational selection rules?

Answer

The effect of nuclear statistics on rotational states is a separate, yet related, concept from the general rotational selection rules. While both influence which rotational states are observed in a spectrum, they operate based on different fundamental principles. The key distinction is that rotational selection rules dictate the possible transitions, while nuclear statistics determine the relative populations of the initial states. A transition must be both “allowed” by the selection rules and originate from a “populated” state to be observed. Therefore, nuclear statistics modify the intensity of the allowed transitions without changing the fundamental rule of which transitions are allowed.

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