Electronic Spectra of Diatomic Molecules: Improved Treatment

Abstract

Term symbols, introduced in the previous chapter, are the notation used to characterize key features of electron spin and orbital angular momentum.

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10.1 Term Symbols for Diatomic Molecules

Term symbols, introduced in the previous chapter, are the notation used to characterize key features of electron spin and orbital angular momentum.

• For an atom, the term symbol is: $^{2S+1}L_{J}$

• For a diatomic, the term symbol is: $^{2S+1}\Lambda _{\Omega }$

Important terms to define are $\vec{S}$, $\vec{\Sigma }$, $\vec{\Lambda }$, and $\vec{\Omega }$:

$\mathbf{\vec{\Lambda }}$ :

projection of orbital angular momentum on the internuclear axis

Its magnitude is $\left \vert \vec{\Lambda }\right \vert = \Lambda \hslash$, where $\Lambda$ is an integer, and the symbols associated with different values of $\Lambda$ are shown in the table below.

Table 1

$\mathbf{\vec{S}}$ :

total electronic spin angular momentum (the net sum of electron spin in unfilled shells). The magnitude is $\left \vert \vec{S}\right \vert = S\hslash$, where S will have 1∕2-integer values.

$\mathbf{\vec{\Sigma }}$ :

projection of $\vec{S}$ onto the internuclear axis (only defined when Λ ≠ 0). The magnitude of this projection is $\left \vert \vec{\Sigma }\right \vert = \Sigma \hslash$, and the allowed values of $\Sigma$ are

$$\displaystyle{\Sigma = S,S - 1,\ldots,-S\ \ \ \ \ \ \mbox{ ($2S + 1$ values)}}$$

$\mathbf{\vec{\Omega }}$ :

sum of projections along the internuclear axis of electron spin and orbital angular momentum

$$\displaystyle\begin{array}{rcl} \vec{\Omega }& =& \vec{\Sigma } +\vec{ \Lambda } {}\\ \Omega & =& \Lambda + S,\Lambda + S - 1,\ldots,\left \vert \Lambda - S\right \vert \ \ \ \ \ \mbox{ ($2S + 1$ values for $\Lambda \geq S$)} {}\\ \end{array}$$

Examples

NO :

The ground state for NO is $X^{2}\Pi$.

S = 1∕2, $\Lambda = 1$, $\Omega = 3/2,1/2$

There are two spin-split sub-states: $^{2}\Pi _{1/2}$, $^{2}\Pi _{3/2}$

Separation: 121 cm⁻¹

CO :

The ground state for CO is $X^{1}\Sigma ^{+}$.

S = 0 and $\Lambda = 0$, therefore $\Omega$ is unnecessary. This is a rigid rotor molecule with no influence from electrons. Easiest case!

O ₂ :

The ground state for O₂ is $X^{3}\Sigma _{g}^{-}$.

S = 1, $\Lambda = 0$

The⁻ and _g are notations about symmetric properties of wave functions. This is an example of a molecule that is modelled by Hund’s case b, discussed below.

10.2 Common Molecular Models for Diatomics

There are four common molecular models that are used to describe diatomic electronic spectra:

Rigid rotor:

$\Lambda = 0,S = 0$

Symmetric top:

$\Lambda \neq 0,S = 0$

Hund’s a :

$\Lambda \neq 0,S\neq 0$

Hund’s b :

$\Lambda = 0,S\neq 0$

The rigid rotor and symmetric top models have no spin, and thus their multiplicity (2S + 1) is one; these states are called “singlets.” For the Hund’s cases, the influence of spin on the electronic state structure must be considered through interactions of $\Lambda$ and $\Sigma$.

10.2.1 Rigid Rotor ($^{1}\Sigma$)

The simplest model for molecular rotation assumes that electron motions do not contribute to the rotational energy. Rotation of the nuclei occurs about an axis perpendicular to the A-axis (i.e., the B-axis). Recall that I _A  ≈ 0 and I _B  = I _C (Fig. 10.1).

Fig. 10.1

Rigid rotor model for molecular motion

When $\Lambda$ and S = 0, the molecule is $^{1}\Sigma$ type and $\Omega$ is not defined. Note that $\Lambda = 0$ means that the projection of the orbital angular momentum onto the A-axis is zero, and that rotation must thus be around the B-axis.

Rotational Energy

The total rotational energy for a rigid rotor, including centrifugal distortion is, as before,

$$\displaystyle{ F(J) = B_{v}J(J + 1) - D_{v}J^{2}(J + 1)^{2} }$$

(10.1)

Total Energy

Rovibronic transitions (those that include electronic, vibrational, and rotational changes in quantum number) have a total energy that includes contributions from each mode, i.e. $\Delta E = \Delta T_{e} + \Delta G + \Delta F$

$$\displaystyle{ \text{where} E(T_{e},v,J) = T_{e} + G(v) + F(J). }$$

(10.2)

Selection Rules

The selection rules for these transitions, as described before, are:

Rotational spectra::

$\Delta J = J' - J'' = +1$

Rovibrational spectra::

$\Delta v = v' - v'' = +1$

$\Delta J = \pm 1$

Rovibronic spectra::

$\Delta v$ determined by Franck–Condon factors

$\Delta J = \pm 1$

Note: An alternate form for the selection rules is used in some texts, i.e. $\Delta \alpha =\alpha _{\text{final}} -\alpha _{\text{initial}}$ where (α = J or v).

Intensity Distribution

Within each band (v′, v″), the intensity distribution follows the Boltzmann distribution for J modified by a J-dependent branching ratio (i.e., for the P and R branches), known as the Hönl–London factor (defined later in this chapter). Similarly, the relative intensities among all the vibrational bands originating from a single initial level v _initial to all possible final levels v _final are given by Franck–Condon factors (a manifestation of the Franck–Condon principle). The relative total emission or absorption from v _initial (i.e., to all values of v _final) depends directly on the Boltzmann fraction in that level, i.e. $n_{v_{\mathrm{initial}}}/n$, and also on an overall Einstein coefficient or oscillator strength for the specific electronic system (i.e., a specific pair of lower and upper electronic states), as presented in Sect. 10.3.

Examples

Most stable diatomics, including CO, Cl₂, Br₂, N₂, H₂ are Rigid Rotors. Exceptions are NO ($X^{2}\Pi$) and O₂ ($X^{3}\Sigma$). (Note that there are no $X\Delta$ states for the diatomics listed in Herzberg—all X states are $\Sigma$ or $\Pi$!) Some linear polyatomics such as CO₂ ($\tilde{X}^{1}\Sigma _{g}^{+}$) and both HCN and N₂O ($\tilde{X}^{1}\Sigma ^{+}$) are Rigid Rotors with $^{1}\Sigma$ ground states. Remember, however, that nuclear spin can have an impact on the statistics of homonuclear diatomic molecules.

10.2.2 Symmetric Top

Symmetric tops have a non-zero projection of orbital angular momentum on the internuclear axis and zero spin ($\Lambda \neq 0,S = 0$). Thus, its ground states are $^{1}\Pi$, $^{1}\Delta$ (although, as mentioned in the previous section, there are no known $X^{1}\Delta$ ground states for diatomics) (Fig. 10.2).

Fig. 10.2

Symmetric top model for molecular motion

The important components are

$\vec{N}$ :

angular momentum of nuclei

$\vec{\Lambda }$ :

A-axis projection of electron orbital angular momentum

$\vec{J}$ :

total angular momentum; $\vec{J} =\vec{ N} +\vec{ \Lambda }$

Only the axial component of orbital angular momentum is used, because only $\vec{\Lambda }$ is a “good” quantum number, i.e. a constant of the motion.

Rotational Energy

The total rotational energy for a symmetric top is

$$\displaystyle{ F(J) = BJ(J + 1) + (A - B)\Lambda ^{2}\ \ \ \ \ \ J = \Lambda,\Lambda + 1,\ldots }$$

(10.3)

The constants A and B are given as before by

$$\displaystyle{ A,B = \frac{h} {8\pi ^{2}cI_{A,B}} }$$

(10.4)

Therefore, the symmetric top energy levels have the same spacing as the Rigid Rotor, but with a constant offset. Note, however, that since I _A is small compared to I _B , A is large compared to B. Lines with $J <\Lambda$ are missing, as $J = \Lambda,\Lambda + 1,\ldots$.

Selection Rules

The selection rules for symmetric top electronic spectra are

$\Delta \Lambda = 0$ :

$\Delta J = \pm 1,0$         ($\Delta J = 0$ may be weak)

$\Delta \Lambda = \pm 1$ :

$\Delta J = \pm 1,0$

As a result of having a Q branch (i.e., $\Delta J = 0$), the bands for a symmetric top will be double-headed, in contrast to the single-headed character of rigid rotor bands.

Spectra

The spectra for the case where $\Delta \Lambda = 0$ are discussed here. The upper- and lower-state energies are described by the following equations:

$$\displaystyle\begin{array}{rcl} T'& =& B'J'(J' + 1) + (A' - B')\Lambda ^{2} + G(v') + T_{ e}'{}\end{array}$$

(10.5)

$$\displaystyle\begin{array}{rcl} T''& =& B''J''(J'' + 1) + (A'' - B'')\Lambda ^{2} + G(v'') + T_{ e}''{}\end{array}$$

(10.6)

$$\displaystyle\begin{array}{rcl} \bar{\nu }_{\infty }& =& \text{upper (for }J' = 0) -\text{lower (for }J'' = 0) = \text{constant}{}\end{array}$$

(10.7)

T _e ″ = 0 for the ground state. Then, for the three branches, the line positions are as follows:

$$\displaystyle\begin{array}{rcl} P(J'')& =& \bar{\nu }_{\infty }- (B' + B'')J + (B' - B'')J^{2}{}\end{array}$$

(10.8)

$$\displaystyle\begin{array}{rcl} Q(J'')& =& \bar{\nu }_{\infty } + (B' - B'')J + (B' - B'')J^{2}{}\end{array}$$

(10.9)

$$\displaystyle\begin{array}{rcl} R(J'')& =& \bar{\nu }_{\infty } + (B' + B'')(J + 1) + (B' - B'')(J + 1)^{2}{}\end{array}$$

(10.10)

Using the rotational number m for the three branches,

$$\displaystyle\begin{array}{rcl} m_{P}& =& -J {}\\ m_{Q}& =& +J {}\\ m_{R}& =& J + 1 {}\\ \end{array}$$

the line positions for the P and R branches become

$$\displaystyle{ \overline{\nu } = \bar{\nu }_{\infty } + am + bm^{2}, }$$

(10.11)

where a = B′ + B″ and b = B′ − B″, and the line positions for the Q branch become

$$\displaystyle{ \overline{\nu } = \bar{\nu }_{\infty } + bm + bm^{2} }$$

(10.12)

Note that the three branches now can lead to two bandheads (see the Fortrat parabola plot below for the case of a $^{1}\Delta \leftarrow \ ^{1}\Delta$ band) (Fig. 10.3).

Fig. 10.3

Symmetric top model for molecular motion, $^{1}\Delta \leftarrow ^{1}\Delta$ case

The Fortrat parabola shows the bandheads in the Q and R branches for the typical case of $B' <B''$. For the $^{1}\Delta \leftarrow \ ^{1}\Delta$ case, J _min = 2 and therefore m _min = 3 for the R branch, m _min = 2 for the Q branch and m = −3 is the first line in the P branch, resulting in multiple missing lines near the origin.

Intensity Distribution

Actual relative intensities depend on n _J ∕n and “relative intensity factors/line strengths,” also known as Hönl–London factors, denoted below by $S_{J}^{P,Q,R}$. Differences in Hönl–London factors indicate the breakdown of the principle of equal probability [1].

Example: Hönl–London factors for symmetric top (p. 208 of  [1])

For $\Delta \Lambda = 0$

$$\displaystyle{\left.\begin{array}{l} S_{J}^{R} = \frac{(J+1+\Lambda )(J+1-\Lambda )} {J+1} \approx J + 1\ \ (J \gg \Lambda )\\ \\ S_{J}^{Q} = \frac{(2J+1)\Lambda ^{2}} {J(J+1)} \approx \frac{2\Lambda ^{2}} {J} \approx 0\ \ \ \mbox{ for large $J$}\\ \\ S_{J}^{P} = \frac{(J+\Lambda )(J-\Lambda )} {J} \approx J\ \ \ \mbox{ for large $J$} \end{array} \right \}\sum S_{J} = 2J+1}$$

Note:

• 1. $\Sigma S_{J} = 2J + 1$, the total degeneracy!

• 2. The R-branch line, for a specified J, is approximately (J +  1)∕J times as strong as the P branch line.

• 3. For $\Delta \Lambda = \pm 1$, and $J \gg \Lambda$

  $$\displaystyle{\left.\begin{array}{l} S_{J}^{R} \approx \frac{(2J+1)} {4}\\ \\ S_{J}^{Q} \approx \frac{(2J+1)} {2}\\ \\ S_{J}^{P} = \frac{(2J+1)} {4}\end{array} \right \}\sum S_{J} = 2J+1}$$

  The Q branch lines are thus twice as strong as the P and R lines! Therefore, the $\Delta \Lambda$ value is important in determining the relative line and branch strengths of rovibronic spectra.

Example: $X =\! ^{1}\Pi$

If $X =\! ^{1}\Pi$, then the following transitions are possible for changes in $\Lambda$ of 0 or ± 1.

Table 2

That is, there are three separate “systems” of bands possible from $X^{1}\Pi$.

10.2.3 Interaction of $\Lambda$ and $\Sigma$

Understanding the interactions and coupling of $\Lambda$ and $\Sigma$ is a key to modelling the influence of spin on the electronic state structure. When $\Lambda \neq 0$ and S ≠ 0, they combine to form a net component of $\Omega$, as shown in Fig. 10.4.

Fig. 10.4

Coupling of $\Sigma$ and $\Lambda$

The presence of $\Lambda \neq 0$ implies that there is an associated magnetic field due to net current about the axis. This field interacts with spinning electrons. This effect is known as spin–orbit coupling (or spin-splitting) (Fig. 10.5).

Fig. 10.5

Spin–orbit splitting

Examples

$^{3}\Delta$ has three components ($^{3}\Delta _{3}$, $^{3}\Delta _{2}$, $^{3}\Delta _{1}$) corresponding to S = 1, $\Lambda = 2$ and $\Omega = 3,2,1$ (since $\Sigma = 1,0,-1$). These states have different electronic energies, which may be represented by

$$\displaystyle{ T_{e} = T_{0} + A\Lambda \Sigma }$$

(10.13)

Here, A is the spin–orbit coupling constant (it’s not exactly the same A as in the symmetric top, but it is similar; keep in mind that these are models). A generally increases with molecular weight and the number of electrons. T ₀ is the energy without interaction.

For $^{3}\Delta$, $\Lambda = 2$, S = 1, and $\Sigma = 1,0,-1$. Therefore,

$$\displaystyle{T_{e} = T_{0}+A(2)\left (\begin{array}{c} 1\\ 0 \\ - 1 \end{array} \right )}$$

Some sample spin–orbit coupling constants are listed below.

$$\displaystyle\begin{array}{rcl} A_{\mathrm{BeH}} \approx 2\,\mathrm{cm^{-1}}& &{}\end{array}$$

(10.14)

$$\displaystyle\begin{array}{rcl} A_{\mathrm{NO}} \approx 124\,\mathrm{cm^{-1}}& &{}\end{array}$$

(10.15)

$$\displaystyle\begin{array}{rcl} A_{\mathrm{HgH}} \approx 3600\,\mathrm{cm^{-1}}& &{}\end{array}$$

(10.16)

$$\displaystyle\begin{array}{rcl} A_{\mathrm{OH}} \approx -140\,\mathrm{cm^{-1}}& &{}\end{array}$$

(10.17)

Note that the spin–orbit coupling constant for OH is negative. (See Herzberg, Vol. I for details; p. 215/216, 232, 558/559, 561.)

Note:

• 1. $Y = \frac{A} {B_{v}}$ is the ratio of the spin–orbit constant and B _v

• 2. Values for A are given in tables in Herzberg, Vol. I

Now, finally, we are ready to consider rotational levels for cases where S ≠ 0. There are two primary cases: Hund’s a and Hund’s b.

10.2.4 Hund’s Case a

Hund’s case a is for $\Lambda \neq 0$ and S ≠ 0 with $\Sigma$ defined as $\Sigma = S,S - 1,\ldots,-S$. Replace $\Lambda$ with $\Omega$ in F(J) for the symmetric top to get [1]

$$\displaystyle\begin{array}{rcl} F(J)& =& BJ(J + 1) + (A - B)\Omega ^{2} \\ \Omega & =& \Lambda + S,\Lambda + S - 1,\ldots,\vert \Lambda - S\vert \\ J& =& \Omega,\Omega + 1,\ldots {}\end{array}$$

(10.18)

Remember that A for the equation above is $A = h/8\pi ^{2}I_{A}c$; it is not the spin–orbit constant.

There are P, Q, and R branches for each value of $\Omega$. For example, with $^{2}\Pi$, we have $\Omega = 3/2$ and 1∕2, i.e. two electronic sub-states, giving a total of 2 × 3 = 6 branches.

10.2.5 Hund’s Case b

Hund’s case b applies when spin is not coupled to the A-axis [1] (Fig. 10.6), e.g.:

• 1. for $\Lambda = 0$ (so $\vec{\Sigma }$ is not defined, and we must use $\vec{S}$)

• 2. at high J, especially for hydrides, even with $\Lambda \neq 0$

Fig. 10.6

Molecular model for Hund’s b

The allowed J are $J = N + S,N + S - 1,\ldots,N - S$, and $J \geq 0$ only. For this case, $\vec{S}$ and $\vec{N}$ couple directly.

Example: O₂

The ground state $X^{3}\Sigma$ has three J’s for each N except N = 0. These energy levels are labelled F ₁(N), F ₂(N) and F ₃(N).

Table 3

There are split rotational levels for N > 0, and each level has a degeneracy of 2J + 1 and a sum of Hönl–London factors of 2J + 1. The minimum J is | N − S | .

Note:

In the N = 0 level, since only spin is active and is not equal to zero (i.e., S = 1), the minimum value of J is 1.

10.2.6 $\Lambda$-Doubling

There is further complexity in the energy levels resulting from a phenomenon known as $\Lambda$-doubling. The two orientations of $\vec{\Lambda }$ ($\pm \Lambda$ along the A-axis) have slightly different energies, owing to different coupling with nuclear rotation (i.e., $\vec{N}$ and $\vec{\Lambda }$ interaction). The result is

$$\displaystyle{F(J) \rightarrow F_{c}(J)\ \text{and}\ F_{d}(J)}$$

That is, there are two components to the energy, F(J) (Fig. 10.7).

Fig. 10.7

Lambda-doubling results in two different energy levels

By definition, $F_{c}(J)> F_{d}(J)$,¹ i.e. the c state is the higher-energy state.

Lambda doubling usually results in a very small change in energy, thus affecting the Boltzmann fraction only slightly (other than adding a state). A more important aspect is found in the change of parity between $\Lambda$-doubled states, which reduces the accessible fraction of molecules for a given transition (due to selection rules).

10.2.7 Comment on Models

• 1. Models are only approximations to real molecules; don’t think of them as exact!

• 2. Coupling may change as J ranges from low to high values.

10.3 Quantitative Absorption

This section is a review of Beer’s Law and spectral absorption as interpreted for molecules with multiplet structure.

$$\displaystyle{ \text{Beer's Law}\ \left ( \frac{I} {I^{0}}\right )_{\nu } =\exp (-k_{\nu }L) }$$

(10.19)

For a two-level system, we had

$$\displaystyle{ k_{\nu } = S_{12}\phi (\nu ) = \left ( \frac{\pi e^{2}} {m_{e}c}\right )n_{1}f_{12}\left (1 -\exp (-h\nu /kT)\right )\phi (\nu ), }$$

(10.20)

where S ₁₂ is the integrated absorption intensity with units of cm⁻¹ s⁻¹ [see Eq. (7.48)]. How do we evaluate n ₁ and f ₁₂ in a complex, multiple level system? There are two issues:

• 1. The Boltzmann fraction, and

• 2. The oscillator strength for a specific transition

10.3.1 Boltzmann Fraction

$$\displaystyle\begin{array}{rcl} n_{1}& =& n_{i}\frac{n_{1}} {n_{i}} {} \\ \text{where }n_{i}& =& \text{the total number density of species i and } \\ n_{1}/n_{i}& =& \mbox{ the fraction of species $i$ in state/level 1} \\ \end{array}$$

(10.21)

The state/level is specified through the quantum numbers, e.g. n (elec), v (vib), $\Sigma$ (spin), $\Lambda$ (orbital), J (total angular momentum), N (nuclear rotation), c or d ($\Lambda$-component).

$$\displaystyle{ \frac{n_{1}} {n_{i}} = \frac{N_{i}(n,v,\Sigma,\Lambda,J,N)} {N_{i}} }$$

(10.22)

$\Lambda$ gives the orientation and distinguishes $\Lambda$-doubling.

10.3.2 Oscillator Strength

It is common to denote the oscillator strength of a specific, single transition (i.e., from one of the $J''$ substates to a specific J′ substate) by $f_{J''J'}$, and to view this transition strength as the product of a system strength, vibrational factor (fraction) and rotational factor (fraction), i.e.,

$$\displaystyle\begin{array}{rcl} f_{12}& =& f_{(m,v'',J'')(n,v',J')}{}\end{array}$$

(10.23)

$$\displaystyle\begin{array}{rcl} & =& f_{J''J'}{}\end{array}$$

(10.24)

$$\displaystyle\begin{array}{rcl} & =& \mathop{\underbrace{f_{\mathrm{el}}}}\limits _{\begin{array}{c} \text{\textquotedblleft system\textquotedblright osc.} \\ \text{strength} \end{array} } \times \mathop{\underbrace{ q_{v''v'}}}\limits _{\begin{array}{c} \text{Franck\textendash Condon} \\ \text{factor} \end{array} } \times \mathop{\underbrace{ \frac{S_{J''J'}} {2J''+1}}}\limits _{\begin{array}{c} \text{normalized H-L}\\ \text{factor or line} \\ \text{strength} \end{array} }{}\end{array}$$

(10.25)

We will show later that the sum of line strengths from a specific substate of $J''$ (i.e., with a specific N and $\Lambda$ component (c or d)) to all possible J′ is simply $2J'' + 1$; further, $\sum _{v'}q_{v''v'} = 1$. Thus, the effective total oscillator strength (i.e., for the sum of all radiative transitions) for each of the substates, i.e. with specific values of $J''$, $N''$, $\Lambda ''$ and $v''$, is f _el.

Note:

• 1. $\sum _{J'}S_{J''J'} = (2J'' + 1)[(2S + 1)\delta ]$

  $\sum _{J'}S_{J''J'}$ is the sum over all allowed J′ (upper states) for the combined lower substates with specific $J = J''$. δ = 1 for $\Sigma - \Sigma$, otherwise δ = 2 (to account for $\Lambda$-doubling!). The term [(2S + 1)δ] = 4 for OH’s $A^{2}\Sigma \leftarrow X^{2}\Pi$ system, since there are four substates with a given $J''$.

• 2. $\sum _{v',J''}f_{J''J'} = [(2S + 1)\delta ]f_{\mathrm{el}}$

  where $\sum _{v',J''}$ is the sum over v′ and all the substates in $J''$. This sum is f _el for a single J″ substate.

Remarks

• 1. It is common to use band oscillator strengths (available in the literature).

  $$\displaystyle{f_{v''v'} = f_{\mathrm{el}}q_{v''v'}}$$

  e.g., f ₀₀ = 0. 001 (OH; $A^{2}\Sigma \leftarrow X^{2}\Pi$)

• 2. Then

  $$\displaystyle{f_{J''J'} = f_{v''v'}\left ( \frac{S_{J''J'}} {2J'' + 1}\right )}$$

  e.g., if only P and R are allowed,

  $$\displaystyle\begin{array}{rcl} S_{J''J'}^{P}& =& J'' {}\end{array}$$

  (10.26)
  $$\displaystyle\begin{array}{rcl} S_{J''J'}^{R}& =& J'' + 1 {}\end{array}$$

  (10.27)

  Therefore, the intensities for R and P transitions from a specific J″ are similar, except for small J ”, but the R-branch transitions are stronger than the P-branch transitions by the ratio $(J\textquotedblright + 1)/J\textquotedblright$.

• 3. In some cases, an additional “correction term” $T_{J''J'}$ is used, e.g. in OH.

  $$\displaystyle{f_{J''J'} = f_{v''v'}\left ( \frac{S_{J''J'}} {2J'' + 1}\right )T_{J''J'},}$$

  where $T_{J''J'}$ is always near 1.

• 4. In terms of A-coefficients, we may also write

  $$\displaystyle\begin{array}{rcl} f_{v''v'}& =& \left (\frac{m_{e}c\lambda ^{2}} {8\pi ^{2}e^{2}} \right )A_{v'v''}\left (\frac{g_{e'}} {g_{e''}}\right ) {}\\ \end{array}$$

• 5. And recalling Eq. (7.54) we may write

  $$\displaystyle\begin{array}{rcl} f_{v''v'}& =& \frac{g_{e'}} {g_{e''}}f_{v'v''} {}\\ \end{array}$$

We now know enough to tackle a real (and important) molecule, OH; see Chap. 14

10.4 Exercises

• 1. A rovibronic transition in nitric oxide (NO) occurs at 226 nm. The oscillator strength for the transition between states 1 and 2 is given by

  $$\displaystyle{f_{12} = f_{\mathrm{el}} \times q_{v''v'} \times \frac{S_{J''J'}} {2J'' + 1} = f_{21}\frac{g'} {g''}}$$

  where f _el is the oscillator strength for transitions between the lower and upper electronic states, $q_{v''v'}$ is the Franck–Condon factor and S _J″J′∕(2J″ + 1) is the normalized Hönl–London factor. For this transition of NO,

  $$\displaystyle\begin{array}{rcl} f_{\mathrm{el}}& =& 3.0 \times 10^{-3} {}\\ q_{v''v'}& =& 1.673 \times 10^{-1} {}\\ \frac{S_{J''J'}} {2J'' + 1}& =& 0.5 {}\\ \end{array}$$

  Consider a gas mixture at 1000 K and 2 atm, with an effective collisional broadening coefficient for this NO transition of $2\gamma = 0.09\,\mathrm{cm^{-1}\,atm^{-1}}$, and a number density in the absorbing state 1 of $n_{1} = 3.7 \times 10^{14}\,\mathrm{cm^{-3}}$.

  • (a) Determine the peak absorption coefficient $k_{\nu _{0}}$ in cm⁻¹ at the above temperature and pressure.

  • (b) Find the fractional absorption at line center and at a detuning of 0.2 cm⁻¹ for path lengths of 1 and 10 cm.

  • (c) Determine the spontaneous lifetime of the upper state. You may assume the electronic degeneracies of the upper and lower states are both 2.

  • (d) What is the mole fraction of NO (in ppm) for this gas mixture, assuming an approximate Boltzmann fraction of NO molecules in the absorption state of 4.2 %. (This is a reasonable estimate, if one neglects issues of $\Lambda$-doubling and assumes that the absorbing state is in $v'' = 0$ and $J''$ near the peak of the rotational distribution.) This step requires no spectroscopic calculation, only a simple use of the ideal gas equation of state, but is useful in giving a sense of the species detection sensitivity of spectrally resolved absorption.