Rotational-vibrational spectroscopy

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https://en.wikipedia.org/wiki/Rotational%E2%80%93vibrational_spectroscopy

Rotational–vibrational spectroscopy is a branch of molecular spectroscopy that is concerned with infrared and Raman spectra of molecules in the gas phase. Transitions involving changes in both vibrational and rotational states can be abbreviated as rovibrational (or ro-vibrational) transitions. When such transitions emit or absorb photons (electromagnetic radiation), the frequency is proportional to the difference in energy levels and can be detected by certain kinds of spectroscopy. Since changes in rotational energy levels are typically much smaller than changes in vibrational energy levels, changes in rotational state are said to give fine structure to the vibrational spectrum. For a given vibrational transition, the same theoretical treatment as for pure rotational spectroscopy gives the rotational quantum numbers, energy levels, and selection rules. In linear and spherical top molecules, rotational lines are found as simple progressions at both higher and lower frequencies relative to the pure vibration frequency. In symmetric top molecules the transitions are classified as parallel when the dipole moment change is parallel to the principal axis of rotation, and perpendicular when the change is perpendicular to that axis. The ro-vibrational spectrum of the asymmetric rotor water is important because of the presence of water vapor in the atmosphere.

Overview

See also: Rotational spectroscopy and Vibronic spectroscopy

Ro-vibrational spectroscopy concerns molecules in the gas phase. There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band.

In the simplest cases the part of the infrared spectrum involving vibrational transitions with the same rotational quantum number (ΔJ = 0) in ground and excited states is called the Q-branch. On the high frequency side of the Q-branch the energy of rotational transitions is added to the energy of the vibrational transition. This is known as the R-branch of the spectrum for ΔJ = +1. The P-branch for ΔJ = −1 lies on the low wavenumber side of the Q branch. The appearance of the R-branch is very similar to the appearance of the pure rotation spectrum (but shifted to much higher wavenumbers), and the P-branch appears as a nearly mirror image of the R-branch. The Q branch is sometimes missing because of transitions with no change in J being forbidden.

The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation.

The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number ${\displaystyle J}$, and depends on temperature. Since there are actually ${\displaystyle 2J+1}$ states with rotational quantum number ${\displaystyle J}$, the population with value ${\displaystyle J}$ increases with ${\displaystyle J}$ initially, and then decays at higher ${\displaystyle J}$. This gives the characteristic shape of the P and R branches.

A general convention is to label quantities that refer to the vibrational ground and excited states of a transition with double prime and single prime, respectively. For example, the rotational constant for the ground state is written as ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }},}$ and that of the excited state as ${\displaystyle B^{\mathrm{\prime }}.}$ Also, these constants are expressed in the molecular spectroscopist's units of cm⁻¹. so that ${\displaystyle B}$ in this article corresponds to ${\displaystyle \overline{B}=B/hc}$ in the definition of rotational constant at Rigid rotor.

Method of combination differences

Numerical analysis of ro-vibrational spectral data would appear to be complicated by the fact that the wavenumber for each transition depends on two rotational constants, ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ and ${\displaystyle B^{\mathrm{\prime }}}$. However combinations which depend on only one rotational constant are found by subtracting wavenumbers of pairs of lines (one in the P-branch and one in the R-branch) which have either the same lower level or the same upper level. For example, in a diatomic molecule the line denoted P(J + 1) is due to the transition (v = 0, J + 1) → (v = 1, J) (meaning a transition from the state with vibrational quantum number ν going from 0 to 1 and the rotational quantum number going from some value J + 1 to J, with J > 0), and the line R(J − 1) is due to the transition (v = 0, J − 1) → (v = 1, J). The difference between the two wavenumbers corresponds to the energy difference between the (J + 1) and (J − 1) levels of the lower vibrational state and is denoted by ${\displaystyle {\mathrm{\Delta }}_2}$ since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by

${\displaystyle {\mathrm{\Delta }}_2^{\mathrm{\prime }\mathrm{\prime }}F(J)=\overline{\nu }(R(J-1))-\overline{\nu }(P(J+1))=(2B^{\mathrm{\prime }\mathrm{\prime }}-3D^{\mathrm{\prime }\mathrm{\prime }})\left(2J+1\right)-D^{\mathrm{\prime }\mathrm{\prime }}{\left(2J+1\right)}^3}$

where ${\displaystyle \overline{\nu }()}$ means the frequency (or wavenumber) of the given line. The main term, ${\displaystyle 2B^{″}(2J+1),}$ comes from the difference in the energy of the ${\displaystyle J+1}$ rotational state, ${\displaystyle B^{″}((J+1)(J+2)),}$ and that of the ${\displaystyle J-1}$ state, ${\displaystyle B^{″}((J-1)J).}$

The rotational constant of the ground vibrational state B′′ and centrifugal distortion constant, D′′ can be found by least-squares fitting this difference as a function of J. The constant B′′ is used to determine the internuclear distance in the ground state as in pure rotational spectroscopy. (See Appendix)

Similarly the difference R(J) − P(J) depends only on the constants B′ and D′ for the excited vibrational state (v = 1), and B′ can be used to determine the internuclear distance in that state (which is inaccessible to pure rotational spectroscopy).

${\displaystyle {\mathrm{\Delta }}_2^{\mathrm{\prime }}F(J)=\overline{\nu }(R(J))-\overline{\nu }(P(J))=(2B^{\mathrm{\prime }}-3D^{\mathrm{\prime }})\left(2J+1\right)-D^{\mathrm{\prime }}{\left(2J+1\right)}^3}$

Linear molecules

Heteronuclear diatomic molecules

Simulated vibration-rotation line spectrum of carbon monoxide, ¹²C¹⁶O. The P-branch is to the left of the gap near 2140 cm⁻¹, the R-branch on the right.

Schematic ro-vibrational energy level diagram for a linear molecule

Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values ${\displaystyle G(v)}$, for an anharmonic oscillator are given, to a first approximation, by

${\displaystyle G(v)={\omega }_e\left(v+\frac{1}{2}\right)-{\omega }_e{\chi }_e{\left(v+\frac{1}{2}\right)}^2}$

where v is a vibrational quantum number, ω_e is the harmonic wavenumber and χ_e is an anharmonicity constant.

When the molecule is in the gas phase, it can rotate about an axis, perpendicular to the molecular axis, passing through the centre of mass of the molecule. The rotational energy is also quantized, with term values to a first approximation given by

${\displaystyle F_v(J)=B_{v}J\left(J+1\right)-D{J}^2{\left(J+1\right)}^2}$

where J is a rotational quantum number and D is a centrifugal distortion constant. The rotational constant, B_v depends on the moment of inertia of the molecule, I_v, which varies with the vibrational quantum number, v

${\displaystyle B_v=\frac{h}{8{\pi }^{2}c{I}_v};I_v=\frac{m_A{m}_B}{m_A+m_B}{d}_v^2}$

where m_A and m_B are the masses of the atoms A and B, and d represents the distance between the atoms. The term values of the ro-vibrational states are found (in the Born–Oppenheimer approximation) by combining the expressions for vibration and rotation.

${\displaystyle G(v)+F_v(J)=\left[{\omega }_e\left(v+\frac{1}{2}\right)+B_{v}J(J+1)\right]-\left[{\omega }_e{\chi }_e{\left(v+\frac{1}{2}\right)}^2+D{J}^2(J+1{)}^2\right]}$

The first two terms in this expression correspond to a harmonic oscillator and a rigid rotor, the second pair of terms make a correction for anharmonicity and centrifugal distortion. A more general expression was given by Dunham.

The selection rule for electric dipole allowed ro-vibrational transitions, in the case of a diamagnetic diatomic molecule is

${\displaystyle \mathrm{\Delta }v=\pm 1(\pm 2,\pm 3,\mathit{\text{etc.}}}$${\displaystyle ),\mathrm{\Delta }J=\pm 1}$

The transition with Δv=±1 is known as the fundamental transition. The selection rule has two consequences.

1. Both the vibrational and rotational quantum numbers must change. The transition :${\displaystyle \mathrm{\Delta }v=\pm 1,\mathrm{\Delta }J=0}$ (Q-branch) is forbidden
2. The energy change of rotation can be either subtracted from or added to the energy change of vibration, giving the P- and R- branches of the spectrum, respectively.

The calculation of the transition wavenumbers is more complicated than for pure rotation because the rotational constant B_ν is different in the ground and excited vibrational states. A simplified expression for the wavenumbers is obtained when the centrifugal distortion constants ${\displaystyle D^{\mathrm{\prime }}}$ and ${\displaystyle D^{\mathrm{\prime }\mathrm{\prime }}}$ are approximately equal to each other.

${\displaystyle \overline{\nu }={\omega }_0+(B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})m^2-2(D^{\mathrm{\prime }}+D^{\mathrm{\prime }\mathrm{\prime }})m^3,{\omega }_0={\omega }_e(1-2{\chi }_e)m=\pm 1,\pm 2etc.}$

Spectrum of R-branch of nitric oxide, NO, simulated with Spectralcalc, showing λ-doubling caused by the presence of an unpaired electron in the molecule

where positive m values refer to the R-branch and negative values refer to the P-branch. The term ω₀ gives the position of the (missing) Q-branch, the term ${\displaystyle (B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m}$ implies an progression of equally spaced lines in the P- and R- branches, but the third term, ${\displaystyle (B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})m^2}$shows that the separation between adjacent lines changes with changing rotational quantum number. When ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ is greater than ${\displaystyle B^{\mathrm{\prime }}}$, as is usually the case, as J increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of carbon monoxide, gives value of ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ of 1.915 cm⁻¹ and ${\displaystyle B^{\mathrm{\prime }}}$ of 1.898 cm⁻¹. The bond lengths are easily obtained from these constants as r₀ = 113.3 pm, r₁ = 113.6 pm. These bond lengths are slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by

${\displaystyle B_v=B_{eq}-\alpha \left(v+\frac{1}{2}\right)}$

where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide r_eq = 113.0 pm.

Nitric oxide, NO, is a special case as the molecule is paramagnetic, with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes lambda-doubling with calculated harmonic frequencies of 1904.03 and 1903.68 cm⁻¹. Rotational levels are also split.

Homonuclear diatomic molecules

The quantum mechanics for homonuclear diatomic molecules such as dinitrogen, N₂, and fluorine, F₂, is qualitatively the same as for heteronuclear diatomic molecules, but the selection rules governing transitions are different. Since the electric dipole moment of the homonuclear diatomics is zero, the fundamental vibrational transition is electric-dipole-forbidden and the molecules are infrared inactive. However, a weak quadrupole-allowed spectrum of N₂ can be observed when using long path-lengths both in the laboratory and in the atmosphere. The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.

Dioxygen is a special case as the molecule is paramagnetic so magnetic-dipole-allowed transitions can be observed in the infrared. The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N, so that each rotational level is split into three states with total angular momentum (molecular rotation plus electron spin) ${\displaystyle \mathrm{J}\hslash }$, J = N + 1, N, and N - 1, each J state of this so-called p-type triplet arising from a different orientation of the spin with respect to the rotational motion of the molecule. Selection rules for magnetic dipole transitions allow transitions between successive members of the triplet (ΔJ = ±1) so that for each value of the rotational angular momentum quantum number N there are two allowed transitions. The ¹⁶O nucleus has zero nuclear spins angular momentum, so that symmetry considerations demand that N may only have odd values.

Raman spectra of diatomic molecules

The selection rule is

${\displaystyle \mathrm{\Delta }J=0,\pm 2}$

so that the spectrum has an O-branch (∆J = −2), a Q-branch (∆J = 0) and an S-branch (∆J=+2). In the approximation that B′′ = B′ = B the wavenumbers are given by

${\displaystyle \overline{\nu }(S(J))={\omega }_0+4BJ+6B={\omega }_0+6B,{\omega }_0+10B,{\omega }_0+14B,...}$

${\displaystyle \overline{\nu }(O(J))={\omega }_0-4BJ+2B={\omega }_0-6B,{\omega }_0-10B,{\omega }_0-14B,...}$

since the S-branch starts at J=0 and the O-branch at J=2. So, to a first approximation, the separation between S(0) and O(2) is 12B and the separation between adjacent lines in both O- and S- branches is 4B. The most obvious effect of the fact that B′′ ≠ B′ is that the Q-branch has a series of closely spaced side lines on the low-frequency side due to transitions in which ΔJ=0 for J=1,2 etc. Useful difference formulae, neglecting centrifugal distortion are as follows.

${\displaystyle {\mathrm{\Delta }}_4^{\mathrm{\prime }\mathrm{\prime }}F(J)=\overline{\nu }[S(J-2)]-\overline{\nu }[O(J+2)]=4B^{\mathrm{\prime }\mathrm{\prime }}(2J+1)}$

${\displaystyle {\mathrm{\Delta }}_4^{\mathrm{\prime }}F(J)=\overline{\nu }[S(J)]-\overline{\nu }[O(J)]=4B^{\mathrm{\prime }}(2J+1)}$

Molecular oxygen is a special case as the molecule is paramagnetic, with two unpaired electrons.

For homonuclear diatomics, nuclear spin statistical weights lead to alternating line intensities between even-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ and odd-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ levels. For nuclear spin I = 1/2 as in ¹H₂ and ¹⁹F₂ the intensity alternation is 1:3. For ²H₂ and ¹⁴N₂, I=1 and the statistical weights are 6 and 3 so that the even-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ levels are twice as intense. For ¹⁶O₂ (I=0) all transitions with even values of ${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ are forbidden.

Polyatomic linear molecules

Spectrum of bending mode in 14N14N16O simulated with Spectralcalc. The weak superimposed spectrum is due to species containing 15N at natural abundance of 0.3%

Spectrum of a perpendicular band from acetylene, C2H2, simulated with Spectralcalc showing 1,3 intensity alternation in both P- and R- branches. See also Hollas p157

Spectrum of the asymmetric stretching (parallel) band of carbon dioxide, 12C16O2 simulated with Spectralcalc. The weak superimposed spectrum is due to absorption of the first vibrationally excited level (0 11 0), which due to its low energy is populated at room temperature

These molecules fall into two classes, according to symmetry: centrosymmetric molecules with point group D_∞h, such as carbon dioxide, CO₂, and ethyne or acetylene, HCCH; and non-centrosymmetric molecules with point group C_∞v such as hydrogen cyanide, HCN, and nitrous oxide, NNO. Centrosymmetric linear molecules have a dipole moment of zero, so do not show a pure rotation spectrum in the infrared or microwave regions. On the other hand, in certain vibrational excited states the molecules do have a dipole moment so that a ro-vibrational spectrum can be observed in the infrared.

The spectra of these molecules are classified according to the direction of the dipole moment change vector. When the vibration induces a dipole moment change pointing along the molecular axis the term parallel is applied, with the symbol ${\displaystyle \parallel }$. When the vibration induces a dipole moment pointing perpendicular to the molecular axis the term perpendicular is applied, with the symbol ${\displaystyle \perp }$. In both cases the P- and R- branch wavenumbers follow the same trend as in diatomic molecules. The two classes differ in the selection rules that apply to ro-vibrational transitions. For parallel transitions the selection rule is the same as for diatomic molecules, namely, the transition corresponding to the Q-branch is forbidden. An example is the C-H stretching mode of hydrogen cyanide.

For a perpendicular vibration the transition ΔJ=0 is allowed. This means that the transition is allowed for the molecule with the same rotational quantum number in the ground and excited vibrational state, for all the populated rotational states. This makes for an intense, relatively broad, Q-branch consisting of overlapping lines due to each rotational state. The N-N-O bending mode of nitrous oxide, at ca. 590 cm⁻¹ is an example.

The spectra of centrosymmetric molecules exhibit alternating line intensities due to quantum state symmetry effects, since rotation of the molecule by 180° about a 2-fold rotation axis is equivalent to exchanging identical nuclei. In carbon dioxide, the oxygen atoms of the predominant isotopic species ¹²C¹⁶O₂ have spin zero and are bosons, so that the total wavefunction must be symmetric when the two ¹⁶O nuclei are exchanged. The nuclear spin factor is always symmetric for two spin-zero nuclei, so that the rotational factor must also be symmetric which is true only for even-J levels. The odd-J rotational levels cannot exist and the allowed vibrational bands consist of only absorption lines from even-J initial levels. The separation between adjacent lines in the P- and R- branches is close to 4B rather than 2B as alternate lines are missing. For acetylene the hydrogens of ¹H¹²C¹²C¹H have spin-1/2 and are fermions, so the total wavefunction is antisymmetric when two ¹H nuclei are exchanged. As is true for ortho and para hydrogen the nuclear spin function of the two hydrogens has three symmetric ortho states and one antisymmetric para states. For the three ortho states, the rotational wave function must be antisymmetric corresponding to odd J, and for the one para state it is symmetric corresponding to even J. The population of the odd J levels are therefore three times higher than the even J levels, and alternate line intensities are in the ratio 3:1.

Spherical top molecules

Upper: low-resolution infrared spectrum of asymmetric stretching band of methane (CH₄); lower: PGOPHER simulated line spectrum, ignoring Coriolis coupling

These molecules have equal moments of inertia about any axis, and belong to the point groups T_d (tetrahedral AX₄) and O_h (octahedral AX₆). Molecules with these symmetries have a dipole moment of zero, so do not have a pure rotation spectrum in the infrared or microwave regions.

Tetrahedral molecules such as methane, CH₄, have infrared-active stretching and bending vibrations, belonging to the T₂ (sometimes written as F₂) representation. These vibrations are triply degenerate and the rotational energy levels have three components separated by the Coriolis interaction. The rotational term values are given, to a first order approximation, by

${\displaystyle F^{+}=B_{\nu }J(J+1)+2B_{\nu }{\zeta }_r(J+1)}$

${\displaystyle F^0=B_{\nu }J(J+1)}$

${\displaystyle F^{-}=B_{\nu }J(J+1)-2B_{\nu }{\zeta }_{r}J}$

where ${\displaystyle {\zeta }_r}$ is a constant for Coriolis coupling. The selection rule for a fundamental vibration is

${\displaystyle \mathrm{\Delta }J=0,\pm 1}$

Thus, the spectrum is very much like the spectrum from a perpendicular vibration of a linear molecule, with a strong Q-branch composed of many transitions in which the rotational quantum number is the same in the vibrational ground and excited states, ${\displaystyle J^{\mathrm{\prime }}=J^{\mathrm{\prime }\mathrm{\prime }}=1,\mathrm{2...}}$ The effect of Coriolis coupling is clearly visible in the C-H stretching vibration of methane, though detailed study has shown that the first-order formula for Coriolis coupling, given above, is not adequate for methane.

Symmetric top molecules

These molecules have a unique principal rotation axis of order 3 or higher. There are two distinct moments of inertia and therefore two rotational constants. For rotation about any axis perpendicular to the unique axis, the moment of inertia is ${\displaystyle I_{\perp }}$ and the rotational constant is ${\displaystyle B=\frac{h}{8{\pi }^{2}c{I}_{\perp }}}$, as for linear molecules. For rotation about the unique axis, however, the moment of inertia is ${\displaystyle I_{\parallel }}$ and the rotational constant is ${\displaystyle A=\frac{h}{8{\pi }^{2}c{I}_{\parallel }}}$. Examples include ammonia, NH₃ and methyl chloride, CH₃Cl (both of molecular symmetry described by point group C_3v), boron trifluoride, BF₃ and phosphorus pentachloride, PCl₅ (both of point group D_3h), and benzene, C₆H₆ (point group D_6h).

For symmetric rotors a quantum number J is associated with the total angular momentum of the molecule. For a given value of J, there is a 2J+1- fold degeneracy with the quantum number, M taking the values +J ...0 ... -J. The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. As with linear molecules, transitions are classified as parallel, ${\displaystyle \parallel }$ or perpendicular,${\displaystyle \perp }$, in this case according to the direction of the dipole moment change with respect to the principal rotation axis. A third category involves certain overtones and combination bands which share the properties of both parallel and perpendicular transitions. The selection rules are

${\displaystyle \parallel }$ If K ≠ 0, then ΔJ = 0, ±1 and ΔK = 0

If K = 0, then ΔJ = ±1 and ΔK = 0

${\displaystyle \perp }$ ΔJ = 0, ±1 and ΔK = ±1

The fact that the selection rules are different is the justification for the classification and it means that the spectra have a different appearance which can often be immediately recognized. An expression for the calculated wavenumbers of the P- and R- branches may be given as

${\displaystyle \overline{\nu }={\overline{\nu }}_0+(B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }}-D_J^{\mathrm{\prime }}+D_J^{\mathrm{\prime }\mathrm{\prime }})m^2}$

${\displaystyle -2(D_J^{\mathrm{\prime }}+D_J^{\mathrm{\prime }\mathrm{\prime }})m^3-(D_J^{\mathrm{\prime }}-D_J^{\mathrm{\prime }\mathrm{\prime }})m^4}$

${\displaystyle +\left\{\left[(A^{\mathrm{\prime }}-B^{\mathrm{\prime }})-(A^{\mathrm{\prime }\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})\right]-\left[D_{JK}^{\mathrm{\prime }}+D_{JK}^{\mathrm{\prime }\mathrm{\prime }}\right]m-\left[D_{JK}^{\mathrm{\prime }}-D_{JK}^{\mathrm{\prime }\mathrm{\prime }}\right]m^2\right\}{K}^2-(D_K^{\mathrm{\prime }}-D_K^{\mathrm{\prime }\mathrm{\prime }})K^4}$

in which m = J+1 for the R-branch and -J for the P-branch. The three centrifugal distortion constants ${\displaystyle D_J,D_{JK}}$, and ${\displaystyle D_K}$ are needed to fit the term values of each level. The wavenumbers of the sub-structure corresponding to each band are given by

${\displaystyle \overline{\nu }={\overline{\nu }}_{sub}+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})J(J+1)-(D_J^{\mathrm{\prime }}-D_J^{\mathrm{\prime }\mathrm{\prime }})J^2(J+1{)}^2-(D_{JK}^{\mathrm{\prime }}-D_{JK}^{\mathrm{\prime }\mathrm{\prime }})J(J+1)K^2}$

${\displaystyle {\overline{\nu }}_{sub}}$ represents the Q-branch of the sub-structure, whose position is given by

${\displaystyle {\overline{\nu }}_{sub}={\overline{\nu }}_0+\left[(A^{\mathrm{\prime }}-B^{\mathrm{\prime }})-(A^{\mathrm{\prime }\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})\right]K^2-(D_K^{\mathrm{\prime }}-D_K^{\mathrm{\prime }\mathrm{\prime }})K^4}$.

Spectrum of the C-Cl stretching band in CH3Cl (parallel band) simulated with Spectralcalc.

Part of the spectrum of the asymmetric H-C-H bending vibration in CH3Cl (perpendicular band), simulated with Spectralcalc

Parallel bands

The C-Cl stretching vibration of methyl chloride, CH₃Cl, gives a parallel band since the dipole moment change is aligned with the 3-fold rotation axis. The line spectrum shows the sub-structure of this band rather clearly; in reality, very high resolution spectroscopy would be needed to resolve the fine structure fully. Allen and Cross show parts of the spectrum of CH₃D and give a detailed description of the numerical analysis of the experimental data.

Perpendicular bands

The selection rule for perpendicular bands give rise to more transitions than with parallel bands. A band can be viewed as a series of sub-structures, each with P, Q and R branches. The Q-branches are separated by approximately 2(A′-B′). The asymmetric HCH bending vibration of methyl chloride is typical. It shows a series of intense Q-branches with weak rotational fine structure. Analysis of the spectra is made more complicated by the fact that the ground-state vibration is bound, by symmetry, to be a degenerate vibration, which means that Coriolis coupling also affects the spectrum.

Hybrid bands

Overtones of a degenerate fundamental vibration have components of more than one symmetry type. For example, the first overtone of a vibration belonging to the E representation in a molecule like ammonia, NH₃, will have components belonging to A₁ and E representations. A transition to the A₁ component will give a parallel band and a transition to the E component will give perpendicular bands; the result is a hybrid band.

Inversion in ammonia

Spectrum of central region of the symmetric bending vibration in ammonia simulated with Spectralcalc, illustrating inversion doubling.

Nitrogen inversion in ammonia

For ammonia, NH₃, the symmetric bending vibration is observed as two branches near 930 cm⁻¹ and 965 cm⁻¹. This so-called inversion doubling arises because the symmetric bending vibration is actually a large-amplitude motion known as inversion, in which the nitrogen atom passes through the plane of the three hydrogen atoms, similar to the inversion of an umbrella. The potential energy curve for such a vibration has a double minimum for the two pyramidal geometries, so that the vibrational energy levels occur in pairs which correspond to combinations of the vibrational states in the two potential minima. The two v = 1 states combine to form a symmetric state (1⁺) at 932.5 cm⁻¹ above the ground (0⁺) state and an antisymmetric state (1⁻) at 968.3 cm⁻¹.

The vibrational ground state (v = 0) is also doubled although the energy difference is much smaller, and the transition between the two levels can be measured directly in the microwave region, at ca. 24 GHz (0.8 cm⁻¹). This transition is historically significant and was used in the ammonia maser, the fore-runner of the laser.

Asymmetric top molecules

Absorption spectrum (attenuation coefficient vs. wavelength) of liquid water (red) ^[39] atmospheric water vapor (green)  and ice (blue line)  between 667 nm and 200 μm. The plot for vapor is a transformation of data Synthetic spectrum for gas mixture "Pure H₂O" (296K, 1 atm) retrieved from Hitran on the Web Information System.

Asymmetric top molecules have at most one 2-fold rotation axis. There are three unequal moments of inertia about three mutually perpendicular principal axes. The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.

The water molecule is an important example of this class of molecule, particularly because of the presence of water vapor in the atmosphere. The low-resolution spectrum shown in green illustrates the complexity of the spectrum. At wavelengths greater than 10 μm (or wavenumbers less than 1000 cm⁻¹) the absorption is due to pure rotation. The band around 6.3 μm (1590 cm⁻¹) is due to the HOH bending vibration; the considerable breadth of this band is due to the presence of extensive rotational fine structure. High-resolution spectra of this band are shown in Allen and Cross, p 221. The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600 cm⁻¹ and 3700 cm⁻¹ are shown in Banwell and McCash, p91.

Ro-vibrational bands of asymmetric top molecules are classed as A-, B- or C- type for transitions in which the dipole moment change is along the axis of smallest moment of inertia to the highest.

Experimental methods

Ro-vibrational spectra are usually measured at high spectral resolution. In the past, this was achieved by using an echelle grating as the spectral dispersion element in a grating spectrometer. This is a type of diffraction grating optimized to use higher diffraction orders. Today at all resolutions the preferred method is FTIR. The primary reason for this is that infrared detectors are inherently noisy, and FTIR detects summed signals at multiple wavelengths simultaneously achieving a higher signal to noise by virtue of Fellgett's advantage for multiplexed methods. The resolving power of an FTIR spectrometer depends on the maximum retardation of the moving mirror. For example, to achieve a resolution of 0.1 cm⁻¹, the moving mirror must have a maximum displacement of 10 cm from its position at zero path difference. Connes measured the vibration-rotation spectrum of Venusian CO₂ at this resolution. A spectrometer with 0.001 cm⁻¹ resolution is now available commercially. The throughput advantage of FTIR is important for high-resolution spectroscopy as the monochromator in a dispersive instrument with the same resolution would have very narrow entrance and exit slits.

When measuring the spectra of gases it is relatively easy to obtain very long path-lengths by using a multiple reflection cell. This is important because it allows the pressure to be reduced so as to minimize pressure broadening of the spectral lines, which may degrade resolution. Path lengths up to 20m are commercially available.

Molecular vibration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecular_vibration

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10¹³ Hz to approximately 10¹⁴ Hz, corresponding to wavenumbers of approximately 300 to 3000 cm⁻¹ and wavelengths of approximately 30 to 3 μm.

Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of parts of the molecule. In general, a non-linear molecule with N atoms has 3N − 6 normal modes of vibration, but a linear molecule has 3N − 5 modes, because rotation about the molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond.

A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is the Planck constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state. When multiple quanta are absorbed, the first and possibly higher overtones are excited.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like a Morse potential or more accurately, a Morse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration–rotation spectra.

Number of vibrational modes

Main article: Degrees of freedom (physics and chemistry)

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates.

An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate.

The number of vibrational modes is therefore 3N minus the number of translational and rotational degrees of freedom, or 3N − 5 for linear and 3N − 6 for nonlinear molecules.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C–H or C–C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups
• Out-of-plane: a change in the angle between any one of the C–H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethylene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as well as the H–C–H angle because the angles at each carbon atom cannot all increase at the same time.

Note that these coordinates do not correspond to normal modes (see § Normal coordinates). In other words, they do not correspond to particular frequencies or vibrational transitions.

Vibrations of a methylene group (−CH₂−) in a molecule for illustration

Within the CH₂ group, commonly found in organic compounds, the two low mass hydrogens can vibrate in six different ways which can be grouped as 3 pairs of modes: 1. symmetric and asymmetric stretching, 2. scissoring and rocking, 3. wagging and twisting. These are shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

See also: Molecular symmetry

Symmetry–adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four (un-normalized) C–H stretching coordinates of the molecule ethene are given by $${\displaystyle \begin{array}{rl}{Q}_{s1}& =q_1+q_2+q_3+q_4\\ Q_{s2}& =q_1+q_2-q_3-q_4\\ Q_{s3}& =q_1-q_2+q_3-q_4\\ Q_{s4}& =q_1-q_2-q_3+q_4\end{array}}$$ where ${\displaystyle q_1-q_4}$ are the internal coordinates for stretching of each of the four C–H bonds.

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch:

• symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. Q = q₁ − q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

• principally C–H stretching with a little C–N stretching; Q₁ = q₁ + a q₂ (a << 1)
• principally C–N stretching with a little C–H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere. $${\displaystyle \mathrm{F}=-kQ}$$ By Newton's second law of motion this force is also equal to a reduced mass, μ, times acceleration. $${\displaystyle \mathrm{F}=\mu \frac{d^{2}Q}{d{t}^2}}$$ Since this is one and the same force the ordinary differential equation follows. $${\displaystyle \mu \frac{d^{2}Q}{d{t}^2}+kQ=0}$$ The solution to this equation of simple harmonic motion is $${\displaystyle Q(t)=A\cos (2\pi \nu t);\nu =\frac{1}{2\pi }\sqrt{\frac{k}{\mu }}.}$$ A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as $${\displaystyle \frac{1}{\mu }=\frac{1}{m_A}+\frac{1}{m_B}.}$$ The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy. $${\displaystyle k=\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{Q}^2}}$$

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies, ν_i, are obtained from the eigenvalues, λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by $${\displaystyle E_n=h\left(n+\frac{1}{2}\right)\nu =h\left(n+\frac{1}{2}\right)\frac{1}{2\pi }\sqrt{\frac{k}{m}},}$$ where n is a quantum number that can take values of 0, 1, 2, ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.

The difference in energy when n (or v) changes by 1 is therefore equal to ${\displaystyle h\nu }$, the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency ${\displaystyle \nu }$ (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, $${\displaystyle \mathrm{\Delta }n=\pm 1}$$but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n = 2 and n = 1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.

When it comes to polyatomic molecules, it is common to solve the Schrödinger Equation using Watson's nuclear motion Hamiltonian. Similar as for diatomics, this can be done within the harmonic approximation as stated above. For the anharmonic calculation of vibrational spectra of polyatomic molecules, more sophisticated approaches are used. Prominent examples in computational chemistry are 2nd order vibrational perturbation theory (VPT2) or vibrational configuration interaction theory (VCI).

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.