Rotational–vibrational spectroscopy

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Rotational–vibrational spectroscopy is a branch of molecular spectroscopy 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

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.^[1]
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, and the P-branch appears as a nearly mirror image of the R-branch.^{[note 1]}

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.

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 ${\displaystyle B^{\prime \prime },}$ and that of the excited state as ${\displaystyle B^{\prime }.}$ Also, these constants are expressed in the molecular spectroscopist's units of cm⁻¹. so that ${\displaystyle B}$ in this article corresponds to ${\displaystyle {\bar {B}}=B/hc}$ 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, ${\displaystyle B^{\prime \prime }}$ and ${\displaystyle B^{\prime }}$. 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.^[2][3] For example, in a diatomic molecule the line denoted P(J + 1) is due to the transition (v = 0, J + 1) → (v = 1, J), 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 denoted by ${\displaystyle \Delta _{2}}$ since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by^[4]

${\displaystyle \Delta _{2}^{\prime \prime }F(J)={\bar {\nu }}[R(J-1)]-{\bar {\nu }}[P(J+1)]=(2B^{\prime \prime }-3D^{\prime \prime })\left(2J+1\right)-D^{\prime \prime }\left(2J+1\right)^{3}}$

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.

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).

${\displaystyle \Delta _{2}^{\prime }F(J)={\bar {\nu }}[R(J)]-{\bar {\nu }}[P(J)]=(2B^{\prime }-3D^{\prime })\left(2J+1\right)-D^{\prime }\left(2J+1\right)^{3}}$

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.^{[note 2]}

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 ${\displaystyle G(v)}$,^{[note 3]} for an anharmonic oscillator are given, to a first approximation, by

${\displaystyle G(v)=\omega _{e}\left(v+{1 \over 2}\right)-\omega _{e}\chi _{e}\left(v+{1 \over 2}\right)^{2}\,}$

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

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

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

${\displaystyle B_{v}={h \over {8\pi ^{2}cI_{v}}};\quad I_{v}={\frac {m_{A}m_{B}}{m_{A}+m_{B}}}d_{v}^{2}}$

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.

${\displaystyle G(v)+F_{v}(J)=\left[\omega _{e}\left(v+{1 \over 2}\right)+B_{v}J(J+1)\right]-\left[\omega _{e}\chi _{e}\left(v+{1 \over 2}\right)^{2}+DJ^{2}(J+1)^{2}\right]}$

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

${\displaystyle \Delta v=\pm 1\ (\pm 2,\pm 3,{\textit {etc.}}}$^{[note 4]}${\displaystyle ),\Delta J=\pm 1}$

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 :${\displaystyle \Delta v=\pm 1,\Delta J=0}$ (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 ${\displaystyle D^{\prime }}$ and ${\displaystyle D^{\prime \prime }}$ are approximately equal to each other.^[5]

${\displaystyle {\bar {\nu }}=\omega _{0}+(B^{\prime }+B^{\prime \prime })m+(B^{\prime }-B^{\prime \prime })m^{2}-2(D^{\prime }+D^{\prime \prime })m^{3},\quad \omega _{0}=\omega _{e}(1-2\chi _{e})\quad m=\pm 1,\pm 2\ etc.}$

Spectrum of R-branch of nitric oxide, NO, simulated with Spectralcalc,^[6] 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 ${\displaystyle (B^{\prime }+B^{\prime \prime })m}$ implies an progression of equally spaced lines in the P- and R- branches, but the third term, ${\displaystyle (B^{\prime }-B^{\prime \prime })m^{2}}$shows that the separation between adjacent lines changes with changing rotational quantum number. When ${\displaystyle B^{\prime \prime }}$ is greater than ${\displaystyle B^{\prime }}$, 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 ${\displaystyle B^{\prime \prime }}$ of 1.915 cm⁻¹ and ${\displaystyle B^{\prime }}$ of 1.898 cm⁻¹. The bond lengths are easily obtained from these constants as r₀ = 113.3 pm, r₁ = 113.6 pm.^[7] 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

${\displaystyle B_{\nu }=B_{eq}-\alpha \left(\nu +{1 \over 2}\right)}$

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.^[8]

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^{[note 5]} with calculated harmonic frequencies of 1904.03 and 1903.68 cm⁻¹. Rotational levels are also split.^[9]

Homonuclear diatomic molecules

The quantum mechanics for 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 these molecules is zero, the fundamental vibrational transition is electric-dipole-forbidden. However, a weak quadrupole-allowed spectrum of N₂ can be observed when using long path-lengths both in the laboratory and in the atmosphere.^[10] 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.^[10] The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N,^{[note 6]} so that each rotational level is split into three states with total angular momentum (molecular rotation plus electron spin) ${\displaystyle \mathrm {J\hbar } \,}$, 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.^[11] 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 spin angular momentum, so that symmetry considerations demand that N may only have odd values.^[12][13]

Raman spectra of diatomic molecules

The selection rule is

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

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

${\displaystyle {\bar {\nu }}(S(J))=\omega _{0}+4BJ+6B=\omega _{0}+6B,\quad \omega _{0}+10B,\quad \omega _{0}+14B,\quad ...}$

${\displaystyle {\bar {\nu }}(O(J))=\omega _{0}-4BJ+2B=\omega _{0}-6B,\quad \omega _{0}-10B,\quad \omega _{0}-14B,\quad ...}$

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.^[14] Useful difference formulae, neglecting centrifugal distortion are as follows.^[15]

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

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

Molecular oxygen is a special case as the molecule is paramagnetic, with two unpaired electrons.^[16]
For homonuclear diatomics, nuclear spin statistical weights lead to alternating line intensities between even-${\displaystyle J^{\prime \prime }}$ and odd-${\displaystyle J^{\prime \prime }}$ 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-${\displaystyle J^{\prime \prime }}$ levels are twice as intense. For ¹⁶O₂ (I=0) all transitions with even values of ${\displaystyle J^{\prime \prime }}$ are forbidden.^[15]

Polyatomic linear molecules

Spectrum of bending mode in 14N14N16O simulated with Spectralcalc.[6] 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[6] 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.[6] 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.^[17] 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.^[18]

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.^[6]

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.^[19] For acetylene the hydrogens of ¹H¹²C¹²C¹H have spin ½ 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.^[20][21]

Spherical top molecules

Upper: low-resolution infrared spectrum of asymmetric stretching band of methane (CH₄); lower: PGOPHER^[22] 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.^[23]

Tetrahedral molecules such as methane, CH₄, have infrared-active stretching and bending vibrations, belonging to the T₂ (sometimes written as F₂) representation.^{[note 7]} These vibrations are triply degenerate and the rotational energy levels have three components separated by the Coriolis interaction.^[24] The rotational term values are given, to a first order approximation, by^[25]

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

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

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

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

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

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, ${\displaystyle J^{\prime }=J^{\prime \prime }=1,2...}$ 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.^[26][27]

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 ${\displaystyle B={h \over {8\pi ^{2}cI_{\perp }}}}$, as for linear molecules. For rotation about the unique axis, however, the moment of inertia is ${\displaystyle I_{\parallel }}$ and the rotational constant is ${\displaystyle A={h \over {8\pi ^{2}cI_{\parallel }}}}$. 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^[28]

${\displaystyle {\bar {\nu }}={\bar {\nu }}_{0}+(B^{\prime }+B^{\prime \prime })m+(B^{\prime }-B^{\prime \prime }-D_{J}^{\prime }+D_{J}^{\prime \prime })m^{2}}$

${\displaystyle -2(D_{J}^{\prime }+D_{J}^{\prime \prime })m^{3}-(D_{J}^{\prime }-D_{J}^{\prime \prime })m^{4}}$

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

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.^[1] The wavenumbers of the sub-structure corresponding to each band are given by

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

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

${\displaystyle {\bar {\nu }}_{sub}={\bar {\nu }}_{0}+\left[(A^{\prime }-B^{\prime })-(A^{\prime \prime }-B^{\prime \prime })\right]K^{2}-(D_{K}^{\prime }-D_{K}^{\prime \prime })K^{4}}$.

Spectrum of the C-Cl stretching band in CH3Cl (parallel band) simulated with Spectralcalc.[6]

Part of the spectrum of the asymmetric H-C-H bending vibration in CH3Cl (perpendicular band), simulated with Spectralcalc[6]

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;^[6] 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.^[29][30]

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.^[6] 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.^[31]

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.^[32]

Inversion in ammonia

Spectrum of central region of the symmetric bending vibration in ammonia simulated with Spectralcalc,[6] 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.^[33]

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).^[34][35] This transition is historically significant and was used in the ammonia MASER, the fore-runner of the LASER.^[36]

Asymmetric top molecules

Absorption spectrum (attenuation coefficient vs. wavelength) of liquid water (red) ^[37] atmospheric water vapor (green) ^[38][39] and ice (blue line) ^[40][41] between 667 nm and 200 μm.^[42] 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.^[43]

Asymmetric top molecules have at most one or more 2-fold rotation axes. 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.^[44] 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.^[45]

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.^[9] This is a type of diffraction grating optimized to use higher diffraction orders.^[46] 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.^[47] 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.^[48] 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.

Are Climate Models Fatally Flawed?

Source:  http://earthobservatory.nasa.gov/IOTD/view.php?id=84499

According to Anthropogenic Global Warming (AGW) theorists, rising global temperatures (resulting from anthropogenic CO2 releases) are causing glaciers and polar ice caps to shrink, increasing the amount of solar energy Earth absorbs, and so raising global temperatures even further by this positive feedback effect.  This is one reason the climate sensitivity factor (the final, total temperature rise due to an initial warming) has been consistently reported to be around 3 (e.g., an initial rise of one degree from rising CO2 yields a final total of three degrees) for years.

If the theorists are correct, the Earth's albedo (the fraction of solar energy the planet reflects back into space) ought to be getting smaller, although perhaps by only a small amount.

Interestingly, we have sound scientific data on changes of Earth's albedo, so we can test this part of AGW theory against reality, at least over a reasonably time period.  This reality is represented by the quote and picture shown below (taken from the source page referenced in the first line):
________________________________________

"The maps [below] show how the reflectivity of Earth—the amount of sunlight reflected back into space—changed between March 1, 2000, and December 31, 2011. This global picture of reflectivity (also called albedo) appears to be a muddle, with different areas reflecting more or less sunlight over the 12-year record. Shades of blue mark areas that reflected more sunlight over time (increasing albedo), and orange areas denote less reflection (lower albedo).

"Taken across the planet, no significant global trend appears. As noted in the anomaly plot below, global albedo rose and fell in different years, but did not necessarily head in either direction for long."

 As the quote admits, and your own eyes can plainly see, the change in Earth's albedo over the 2000-2012 time span is either zero or so close to zero as to make no practical difference.  And it is clearly in contradiction to what AGW theory, which claims a significant amount of polar and glacial ice loss during this time.

It seems likely, however, that the planet's albedo has been subject to several forces that would either raise or lower it, but that these forces have canceled each other out over this time period.  If so, it is reasonable to suggests overall albdedo might decline in the future.  But there is no clear evidence that we should expect this.  I hold that the steadiness of albedo from 2000-2012 is clear evidence that this positive feedback simply does not exist, or require much larger rises in planetary temperature.

Key Uncertainties in IPCC Climate Science – Buried in Back Pages of Technical Summary

January 23, 2016January 23, 2016 / 

David McGruer is a financial advisor who has taken a passionate interest in philosophy and climate science. He is an advocate for freedom/capitalism, reason and rational self-interest. 

~~~~

 Governments, policy-makers and eco-activists rely on the IPCC Summary for Policy Makers (SPM) as their reference point for ‘what the science says’ – but this is a fallacy.  As Dave McGruer explains, the science lies buried in the back pages of the Technical Summary, and it is far from catastrophic in nature. Why doesn’t anyone report on that? Uncertainty won’t make headlines…won’t scare people into funding eco-charities to save the planet…won’t make people say ‘yes’ to public policies on renewables and carbon taxes that will make them poor.  Catastrophic predictions will.

Contributed by David McGruer, Ottawa @2016

For anyone who still believes man’s activities are causing a dangerous rise in global temperature that warrants a dismantling of civilization, please do some reading. The IPCC reports are highly biased due to the very mandate of the IPCC, which is to to focus on man-made changes and almost ignores powerful, nay, dominant natural cycles, and is politically driven and politically funded. The IPCC Summary For Policymakers (SPM) documents are patently ridiculous and contradict the Technical Summaries (TS) on which they are supposed to be based. The policy wonks who have their minds made up before they start writing the SPM documents massively distort the already distorted IPCC technical work.  To have a chance of understanding what the IPCC science report itself is saying, you have to read the technical documents and then go to the “Key Uncertainties” section – see page 114-115 in the 2013 Fifth Assessment Report, Technical Summary. There, you will read statements such as:

“There is only medium to low confidence in the rate of change of tropospheric warming and its vertical structure. Estimates of tropospheric warming rates encompass surface temperature warming rate estimates. There is low confidence in the rate and vertical structure of the stratospheric cooling.”
“Substantial ambiguity and therefore low confidence remains in the observations of global-scale cloud variability and trends.”

“There is low confidence in an observed global-scale trend in drought or dryness (lack of rainfall), due to lack of direct observations, methodological uncertainties and choice and geographical inconsistencies in the trends.”

“There is low confidence that any reported long-term (centennial) changes in tropical cyclone characteristics are robust, after accounting for past changes in observing capabilities.”

“Robust conclusions on long-term changes in large-scale atmospheric circulation are presently not possible because of large variability on interannual to decadal time scales and remaining differences between data sets.”

“Different global estimates of sub-surface ocean temperatures have variations at different times and for different periods, suggesting that sub-decadal variability in the temperature and upper heat content (0 to to 700 m) is still poorly characterized in the historical record.”

“In Antarctica, available data are inadequate to assess the status of change of many characteristics of sea ice (e.g., thickness and volume).”

“On a global scale the mass loss from melting at calving fronts and iceberg calving are not yet comprehensively assessed. The largest uncertainty in estimated mass loss from glaciers comes from the Antarctic, and the observational record of ice–ocean interactions around both ice sheets remains poor.”

“In some aspects of the climate system, including changes in drought, changes in tropical cyclone activity, Antarctic warming, Antarctic sea ice extent, and Antarctic mass balance, confidence in attribution to human influence remains low due to modelling uncertainties and low agreement between scientific studies.”

“Based on model results there is limited confidence in the predictability of yearly to decadal averages of temperature both for the global average and for some geographical regions. Multi-model results for precipitation indicate a generally low predictability. Short-term climate projection is also limited by the uncertainty in projections of natural forcing.”

“There is low confidence in projections of many aspects of climate phenomena that influence regional climate change, including changes in amplitude and spatial pattern of modes of climate variability.”

To summarize:

Low confidence in atmospheric temperature change
Low confidence in the understanding of clouds
Low confidence in drought cycle trends
Low confidence in storm cycle changes
Low confidence in atmospheric circulation modeling
Poor characterization of ocean temperature cycles
Inadequate data to assess Antarctic ice changes
Low confidence in attribution of climate change to human activities
Low confidence in predictive value of models.

Surely you can see that even the IPPC itself has no basis to claim there is an urgent need for massive government force of any kind, never mind the many scientists whose work contradicts much of what the IPCC claims as factual.

Rats, now that we see their true nature, we will have to label the IPCC and all its supporters as an evil climate denier organization, attack their spokespersons, attack them for refusing to conform to the consensus of the so-called smart people, vilify their professional work, prevent them from achieving tenure, prevent them from receiving research grants, try to keep them from being able to publish in scientific journals, refuse to debate them in public and suppress their right to free speech.

Optical depth

From Wikipedia, the free encyclopedia

 

In physics, optical depth or optical thickness, is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material.^[1] Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.^[1]

In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material, that is the optical depth divided by ln 10.

Mathematical definitions

Optical depth

Optical depth of a material, denoted τ, is given by:^[2]

where

• Φ_e^t is the radiant flux transmitted by that material;
• Φ_eⁱ is the radiant flux received by that material;
• T is the transmittance of that material.

Absorbance is related to optical depth by:

where A is the absorbance.

Spectral optical depth

Spectral absorbance in frequency and spectral absorbance in wavelength of a material, denoted τ_ν and τ_λ respectively, are given by:^[1]

where

• Φ_e,ν^t is the spectral radiant flux in frequency transmitted by that material;
• Φ_e,νⁱ is the spectral radiant flux in frequency received by that material;
• T_ν is the spectral transmittance in frequency of that material;
• Φ_e,λ^t is the spectral radiant flux in wavelength transmitted by that material;
• Φ_e,λⁱ is the spectral radiant flux in wavelength received by that material;
• T_λ is the spectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by:

where

• A_ν is the spectral abosrbance in frequency;
• A_λ is the spectral absorbance in wavelength.

Relationship with attenuation

Attenuance

Optical depth measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by absorption, but also reflection, scattering, and other physical processes. Optical depth of a material is approximately equal to its attenuance when both the absorbance is much less than 1 and the emittance of that material (not to be confused with radiant exitance or emissivity) is much less than the optical depth:

where

• Φ_e^t is the radiant power transmitted by that material;
• Φ_e^att is the radiant power attenuated by that material;
• Φ_eⁱ is the radiant power received by that material;
• Φ_e^e is the radiant power emitted by that material;
• T = Φ_e^t/Φ_eⁱ is the transmittance of that material;
• ATT = Φ_e^att/Φ_eⁱ is the attenuance of that material;
• E = Φ_e^e/Φ_eⁱ is the emittance of that material,

and according to Beer–Lambert law,

so:

Attenuation coefficient

Optical depth of a material is also related to its attenuation coefficient by:

where

• l is the thickness of that material through which the light travels;
• α(z) is the attenuation coefficient or Napierian attenuation coefficient of that material at z,

and if α(z) is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes:

Sometimes the relation is given using the attenuation cross section of the material, that is its attenuation coefficient divided by its number density:

where

• σ is the attenuation cross section of that material;
• N(z) is the number density of that material at z,

and if N(z) is uniform along the path, the relation becomes:

Applications

Atomic physics

In atomic physics, the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by

where

• d is the transition dipole moment;
• N is the number of atoms;
• ν is the frequency of the beam;
• c is the speed of light;
• ħ is Planck's constant;
• ε₀ is the vacuum permittivity;
• σ the cross section of the beam;
• γ the natural linewidth of the transition.

Atmospheric sciences

In atmospheric sciences, one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. The optical depth for a slant path is τ = mτ′, where τ′ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. Therefore,

The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols, and gaseous absorption. The optical depth of the atmosphere can be measured with a sun photometer.

Astronomy

In astronomy, the photosphere of a star is defined as the surface where its optical depth is 2/3. This means that each photon emitted at the photosphere suffers an average of less than one scattering before it reaches the observer. At the temperature at optical depth 2/3, the energy emitted by the star (the original derivation is for the Sun) matches the observed total energy emitted.^{[citation needed][clarification needed]}

Note that the optical depth of a given medium will be different for different colors (wavelengths) of light.

For planetary rings, the optical depth is the (negative logarithm of the) proportion of light blocked by the ring when it lies between the source and the observer. This is usually obtained by observation of stellar occultations.

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	watt	W or J/s	M⋅L2⋅T−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux	Φe,ν[nb 3] or Φe,λ[nb 4]	watt per hertz or watt per metre	W/Hz or W/m	M⋅L2⋅T−2 or M⋅L⋅T−3	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity	Ie,Ω[nb 5]	watt per steradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν[nb 3] or Ie,Ω,λ[nb 4]	watt per steradian per hertz or watt per steradian per metre	W⋅sr−1⋅Hz−1 or W⋅sr−1⋅m−1	M⋅L2⋅T−2 or M⋅L⋅T−3	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity.
Radiance	Le,Ω[nb 5]	watt per steradian per square metre	W⋅sr−1⋅m−2	M⋅T−3	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance	Le,Ω,ν[nb 3] or Le,Ω,λ[nb 4]	watt per steradian per square metre per hertz or watt per steradian per square metre, per metre	W⋅sr−1⋅m−2⋅Hz−1 or W⋅sr−1⋅m−3	M⋅T−2 or M⋅L−1⋅T−3	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Irradiance	Ee[nb 2]	watt per square metre	W/m2	M⋅T−3	Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance	Ee,ν[nb 3] or Ee,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m−2⋅Hz−1 or W/m3	M⋅T−2 or M⋅L−1⋅T−3	Irradiance of a surface per unit frequency or wavelength. The terms spectral flux density or more confusingly "spectral intensity" are also used. Non-SI units of spectral irradiance include Jansky = 10−26 W⋅m−2⋅Hz−1 and solar flux unit (1SFU = 10−22 W⋅m−2⋅Hz−1).
Radiosity	Je[nb 2]	watt per square metre	W/m2	M⋅T−3	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity	Je,ν[nb 3] or Je,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m−2⋅Hz−1 or W/m3	M⋅T−2 or M⋅L−1⋅T−3	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Radiant exitance	Me[nb 2]	watt per square metre	W/m2	M⋅T−3	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance	Me,ν[nb 3] or Me,λ[nb 4]	watt per square metre per hertz or watt per square metre, per metre	W⋅m−2⋅Hz−1 or W/m3	M⋅T−2 or M⋅L−1⋅T−3	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Radiant exposure	He	joule per square metre	J/m2	M⋅T−2	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure	He,ν[nb 3] or He,λ[nb 4]	joule per square metre per hertz or joule per square metre, per metre	J⋅m−2⋅Hz−1 or J/m3	M⋅T−1 or M⋅L−1⋅T−2	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
Hemispherical emissivity	ε			1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	εν or ελ			1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ			1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	εΩ,ν or εΩ,λ			1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	A			1	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	Aν or Aλ			1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ			1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν or AΩ,λ			1	Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	R			1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	Rν or Rλ			1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ			1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν or RΩ,λ			1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T			1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	Tν or Tλ			1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ			1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν or TΩ,λ			1	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	μ	reciprocal metre	m−1	L−1	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	μν or μλ	reciprocal metre	m−1	L−1	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	μΩ	reciprocal metre	m−1	L−1	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μΩ,ν or μΩ,λ	reciprocal metre	m−1	L−1	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.