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| '''Rotational-vibrational spectroscopy''' is a branch of molecular [[spectroscopy]] concerned with the [[Infrared spectroscopy|infrared]] and [[Raman spectroscopy|Raman spectra]] of [[molecule]]s in the [[gas phase]]. Changes in rotational state give fine structure to the [[Molecular vibration|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.
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| == Overview ==
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| {{see also|Rotational spectroscopy|Vibronic spectroscopy}}
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| 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''.<ref name="Hollas p101">Hollas p101</ref>
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| 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.<ref group=note>Traditionally, infrared spectra are shown with the wavenumber scale decreasing from left to right, corresponding to increasing wavelength. More modern texts may show the wavenumber scale increasing from left to right. The P-branch is always at lower wavenumbers than the Q-branch.</ref>
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| The appearance of rotational fine structure is determined by the [[Rotational spectroscopy#Classification of molecular rotors|symmetry of the molecular rotor]]s 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 [[rotational spectroscopy|pure rotation]].
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| 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 [[Rigid rotor#The quantum mechanical linear rigid rotor|rotational constant]] for the ground state is written as <math>B^{\prime\prime} ,</math> and that of the excited state as <math>B^\prime .</math>
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| === Method of combination differences ===
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| 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, <math>B^{\prime\prime}</math> and <math>B^\prime</math>. 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.<ref>Hollas p132</ref><ref>Atkins and de Paula p458 with diagrams</ref> 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 <math>\Delta_2</math> since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by<ref>Allen and Cross, p 116</ref>
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| :<math>\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</math>
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| 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 [[Rotational spectroscopy#Linear molecules|pure rotational spectroscopy]]. (See [[#Appendix|Appendix]])
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| 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).
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| :<math>\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</math>
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| == Linear molecules ==
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| === Heteronuclear diatomic molecules===
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| [[File:Vib rot CO.png|thumb|left|250 px|Simulated vibration-rotation line spectrum of [[carbon monoxide]], <sup>12</sup>C<sup>16</sup>O. The P-branch is to the left of the gap near 2140 cm<sup>−1</sup>, the R-branch on the right.<ref group=note>Observed spectra of the fundamental band (shown here) and the first [[overtone band]] (for double excitation near 2 × 2140 cm<sup>−1</sup>) can be seen in Banwell and McCash, p 67</ref>]][[File:Vibrationrotationenergy.svg|right|thumb|250px|Schematic ro-vibrational energy level diagram for a linear molecule]]
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| Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values <math> G(v)</math>,<ref group=note>Term value is directly related to energy by <math>E=hc G(v)</math></ref> for an [[anharmonicity|anharmonic]] oscillator are given, to a first approximation, by
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| :<math> G(v) = \omega_e (v+{1 \over 2}) - \omega_e\chi_e (v+{1 \over 2})^2\,</math>
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| where ''v'' is a [[Molecular vibration#Quantum mechanics|vibrational quantum number]], ω<sub>e</sub> is the harmonic wavenumber and χ<sub>e</sub> is an anharmonicity constant.
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| 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
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| :<math> F_v(J) = B_v J \left( J+1 \right) - D J^2 \left( J+1 \right)^2 </math>
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| where ''J'' is a rotational quantum number and ''D'' is a [[Rotational spectroscopy#Centrifugal distortion|centrifugal distortion constant]]. The rotational constant, ''B''<sub>v</sub> depends on the moment of inertia of the molecule, ''I''<sub>v</sub>, which varies with the vibrational quantum number, ''v''
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| :<math> B_v = {h \over{8\pi^2cI_v}}; \quad I_v=\frac{m_A m_B}{m_A+m_B}d_v^2</math>
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| where ''m''<sub>A</sub> and ''m''<sub>B</sub> 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.
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| :<math> G(v)+F_v(J) = \left[ \omega_e (v+{1 \over 2}) + B_v J (J+1) \right]- \left[ \omega_e\chi_e (v+{1 \over 2})^2 + D J^2 (J+1)^2 \right]</math>
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| 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 expansion| Dunham]].
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| The [[selection rule]] for electric dipole allowed ro-vibrational transitions, in the case of a diamagnetic diatomic molecule is
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| :<math> \Delta v = \pm 1 \ (\pm 2, \pm 3, etc.</math><ref group=note>Transitions with ∆''v''≠1 are called overtones. They are forbidden in the harmonic approximation but can be observed as weak bands because of anharmonicity.</ref><math>),\Delta J = \pm 1 </math>
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| The transition with Δv=±1 is known as the fundamental transition. The selection rule has two consequences.
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| # Both the vibrational and rotational quantum numbers must change. The transition :<math> \Delta v = \pm 1, \Delta J = 0 </math> (Q-branch) is forbidden
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| # 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.
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| The calculation of the transition wavenumbers is more complicated than for pure rotation because the rotational constant ''B''<sub>ν</sub> is different in the ground and excited vibrational states. A simplified expression for the wavenumbers is obtained when the centrifugal distortion constants <math>D^\prime</math> and <math>D^{\prime\prime}</math> are approximately equal to each other.<ref>Allen and Cross, p113</ref>
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| :<math> \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,
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| \quad \omega_0=\omega_e(1-2\chi_e)\quad m=\pm 1, \pm 2 \ etc. </math>
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| [[File:NO R-branch.png|thumb|300px|Spectrum of R-branch of [[nitric oxide]], NO, simulated with Spectralcalc,<ref name=SpectraCalc/> showing λ-doubling caused by the presence of an unpaired electron in the molecule]]
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| where positive ''m'' values refer to the R-branch and negative values refer to the P-branch. The term ω<sub>0</sub> gives the position of the (missing) Q-branch, the term <math>(B ^\prime+B^{\prime\prime})m</math> implies an progression of equally spaced lines in the P- and R- branches, but the third term, <math>(B^\prime-B^{\prime\prime})m^2</math>shows that the separation between adjacent lines changes with changing rotational quantum number. When <math>B^{\prime\prime}</math> is greater than <math>B^\prime</math>, 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 <math>B^{\prime\prime}</math> of 1.915 cm<sup>−1</sup> and <math>B^{\prime}</math> of 1.898 cm<sup>−1</sup>. The bond lengths are easily obtained from these constants as ''r''<sub>0</sub> = 113.3 pm, ''r''<sub>1</sub> = 113.6 pm.<ref>Banwell and McCash, p 70</ref> 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
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| :<math>B_\nu=B_{eq}-\alpha(\nu+{1\over 2})</math>
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| 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<sub>eq</sub> = 113.0 pm.<ref>Banwell and McCash, p69</ref>
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| [[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''<ref group=note>Another example of lambda-doubling is found in the energy levels of the [[hydroxyl radical]].</ref> with calculated harmonic frequencies of 1904.03 and 1903.68 cm<sup>−1</sup>. Rotational levels are also split.<ref name=G>{{cite journal| last=Gillette|first=R. H.|coauthors=Eyster,Eugene H.|title=The Fundamental Rotation-Vibration Band of Nitric Oxide|journal=Phys. Rev.|year=1939|volume=56|issue=11|pages=1113–1119|doi=10.1103/PhysRev.56.1113}}</ref>
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| ===Homonuclear diatomic molecules===
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| The quantum mechanics for diatomic molecules such as [[dinitrogen]], N<sub>2</sub>, and [[fluorine]], F<sub>2</sub>, 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 N<sub>2</sub> can be observed when using long path-lengths both in the laboratory and in the atmosphere.<ref name=Goldman>{{cite journal|last=Goldman|first=A.|coauthors=Reid, J.; Rothman, L. S.|title=Identification of electric quadrupole O<sub>2</sub> and N<sub>2</sub> lines in the infrared atmospheric absorption spectrum due to the vibration‐rotation fundamentals|journal=Geophysical Research Letters|year=1981|volume=8|issue=1|pages=77|doi=10.1029/GL008i001p00077}}</ref> The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.
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| [[Dioxygen]] is a special case as the molecule is [[paramagnetic]] so [[selection rule|magnetic-dipole-allowed]] transitions can be observed in the infrared.<ref name=Goldman/> The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N,<ref group=note>Some texts use the symbol K for this quantum number</ref> so that each rotational level is split into three states with total angular momentum (molecular rotation plus electron spin) <math>\mathrm {J\hbar} \,</math>, 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.<ref>Hollas p116</ref> 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 <sup>16</sup>O nucleus has zero nuclear spin angular momentum, so that symmetry considerations demand that N may only have odd values.<ref>{{cite journal|last=Strandberg,|first=M. W. P.|coauthors=Meng, C. Y.;Ingersoll, J. G.|title=The Microwave Absorption Spectrum of Oxygen|journal=Phys.Rev.|year=1949|volume=75|issue=10|pages=1524–1528|doi=10.1103/PhysRev.75.1524}}[http://dspace.mit.edu/bitstream/handle/1721.1/4963/RLE-TR-087-14236979.pdf pdf]</ref><ref>{{cite journal|last=Krupenie|first=Paul H.|title=The Spectrum of Molecular Oxygen|journal=J. Phys. Chem. Ref. Data 1, 423 (1972);|year=1972|volume=1|issue=2|pages=423–534|doi=10.1063/1.3253101|url=http://www.nist.gov/data/PDFfiles/jpcrd8.pdf}}</ref>
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| === Raman spectra of diatomic molecules ===
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| The selection rule is
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| :<math>\Delta J = 0, \pm 2</math>
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| 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
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| :<math>\bar \nu (S(J))= \omega_0 +4BJ+6B = \omega_0 + 6B, \quad\omega_0 + 10B, \quad\omega_0 + 14B, \quad ...</math>
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| :<math>\bar \nu (O(J))= \omega_0 -4BJ+2B = \omega_0 - 6B, \quad\omega_0 - 10B, \quad\omega_0 - 14B, \quad ...</math>
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| since the O-branch starts at J=0 and the S-branch at J=2. So, to a first approximation, the separation between ''S''(0) and ''O''(2) is 12''B'' and the separation between adjacent lines in both O- and S- branches is 4''B''. 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 Δ''J''=0 for ''J''=1,2 etc.<ref>Hollas, pp 133–135. The Stokes-side spectrum of [[carbon monoxide]] is shown on p134</ref> Useful difference formulae, neglecting centrifugal distortion are as follows.<ref name="Hollas, p135">Hollas, p135</ref>
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| :<math>\Delta_4^{\prime\prime}F(J) = \bar \nu [S(J-2)] - \bar \nu [O(J+2) ] = 4B^{\prime\prime}(2J+1)</math>
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| :<math>\Delta_4^{\prime}F(J) = \bar \nu [S(J)] - \bar \nu [O(J)] = 4B^{\prime}(2J+1)</math>
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| Molecular oxygen is a special case as the molecule is paramagnetic, with two unpaired electrons.<ref>{{cite journal|last=Fletcher|first=William H.|coauthors=Rayside, John S.|title=High resolution vibrational Raman spectrum of oxygen|journal=Journal of Raman Spectroscopy|year=1974|volume=2|issue=1|pages=3–14|doi=10.1002/jrs.1250020102}}</ref>
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| For homonuclear diatomics, nuclear spin statistical weights lead to alternating line intensities between even-<math>J^{\prime\prime}</math> and odd-<math>J^{\prime\prime}</math> levels. For nuclear spin ''I'' = 1/2 as in <sup>1</sup>H<sub>2</sub> and <sup>19</sup>F<sub>2</sub> the intensity alternation is 1:3. For <sup>2</sup>H<sub>2</sub> and <sup>14</sup>N<sub>2</sub>, ''I''=1 and the statistical weights are 6 and 3 so that the even-<math>J^{\prime\prime}</math> levels are twice as intense. For <sup>16</sup>O<sub>2</sub> (''I''=0) all transitions with even values of <math>J^{\prime\prime}</math> are forbidden.<ref name="Hollas, p135"/>
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| === Polyatomic linear molecules ===
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| {|
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| |[[File:Nu2 nitrous oxide.png|thumb|270 px|Spectrum of bending mode in <sup>14</sup>N<sup>14</sup>N<sup>16</sup>O simulated with Spectralcalc.<ref name=SpectraCalc/> The weak superimposed spectrum is due to species containing <sup>15</sup>N at [[isotopes of nitrogen|natural abundance]] of 0.3%]]
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| |[[File:Acetylene 730.png|thumb|270 px|Spectrum of a perpendicular band from [[acetylene]], C<sub>2</sub>H<sub>2</sub>, simulated with Spectralcalc<ref name=SpectraCalc/> showing 1,3 intensity alternation in both P- and R- branches. See also Hollas p157]]
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| |[[File:Nu3 carbon dioxide.png|thumb|270 px|Spectrum of the asymmetric stretching (parallel) band of [[carbon dioxide]], <sup>12</sup>C<sup>16</sup>O<sub>2</sub> simulated with Spectralcalc.<ref name=SpectraCalc/> The weak superimposed spectrum is due to species <sup>13</sup>C<sup>16</sup>O<sub>2</sub> at [[isotopes of carbon|natural abundance]] of 1%]]
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| |}
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| These molecules fall into two classes, according to [[molecular symmetry|symmetry]]: centrosymmetric molecules with [[Point groups in three dimensions|point group]] D<sub>∞h</sub>, such as [[carbon dioxide]], CO<sub>2</sub>, and [[ethyne]] or acetylene, HCCH; and non-centrosymmetric molecules with point group C<sub>∞v</sub> such as [[hydrogen cyanide]], HCN, and [[nitrous oxide]], NNO. Centrosymmetric linear molecules have a [[Dipole#Molecular dipoles|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.
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| 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 <math>\parallel</math>. When the vibration induces a dipole moment pointing perpendicular to the molecular axis the term ''perpendicular'' is applied, with the symbol <math>\perp</math>. 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.<ref>Straughan and Walker, vol2, p185</ref> 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.<ref>The spectrum of this vibration mode, centered at ca. 3310 cm<sup>−1</sup> is shown in Banwell and McCash, p76, and also in Hollas, p156</ref>
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| For a perpendicular vibration the transition Δ''J''=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<sup>−1</sup> is an example.<ref name=SpectraCalc>Simulated spectrum created using [http://www.spectralcalc.com/ infrared gas spectra simulator]</ref>
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| 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 <sup>12</sup>C<sup>16</sup>O<sub>2</sub> have spin zero and are [[boson]]s, so that the total wavefunction must be symmetric when the two <sup>16</sup>O 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.<ref>Straughan and Walker, vol2 p186</ref> For acetylene the hydrogens of <sup>1</sup>H<sup>12</sup>C<sup>12</sup>C<sup>1</sup>H have spin ½ and are [[fermion]]s, so the total wavefunction is antisymmetric when two <sup>1</sup>H nuclei are exchanged. As is true for [[Spin isomers of hydrogen|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.<ref>Hollas p155</ref><ref>Straughan and Walker vol2, pp 186−8</ref>
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| == Spherical top molecules ==
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| [[File:Methane rotational-vibrational spectrum.png|right|thumb|300px|Upper: low-resolution infrared spectrum of asymmetric stretching band of methane (CH<sub>4</sub>); lower: PGOPHER<ref>[http://pgopher.chm.bris.ac.uk PGOPHER a Program for Simulating Rotational Structure, C. M. Western, University of Bristol]</ref> simulated line spectrum, ignoring Coriolis coupling]]
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| These molecules have equal moments of inertia about any axis, and belong to the point groups T<sub>d</sub> (tetrahedral AX<sub>4</sub>) and O<sub>h</sub> (octahedral AX<sub>6</sub>). Molecules with these symmetries have a dipole moment of zero, so do not have a pure rotation spectrum in the infrared or microwave regions.<ref>A very weak spectrum can be observed due to an excited vibrational state being polar. See [[rotational spectroscopy]] for more details.</ref>
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| Tetrahedral molecules such as [[methane]], CH<sub>4</sub>, have infrared-active stretching and bending vibrations, belonging to the T<sub>2</sub> (sometimes written as F<sub>2</sub>) representation.<ref group=note>The term ''representation'' is used in [[group theory]] to classify the effect of symmetry operations on, in this case, a molecular vibration. The symbols for the representations are to be found in the first column of the [[List of character tables for chemically important 3D point groups|character table]] that applies to the particular molecular symmetry.</ref> These vibrations are triply degenerate and the rotational energy levels have three components separated by the [[Coriolis effect|Coriolis interaction]].<ref>Straughan and Walker vol2, p199</ref> The rotational term values are given, to a first order approximation, by<ref>Allen and Cross, p67</ref>
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| :<math>F^+=B_\nu J(J+1) +2B_\nu\zeta_r (J+1)</math>
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| :<math>F^0=B_\nu J(J+1) </math>
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| :<math>F^-=B_\nu J(J+1) -2B_\nu\zeta_r (J+1)</math>
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| where <math>\zeta_r</math> is a constant for Coriolis coupling. The selection rule for a fundamental vibration is
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| :<math>\Delta J = 0, \pm 1</math>
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| 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, <math>J^\prime =J^{\prime\prime} = 1, 2 ...</math> 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.<ref>{{cite journal|last=Jahn|first=H.A.|journal=Proceedings of the Royal Society|year=1938|volume=A168|pages=469, 495}};{{cite journal|year=1939|volume=A171|pages=450}}</ref><ref>{{cite journal|last=Hecht|first=K.T.|journal=J. Mol. Spectrosc.|year=1960|volume=5|pages=335, 390|title=Vibration-rotation energies of tetrahedral XY<sub>4</sub> molecules: Part II. The fundamental ν3 of CH<sub>4</sub>|doi=10.1016/0022-2852(61)90103-5}}</ref>
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| == Symmetric top molecules ==
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| These molecules have a unique [[Molecular symmetry#Elements|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 <math>I_{\perp}</math> and the rotational constant is <math>B = {h \over{8\pi^2cI_{\perp}}}</math>, as for linear molecules. For rotation about the unique axis, however, the moment of inertia is <math>I_{\parallel}</math> and the rotational constant is <math>A = {h \over{8\pi^2cI_{\parallel}}}</math>. Examples include [[ammonia]], NH<sub>3</sub> and [[methyl chloride]], CH<sub>3</sub>Cl (both of [[molecular symmetry]] described by point group C<sub>3v</sub>), [[boron trifluoride]], BF<sub>3</sub> and [[phosphorus pentachloride]], PCl<sub>5</sub> (both of point group D<sub>3h</sub>), and [[benzene]], C<sub>6</sub>H<sub>6</sub> (point group D<sub>6h</sub>).
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| 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 2''J''+1- fold [[Degenerate energy levels|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'', <math>\parallel</math> or ''perpendicular'',<math>\perp</math>, in this case according to the direction of the dipole moment change with respect to the principal rotation axis. A third category involves certain [[Overtone band|overtones]] and combination bands which share the properties of both parallel and perpendicular transitions. The selection rules are
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| :<math>\parallel</math> If ''K'' ≠ 0, then Δ''J'' = 0, ±1 and Δ''K'' = 0
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| ::If ''K'' = 0, then Δ''J'' = ±1 and Δ''K'' = 0
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| :<math>\perp</math> Δ''J'' = 0, ±1 and Δ''K'' = ±1
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| 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.
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| An expression for the calculated wavenumbers of the P- and R- branches may be given as<ref>Allen and Cross, p131</ref>
| |
| | |
| :<math>\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</math>
| |
| ::<math>-2(D_J^\prime+D_J^{\prime\prime})m^3-(D_J^\prime-D_J^{\prime\prime})m^4</math>
| |
| ::<math>+\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</math>
| |
| in which ''m'' = ''J''+1 for the R-branch and -''J'' for the P-branch. The three centrifugal distortion constants <math>D_J, D_{JK}</math>, and <math>D_K</math> are needed to fit the term values of each level.<ref name="Hollas p101"/> The wavenumbers of the sub-structure corresponding to each band are given by
| |
| :<math>\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
| |
| </math>
| |
| <math>\bar \nu_{sub}</math> represents the Q-branch of the sub-structure, whose position is given by
| |
| :<math>\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
| |
| </math>.
| |
| {|
| |
| |[[File:Nu C-Cl in MeCl.png|thumb|250px|Spectrum of the C-Cl stretching band in CH<sub>3</sub>Cl (parallel band) simulated with Spectralcalc.<ref name=SpectraCalc/>]]
| |
| |[[File:Asymmetric bend methyl chloride.png|thumb|250px|Part of the spectrum of the asymmetric H-C-H bending vibration in CH<sub>3</sub>Cl (perpendicular band), simulated with Spectralcalc<ref name=SpectraCalc/>]]
| |
| |}
| |
| | |
| === Parallel bands ===
| |
| The C-Cl stretching vibration of [[methyl chloride]], CH<sub>3</sub>Cl, 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;<ref name=SpectraCalc/> in reality, very high resolution spectroscopy would be needed to resolve the fine structure fully. Allen and Cross show parts of the spectrum of CH<sub>3</sub>D and give a detailed description of the numerical analysis of the experimental data.<ref>Allen and Cross pp 134–148</ref><ref>{{cite journal|last=Allen|first=H.C. jr.|coauthor=E.K.Pyler|title=Some Vibrational-Rotational Bands of Deuterated Methanes |journal=J. Research Nat. Bur. Standards|year=1959|volume=63A|issue=2|pages=145–153}}[http://cdm16009.contentdm.oclc.org/cdm/ref/collection/p13011coll6/id/72772 ]</ref>
| |
| | |
| === 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.<ref name = SpectraCalc/> 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.<ref>Allen and Cross, pp 149–164 has a detailed analysis.</ref>
| |
| | |
| === 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, NH<sub>3</sub>, will have components belonging to ''A''<sub>1</sub> and ''E'' representations. A transition to the ''A''<sub>1</sub> component will give a parallel band and a transition to the ''E'' component will give perpendicular bands; the result is a hybrid band.<ref>Allen and Cross, pp 164–70.</ref>
| |
| | |
| === Inversion in ammonia ===
| |
| {|
| |
| |[[File:Ammonia nu2.png|thumb|300px|Spectrum of central region of the symmetric bending vibration in ammonia simulated with Spectralcalc,<ref name=SpectraCalc/> illustrating inversion doubling.]]
| |
| |[[Image:Nitrogen-inversion-3D-balls.png|thumb|300px|Nitrogen inversion in ammonia]]
| |
| |}
| |
| | |
| For ammonia, NH<sub>3</sub>, the symmetric bending vibration is observed as two branches near 930 cm<sup>−1</sup> and 965 cm<sup>−1</sup>. This so-called inversion doubling arises because the symmetric bending vibration is actually a large-amplitude motion known as [[Nitrogen inversion|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<sup>+</sup>) at 932.5 cm<sup>−1</sup> above the ground (0<sup>+</sup>) state and an antisymmetric state (1<sup>-</sup>) at 968.3 cm<sup>−1</sup>.<ref>Hollas p167 gives 0.79 cm<sup>−1</sup> for the 0<sup>+</sup>−0<sup>-</sup> energy difference, 931.7 cm<sup>−1</sup> for 0<sup>-</sup>−1<sup>+</sup>, and 35.8 cm<sup>−1</sup> for 1<sup>+</sup>−1<sup>-</sup>.</ref>
| |
| | |
| 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<sup>−1</sup>).<ref>{{cite book|last=Harris|first=Daniel, C.|title=Symmetry and Spectroscopy|year=1978|publisher=OUP|isbn=0-19-502001-4|pages=168–170|coauthors=Bertolucci, Michael, D.}}</ref><ref>Spectra are shown in Allen and Cross, pp 172–174</ref> This transition is historically significant and was used in the ammonia [[MASER]], the fore-runner of the [[LASER]].<ref>Straughan and Walker p124</ref>
| |
| | |
| ==Asymmetric top molecules==
| |
| [[File:Water infrared absorption coefficient.gif|thumb|350px|link=File:Water infrared absorption coefficient large.gif|Absorption spectrum ([[attenuation coefficient]] vs. wavelength) of liquid water (red) <ref name="Bertie96">{{cite journal |author=Bertie J. E., Lan Z. |year=1996 |title=Infrared Intensities of Liquids XX: The Intensity of the OH Stretching Band of Liquid Water Revisited, and the Best Current Values of the Optical Constants of H2O(l) at 25°C between 15,000 and 1 cm<sup>−1</sup> |journal=Applied Spectroscopy |volume=50 |issue=8 | pages=1047–1057 |url=http://www.opticsinfobase.org/as/abstract.cfm?uri=as-50-8-1047 |accessdate=2012-08-08 |bibcode = 1996ApSpe..50.1047B |doi = 10.1366/0003702963905385 }}</ref> atmospheric [[water vapor]] (green) <ref name=spectraiaoru>{{cite web |url=http://spectra.iao.ru/1024x563/en/mol/survey/1/ |title=Spectroscopy of Atmospheric Gases (spectral databases) |publisher=V.E. Zuev Institute of Atmospheric Optics SB RAS |accessdate=August 8, 2012 |quote=... various data sources: HITRAN and GEISA spectral databanks, original data obtained by IAO researchers in collaboration with other scientists, H2O spectra simulated by Partridge and Schwenke etc...
| |
| ...}}</ref><ref name="Aringer02">{{cite journal |author=Aringer B., Kerschbaum F., Jørgensen U. G. |year=2002 |title=H<sub>2</sub>O in stellar atmospheres |journal=A&A |volume=395 |pages=915–927 |publisher= EDP Sciences |doi=10.1051/0004-6361:20021313 |url=http://www.aanda.org/articles/aa/pdf/2002/45/aah3665.pdf |accessdate=2012-08-08 |bibcode = 2002A&A...395..915A }}</ref> and ice (blue line) <ref name="Warren84">{{cite journal |author=Warren S. G. |year=1984 |title=Optical constants of ice from the ultraviolet to the microwave |journal=Applied Optics |volume=23 |pages=1206 |url=http://www.atmos.washington.edu/~sgw/PAPERS/1984_Icemcx.pdf |accessdate=2012-08-08 |bibcode = 1984ApOpt..23.1206W |doi = 10.1364/AO.23.001206 }}</ref><ref name="AringerWarren02">{{cite journal |author=Warren S. G., Brandt R. E. |year=2008 |title=Optical constants of ice from the ultraviolet to the microwave: A revised compilation |journal=J. Geophys. Res. |volume=113 |doi=10.1029/2007JD009744 |url=http://www.atmos.washington.edu/ice_optical_constants/Warren_and_Brandt_2008.pdf |accessdate=2012-08-08 |bibcode = 2008JGRD..11314220W }}</ref> between 667 nm and 200 μm.<ref name="WozniakDera07">{{cite book | author=Wozniak B., Dera J. | year = 2007 | title = Atmospheric and Oceanographic Sciences Library | publisher = Springer Science+Business Media. LLC | location=New York | url=http://www.springer.com/cda/content/document/cda_downloaddocument/9780387307534-c2.pdf | accessdate=August 4, 2012 | isbn = 978-0-387-30753-4}}</ref> The plot for vapor is a transformation of data ''Synthetic spectrum for gas mixture "Pure H<sub>2</sub>O"'' (296K, 1 atm) retrieved from [[HITRAN|Hitran]] on the Web Information System.<ref name=hitraniaoru>{{cite web |url=http://hitran.iao.ru/ |title=Hitran on the Web Information System |publisher=Harvard-Smithsonian Center for Astrophysics (CFA), Cambridge, MA, USA; V.E. Zuev Institute of Atmosperic Optics (IAO), Tomsk, Russia |accessdate=August 11, 2012 }}</ref>]]
| |
| | |
| Asymmetric top molecules have at most one or more 2-fold rotation axes. There are three unequal moments of inertia about three mutually perpendicular [[Moment of inertia#Principal axes|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<sup>−1</sup>) the absorption is due to pure rotation. The band around 6.3 μm (1590 cm<sup>−1</sup>) 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.<ref>{{cite journal|last=Dalby|first=F.W.|coauthors=Nielsen, H.H.|title=Infrared Spectrum of Water Vapor. Part I—The 6.26μ Region |journal=J. Chem. Phys.|year=1956|volume=25|issue=5|pages=934–940|doi=10.1063/1.1743146 }}</ref> 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<sup>−1</sup> and 3700 cm<sup>−1</sup> 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.<ref>Allen and Cross, chapter 8.</ref>
| |
| | |
| ==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 [[Dispersion (optics)|spectral dispersion]] element in a grating [[spectrometer]].<ref name=G/> This is a type of [[diffraction grating]] optimized to use higher diffraction orders.<ref>Hollas, pp. 38−41. Worked axample 3.1 shows how resolving power is related to diffraction order and line spacing on the grating</ref> The resolving power of an [[fourier transform infrared spectroscopy|FTIR]] spectrometer depends on the maximum retardation of the moving mirror. For example, to achieve a resolution of 0.1 cm<sup>−1</sup>, 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 CO<sub>2</sub> at this resolution.<ref>{{cite journal|last=Connes|first=J.|coauthors=Connes, P.|year=1966|title=Near-Infrared Planetary Spectra by Fourier Spectroscopy. I. Instruments and Results|journal=Journal of the Optical Society of America|volume=56|issue=7|pages=896–910|doi=10.1364/JOSA.56.000896}}</ref> A spectrometer with 0.001 cm<sup>−1</sup> 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 [[monochromator#Czerny-Turner monochromator|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.<ref>{{cite book|title=Practical Sampling Techniques for Infrared Analysis|year=1993|publisher=CRC PressINC|isbn=0-8473-4203-1|editor=Patricia B. Coleman}}</ref> This is important because it allows the pressure to be reduced so as to minimize [[Spectral line|pressure broadening]] of the spectral lines, which may degrade resolution. Path lengths up to 20m are commercially available.
| |
| | |
| == Notes ==
| |
| {{reflist|group=note}}
| |
| | |
| == Appendix ==
| |
| The method of combination differences uses differences of wavenumbers in the P- and R- branches to obtain data that depend only on rotational constants in the vibrational ground or excited state. For the excited state
| |
| :<math>\Delta_2^{\prime}F(J)^{observed} = \bar \nu [R(J) ] - \bar \nu [P(J) ] </math> | |
| This function can be fitted, using the [[least squares|method of least-squares]] to data for carbon monoxide, from Harris and Bertolucci.<ref name=Harris>{{cite book|last=Harris|first=Daniel, C.|title=Symmetry and Spectroscopy|year=1978|publisher=OUP|isbn=0-19-502001-4|page=125|coauthors=Bertolucci, Michael, D.}}</ref> The data calculated with the formula
| |
| :<math>\Delta_2^{\prime}F(J)^{calculated} = 2B^{\prime\prime} \left(2J+1\right)</math>
| |
| in which centrufugal distortion is ignored, are shown in the columns labelled with (1). This formula implies that the data should lie on a straight line with slope 2B′′ and intercept zero. At first sight the data appear to conform to this model, with a [[Root-mean-square deviation|root mean square]] residual of 0.21 cm<sup>−1</sup>. However, when centrifugal distortion is included, using the formula
| |
| :<math>\Delta_2^{\prime}F(J)^{calculated} = (2B^{\prime\prime}-3D^{\prime\prime}) \left(2J+1\right)-D^{\prime\prime}\left(2J+1\right)^3</math>
| |
| the least-squares fit is improved markedly, with ms residual decreasing to 0.000086 cm<sup>−1</sup>. The calculated data are shown in the columns labelled with (2).
| |
| | |
| [[File:Combination plot CO.png|thumb|300px|Plot of the differences R(J)-P(J) as a function of 2J+1 taken from observed data for the fundamental vibration of carbon monoxide<ref name=Harris/>]]
| |
| {| class="wikitable"
| |
| |+Wavenumbers /cm<sup>−1</sup> for the fundamental ro-vibrational band of carbon monoxide
| |
| |-
| |
| ! J!!2J+1 !! <math>\Delta_2^{\prime}F(J)</math> !! Calculated (1) !! Residual (1) !! Calculated (2) !! Residual (2)
| |
| |-
| |
| |1||3||11.4298||11.3877||0.0421||11.4298||−0.000040
| |
| |-
| |
| |2||5||19.0493||18.9795||0.0698||19.0492||0.000055
| |
| |-
| |
| |3||7||26.6680||26.5713||0.0967||26.6679||0.000083
| |
| |-
| |
| |4||9||34.2856||34.1631||0.1225||34.2856||0.000037
| |
| |-
| |
| |5||11||41.9020||41.7549||0.1471||41.9019||0.000111
| |
| |-
| |
| |6||13||49.5167||49.3467||0.1700||49.5166||0.000097
| |
| |-
| |
| |7||15||57.1295||56.9385||0.1910||57.1294||0.000089
| |
| |-
| |
| |8||17||64.7401||64.5303||0.2098||64.7400||0.000081
| |
| |-
| |
| |9||19||72.3482||72.1221||0.2261||72.3481||0.000064
| |
| |-
| |
| |10||21||79.9536||79.7139||0.2397||79.9535||0.000133
| |
| |-
| |
| |11||23||87.5558||87.3057||0.2501||87.5557||0.000080
| |
| |-
| |
| |12||25||95.1547||94.8975||0.2572||95.1546||0.000100
| |
| |-
| |
| |13||27||102.7498||102.4893||0.2605||102.7498||−0.000016
| |
| |-
| |
| |14||29||110.3411||110.0811||0.2600||110.3411||0.000026
| |
| |-
| |
| |15||31||117.9280||117.6729||0.2551||117.9281||−0.000080
| |
| |-
| |
| |16||33||125.5105||125.2647||0.2458||125.5105||−0.000041
| |
| |-
| |
| |17||35||133.0882||132.8565||0.2317||133.0882||0.000035
| |
| |-
| |
| |18||37||140.6607||140.4483||0.2124||140.6607||0.000043
| |
| |-
| |
| |19||39||148.2277||148.0401||0.1876||148.2277||−0.000026
| |
| |-
| |
| |20||41||155.7890||155.6319||0.1571||155.7891||−0.000077
| |
| |-
| |
| |21||43||163.3443||163.2237||0.1206||163.3444||−0.000117
| |
| |-
| |
| |22||45||170.8934||170.8155||0.0779||170.8935||−0.000053
| |
| |-
| |
| |23||47||178.4358||178.4073||0.0285||178.4359||−0.000093
| |
| |-
| |
| |24||49||185.9713||185.9991||−0.0278||185.9714||−0.000142
| |
| |-
| |
| |25||51||193.4997||193.5909||−0.0912||193.4998||−0.000107
| |
| |-
| |
| |26||53||201.0206||201.1827||−0.1621||201.0207||−0.000097
| |
| |-
| |
| |27||55||208.5338||208.7745||−0.2407||208.5338||−0.000016
| |
| |-
| |
| |28||57||216.0389||216.3663||−0.3274||216.0389||0.000028
| |
| |-
| |
| |20||59||223.5357||223.9581||−0.4224||223.5356||0.000128
| |
| |-
| |
| |30||61||231.0238||231.5499||−0.5261||231.0236||0.000178
| |
| |}
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| == Bibliography ==
| |
| *{{cite book|last=Allen|first=H.C.|title=Molecular vib-rotors; the theory and interpretation of high resolution infra-red spectra |year=1963|publisher=Wiley|location=New York|coauthors=Cross, P.C.}}
| |
| *{{cite book|last=Atkins|first=P.W.|title=Physical Chemistry|year=2006|publisher=Oxford University Press|edition= 8th|isbn=0198700725|pages=431–469|coauthors=de Paula, J.}} Chapter (Molecular Spectroscopy), Section (Vibration-rotation spectra) and page numbers may be different in different editions.
| |
| *{{cite book |last1=Banwell |first1=Colin N. |last2=McCash |first2=Elaine M. |title=Fundamentals of molecular spectroscopy |edition= 4th |year=1994 |publisher=McGraw-Hill|isbn=0-07-707976-0 |page=40 }}
| |
| *{{cite book|last=Hollas|first=M.J.|title=Modern Spectroscopy|edition=3rd|year=1996|publisher=Wiley|isbn=0471965227}}
| |
| *{{cite book |last1=Straughan|first1=B.P. |last2=Walker |first2=S. |title=Spectroscopy vol.2 |edition= 3rd |year=1976 |publisher=Chapman and Hall|isbn=0-470-15032-7 |pages=176–204 }}
| |
| | |
| ==External links==
| |
| *[http://www.spectralcalc.com/ infrared gas spectra simulator]
| |
| *[http://www.nist.gov/pml/data/msd-di NIST Diatomic Spectral Database]
| |
| *[http://www.nist.gov/pml/data/msd-tri/index.cfm NIST Triatomic Spectral Database]
| |
| *[http://www.nist.gov/pml/data/msd-hydro/index.cfm NIST Hydrocarbon Spectral Database]
| |
| | |
| {{BranchesofSpectroscopy}}
| |
| | |
| [[Category:Chemical physics]]
| |
| [[Category:Spectroscopy]]
| |