Introduction

From the interpretations specified to thermodynamic properties of gases, gas molecules are simultaneously rotating and vibrating. It follows that an absorption spectrum or a Raman spectrum of a gas might illustrate the consequences of transforms in both rotational and vibrational energies.

Definition of Vibrational and Rotational Transitions of diatomic Molecules

High-resolution gas-phase IR spectra illustrate information about the vibrational and rotational behaviour of heteronuclear diatomic molecules (NIST, 2009). Vibrational transitions of HCl and DCI may be modelled by the harmonic oscillator when the bond length is near Re. In this region, the potential energy can be calculated as:

Fig: High-Resolution Gas-phase IR Spectra Showing Information about the Vibrational and Rotational Behavior of Heteronuclear Diatomic Molecules

E=1/2K(R-R₀)²

Where k is the force constant of the bond. The Schrödinger equation for a particle undergoing harmonic motion can be modified to provide an equation to compute the allowed vibrational energy levels:

E (v) = (v + 1/2)h     

Where h is Planck's steady, is the vibrational frequency, and the vibrational quantum number v = 0, 1, 2,....

Of course, diatomic molecules don't continue stationary as they undergo vibration; they as well rotate through space. The stiff rotor model might be utilized to approximate the rotational contribution to the IR spectrum of a diatomic molecule (NIST, 2009). For a stiff rotor, the permitted energy levels might be computed as:

E(j) = h2/8  π2 I* J(J+1)

Where

I = μr² , μ = m₁m₂/(m₁+m₂)

 J is the rotational quantum number (by integer values 0, 1, 2, ...),  h  is  Planck's  constant, and I is the moment of inertia for the molecule (computed as revealed using the reduced mass, , and through  r = re). As we know that the equation for E(J) isn't necessarily in cm^-1; we might have to utilize a conversion factor to obtain the desired units.

Adding such vibrational and rotational energy terms provides a 1^st approximation of the value of its energy levels. Though, we should as well get in terms that account for anharmonicity,   centrifugal distortion (stretching), and the interaction between vibration and rotation. A term for the energy levels for the heteronuclear diatomic molecule in expressions of wave numbers (cm^-1) is:

G(V.I) - V_a(V+1/2) - v_ex_e(V+1/2)² + BJ(J+1) - DJ²(J+1)² - α_e (v+1/2)J(J+1)

Where v_e is the frequency (which we, from now, will state in cm^-1, tilda is usually utilized on the top of v_e but we will omit it) for the molecule vibrating about the equilibrium bond distance re, and

        Be = h/8 π2 I_eC

The difference in energy, and thus the separation between adjacent lines (of the same isotope) in each branch of the IR spectrum is related to Be. The first and third terms of the equation for G account for the harmonic oscillator and stiff rotor behavior of the diatomic molecule; the 2^nd term accounts for anharmonicity (note it contains a constant, xe); the fourth term obtains into account centrifugal stretching, and the last term accounts for the interaction between vibration and rotation. The centrifugal stretching term might be neglected in this experiment since De is small, making this term important only at high J values. The last term accounts for the interaction between the vibration and rotation of the molecule; as the molecule vibrates, the moment of inertia transforms and the rotation of the molecule is influenced. The term is huge sufficient that  the  final  term  in  the  equation  for  G  can't  be neglected.

Selection rules

For a harmonic oscillator, the selection rules need that = ± 1 and = ± 1. That is, when the vibrational transition (represented as v + 1 <-- v) occurs, J changes by +1 for the R branch and -1 for the P branch. As we know that= 0 is a forbidden transition for the diatomic species we are examining (as having no net spin or orbital angular momentum), so you won't see the Q branch corresponding to such a change. As well, since molecules aren't exact harmonic oscillators, for example. They possess several anharmonic character, weak overtones consequential from = ± 2, ± 3, ± 4,... transitions are present, but we will not be concerned through such for the moment. Such overtones are significant to the blue colour of water.

We are most interested in the fundamental conversions from the J" levels of the vibrational ground state (v"=0) to the first excited state J' levels corresponding to v' = 1 (see figure). The frequency in wave numbers, v, might be computed for the R and P branches using the subsequent equations:

v₀ = v₀ + (2B_e - 3a_e) + (2B_e - 4a_e)J - a_eJ'²              for J'' = 0,1,2,3..

v₀ = v₀ - (2B_e - 2a_e)J''- a_eJ^m2                           for  J'' = 1,2,3.....

Forbidden transition, vo, shows at a wave number between the R(0) and P(1) transitions (see figure below; the numbers in parentheses are  J" values). It isn't, though, precisely between the 2 transitions. As we know that the division between the lines in the P branch rises as the J values enhance while the division between R branch lines decreases. This consequence results from the interaction between the vibration and rotation; if there was no interaction, would be zero and the division between lines would be 2Be. The energy of the forbidden transition,, must be computed using its relation to Be and using the equation

V = V_e + (2B_e - 2a_e)m - a_em²

Where m is an integer and is described as m = J" + 1 for the R branch and m = - J'' for the P branch. The division between adjacent lines of the similar isotope in the IR spectrum is consequently.

The values of B_e and can be computed from a plot of (m) versus m. Once such 2 values

Δv (m) = v(m+1) - v(m) = (2B_e - 3a_e) - 2a_em

Are computed, can be estimated using any value of m and the Eq.

Fig: The Fundamental Absorption Band under High Resolution

Isotope effect

We will talk about the isotope consequence through reference to HCl; though, the other gases also exhibit isotope effects and we should analyze them as well. The most abundant form of HCl is ¹H³⁵Cl. Another isotope of chlorine, ³⁷Cl, has a high natural abundance, however, and the lines for ¹H³⁷Cl are obvious in a high-resolution spectrum of HCl, right next to the ¹H³⁵Cl lines. In fact, the isotopic abundance of ³⁵Cl and ³⁷Cl may be computed from the relative absorbance values in the IR spectrum (since absorbance is proportional to concentration). Though the change of an isotope (for example,³⁵Cl to ³⁷Cl) doesn't influence the equilibrium bond length re, or the force constant k for the molecule, fluctuating an isotope does transform, the decreased mass. Since the decreased mass affects the vibrational and rotational behaviour or a molecule, the energy of its transitions are affected. For the harmonic oscillator, the vibrational transition happens at the frequency V_harmonic, which is specified via the equation

 V_harmonics = 1/2π √k/μ

The consequence of the decreased mass on V_harmonic (disregarding anharmonicity) can be presented in the form of the ratio

V*_harmonics / V_harmonics = √ μ/ μ*

Where the asterisk simply signifies a different isotope (for convenience, utilize the asterisk to indicate the heavier isotope). For the consequence of different isotopes on rotation, a similar connection can be attained as

B_e*/B_e = μ/ μ*

Of course, such results apply to isotopes of hydrogen in addition to chlorine.

The Vibrational and Rotational Transitions of Polyatomic Molecules and Acetylene

Vibrational levels and wave functions. Acetylene is recognized to be a symmetric linear molecule by D_{oo h} point group symmetry and 3N - 5 = 7 vibrational normal modes, as depicted in Table. Symmetry is established to be an invaluable aid in understanding the motions in polyatomic molecules. Group theory illustrates that each vibrational coordinate and each vibrational energy level, along through its connected wave function, must have a symmetry analogous to one of the symmetry species of the molecular point group. The D_ooh symmetry species analogous to the different kinds of atomic motion in acetylene are specified in the table. Motions that hold the centre of inversion symmetry, these as the v_l, v₂, and v₄, modes of Table, are labelled g (gerade, German for even) while those for that the displacement vectors are repealed on inversion are labelled u (ungerade, odd). Modes involving motion along the  molecular axis (z) are called parallel vibrations and  labeled while those involving  perpendicular motion  are  labelled and  are  doubly  degenerate  since  equivalent bending can take place in either x or y directions. From the look of the nuclear displacements, it can be seen that only the v₃ and v₅ modes create an oscillating transform in the zero dipole moment of the molecule and therefore provide increase to infrared absorption.

Table: Fundamental Vibrational Modes of Acetylene

From the harmonic-oscillator model of quantum mechanics, the expression value G for the vibrational energy levels for a linear polyatomic molecule can be written as

G(v_l,v₂, . . .) = ^3N-5 Σ _{i =1} vi  (vi + ½)  

where v_i is the  vibrational  frequency  of  mode  i  computed  in  cm^-1. Additional anharmonicity corrections, analogous to v_ex_e, for diatomic molecules, can be added; but these are hopefully small (1-5% of v_i) and will be neglected in this discussion. The energy levels of some of the states of acetylene are shown in Figure to the right. Each level is characterized by a set of harmonic oscillator quantum numbers v_lv₂v₃v₄v₅, revealed at the left of the figure. The fundamental conversions from the ground state are those in that only one of the 5 quantum numbers amplifies from 0 to 1; the two infrared active fundamentals v₃ and v₅ are specified through bold arrows in the figure.

Fig: Indicated with bold arrows

 The Vibrational Wave Functions and Their Properties

The set of quantum numbers of a level as well labels the analogous wavefunction, which,  approximately at low vibrational energy, can be approximated as a product of harmonic oscillator wavefunctions in the separable 'normal coordinates', Q:

Ψ = Ψv1(Q₁) Ψv2(Q₂) Ψv3(Q₃)...

Where each of the one dimensional wavefunctions has the conventional SHO

 Ψ₀ = (γ_i / π ) exp[-γ_i Q_i² /2]

Ψ₁  = (4 γ_i / π) exp[-γ_i Q_i² /2] γ_i Q_i

Ψ₂  = (γ_i /4 π ) exp[-γ_i Q_i² /2](2 γ_i Q_i²-1)

Form where =  k_i/h and  k_i is the  quadratic force  steady and  vi  is the vibrational  frequency in each of the normal coordinate directions. The function Ψ is even or odd depending on the parity of the Hermite polynomial, that is of order v in the displacement. In common, this displacement might be a amalgamation of bond stretches and bends that all happen at the similar frequency through a specified phase relation to each other.

The accurate combination that characterizes a 'normal coordinate' displacement is attained via solution of Newton's equations.

Symmetry relations

Each usual coordinate Q_i, and every wave function involving products of the normal   coordinates, must change under the symmetry operations of  the molecule as one of the  symmetry  species of  the molecular  point  group. The ground-state function in Eq. is a Gaussian exponential function that is quadratic in Q, and examination + illustrates that this is of g symmetry for each normal coordinate, because it is unchanged via any of the D_{oo h} symmetry operations. From group theory, the symmetry of a product of 2 functions is realized from the symmetry species for each function via a systematic process discussed in detail in. The consequences for the D_{oo h} point group appropriate to acetylene can be summarized as follows:

g xg = uxu = g                   gxu = u x g = u

Σ⁺ xΣ⁺  = Σ^-  xΣ^- = Σ⁺                 Σ⁺ x Σ^- = Σ^-

Σ⁺ x ∏ = Σ^- x ∏ = ∏              Σ⁺ x Δ = Σ^- x Δ = Δ

∏ x ∏ = μ + Σ^-  + Δ              Δ x Δ =  Σ⁺ + Σ^- + ∏

∏ x  Δ  = ∏ + Ψ

Application of such rules illustrates that the product of two or more Σ_g+ functions has symmetry Σ_g+, hence the product function for the ground state level (00000) is of Σ_g+ symmetry.

From Eq , it is apparent that the symmetry species of a level through vi = 1 is the similar as that of the coordinate Qi. In the case of a degenerate level these as (00001), there are two wave functions involving the degenerate Q_5x, Q_5y, pair of symmetry ∏_u,. The symmetry of combination levels involving 2 different degenerate modes is attained according to the above rules and, for instance, for the (00011) level, one obtains Σ_g⁺ * Σ_g⁺ * Σ_g⁺ * ∏_g * ∏_d = Σ_d⁺ + Σ_u + Δ_u . Therefore one sees that the product of 2 degenerate functions provides increase to multiplets of different symmetries. For overtone levels of degenerate modes, a more detailed analysis is needed  in that it is originate that levels these as (00020), (00003), and (00004) consist of multiplets of symmetry and (∏ + Ψ ), (Σ⁺+ Δ), and (Σ⁺ + Δ + ∏), respectively.

From such considerations, the symmetry species of each wave function connected through an energy level is computed, and these are indicated at the right in the previous Figure. It is important to realize that this symmetry label is the accurate one for the true wave function, even though deduced from an approximate harmonic oscillator model. This is important since transition selection rules depend on symmetry are exact whereas, for instance, the common harmonic oscillator constraint that v = ± l is only approximate for real molecules.

The set of quantum numbers of a level as well labels the analogous wave function, that, just about at low vibrational energy, can be approximated as a product of harmonic oscillator wave functions in the separable 'normal coordinates', Q:

Ψ = Ψ Ψ1(Q1) Ψv2(Q2) ΨV3(Q3)                                                   

where each of the one dimensional wave functions have the conventional SHO form

Ψ₀ = (γ_i / π ) exp[-γ_i Q_i² /2]

Ψ₁ = (4 γ_i / π) exp[-γ_i Q_i² /2] γ_i Q_i

Ψ₂ = (γ_i /4 π ) exp[-γ_i Q_i² /2](2 γ_i Q_i²-1)

Where = k_i/h and ki is the quadratic force steady and  vi is the vibrational  frequency in each of the usual coordinate directions. The function Ψ is even or odd depending on the parity of the Hermite polynomial, which is of order v in the displacement. In common, this displacement might be a combination of bond stretches and bends that all take place at the similar frequency through a specified phase relation to each other.

The specific amalgamation that characterizes a 'normal coordinate' displacement is attained via solution of Newton's equations.

Symmetry relations

Each normal coordinate Qi, and every wave function involving products of the normal   coordinates, must  transform  under  the  symmetry operations  of  the  molecule  as  one  of  the  symmetry species of  the molecular point group. The ground-state functions in Eq. is a Gaussian exponential function that is quadratic in Q, and examination Σ_g⁺illustrates that this is of g  symmetry for each normal coordinate, since it is unchanged via any of the D_{oo h} symmetry operations. From group theory, the symmetry of a product of 2 functions is deduced from the symmetry species for each function via a systematic process discussed in detail in. The consequences for the D_{oo h} point group suitable to acetylene can be summarized as follows:

g xg = uxu = g                   gxu = u x g = u

Σ⁺ xΣ⁺  = Σ^-  xΣ^- = Σ⁺                   Σ⁺ x Σ^- = Σ^-

Σ⁺ x ∏ = Σ^- x ∏ = ∏                  Σ⁺ x Δ = Σ^- x Δ = Δ

∏ x ∏ = μ + Σ^-  + Δ                  Δ x Δ =  Σ⁺ + Σ^- + ∏

∏ x  Δ  = ∏ + Ψ

Application of these rules illustrates that the product of 2 or more +functions has symmetry Σ_g⁺, therefore the product function for the ground Σ_g⁺ state level (00000) is of Σ_g symmetry.

From Eq., it is apparent that the symmetry species of a level through vi = 1 is the similar as that of the coordinate Qi. In the case of a degenerate level such as (00001), there are two wave functions including the degenerate Q_5x, Q_5y, pair of symmetry ∏_u,. The symmetry of combination levels involving 2 dissimilar degenerate modes is gained according to the above rules and, for example, for the (00011) level, one attains Σ_g⁺ * Σ_g⁺ * Σ_g⁺ * ∏_g * ∏_d = Σ_d⁺ + Σ_u + Δ_u. Consequently one sees that the product of 2 degenerate functions provides increase to multiplets of different symmetries. For  overtone  levels of degenerate modes, a more detailed analysis is essential in which it is found that levels these as (00020), (00003), and (00004) consist of multiplets of symmetry and (  ∏+φ  ), (sΣ⁺ + Δ), and (Σ⁺ + Δ + ∏ ), correspondingly.

From such considerations, the symmetry species of each wave function connected through an energy level is determined, and such are designated at the right in the previous Figure. It is  significant to  realize  that  this symmetry  label  is  the  correct one  for  the true  wave  function, even though deduced from an approximate harmonic oscillator model. This is significant because transition selection rules depend on symmetry are exact whereas, for instance, the usual harmonic oscillator constraint that v = ± l is only approximate for real molecules.

Selection rules

The probability of a transition between 2 levels i and j in the presence of infrared (electric dipole) radiation is specified via the transition moment P_ij

P_ij =   ∫ Ψ_i μ Ψ_j dτ                                                                 

For a given molecule, Pij is a physical quantity through a unique numerical value that must remain unchanged through any molecular symmetry operation such as rotation or inversion. Therefore to have a nonzero value, Pij must be totally symmetric, for example ∏( Ψ_i)x ∏(μ)x∏(Ψ_j)- Σ_a⁺ where ∏(Ψ_i) signifies the symmetry of Ψ_i, and so on. The dipole moment component u_z, and the +equivalent pair u_x and u_y are of symmetries Σ_u and ∏_u, correspondingly, for the D_ooh  point group and are usually indicated at the far right in point group (or character) tables. From this and the rules of Eq. it follows that, for a evolution between 2 levels to be infrared-allowed, it is needed that the symmetry species of the product of the 2 wave functions be the similar as one of the dipole components.

Therefore from the Σ_g ground vibronic state of acetylene, transition to the or Σ_u or Δ_u members the (00011) multiplet is forbidden while that to the Σ_u⁺ level is permitted via the μ_z dipole component. Transitions involving μ_z, are termed parallel bands while those involving μ_x and μ_y termed perpendicular bands, because of the angle the dipole moment builds through the symmetry axis of the molecule.

In the case of a Raman conversion, the similar symmetry arguments are relevant, except that the dipole function u must be replaced via the polarisability tensor components a_zz, a_xx, a_xy, etc. For molecules of D_∞h symmetry, such components belong to the symmetry species Σ_g⁺,∏_g , and Δ_g  so  that  the condition for a Raman-active transition is that the product  ∏(Ψ_i) × ∏(Ψ_j) comprise one  of  such species. Consequently from  the  Σ_g⁺ ground state of Σ_g⁺ acetylene,  Raman  transitions  to  the  (10000)  g , (01000)g , and (00010)g  levels are permitted and can be utilized to find out the v₁ , v₂, and v₄  fundamental frequencies correspondingly. As can be seen in Table, such three modes don't generate a dipole transform as vibration occurs and therefore such transitions are absent from the infrared spectrum. This is an instance of the 'rule of mutual exclusion,' that applies for IR/Raman transitions of molecules through a centre of symmetry.

Even though direct access to the (10000), (01000), and (00010) levels from the (00000) ground state level via infrared absorption is therefore thoroughly forbidden via symmetry, access from molecules in the (00010) or (00001) levels can be symmetry-permitted.  For instance, (00001) × (10000) = u × += = ( ) and so the transition between such levels, sometimes termed a difference band, v₁ - v₅, isn't formally forbidden. As can be seen in Figure above, the frequency (v₁  - v₅) can be added to the fundamental frequency v₅  to provide the exact value of v₁, the (10000) - (0000)  spacing. Similarly the v₂-v₅ and v₃ - v₄ difference bands are infrared-active and can be combined through v₅ and v₃, to deduce v₂ and v₄, correspondingly. Such difference bands are detectable for acetylene but will, of course, have low intensity because they originate in vibrationally excited levels that have a small Boltzmann population at room temperature. The intensity of such bands rises by temperature, hence they are as well termed 'hot band' transitions.

Other non-fundamental bands frequently show in infrared spectra and can be utilized to obtain an estimate of the fundamental frequencies. For instance, from the ground state of acetylene, an infrared transition to the (00011) level is permitted and is termed the v₄ + v₅ combination band. The difference (v₄ + v₅) - v₅  can be used as an estimate of v4, but it should be noted that this is actually the separation between  levels (00011) and (00001) and not the separation between (00010) and (00000), which is a better measure of the frequency of the v4  normal mode in the harmonic approximation. Because of anharmonicity effects, such 2 divisions aren't identical and therefore the determination of basic frequencies from difference bands is to be preferred.

Force Constants of Acetylene

From the vibrational frequencies of the normal modes one can compute the force constants for the dissimilar bond stretches and angle bends in the C₂H₂ molecule. In the most complete valence-bond, harmonic- oscillator approximation, the potential energy for C₂H₂ can be written as

U = 1/2k_r(r²₁ + r²₂) + 1/2k_rR² + 1/2 k_δ (δ₁² + δ₂²) + k_rrr1r2 + k_RrR(r₁ +r₂) + k_δδδ₁δ₂    

Where r and R refer, correspondingly, to the stretching of the CH and CC bonds and represents bending of the H-C-C angle from its equilibrium value.  The interaction constants k_rr, k_rR, and characterize the coupling between the dissimilar vibrational coordinates and are generally small compared to the principal force constants k_r, k_R and k_g.

The normal modes are amalgamations of r, R, and coordinate that give an accurate explanation of the atomic motions as vibration occurs. Such combinations must be chosen to have a symmetry corresponding to the symmetry species of the vibration. As a result, for instance, there is no mixing between the orthogonal axial stretches and the perpendicular bending modes and U encloses no cross terms these as r_δ or R_δ. The procedure of discovering the accurate combination of coordinates, termed a normal coordinate analysis, basically involves the solution of Newton's equations of motion in the form of a normal coordinate analysis. This solution as well provides the vibrational frequencies in terms of the force constants, atomic masses, and geometry of the molecule.

These analysis yields the subsequent consequences for this case (linear HCCH)

4 π²v₁² = (k_r + k_rr)(1/mH  + 1/mC) + 2(k_R  - k_rR)/mC     

4 π²v₁² 4 π²v₂² = 2[(k_r + k_rr)k_R  - 2k_r  2)/m m  

4 π²v₂²= (k_r  - k_rr)(1/m_H  + 1/m_C)  

4 π²v₄²= (k_δ - k_δδ ) [(1/R_CH m²_H + (1/R_CH + 2/R_CC)² /m_C]

4 π²v₆² = (k_δ - k _δδ ) (1/m_H + 1/m_C)/R_CH²   

When C₂D₂ frequencies are used, m_H should be replaced by m_D. The force  constants  for  acetylene  can  be  calculated  from  these  relations using the measured vibrational frequencies and the bond lengths can be determined  from  the  rotational  analysis  explained  below.  If one expresses the frequencies in cm^-1 units and the masses in atomic mass units, the factors 4π² should be replaced by 4 π²c²/N₀ = 5.892 x 10^-5. This substitution gives the force constants kr, k_R, kr_r, and k_rR in N/m units and the bending constants k_δ and k_δδ in units of N m.

Rotational Levels and Transitions

The vibrational-rotational energy levels for a linear molecule are similar to those for a diatomic molecule and, to a good approximation, are given in cm-1 units by the sum G(v1,v2,...) + Fv  (J) where

F_v(J) = B_v[j(j+1)-l²] = D_v[j(j+1)-l²]²

The general label v characterizes the set v₁v₂v₃... and is added to F_v to account for the fact that the rotational constant B and centrifugal distortion constant D change slightly through vibrational level. B_v is related to the moment of inertia I_v via the equation:

B_v  = h/ 8 π²I_vc

Where

    I_v = Σ ^N_i=1 m_i r²_i

and the sum is over all atoms in the molecule, having mass  m_i  and located  a  distance  ri    from  the centre-of-mass of the molecule.  The quantum number l characterizes the vibrational angular momentum about the linear axis and is 0, 1, 2,... for levels of symmetries, , , ..., correspondingly. This angular momentum derives from a rotary motion created about the linear axis via an amalgamation of the degenerate x and y bending motions. For acetylene, there are 2 bending modes, requiring l4 and l5 quantum numbers that are sometimes revealed as superscripts to the v₄ and v₅ labels. The permitted transforms in the rotational quantum number J are ΔJ = ± l for parallel ( u ) transitions and  ΔJ  =  0,  ±  l for perpendicular  ( u) transitions. Parallel transitions these as v₃ for acetylene therefore have P (ΔJ = -1) and R (ΔJ = + 1) branches through a trait minimum or 'missing line', between them, as exposed for diatomic molecules these as HCl. However, perpendicular transitions such as v₅ for acetylene and v₂ for HCN have a strong central Q branch (J = 0) along with P and R branches. This characteristic PQR versus PR band shape is quite obvious in the spectrum and is a helpful aid in assigning the symmetries of the vibrational levels included in the infrared transitions of a linear molecule.

The individual lines in a Q branch are resolved only under extremely elevated resolution, but the lines in the P and R branches are easily discerned at a resolution of 1 cm^-1 or better. As discussed in relation to the IR spectrum of HCl, it is possible to symbolize both P and R transition frequencies through a single relation:

 v_m = v_o + B''l''²   B'l'² + (B'' + B')m + (B'  B'')m²   4D_em³

Here v_o is the rotationless transition frequency analogous to Δ_G, the spacing between the 2 vibrational levels through J = 0. B' and B" are the rotational constants of the upper and lower states, correspondingly, and the index m = - J for P branch lines, m = J + 1 for R branch lines. The centrifugal distortion constants are neglected in this analysis because they are extremely small (typically 10^-6 cm^-1), i.e. De" = De' = De.

Intensities and Statistical Weights

The absolute absorption intensity of a vibrational-rotational transition is proportional to the square of the transition moment P_ij times the population in the lower state.  P_ij varies only slightly for dissimilar rotational levels so that the principal factors determining the relative intensity are the degeneracy and the Boltzmann weight for the lower level

I_J ~ g_Ig_J  exp [ hcBJ(J +1)/kT ]  

The rotational degeneracy g_J is 2J + 1, and the nuclear-spin degeneracy g_I, varies by rotational level only when the molecule encloses symmetrically equivalent nuclei (NIST, 2009). Briefly, the total wave function, ψ_tot, for molecules with equivalent nuclei must obey certain symmetry requirements upon exchange, as determined by the Pauli principle. Exchange of nuclei with half-integral spin (Fermions), these as protons (I = 1/2), must create a sign change in ψ_tot. Nuclei by integral nuclear spin, such as deuterium (I = 1), obey Bose-Einstein statistics and are termed bosons; for these the sign of tot, is unchanged via interchange of the equal components. The total wave function can be written, just about, as a product function

Ψ _tot = ψ _elec ψ _vib ψ _rot ψ _{n spin}

For the ground vibrational state of acetylene, ψ _elec ψ _vib is symmetric with respect to nuclear exchange so that ψ _rot ψ _{n spin} must be antisymmetric for C₂H₂, symmetric for C₂D₂. For linear molecules the ψ _rot functions arespherical harmonics that are symmetric for even J, antisymmetric for odd J. The ψ_{n spin} spin-product functions for two protons consist of three that are symmetric (α, α, β β, αβ+ βα) and one that is antisymmetric (αβ - βα) where α and β are the functions corresponding to M_I values of +1/2 and -1/2 therefore it follows that for C₂H₂, g_I is 1 for even J, 3 for odd J and the P and R branch lines will alternate in intensity. For C₂D₂, through spin functions αβ and γ representing the MI values of +1, 0, -1, there are 6 symmetric nuclear spin combinations and 3 that are antisymmetric to exchange (can we write these?). Consequently the even J rotational lines are stronger in this case.  The experimental examination of these intensity alternations confirms the D_h symmetry of acetylene, and in the present experiment provides as a helpful check on the assignment of the J values for the P and R branch conversions.

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