9 The vibrating Diatomic Molecular :Anharmonic Oscillator
Contents
1. Molecule as Anharmonic Oscillator
2. Vibrational Frequency and Force Constant for Anharmonic Oscillator
3. Isotope Effect on Vibrational Levels
4. Molecule as Vibrating Rotator
The students will be able to learn The Vibrating molecule and the anharmonicity associated with it, its force constant, Isotope effects on vibrational levels and fine structure of Infra red bands.
1. Molecule as Anharmonic oscillator
A comparison of an observed near infra-red spectrum with that expected from a diatomic molecule treated as harmonic oscillator reveals a disagreement.
The harmonic oscillator gives a single band at wave number ω that is the classical frequency of vibration of the molecule. The actual infra-red spectrum consists of an intense (fundamental) band at ω, plus a number of weak bands (overtones) at wave number slightly lesser than 2ω, 3ω, ……… . The observation of overtones indicates that the selection rule ∆v = ±1 is not strictly obeyed, and transitions corresponding to ∆v > 1 do take place. This is attributed to the fact that the dipole moment of the molecule is not strictly linear with respect to the internuclear displacement x= r – re. This is referred to ‘electrical anharmonicity’of the molecule. The observation that the overtones does not appear exactly at 2ω, 3ω, ……. but at lesser and lesser values which implies that the vibrational energy levels are not equally-spaced and converge slowly and it is understood due to the fact that for an actual molecule the potential energy curve is not strictly parabolic, except near the minimum. That is, the potential energy function V (r ) is not harmonic and there is a need to include terms higher than quadratic in the Taylor’s series expansion of V (r) which is expressed as ‘mechanical anharmonicity’ of the molecule.
The tailor series for the potential energy near equilibrium position is
Here V(0) is a constant set arbitrarily to zero corresponding to energy x=0.
The First derivative of V is 0 because the slope is zero at the minimum and for small displacements all high terms are ignored.
According to the first approximation to a molecular potential energy curve is a parabolic potential.
K is large when V(x) is sharply curved,
And it is small if V(x) is wide and shallow
and solving by perturbation method, the eigenvalues of the wave equation, that is, the energy values of the anharmonic oscillator are given by
The quantity ωe is the wave-number spacing of energy levels that occurs if the potential curve is a parabola, ωexe is the ‘anharmonicity constant’ that is much smaller than ωe (ωexe << ωe ) and is always positive. The eq. (b) indicates that the energy levels of the anharmonic oscillator are not equidistant and their separation decreases slowly with increasing v
The molecule dissociates into atoms when it receives energy more than that corresponding to the uppermost vibrational level. The excess energy appears as kinetic energy of these atoms and this energy is unquantised. The continuum joins the uppermost level.
The zero-point energy of the anharmonic oscillator is obtained from eq. (b) at v = 0 Thus
Re-writing eq. (b) for the energy levels considered at the lowest level as zero
G0 (v) = ω0v – ω0x0v2 + …………… …(d)
Equating the coefficients of like powers of v in eqs. (b) and (d), one obtains :
Now investigating the infra-red spectrum for the anharmonic oscillator. The eigenfunctions of the anharmonic oscillator following the selection rule ∆v = ± 1 still holds giving the most intense transition. Also, for the anharmonic oscillator, transitions corresponding to ∆v = ± 2, ±3, ……. appear, but with rapidly decreasing intensity. The possible transitions in absorption when all the molecules are initially in the state corresponding to v = 0 are shown in the figure. This explains the appearance of observed weak overtone bands, together with the intense fundamental band.
The transition with ∆v = 2, 3, 4, ……..have approximately, two, three, four, …….. times the wave number of the transition ∆v = 1, is in good agreement with observation. Further, the wave-number separation between two successive absorption bands is given by
Thus, as v increases, the separation between successive bands (or levels) decreases linearly that is in agreement with observations.
The second differences are given by
Thus, the second difference directly determines the anharmonicty constant ( 0 0) .
The values of ωe and ω0 are obtained from the observed wave number of the fundamental band
As and 0 0 have been determined already, ωeand ω0can be evaluated. Hence the vibrational constants ωe and ωe xe(or ω0and ω0x0) from the observation of the infra-red absorption bands of a diatomic molecule are determined.
2. Vibrational Frequency and Force Constant for Anharmonic Oscillator
The classical vibrational frequency for a harmonic oscillator is
where k is the force-constant and µ is the reduced mass. The separation of successive vibrational levels is constant and is equal to ? = ???? /c that is the wave-number frequency.
In case of anharmonic oscillator, the classical frequency as given above holds for very small amplitudes only. It decreases slowly as the amplitude (that is, v) increases.
The exact classical expression for the vibrational frequency of anharmonic oscillator in the state v is given by
Thus, as v increases, the classical vibrational frequency ????(?) decreases.
Considering a hypothetical state with v= − 1/2
The vibrational energy for this state is zero and the frequency for this state is given by
ωe represents the vibrational frequency (in wave number) that the anharmonic oscillator would have classically for an infinitesimal amplitude, that is, in the imaginary state = − 1/2 at the very bottom of the potential curve.
The force-constant K of the anharmonic oscillator for the infinitesimal displacement from the vibrational frequency ωe for infinitesimal amplitude, can be determined
Assignment;
Consider the molecule HCI
µ = 1.61 x 10-27 kg and ωe = 2989 cm -1 = 298900m-1.
So, ke = 4 x (3.14)2 x (1.61 x 10-27kg) x (3.0 x 108 ms-1)2 x (298900m-1)2 = 510 N/m.
This value is near the value of the force-constant (K = 480 N/m) obtained from the harmonic oscillator model of HCI, using k = 4π2 µ c2 ω2 where ω = 2886 cm-1.
3. Isotope Effect on Vibrational Levels
Different isotopic molecules have different vibrational levels, and hence different vibrational frequencies. The classical frequency of a molecule assumed as harmonic oscillator is given by
The force-constant ‘K’ is determined by the electronic motion only and is therefore exactly the same for different isotopic molecules.
The reduced mass is different for different isotopes. If ωi is vibrational constant for the heavier isotope, then
The shifting in levels results in the doubling of the vibrational bands. The band-shift is (ignoring anharmonicity)
The shift increases as the order of the band (value of v′)increases.
It is to note that in the infra-red spectrum of HCI each band is a double band, one corresponding to HCI35and the other to HCI37. The band belonging to HCI37is shifted by a small amount toward shorter wave number with respect to the corresponding band belonging to HCI35, and the shift increases with the order of the band.
4. Molecule as Vibrating Rotator: Fine Structure of Infra-red Bands
The near infra-red spectra of molecules consist of ’bands’ and not lines, rather each band is composed of close lines arranged in a particular manner. A line is missing at the centre of the band in the series of lines that are not equidistant. The missing line is known as the ‘null line’ or ‘zero gap’. These lines show a poor tendency of convergence toward the high-wave number side, and the band is said to be degraded toward the low-wave number side (towards the red).
The observed fine structure in infra-red band suggests that in a vibrational transition the molecule also changes its rotational energy state and therefore is considered as a vibrating-rotator.
Let any interaction between vibration and rotation of the molecule is ignored so that the term values of a vibrating-rotator are given by the sum of the term values of the anharmonic oscillator and the (rigid) rotator, that is,
is the reduced mass of the molecule. That gives a set of rotational levels, with similar spacings, associated with each vibrational level.
In case of HCI 50 rotational levels are associated with each vibrational level. A transition between two vibrational levels is accompanied by a number of transitions between the two corresponding sets of rotational levels. This results in a number of lines (rotational) in the band. These lines form two branches of equidistant lines. The spacing between the lines of one branch slowly decreases, and of the other branch slowly increases as we move toward higher and higher lines of the branch. This is attributed to ‘vibration-rotation interaction’.
It is clear from the shape of the potential curve, the equilibrium internuclear separation re, and hence the moment of inertia of the molecule increases as v increases, so that the rotational constant B decreases. The, rotational constant, Bv,
The internuclear distance and hence the rotational constant is changing during the vibration in a given vibrational state. Therefore, a mean value for the rotational constant in a given vibrational state is used
The eigenfunctions of the vibrating-rotator are the products of the eigenfunctions of the oscillator and that of the rotator, the selection rules remains the same as for these systems individually, i.e.,
∆v = ± 1 , ±2 … … …
∆ v= ± 1 .
∆v= + 1 represents absorption as the two J-levels involved belong to different vibrational levels. For a particular vibrational transition, the rotational transitions ∆j = + 1 give one set of lines reffered to the ‘R-branch’, while the rotational transitions ∆j = + 1 give the other set of lines referred to the ‘P-branch’. All the lines of both branches form a vibration-rotation band.
The wave numbers of the branch-lines of a particular band
P-branch: For ∆J = J’ – J’’ = – 1 , so that J’ = j’’ – 1,we obtain the lines of P-branch with wave numbers given by
Here J’’ , the lower rotational quantum number, takes the values 1, 2, 3,………
It is noticeable that J’’ cannot be zero as the level J’= -1 does not exist. Therefore, the P-branch consists of a series of lines P (1) , P(2), P(3),……….. corresponding to J’’ = 1, 2, 3,……….. on the low wave-number side of the band-origin 0 .
Further ??′ < ??′′, ??′ − ?v”is negative and therefore both the linear and the quadratic terms in the equation are of the same sign so that the lines of this branch draw farther apart with the increasing values of J’’.
Neglecting the vibration-rotation interaction so that
have equispaced lines.
The exact equations for both branches are the equations for parabolas. These are inter-related as the two branches have common upper and lower states. They can be fitted to the same parabolic equation:
A plot of the common parabolic equation is shown in the figure wherein the dashed line indicates the corresponding plot for Bv′ = Bv′′ .
As the constant is very small, the difference ( Bv′ − Bv′′) is also very small. Hence the curve is only very slightly deviated from the straight line because the vibration-rotation bands show a very poor tendency of head formation.
Assignment:
Calculate the equilibrium frequency, anharmonicity constant, force constant, De and Do of the HCl molecule given that the spectrum of HCI shows a very intense absorption at 2886cm-1, a weaker one at 5668 cm-1and a very weak at 8347cm-1.
2. Calculate the fundamental and first overtone transition. The equilibrium vibrational frequency and anharmonicity constant for HI molecule are 2309.5cm-1and 0.0172cm-1, respectively. The fundamental transition from Using 
Summary:
The Vibrating molecule and the anharmonicity associated with it, has been discussed. Its force constant has also been calculated. Isotope effects on vibrational levels and fine structure of Infra red bands.
Both the involved vibrational levels, for vibration-rotation bands, belong to the same electronic state. Therefore, the rotational constant for the upper vibrational state, Bv′ is always smaller than that for the lower vibrational state, Bv′′, Hence, the line-spacing decreases as m takes on increasing positive values.
There is a tendency of head formation in the R-branch. On the other hand, the degradation of band is observed in the P-branch. This means that rotation-vibration bands are always degraded toward the lower wave-number side ( toward the red).
In contrast, electronic bands show a strong tendency of head formation, and may be degraded either way.
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