A Raman spectrum is excited by electromagnetic radiation in the visible region of the spectrum, 15000- 20000 cm−1, 700 - 400 nm, a long way from energies associated with rotational transitions (far infrared). To emphasize this difference the spectrum is shown as it might actually appear in the upper half of the diagram and is plotted as a function of wavelength. The unchanged Rayleigh line always appears very strongly (more strongly than shown) at the centre of the spectrum. Since this is a pure rotation spectrum only the ground state value of B affects the spectrum. A slider to vary the centrifugal distortion constant is included with values in the range typical of these types of molecules, but the effect, which is just visible, only really shows up in quantitative calculations. Suggested "experiments" are
(i) The B (lower) slider controls the value of B. You will be able to observe that the spacing of the lines is approximately 4B. You should write out the formula for the lines to see why this is.
(ii) With the value of B fixed vary the temperature and note how the maxima in the O and S branches move further from the band origin as the temperature increases. The reason for the maximum is that as J increases the increasing degeneracy (2J + 1) favours a higher population but the decreasing Boltzmann factor favours a lower one. The different rate at which these factors change generates the maximum.
(iii) The spin statistics are controlled by the right hand slider and all the common cases are included. Note that, when one set of lines is completely missing (slider moved to zero) the pattern of line spacings is different from that when odd and even J are equally weighted. The first S line, S(0), is present but the first O line, O(1), is absent.
(i) Note that the gap between the Rayleigh line and the first rotational line is 6B (if all lines are present) and is 4B between successive lines and make sure that you understand the reasons. These are consequences of the ΔJ = ± 2 selection rule.
(ii) The first four values of the spin statistics are the same as for the infrared spectrum. Test the relative spacings for odd J: even J = 0:1 (right hand slider fully to the left); the first line is still at 6B but the spacing between successive lines has increased to 8B. Here the additional possibility, that occurs in oxygen, of even J lines being missing is included (right hand slider fully to the right). This happens because the oxygen nuclei are bosons but the ground state electronic wavefunction is antisymmetric. Verify that the first line is now at 10B not 6B while the spacing between successive lines is 8B just as for the case of odd J: even J = 0:1.
The applet shows all the four vibrational modes of a linear symmetric triatomic. Press "Start" to set the vibrations in motion. The "frequency" slider should be used to adjust the speed of the motion so that it appears reasonably smooth (this will be computer dependent). The masses of the atoms can be adjusted over a wide range and, because this will also cause large frequency changes, you may need to adjust the "frequency" slider again. The three vibrational modes can be switched on and off by the buttons at the top. Switching them on and off allows you to examine the effects of masses on the different modes more clearly. The relative phase button also changes the phase of the in and out of plane bending motions. In certain positions, e.g. 90o and 180o the combination of the two degenerate bending vibrations generates a rotation about the molecular axis and, for these two angles the rotation is in opposite directions.
The normal modes in XY2 do not change character as the masses are varied, but the actual motions change enormously. Note that the central atom never moves in the symmetric stretch however light the central atom (this is required by symmetry) whereas the central atom does nearly all the moving in the antisymmetric stretch and bend when it is the light atom.Note also that the antisymmetric stretch is always the higher of the two stretching vibrations (this can be observed by watching the beating of the motions when the bend is switched off.
I have found that this applet is very sensitive to the computer it is on and the version of java being used. If you download it with the controlling html program (VibrationJ.html) there is some scope for altering the running parameter, in particular the "rest" parameter which controls how long the calculation pauses to allow the graphics time to catch up. Altering the size of the display will also have an effect (it is the graphics that is slow not the calculation).
The interactive applet below calculates the infrared vibration rotation spectrum associated with a stretching vibration of a linear molecule. Click here for information concerning applets.
The calculation of the infrared spectrum is complete and includes the effects of centrifugal distortion, although these cannot be adjusted. If you download the programs the parameters of the control program can be changed to calculate any chosen spectrum for stretching of a linear molecule. Just set the parameter "control" to 1 and the remaining parameters are obvious.
The following are suggested "experiments" for the infrared spectrum.
(i) The B (lower) slider controls the value of B in the lower state and the B (upper) slider controls the value of the upper state B relative to B (lower). Leaving the other sliders at their initial setting vary these two sliders. You should be able to observe that the spacing of the lines is approximately 2B and that a difference between the upper and lower state Bs leads to a convergence of the lines in one branch and a divergence in the other. You should write out the formula for the lines to see why this is. The most common difference in the two B values leads to a the convergence being in the R branch. Note that the maxima in the P and R branches occur at a similar distance in cm−1 from the band origin (2000 cm−1) regardless of the value of B.
(ii) With the values of B lower and upper fixed vary the temperature and note how the maxima in the P and R branches move further from the band origin as the temperature increases. The reason for the maximum is that as J increases the increasing degeneracy (2J + 1) favours a higher population but the decreasing Boltzmann factor favours a lower one. The different rate at which these factors change generates the maximum. This gives a very useful means of estimating temperatures, especially in inaccessible situations such as in planetary atmospheres. If you keep the temperature fixed and increase B (lower) to a moderately high value you will realize that the discreteness of the lines makes it less and less accurate to determine temperature from the line of maximum intensity.
This applet shows how the vibrational sub-structure of an electronic transition varies with various parameters, principally the difference in bondlengths between ground and excited states and, to a lesser extent, the force constant and dissociation energies of the upper state (the parameters of the ground state being held fixed). The calculation of the intensities is based on the overlap integrals of the wavefunctions for a harmonic oscillator, also shown for selected states in the diagram. These are taken to be centred not on re, which would be strictly correct, but on the mid point of the motion at the appropriate potential energy. In this way some account is taken of the gradual lengthening of the mean bondlength by anharmonicity. The two potential energy curves are Morse functions but are calculated in terms of the dissociation energy and force constants rather than the usual pair of parameters (D and α). The value of v at which the molecule would dissociate is also given. There are some small numerical inaccuracies around the Δr = 0 point. This is because I felt it necessary to effect some economies in what is quite a large calculation.
The difference between this applet and the previous one is that the values of B in the ground and excited states can be changed over a wide enough range that bandheads can be made to appear in either the R branch (the most common case) or the P branch. The first effect of decreasing B upper is to cause convergence of the R lines. Eventually these get close enough that they overlap and form a bandhead. At still smaller values of the upper B the higher J lines of the R branch start to appear in amongst the P branch lines.
Since excited electronic states often involve states with electronic angular momentum, it is relatively common also to have a Q branch and three examples are shown (press the Q branch button). When a Q branch appears in the infrared the lines are heaped on top of each other to porduce more or less a single peak. However, for large differences between B1 and B0 the quadratic term in J will cause them also to be widely spread out. When the Q branch is intense and the change in B large the spectrum can become quite complicated in appearance.
Note that when the electronic angular momentum is 1, the toal angular momentum canot be less than 1. Thus, depending on the direction of the transition the P(1) and/or the R(0) lines may be missing. This can be seen in the applet by examining the spectra when the upper and lower values of B are identical.