11 Franck-Condon principle

Contents:

1.     Fine/ Rotational Structure of Electronic Bands

2.     Fortrat parabola

3.     Franck-Condon principle

4.     Use of Franck-Condon principle

5.     Explanation of Intensity Distribution in Absorption Spectrum

The students will able to learn about Fine/ Rotational Structure of Electronic Bands, Fortrat parabola, Franck-Condon principle, Use of Franck-Condon principle, Explanation of Intensity Distribution in Absorption Spectrum

1. Fine/ Rotational Structure of Electronic Bands

It is understood that the vibrational and rotational constants for two different states (Te’ and Te’’) are different as these are associated with different electronic levels. So  ?̅= ?̅e+?̅v+?̅r

can be written as  ?̅= ?̅0 + ?̅? = ?̅0 + ?′(?′) − ?′′(?′′)

that is due to possible changes in the rotational state for any given vibrational transition.

Here  ?̅0 = ?̅? + ?̅_v is constant for a particular vibrational transition and is called the  band  origin  or  zero  line.  As  ?̅? depends  upon  the  different  values  of  the rotational quantum number in the upper and lower state and F’(J’) and F’’(J’) are the rotational terms of upper and lower electronic states, respectively.

∆J = J’- J’’ = 0, ±1

In case, both the electronic states have ᴧ= 0, then ∆J = 0 is forbidden ;as an example 1∑ – 1∑ transition.

There are three series of lines or branches; R, Q and P corresponding to ∆J = +1, 0 and -1, respectively for allowed transitions.

Neglecting small term in Dv

In the absence of Q branch, there is one simple series of lines in which the separation between lines changes regularly.

For singlet Π state, the possible values of J are ᴧ, ᴧ +1, ᴧ + 2,……….., that is, 1, 2, 3,…….

For singlet ∑ state, J values are ᴧ, ᴧ +1, ᴧ + 2,……….., that is, 0, 1, 2,.

Consequently, the figure depicts the lowest level of upper state corresponds to J = 1 while for the lower state it is J = 0. The first lines in the R, Q and P branches are those having J = 0, 1, and 2, respectively.

The P branch start from J = 2 and there is no transition corresponding to J = 0 and J=1.

There are equatiuon missing  lines ?̅= ?̅0 and ?̅= ?̅0 − 2??′′  as is evident from the equatiuon ?̅? = ?̅0 − (??′ + ??′′)? + (??′ − ??′′)?2 = ?(?) values of J.

The first line of Q branch lies at

As   ??′ and??′ are not very different and hence Q branch starts neighbouring to  ?̅0 .The gap at ?̅= ?̅₀ is not so apparent.

2.  Fortrat Diagram:

Fortrat diagram or parabola are the plots of J against frequency of the rotational lines and is shown in the figure. As in the equation

?̅?,? = ?̅0 + (??′ − ??′′)? + (??′ − ??′′)?^{2 there is the quadratic terms so one of the two branches turn back and thus form a band head.}

(a) Considering  ??′ < ?? ′ There are the linear and quadratic terms in the above equation for  ?̅_?,Rthose have opposite sign for R branch i.e. m = J + 1 due to  , which the separation between lines goes on decreasing as J increases. For a certain value of J, the quadratic term contribution exceeds the linear term contribution and the lines of the R branch turn back towards lower value of   ̅(higher value of wavelength) with the increase in J value. Therefore, the band head lies on the short wavelength side of the zero line and the band is shaded (degraded) towards the red (larger wavelength). This further suggest that the internuclear distance in vibrational state of upper electronic state is greater than that of vibrational state of the lower electronic state.

For P branch (m = -J) the linear and quadratic terms of the equation have same negative sign. Hence  ?̅_p decreases as the value of J increases. For P branch the spacing between lines increases at longer wavelength side as has been shown in the figure and no band head is formed.

For Q branch (q = J) and both linear and quadratic term of  ?̅? = ?̅0 +(?_?′ − ?_?′′)? + (?_?′ − ?_?′′)?² have negative sign. Hence the Q branch extends to lower wavenumber (higher wavelength side) of ?̅0 as the J increases. It is pointed out that the lines in Q branch are not be separately resolved for small value of J for ?? ????? ?????????? ?? ?_?′ − ?_?′. A single broad line may appear at  ?̅0 that is shaded towards the red.

(b)     Considering ??′ > ??′′ :This indicates that internuclear distance in upper electronic state is smaller than that of lower electronic state. The coefficient ?_?′ − ?’_? is positive. For R branch the linear and quadratic terms of Eq.(14.39) have positive contribution. Thus ?̅_R goes on increasing as the J value increases. The separation between lines goes on increasing on the shorter wavelength side.

For P branch, the linear term gives negative contribution to   ?̅_p whereas quadratic term gives positive contribution for a J value. For small J, the linear term is prominent and   therefore,?̅_p  shifts  towards lower  wave number side    (Longer wavelength). However, for large values of J, the contribution of quadratic term to ?̅_p plays  a  significant  role  and  due  competition,  the  separation  between  lines decreases  and  ?̅_p also  decreases.  But,  after  a  particular  value  of  J, ?̅_p stops decreasing and starts increasing and contribution of the quadratic term exceeds the contribution of linear term and lines of P branch turn towards higher value of   ?̅_p (lower wavelength) as J is further increased. Hence, in this case the band head lies on the longer wavelength side of the zero line and band is shaded towards violet.

In Q branch, the linear and quadratic terms have positive contribution and thus the spacing between the lines goes on increasing with J towards higher side of ?̅_p

The band head in P and R lies at a great distance from the zero line for small difference of ??′ − ??′′.

Further, in case ??′ − ??′′. , the head in P and R branch may lie at such a large distance from the band origin that it is not observed as for the corresponding m values the intensity of lines must have decreased to zero.

Also for Q branch the linear term is small for ??′ − ??′′. and Q branch parabola intersect the abscissa axis at about right angle.

Note: 1. A large difference in ??′ and ??′  causes the spreading out the Q branch components and hence the sharp Q branch is not obtained.

2.The bands having a Q branch show two heads either in P and Q branch (  ?_?′ > ?_?′  ) shaded to violet or in R and Q branch (  ?_?′ < ?_?′)) shaded to red.

The m value corresponding to the band head also called vertex of the Fortrap parabola is obtained by

3. Intensity Distribution in Band Systems: Franck-Condon Principle

Electronic transition in a molecule takes place so rapidly compared to the vibrational motion of the nuclei that the instantaneous inter- nuclear distance is considered as unchanged during the electronic transition.

According to Franck-Condon principle, an electronic transition between the states A and B is represented by vertical line from one level to the other.

The principle can be combined with probability density IᴪI2 of linear harmonic oscillator to understand the intensity distribution among the bands.

Consider a molecule in the lowest vibrational level of an electronic state; the probability distribution function IᴪI² is maximum at the midpoint while this probability distribution function IᴪI² is maximum at the end points, in the exited vibrational states

The figure above depicts that the IᴪI² is maximum at the midpoint of the υ=0 level, and near the points of the higher vibrational levels.

Therefore, the most probable vibrational transitions are those in which one of two turning points of a vibrational level of one electronic state lies at the same inter-nuclear distance as one of the two turning points of a level of the other electronic state, except in case of υ=0 level for which midpoint rather than the turning point is substituted. Quantum mechanically, the transition is governed by Frank -Condon factor and is represented by

F_v’,v’’=[∫ ᴪ∗ _v′ ᴪ _v ′′ ]²

The most intense transitions will be those for which the overlap between ᴪv’’(determined from the lower electronic state) and ᴪv’(determined from the upper electronic state) is maximum.

4. Use of Frank-Condon principle:

(a) Intensity distribution in absorption spectrum

All the absorption transitions start from the lowest υ̋ =0 level of the ground electronic state.

The figure depicts the absorption transitions between two electronic states in a diatomic molecule wherein three typical situations:

(i) The equilibrium internuclear distance are almost equal in both electronic states: The transition υ̋=0→υ׳=0 is most probable because it connects configurations of high probability and also appears as a vertical ̓ line asto follow Frank –Condon Principle and in this case ‘r’ does not change during electronic transition . Transitions to level υ׳ =1,2, …….also occur, but rarely as these involve a change in ‘r’ and thus deviate from the Frank –Condon Principle and lead to weak absorption bands . Thus in the band system, the (0,0) band appears with maximum intensity , and the intensity decreases rapidly for the higher bands. As an example, the intensity distribution observed in O2 absorption bands.

(ii)  The internuclear distance is slightly greater in the upper state:

The most probable transition is υ̋ =0→υ׳=2 that connects the configurations of maximum probability and also satisfies the Frank –Condon principle. There are weak bands corresponding to transitions to levels υ׳ =0,1,3,4 …….. Hence starting from the (0,0) band the intensity first rises to a maximum and then decreases . As an example, this distribution is observed in the absorption band system of CO molecule.

(iii) The internuclear distance is much greater in the upper state :

A vertical electronic transition from the mid point of υ̋ =0level is most likely to terminate in the continuum of the upper electronic state, thus dissociating the molecule. The spectrum is expected to consist of a progression of weak bands joined by a continuum of maximum absorption intensity. As an example, this is the case with the absorption spectrum of I2 molecule.

5. Intensity Distribution in Emission Bands: Condon Parabola

According to the Frank-Condon principle, the distribution of intensity in a band progression with υ’=0 in emission corresponds exactly to that of a progression with υ̋ =0 in absorption. There is an intensity maximum at a υ̋ value depending on relative position of the minima of the two potential curves. The greater the difference re׳-re̋, the larger is the υ̋value for the intensity maximum.

However, in emission, the initial level is a υ׳ level. These are well populated depending upon the means of excitation and therefore a number of band – progressions are seen in emission but only one progression with υ̋ =0 is generally seen in absorption. The intensity distribution for band progressions in emission with υ׳>0 is different from that for the progression with υ׳=0. It will be of interest to note that the intensity distribution for band progressions in absorption with υ”>0 as in case of heavy molecules.

Referring to the figure let a molecule occupies the level AB of upper electronic state after excitation, the molecule spend more time at the turning points A and B from which transitions are most probable .

The molecule will be in its new vibrational level FE or CD according to Franck – Condon principle as it will be either at F( vertically below A) or at C (vertically below B) immediately after transition. Therefore, there are two υ̋levels to which probability of transition from a given υ׳ level is a maximum and hence there will be two intensity maxima in a υ̋ progression (υ׳=constant); one at smaller υ̋ and other at larger υ̋.The point C moves up more rapidly than the point F with the increase in υ׳. Thus the two intensity maxima would separate more and more from each other and also would go to higher υ̋values. This is in agreement with the experimental observations. The plot of the intensities of the bands of a system in a υ׳ -υ̋array similar to a Deslandres table and joining up the most intense bands, a parabolic curve is obtained whose axis is the principal diagonal. This is called the ‘Condon Parabola’ as shown in the figure.

There are two intensity maxima in all the horizontal rows with the exception of that with υ׳=0 which has only one maxima. The two maxima (limbs of parabola) separate further with the higher υ׳ values.

Conclusively, the separation of the two intensity maxima , and therefore the width of the Condon Parabola, increases with increasing difference of r_e׳ – r e̋( minima of the potential curves). This is the reason that the ‘Condon Prabola’ for SiN is almost a straight line , while that for O2+ is so much open as to have its limbs almost at right angle to each other. In case of SiN ( r_e׳ = r” _e ), where the two maxima in each progression (υ׳=constant) coincide , the Δυ=0 sequence ( principal diagonal of the Deslandres table ) is the strongest and traceable to quite high value of υ׳.The Δυ=±1 sequences are either much weaker or not observed at all.

In case of C2 and CO, the Δυ=0 sequence is very strong but with only first few detectable bands, while in other sequences Δυ=±1, ±2……a larger number of bands are detectable . The first band (0,0) is the strongest in the Δυ=0 sequence and in the higher sequences the maximum intensity shifts to higher bands. It is noticed that CO has a wider parabola than C2 as a large number of bands appear for CO.

The difference r_e׳ – r e̋ is appreciably large for O2+. The ‘Condon parabola’ is so much widening that its two limbs are 900 apart and follow the low υ׳and υ̋ progressions.

There is one general feature of intensity distribution in all the cases that the potential curves become increasingly asymmetrical with increasing r. This infers that the nuclei spend more time in the extended phase (r >re) than in the contracted phase ( r < re).

According to Franck – Condon principle’ for a system degraded to red ( r׳ > r ̋) the right limb of the Condon parabola will be the stronger; whereas the left limb will be stronger for the system degraded to the violet ( r׳ < r ̋). Thus,Condon parabola provides much information about a molecule.

Summary

Fine/ Rotational Structure of Electronic Bands

Fortrat parabola

Franck-Condon principle

Use of Franck-Condon principle

Explanation of Intensity Distribution in Absorption Spectrum