I Stephen E. Schwmrtz State University of N e w York Stony Brook, 11790
Textbook Errors, 773
The Franck-Condon Principle and the Duration of Electronic Transitions
The Franck-Condon principle was formulated first by Franck ( I ) and shortly thereafter by Condon (2) in a lanmaze that woke of "vertical" transitions, i.e., electronic haisitions that occur without change of nuclear positions and momenta. In a second article (3) Condon presented a wave-mechanical statement and proof of what we now call the Franck-Condon principle and a t the same time rejected as erroneous, on grounds of violation of the Heisenherg uncertainty principle, the previously given arguments that transitions take place in a strictly "vertical" manner. Despite Condon's rejection of the rather heuristic argument previously given, the Franck-Condon principle is fre~uentlvstated and iustified in textbooks of physical chemistry, spectroscopy, and even quantum mechanics on an argument that runs something like this: Because of the large mass disparity between theelectrons and the nuclei, electronic motion is fast compared to nuclear motion and hence an electronic transition takes place on a time scale short compared to that of nuclear motion. As a consequence, neither the positions nor the momenta of the nuclei may appreciably change during the transition. The duration of the electronic transition has been proposed by various authors as 10-15 or even 10W18 sec. Since an argument such as this is widely given, it is perhaps worthwhile to review briefly the quantum mechanical formulation of the Franck-Condon principle, and also, from a rather different point of view than that taken by Condon (3) to consider some uncertainty principle implications of the short electronic transition times proposed. First we consider the Franck-Condon principle and stress that the derivation that we shall develop, while relvine unon the mass disnaritv of the nuclei and elect r o k t o perform the om-bppe~heimer separation of the total wavefunction into a product of electronic and nuclear functions, does not require any statements concerning the duration of the electronic transition. The electric dipole transition moment integral (4) is given by R
=
J$'Yr,r)(R,, +R&"(r,r)drdr
R,
=
C z,R:.
R,
=
-CR~
and
where the operators RI and R, are position vectors. (The Z1 are the nuclear charges.) We now perform the Born-Oppenheimer separation of the wavefunction into a product of electronic and nuclear functions. Thus
Here we emphasize the parametric dependence of $,' upon the instantaneous nuclear configuration, and also the dependence of $', upon the particular electronic configuration, here $,.' We neglect any "finer interaction" of the electronic and nuclear motion (5). Similarly we write $,(r) as a product of vibrational and rotational wavefunctions, again neglecting the interaction of these functions: $,,(r) = Jiror(r)$"(r). Finally, since we are not interested in the rotational structure of the spectrum, but only in the relative intensities of the various vibrational bands within an electronic spectrum, we may ignore the rotational wavefunctionsl and write
We shall specialize our discussion to diatomic molecules a t this point, as the arguments presented may easily he generalized to molecules having more than one vibrational mode. The integral in eqn. (2) may be written as a sum of the two terms corresponding to the nuclear and electronic dipole operators
(1)
where the $ ' ( ~ , r ) and $"(r,r) are the wavefunctions describing the unner .. and lower Quantum states, respectively, indicated as functions of electronic coordinates r and nuclear coordinates r. Here the dipole moment operators Suggestions of material suitable far this column and guest columns suitable for publication directly should be sent with as many details as possible, and particularly with reference to modem textbooks, to W. H. Eberhardt, School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. In order to be presented, an error must occur in at least two independent recent standard books. 1 An instructive discussion of the rotational selection rules, and the change in coordinates from laboratory fixed to molecule fixed in going from eqn. (1)to eqn. (2) is in reference (51,p. 131. 608 /Journal of Chemical Education
are given by sums over nuclei and electrons, respectively,
By the orthogonality of the electronic wavefunctions $,' and $e", the first term vanishes. Denoting the electronic integral in the second term
then we may express the functional dependence of this integral in a Taylor series a b o u t a point ro that we may choose conveniently as the r value corresponding to the maximum of the product $,'*$,".
Finally, following Condon's argument (3) we may neglect all but the constant term in the power series. (We assume here that this term does not vanish and we are dealing with an electronically allowed transition) and obtain
The vihrational overlap integral appearing in eqn. (5) does not vanish by the argument given for the electronic overlap.integral in eqn. (3), hecause, as noted above, the $,' and the $," are vihrational functions belonging to different electronic states. The Einstein B and A coefficients, and in turn ahsorption and emission intensities are proportional to the ahsolute square of the transition moment integral (4) IRIS = lR,(ro)lslS$,'*(r)$."(r)drlz
(6)
The second factor on the right-hand side is known as the Franck-Condon factor, and i t is this factor that is largely responsible for the relative intensities of the vihrational hands within an electronic spectrum. Equation (6) embodies the wave mechanical statement of the Franck-Condon principle: The intensity of a vihrational hand in an electronically allowed transition is proportional to the ahsolute square of the overlap integral of the vihrational wavefunctions of the initial and final states. Phrased in this way, the Franck-Condon principle does not specify a "verticality" of electronic transitions, hut rather indicates that transitions will he favored when there is large overlap between the two vihrational wavefunctions of the transition. This conditions favors, hut does not require, transitions in which the relative "position" of the nuclei (as represented by large vihrational amplitudes) is the same in the initial and final states. The same statement may he made for the vihrational momentum, as we might just as well have used in our derivation the momentum representation wavefunctions. However, as Condon pointed out (3), the uncertainty principle nreclndes the orecise suecification of the nuclear position Hnd momentum. For this reason Condon rejected the earlier statement of the Franck-Condon principle. It is instructive to consider the &cumstance in which the constant term R,(ro) vanishes, e.g., for reason of symmetry. Then the leading term in the transition moment expression is given by
(Here we have chosen the origin of our coordinate system, ro = 0.) Now the expression is not that of a "vertical" transition hut an expression that is somewhat weighted against a vertical transition by the factor r. Indeed in this circumstance the intensity maximum of the transition may well correspond to a "non-vertical" transition (6). Let us summarize the discussion thus far: The FranckCondon principle follows, without any arguments based upon the duration of an electronic transition, from the Bom-Oppenheimer product separation of electronic and nuclear wavefunctions and from elementary quantum mechanical considerations. We now turn our attention to the argument that electronic transitions take place in a time short compared to the period of molecular vihrations. T o set the time scale let us consider a vihrational frequency t = 1000 cm-I or u = tc x 3 x 1013 Hz, corresponding to a period rvlb = 3 x 10-14 sec. Let us assume that we wish to measure the position of the nuclei on a time scale T 5 7uLb. The energy uncertainty resulting from such a measurement is related to T by the Heisenherg time-energy uncertainty expression ; h7. Since E = hu, then r A u 1. (This is the rAE ; "classical" uncertainty principle that relates the accuracy with which the frequency of a wave train may he determined in a time 7 . ) If we wish to estimate the energy un2 It is doubtful whether unit mmllator strength is achieved in molecular systems, so that the radiative lifetime calculated here is a lower limit to an expected natural lifetime. Measured fluorescence lifetimesoften reflect non-radiativedecay mechanisms.
certainty (cm-1) introduced by our measurement in a time T , then r A t a l / c = 3 X 10-1' cm-I sec. For r 5 3 x 10-14 sec, Au > 1000 cm-', just the vihration frequency that we have assumed. The proposed measurement will introduce an energy uncertainty that leads to complete indeterminateness of the phase of the vihration. Thus the statement that an electronic transition takes place in a time short compared to the period of a vihration cannot he tested by experiment. Consequently such a statement is in Heisenberg's phrase (71, "just as meaningless as the use of words whose sense is not defined." Finally, if we may not, according to the uncertainty principle, speak of the period of time in which an electronic transition "takes place" we ask whether it is possible to determine the duration of the interaction of the molecules with the excitation radiation. The answer is yes, hut that one must specify carefully the experimental conditions. Further, the answer will be valid only in a statistical sense (8). Thus to consider gas phase molecules ahsorhing light in the absence of perturbations, we must he careful not to disturb the system by any measurement during the time of interaction of the molecules with the excitation radiation. In this case we may infer from a study of the natural width of an absorption line in the spectrum (here we use the term "natural" to rule out contributions to the width due to collisions, Doppler effect, etc.) that the duration of the average molecular interaction with the excitation radiation is r = (A")-' where Au is the width, in Hz, of the absorption line. But according to radiation theory the natural line width (A") is equal to the inverse of the radiative lifetime, m (9). Thus we infer that, in this circumstance, the molecules have interacted with the radiation field for a time equal, on average, to their radiative lifetime. This radiative lifetime may he measured (in flash-decay experiments). A lower limit to such a lifetime may he set by the classical expression
For a transition in the near ultraviolet, say 30,000 cm-', m 2 1.5 x 10W9 sec which we see to he several orden of magnitude longer than the period of molecular vihrations, r.ib 3 x 10-14 sec. For a molecule ahsorhing light in the nresence of a disturbance (exuerimental intervention. collision, etc.) the absorption ] b e may he hroadened. Similarlv. other exit channels for the excited state h e dissocia&on, dissociation) may he present and broaden the absorption line. In such situations, the molecule may interact with the exciting radiation for a shorter period of time. This ueriod of time mav still be long- comuared to a vihration period (pre-dissociation-vibrational structure evident) or may he the order of a vihrational period or shorter, (dissociat~on-no evident vihrational structure) (11). In summary, the statement of the Franck-Condon principle in terms of vertical transitions should he replaced by a statement that speaks of the overlap integrals of vihrational wavefunctions. Further, the justification of the Franck-Condon principle, that electronic transitions take place fast compared to nuclear motions suggests the possibility of a measurement that violates the Heisenherg uncertainty principle. Finally, for the case of hound-bound transitions, to which the Franck-Condon principle is frequently applied, we infer from absorption line width measurements that the interaction of the molecule with the excitation radiation takes place in an average time that is long, not short, compared to the period of a molecular vihration.
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Literature Cited
Volume 50. Number 9. September 1973 / 609
(4) Sm, for example, Eyring. H..Walter. J., and Kimbsll. G. E.. "Quantum Chcmib try:John WileyandSon8,Inc.. New York, 1944. Chapter W I . (5) Herzbeq, G., "Molecular S a e t r a and Molecular Structure," Vol, IlI. D. Van Nos. trandCo..hc..Prinectm. N. J.. 1966.Chspter11. (6) ReL (51, p. 172. (7) Heisenberg. W.. "The Physical Prineiplca of the Quanfvm Theory: Dover Publicstmns, Inc. New York. 1949, Cbspter U (reprint of 1930Editianl. (8) Gmnevold. H.J.. ~'QuanrslOhsenation in Statistical Intemmtatim" in "Quan-
610
I Journal of Chemical Education
tum Thwry and Beyond: Editor Bsstin, T.. Cambridge University Pmm, Cambridge. 1911, p. U . I em indebted to D. M. Hanmn for valuable diaeussion of this paint. (9) Heifler. W., "The Quantum Theory of Rsdiation." (3rd Ed.), Oxford University Pleas, London. 1954. Chapter V. Measiah, A,. "Quantum Mechanio." Vol. I. John Wiley and Som, he., NewYork. 1-1. Chapter4. Sac. 10. (101 Mitchell, A. C. G., and Zcmsnaky. M. W.. "Resonance Radiation and Excited ~fomn: cambride. University Press, cambridge, 1934. Chapter m. (11) Rof. (51. Chapter LV. Seefiona2.3.
