Calculation of Franck-Condon Factors for Undergraduate Quantum Chemistry Roland L. Dunbrack, Jr.' Haward University, Cambridge, MA 02138 Numerous theories exist for the calculation of potential surfaces for diatomic and polyatomic molecules, and a number for use in of these have been presented in THIS JOURNAL undergraduate quantum chemistry courses ( I , 2). I t is useful, however, for students to see how such computations are used in relation to experimental data such as ultraviolet absorption and fluorescence intensities. One important step in such a comoarison is to use the wotential surfaces to calculate vihrational wavefunctions and the transition p r o b dl1ili1ic.ibetween vibrational levels of two electronic states. Thesf transition probabilities are proportional t o the Franck-Condon factors (3), which are the squares of the overlap integrals between vibrational wavefunctions, P,,al[S J.,m*(s)J..,(dds112
(1)
where q is the coordinate of motion along the potential surface, is the mth vibrational wavefunction of the is the nth vibrational wavefunction of ground state, and an excited electronic state. The derivation of the FranckCondon factor formula in eq 1is given in a number of standard quantum chemistry texts (4). We may explain eq 1in a qualitative way by stating that an electronic transition is more likely to occur when probable ground state nuclear configurations (i.e., large values for l+,,(q)l) coincide with robable excited state nuclear configurations (large ,,(q)l). This is what is meant by vertical excitation: the nucle~have very little time to move during the excitation of an electron, and thus the transition is more likely to occur when there is large overlap between the ground and excited state vibrational motions. There are several techniques used to calculate vibrational wavefunctions, some based on numerical methods such as the Noumerov method (5).In this paper, we present a different method, widely used for polyatomic molecules that can be oroarammed bv advanced undermaduates. The calcula. tion involves analytic representation of one-dimensional potentials, exoansion of the wavefunctions in an orthonormal basis, and simple computation of the Franck-Condon factors between levels of different electronic states. A nvmber of different assignments are possible, depending on the level of students and the amount of time that the project is to consume. The calculations require that a digital computer with matrix diagonalization routines be made available to the students. We have used this assignment in an experimental physical chemistry course taken by third- and fourth-year undergraduates. We have found it successful in showing our underaraduates the usefulness of a comnuter in auantum chekistry and in teaching them the concepts of eigenfunction ex~ansionsof wavefunctions and the orincioles of soectroscopy.
+,
function and a potential and calculate eigenvalues and eigenfunctions. A set of functions is orthonormal on [a,b] if
We wish to calculate theeigenvalues and eigenfunctions of the following: where I? is the Hamiltonian of a one-dimensional system,
and E is the energy. Wr use a varia;innal method (6) to approximate $ as a finite linear e x p a n h n in the orthonormal functions
@,
+.,
L
a
+
where the c, are (in general) complex. is exact if N- a,hut we are looking for an approximation by keeping only a partial sum. The c, are determined by finding the solutions to the matrix equation (6) where
H is an NXN-matrix and C is an N-element column vector. The E's and C's can be found by diagonalizi~gthe H matrix. The eigenvalues En and the eigeuvectors C, give approximate energies and wavefunctions for the N lowest levels of the potential V(q). The difficulty lies in calculating the matrix elements H,. Ordinarily, we try to expand the potential V in the same orthonormal basis @,as we used for the wavefunctions. In most actual cases, only a few points of the potential are known and we must make a least-squares fit or guess the coefficients in the expansion
The key to choosing a basis is to make the integrals in eq 7 easy to compute. Once the eigenfunctions have heenobtained, we can calculate the probability of a transition from a ground electronic state level to an excited electronic state level by calculating the overlap integrals between the appropriate vibrational wavefunctions. If
Theory
Many calculations of wavefunctions involve the use of orthonormal functions which are used to expand a wave-
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Present address: University of Cambridge, Emmanuel College. Cambridge CB2 3AP. England. Volume 83
Number 11 November 1986
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Table 1. Eigenvalues for Ground State Torsional Potential 01 Tolane
dihromo ethylene we would calculate the moment of inertia, II, of the Br-C-Br group rotating about the C=C double bond, and the moment of inertia, 12,of the H-C-H group. We find the reduced moment of inertia in an analogous fashion to finding reduced masses:
If we write the energy in wavenumbers, and the moment of inertia, I , in amu A2, then B is given by:
The wavefunctions are expressed in an orthonormal basis of rieid rotor functions (a Fourier series. with each term normalized~The wavefuiictions of an even potential are either even or odd (hut not mixed) so we can s ~ l ithe t Fourier series into even and odd parts, 'All levels are doubly & g e m f a . Even qvanta sigentunctlons are mslne Jerles. and
ow quanta bansitions are sine aeries. Each oerieo is 50 t e r m l a g (1.a.. N = 50) %!I en~rglesare In cm-' and are relative b me lowest energy vibrational level. Expaimenfal energy levels are from ref 8. rCal~ulatedwllh B = 0.379 cm-' and pdantsls glven In me text and In ref 8.
where
Consequently, twosetsof wnvefunctima must hecalculated: first, where the matrix H is calculated using the even hasis functions of eq 18; and second, where H iscalculated with the odd basis functions of ea 19. The integrals - involved are tabulated in the appendix. The program we ask our students to write calculates the matrixelements and then diagonalizes the matrix H with a packaged routine. Such routines are available on most college and university mainframe computers and are also available for personal computer^.^ The appropriate Franck-Condon factors are calculated and comnared with exnerimental absorption and fluorescence spectra. As an examnle. we calculate the enerw levels and FranckCondon factors for tolane (1,2-diphe& acetylene or PhC=C-Ph). This molecule has a torsion mode in its around state (SO)and in an excited state (SI), that involve rotation of the phenyl groups about the carbon-carbon triple-bond axis. he technique of dispersed fluorescence spec~roscopy has been used by Okuyama, Hasegawa, Ito, and Mikami (8) t o determine the energy levels of these modes. The spectra were ohtained by focusing a tunahle laser a t the frequency of a particular absorption hand (o' = 1,2,3,4 in the notation of ref 8) of the excited state, and dispersing the released light to measure the freauencies and intensities of the fluorescence down to the state levels (u = 0,1,2,3, etc.). Okuyama et al.. usine a similar nrocedure to the one described here. measured the torsional spacings of the So and S1 states through fluorescence, and fitted the spacines to notentials of the foim of eq 12. For the ground staie, they foind that the best fit was ohtained when V, = 202.0 em-'. For the excited state, the best fit was Vo = 1600 cm-I, and the kinetic energy constant B was taken as 0.379 cm-'. With these potentials, they calculated the Franck-Condon factors for the fluorescence transitions and compared them with the results of the experimental spectra. Their calculated results are reproduced here using a program written for our experimental ~hvsical chemistrv course. . . The eigenvalues for the ground state potential are summarized in Table 1. For a notential with onlv one adiustahle parameter, the agreement is quite good. w i t h the eigenva-
I is the reduced moment of inertia for two groups rotating about a common axis. That is, in a molecule such as 1,l-
Mathematical routines for mainframes and personal computers are available from IMSL, NBC Building, 7500 Beliaire Boulevard. Houston, TX 77036.
we have
or just the square of the complex dot product of B and C, I,, = IB .CIZ
(11)
The I,, are called Franck-Condon factors and the unsquared integral is called a Franck-Condon or overlap integral. The transition probabilities P,, are proportional to the Franck-Condon factors (4). Application to Torsional Potentials
Perhaps the most common example of this method is the calculation of wavefunctions and Franck-Condon factors for periodic potentials (7). Usually such potentials are found in molecules with internal rotations such as ethane and ethylene. For ethylene, we might have a potential of the form,
where T is the angle between the HCH planes. VOfor the ground state of ethylene is -65 kcal/mol. More generally, the potential might he fit to a finite cosine series, N
V(d
=
V, t
2 Vncos(nd
(13)
"=I
We can write the Hamiltonian as
954
Journal of Chemical Education
Table 2. Intenrilles for tha Dispersed Fluorescence Spectra of Tolane
program that calculates eigenvalues, eigenvectors, and Franck-Condon factors for torsional potentials such as 9phenyl anthracene and its derivatives ( 9 , 1 0 ) , biphenyl (11, 121, ethylene (13) and toluene (14). In addition, other kinds of potentials and basis sets for the wavefunctions can he used. For instance, for diatomics we can use one-dimensional potentials that are harmonic with cubic and quartic terms added to account for anharmonicity. The wavefunctions for this kind of potential are linear combinations of harmonic oscillator functions. with the freauencv of the oscillator tak. en to correspond &the harmonic term of the potential (6). A copy of the computer program used here to calculate torsional wavefunctions is available from the author upon request.
.
Acknowledgment I would like to thank Cynthia Friend for helpful comments, and the Department of Chemistry, H a n a r d University, for support of this work. Literature Clted
,67
Ill M u a h m i . J , ho. M . Kaya. K .ICnsm rh,a 1981.74.6w5. 121 Rarf8enen.G: Samda1.i. J .lol.Srrurr. 19R5.18.I15 131 18nrongoln.C.. Bucnkn R .I Peyer#mhuff.S 1) J Chem I'hya. 1982. 76.16Sj. 14. Murnhnml..l.lm M Kava. ti Chem I'h,% Lett. 1581.bb.'202
Appendlx Integrals for torsional wavefunction calculations: lues and eiaenvectors of both ootentials in hand. we can calculate the Franck-Condon factors and compare these results with theexperimental intensities.The measured intensitities and ca1c;lated Franck-Condon factors are presented in Table 2. The calculated Franck-Condon factors approximate the experimental intensities quite well for a crude onedimensional approximation. The one-dimensional potentials are able to reoroduce even the laree fluctuations in intensity in the u' = 3 and o' = 4 fluorescence. The errors that do occur are due mostlv t o the crude aoaroximation of the .. potentials. In addition, there is the Condon approximatiun of a vertical transition. Calculations involving non-Condon effects are more difficult than the simple ~alculationdescribed here. Other Applications A number of other molecules may be investigated using a
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