J. Phys. Chem. 1985,89, 917-925 of the molecular complexes in solution. The structure proposed for the molecular complex based on the IH and I3Cchemical shifts has the TNF molecule disposed in the region of rings I, 11, and I11 of the donor. This structure is analogous to that proposed for aggregated Chl and Such structures indicate a planeto-plane overlap of the donor and acceptor separated by the van der Waals distance (-4.0 A). The overlap criterion is one of the important parameters for light-induced electron-transfer reactions.
917
Whether similar features exist in the natural photosynthetic system is an interesting aspect of further studies.
Acknowledgment. The authors thank the Department of Science and TwhnobY, C h " m e n t Of India for Suppod. They are also thankful to the referee for his valuable sW3estions. Registry No. TNF, 129-79-3; ZnPheo, 15739-1 1-4; NiPheo, 1573910-3; Chl, 479-61-8; Phw, 603-17-8.
Direct Deconvolutlon of Extensively Perturbed Spectra: The Singlet-Triplet Molecular Eigenstate Spectrum of Pyrazlne Warren D. Lawrancet and Alan E. W. Knight* School of Science, Griffith University, Nathan, Queensland, 41 1 I Australia (Received: June 20, 1984; In Final Form: October 4, 1984)
A method is presented for analyzing spectra that arise when the final state of the transition is a mixed eigenstate with respect to some zero-order basis, as occurs, for example, in vibronic coupling between two electronic manifolds. The method involves a Green's function approach that permits a direct de-diagonalization of the mixed eigenstates in circumstances where only one of the component states of the mixed eigenstate carries oscillator strength with respect to the initial state. Specific attention is focussed on the case where the perturbed level structure is discrete, Le., the spacing between levels is in excess of their energy widths. The IBju-IAg fluorescence excitation spectrum of pyrazine in the region of the electronic origin, measured at ultrahigh resolution, reveals such discrete level structure associated with singlet-triplet mixing. The method described is used to analyze the P(1) band in the pyrazine (+TI) So molecular eigenstate spectrum. The computation of the energies of the zero-order eigenstates and the associated coupling matrix elements is demonstrated to be efficient and accurate.
-
1. Introduction Radiationless transitions have commanded attention from theoreticians, spectroscopists, and photochemists for more than 2 decades. The conceptual advances of Ross,' Robinson? and their co-workers, followed by the elegant formal theory developed by Bixon and J ~ r t n e r ,stimulated ~ intense activity among experimentalists: classical high-resolution spectroscopy found new applications in characterizing stationary states of advances in light source and detector technology facilitated direct measurements of dynamical evolution of electronically excited polyat~mics.'-'~ The intermediate case, where the density of states in the final state manifold is relatively sparse, has served as a crucial testing ground for the quantitative links between formal theories and laboratory measurements of molecular behavior. The observation of quantum beats in molecules such as biacetyl,I3 methylgly~xal,'~ and pyrazineI4-l6 have substantiated at least the essence of the theoretical models that have been propo~ed.~J'**~ However, the mesh of theory and experiment has in the past remained relatively imprecise. Experiments have normally only yielded estimates of average interelectronic state coupling matrix elements and rovibronic state densities. The quest for exact theoretical descriptions continue to be confounded by the large scale of computations associated with potential energy surface calculations of even smaller polyatomics. The recent experiments of Kommandeur and co-workers have provided a new and spectacular view of the intermediate case of radiationless transition^.^^*^^ These authors have measured the fluorescence excitation spectrum near the IB3,, IA, 0,O band in pyrazine with a 200-kHz bandwidth C W laser. The use of a molecular beam for their experiment reduces the Doppler width of spectral lines to -30 M H z and enables individual rotational bands in the electronic spectrum to be distinguished. The astonishing but not unexpected result is that each rovibronic "line"
-
'Currently Miller Research Fellow, Department of Chemistry, University of California, Berkeley, CA 94720.
0022-3654/85/2089-0917$01.50/0
is composed of a number of bands. For example, the P ( l ) transition appears as a group of 12 lines spread over ca. 4 GHz. The groups of transitions are identified as the stationary molecular eigenstates that stem from the mixing of one zero-order singlet state with a bunch of near-degenerate zero-order triplet states. The intermediate case character of pyrazine is exemplified by the sparsity of the level structure. Kommandeur et al.'9920have explored the question of whether their high-resolution spectra are representative of the true molecular eigenstates of the isolated pyrazine molecule. Their at(1) Hunt, G. R.; McCoy, E. F.; Ross, I. G. Aust. J . Chem. 1962, 25, 591. Byrne, J. P.; McCoy, E. F.; Ross, I. G. Ausr. J . Chem. 1965, 18, 1589. (2) Robinson, G. W.; Frosch, R. P. J . Chem. Phys. 1962,37, 1962; 1963, 38, 1187. Robinson, G. W. J. Chem. Phys. 1967, 47, 1967. (3) Bixon, M.; Jortner, J. J . Chem. Phys. 1968, 48, 715. Bixon, M.; Jortner, J. J . Chem. Phys. 1969, 50, 3284, 4061. (4) Callomon, J. H.; Parkin, J. E.; Lopez-Delgado, R. Chem. Phys. Lett. 1972, 13, 125. (5) Brand, J. C. D.; Stevens, C. G. J . Chem. Phys. 1973, 58, 3331. (6) Clouthier, D. J.; Ramsay, D. A. Annu. Reu. Phys. Chem. 1983,34,31. (7) Smalley, R. E. Annu. Rev. Phys. Chem. 1983, 34, 129. (8) Parmenter, C. S.J . Phys. Chem. 1982,86, 1735. (9) Tramer, A.; Voltz, R. In "Excited States"; Lim, E. C., Ed.; Academic Press: New York, 1979; p 281. (10) Lee, E. K. C.; Loper, G. L. In "Radiationless Transitions"; Lin, S. H., Ed.; Academic Press: New York, 1980; p 2. (11) Jortner, J.; Levine, R. Adu. Chem. Phys. 1981, 48, 1. (12) Zewail, A. H. Discuss. Faraday SOC.1983, 75, (13) Chaiken, J.; Gurnick, M.; McDonald, J. D. J. Chem. Phys. 1981, 74, 106, 117, 123. (14) van der Meer, B. J.; Jonkman, H. Th.; ter Horst, G. M.; Kommandeur, J. J . Chem. Phys. 1982, 76, 2099. (15) Felker, P. M.; Lambert, W. R.; Zewail, A. H. Chem. Phys. Lett. 1982, 89, 309. (16) Okajima, S.;Saigusa, H.; Lim, E. C. J. Chem. Phys. 1982, 76, 2096. (17) Tric, C. Chem. Phys. Lett. 1973, 21, 83. Lahmani, F.; Tramer, A.; Tric, C. J . Chem. Phys. 1974, 60, 4431. (18) Avouris, P.; Gelbart, W. M.; El-Sayed, M. A. Chem. Reu. 1977,77, 793. (19) van der Meer, B.J.; Jonkman, H. Th.; Kommandeur, J.; Meerts, W. L.; Majewski, W. A. Chem. Phys. Len. 1982, 92, 565. (20) Kommandeur, J. Red. Trau. Chim. Pays-Bas 1983, 102,421. van der M e r , B. J.; Jonkman, H. Th.; Kommandeur, J. Laser Chem. 1983,2,77.
0 1985 American Chemical Society
918
The Journal of Physical Chemistry, Vol. 89, No. 6,1985
Lawrance and Knight
tention is focussed principally on the J’ = 0 level, accessed via we re-address the issue of how best to extract the constituent the P( 1) transition. Here, since K’ = 0, the parallel-band selection zero-order states and coupling matrix elements associated with rule, AK = 0, ensures that there will be no K subbranch structure. a perturbed spectrum. We draw inspiration from Berg’s earlier A reasonably satisfying proof that the P ( l ) group of 12 lines analysisz4 of the naphthalene S2-So origin band. corresponds to the molecular eigenstate spectrum emerges from Berg24and independently Ziv and RhodesZShave shown that the beat structure in the fluorescence decay curve following cothe “spectrum” of zero-order dark states, with “intensities” related herent excitation of this group of transition^.'^,^^ The Fourier to the strength of coupling to the light state, may be extracted transform of the time profile of fluorescence decay matches the directly from the absorption spectrum. The extraction procedure spectral line spacings observed directly in the molecular eigenstate requires the calculation of the real and imaginary parts of the spectrum. Green’s function derived from the Hamiltonian pertaining to the Some reservations must be expressed, however, concerning the system. In short, the imaginary part of the Green’s function is precise identity of the individual bands in the molecular eigenstate proportional to the absorption spectrum; the real part of the spectrum. Kommandeur has addressed the possibility that each Green’s function may be calculated from the imaginary part by member of the group of 12 or so bands may itself comprise a set using the rigorous integral relations that connect real and imagof eigenstates that a t still higher resolution would be seen to be inary parts of continuous functions; the so-called zero-order split by rotational spatial orientation (mJ)and nuclear spin hyweighted density of states function may be evaluated from a simple perfine interactions. Furthermore, Pratt and co-workers21.22have combination of the real and imaginary parts. Application of the demonstrated that the magnetic field dependence of J’ = 0 differs Green’s function inversion procedure is restricted to situations from higher levels in which J’> 0. They argue convincingly that, where only one zero-order state among the mixed states carries while the set of triplets interacting with the J’ = 0 singlet level the oscillator strength responsible for the absorption of a photon. We examine afresh the Green’s funtion inversion method. In may be true zero-order states (in the Born-Oppenheimer sense), particular, we place emphasis on providing a framework that is the sets that couple with J’ > 0 are most probably themselves intermixed within the triplet manifold. The manifestations of such accessible to the experimentalist. It emerges from our analysis mixing, as seen in the experiments of Pratt and co-workers,2i are that it is advantageous to divide perturbed spectra into two classes. characteristic of Coriolis-induced coupling between two categories Congested spectra, as typified by the naphthalene S2-S0origin of triplet manifolds. One manifold is strongly coupled to the band system, occupy one category. The molecular eigenstate spectra of pyrazine typify the other extreme case where the zero-order singlet state, the other weakly coupled by virtue of structure is discrete, i.e., the spacing between lines is in excess inappropriate symmetry or selection rules. While recognizing these caveats, it will suffice for our purposes of the line widths. The Green’s function inversion is applicable to categorize at least the J’ = 0 molecular eigenstate spectrum to both cases but there are important differences in the philosophy as arising from the interaction of one zero-order singlet state with of approach. Discrete spectra are especially amenable to the a set of zero-order triplet states. Within this framework, there Green’s function method for extracting directly the zero-order arises a pressing need to determine the energy spectrum of the states and coupling strengths from the absorption spectrum. We illustrate how the molecular eigenstate spectra measured so-called zero-order states that give rise to the observed singletby Kommandeur and co-workersi9succumbs readily to a simplified triplet mixed level structure, and to establish the magnitudes of inversion method applicable to discrete spectra. The subtle the coupling matrix elements responsible for distributing the singlet economy of the deconvolution method that we advance contrasts state character among the nearby triplet states. The justification with other more extravagant de-diagonalization procedures and for de-diagonalizing the molecular eigenstate spectrum in this way is certainly more in keeping with the elegance of Kommandeur’s is to some extent pragmatic: theory copes better with the comexperiments. Moreover, the uniqueness of the solution for the putation of molecular level structures associated with specific zero-order states and coupling strengths emerges as a natural electronic configurations; the spectroscopist is normally more consequence of the direct Green’s function deconvolution method. comfortable if a perturbed spectrum can be viewed in terms of We discover that the simplified inversion method is astonishingly component “zero-order” states whose description is secured from accurate: the accuracy is limited only by the uncertainties in the unperturbed regions of molecular potential surfaces. In addition, measurement of the experimental data. however, the formalism of quantum mechanics facilitates calcuThe paper is set out as follows. We first examine the relalation of the time evolution of compound states constructed from tionship between the Green’s function of the Hamiltonian and the zero-order basis sets. Indeed, it is a natural consequence of the molecular absorption spectrum and identify the procedures needed division of the molecular Hamiltonian to express the molecular to extract zero-order energies and coupling strengths. Application eigenstate of an intermediate case molecule in terms of a linear of the procedure for a discrete spectrum is then illustrated with combination of a zero-order singlet state that can couple with the the molecular eigenstate spectrum of the lB3,,-lAg 0,O band of radiation field and an ensemble of triplet states that are essentially pyrazine. inaccessible through photon absorption. We consider the ensemble of triplets to be a diagonal set and indeed the Hamiltonian may 2. Theory always be so formulated. The separate issue of triplet-triplet interactions is not considered here. We seek to relate the absorption spectrum directly, via analytical A de-diagonalization of the pyrazine MES spectrum has been expressions, to the energies and couplings of the zero-order states. carried out by Kommandeur et a1.20 Their approach is to construct As we have outlined in the Introduction, we consider only that a set of simultaneous equations relating the unknown zero-order situation in which one of the zero-order states carries absorption energies and coupling matrix elements to the observable energies oscillator strength. It will become clear in the development that and intensities. While this approach is an advance over the follows that the Green’s function approach presented here cannot trial-and-error philosophy pioneered by Wessel and M c C l ~ r e ~ ~ be extended to include cases where more than one zero-order state in their related analysis of the vibronically perturbed S2-So origin is active in carrying absorption intensity. We also require that band of naphthalene, it remains laborious and is useful only for the manifold of “dark” states to which this “light” state is coupled spectra consisting of discrete, well-separated bands. In this paper do not interact among themselves.26 A schematic of the inter(21) Matsumoto, Y.; Spangler, L. H.; Pratt, D. W. J. Chem. Phys. 1984, 80, 5539. (22) Matsumoto, Y.; Spangler, L. H.; Pratt, D. W. Laser Chem. 1983, 2, 91. Matsumoto, Y.; Spangler, L. H.; Pratt, D. W. Chem. Phys. Lett. 1983, 95, 343. (23) McClure, D. S. J. Chem. Phys. 1956, 25, 481. Wessel, J. E. Ph.D. Thesis, University of Chicago, 1971. Langhoff, C. A.; Robinson, G. W. Chem. Phys. 1974, 6, 34.
(24) Berg, J. 0. Chem. Phys. Lett. 1916, 41, 547. (25) Ziv, A. R.; Rhodes, W. J. Chem. Phys. 1976, 65, 4895. See also: Cable, R.; Rhodes, W. J. Chem. Phys. 1980, 73, 4736. (26) An alternative statement of this assumption is that we consider the basis set of dark states (e.g., the triplet states) to be diagonal with respect to the chosen zero-order Hamiltonian, Le., H oexcludes only the interaction between “light” and “dark” states (e.g., spin-orbit coupling between a singlet state and a manifold of diagonal triplet states).
The Journal of Physical Chemistry, Vol. 89, No. 6,1985 919
Deconvolution of Molecular Eigenstate Spectra ZERO ORDER STATES
t;
I
E\-+
En-1
* -,
;', # ? En.? y
. : .
..'.fni, 11,' I.
,,,l
.:"' fn-2
,$,,,
,,
.
-
~i
En
MOLECULAR EIGENSTATES Cda> bl,nlPn>
+X
CI-I
IW+~~I-~,~IP+
CI-2I++6bl-2,nlPn)
~1'-2-
I
f ;.,
'i
E1
{I Pn>}
(7)
= f ~ ~ l ( l I P l g ) l Z- ~E() ~ I
diagonalize
Ea
-$
= 6(el - E ) Substitution of eq 7 into eq 4 yields
-
CI
++$bj,nlPn>
Figure 1. Schematic illustrating the intermediate case category of in-
terelectronic state coupling. Only the state la),energy ea, carries oscillator strength with respect to an initial state. The states (I&)} are considered prediagonalized among themselves, with eigenvalues E,. The jiare the matrix elements that couple la)with each I&,). The molecular eigenstates, eigenvalues e,, for 1 = 1, 2, ..., n + 1, each contain a proportion lcllz of the light state ICY), as indicated schematically by the cross-hatching. The molecular eigenstate spectrum would display line spacings and relative intensities in proportion to the representation by the cross-hatched bars in the schematic. mediate case category of this model is provided in Figure 1. 2.1, The Absorption Spectrum Expressed Using Green's Functions. The absorption cross section for dipole-induced transitions from the ground state Ig) to an energy E is given by2'
M )= € E Im ((glrG(E)Mu(g)I
(1)
where is a constant, p denotes the dipole moment operator, and G ( E ) is the Green's function, defined by G(E) = lim [ H - ( E + iq)Il-l
e
(2)
Here Hdenotes the molecular Hamiltonian and l i s the identity matrix. By making use of the closure relationship
(8)
We can see that this expression contains the essential features of the absorption spectrum. Peaks occur only a t the energies of the molecular eigenstates, as ensured by the 6 function, and the "intensity" of these peaks is proportional to the square of the dipole matrix element connecting the initial, ground state lg) with the final molecular state 11). The situation depicted by eq 8 is, however, unphysical in that the transitions are infinitely narrow. This corresponds to the case where the molecule, once placed in an eigenstate, remains there until it is disturbed by some external influence. In reality of course, molecules are constantly subjected to external perturbations; this could come, for example, from the vacuum states of the radiation field. Such effects can be incorporated into the present formulation by allowing the states to have complex energy.28 For example, the energy of state 11) would be written as e, - irl,where rlcorresponds to the lifetime of the state 11). (The validity of this procedure is established by performing the Fourier transform from energy space to time space.) The use of complex energy for the states will change the imaginary part of the Green's function from a 6 function to a Lorentzian of width rl (half-width at half-maximum). Equation 8 becomes
..
(9)
Equation 9 relates the absorption spectrum, in a particularly explicit fashion, to the molecular eigenstates, Le., the eigenstates of H . We desire to express the spectrum in terms of the zero-order states, Le., the eigenstates of Ho.(To avoid confusion, we shall denote the eigenstates of H ousing Greek symbols.) Using the eigenstates la) of Hoas the basis set for the closure relations in eq 3, we obtain an expression for the absorption spectrum in terms of the zero-order basis:
4 E ) = FEC(glpla)(PI4g) Im G a B W
(10)
a.B
Clk)(kl = I
Gap can be expressed in terms of the eigenstates of H , since the eigenstates of H c a n be used as a basis set for the eigenstates of Ho:
k
we have
49 = EE Im IC(gl~lk)(klG(E)I1)(1I~lg) =I
la) = CcaiIl) 1
k,I
{ E C(glklk)(ll&) Im Gkl(E) (3) k,l
where we have written (klG(E)(l) as Gkl(E). The basis states used for the closure relationship can be any basis that forms a complete orthonormal set in the space of H. Let us now consider the form of eq 3 and establish that it possesses the features required to provide a physically reasonable representation of the absorption spectrum. For the case where the states used in the closure relationship are in fact the eigenstates of H (kldl) =
6klel
and we have that
Thus
The asterisk here denotes the complex congugate. Equation 12 can also be derived directly from eq 9 by expanding 11) in terms of the Hobasis set. From eq 12 one sees immediately the complications that prevail if more than one of the zero-order basis states carries oscillator strength in absorption. However, when only one of these states, say la), has a nonzero dipole matrix element connecting it with the ground state, eq 10 reduces to a(E) = EEI(aIPIg)I* Im Gaa(E)
= €ECl(llrlg)12 Im G I I W I
(4)
(1 1)
(13)
where (see eq 12)
From the definition of G(E) given in eq 2 it follows that 1 Gll(E) = lim +q-E-iq
Equation 13 shows that the Green's function for the zero-order state that carries the oscillator strength defines the absorption
and thus (27) Harris, R. A. J . Chem. Phys. 1963, 39, 978.
(28) See for example: Ziman, J. M. "Elements of Advanced Quantum Theory"; Cambridge University Press: London, 1969; pp 108-1 10. Reinhardt, W. P. Annu. Rev. Phys. Chem. 1982, 33, 223.
920
Lawrance and Knight
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
spectrum (compare with eq 9). We thus require an explicit expression for Im G,,(E) in terms of the coupling strengths and zero-order energies. Before pursuing this direction, however, we pause to establish some results which will be used later in the deconvolution procedure. First, we examine eq 14. This equation shows that the intensity of a particular absorption transition is proportional to the square of the coefficient of the light state la) in the optically excited compound molecular ~ t a t e . 2 These ~ coefficients can thus be obtained directly from the absorption spectrum, and normalized by using
CIC,Il2 = 1
This identity may be expressed in the form G,,(E) = [ A ( E ) - iB(E)]-’
(20)
where
A ( E ) and B(E) are simple functions of energy
I
This is only practical of course in the case of an uncongested spectrum. We next note that Re G,,(E) can be written in a form similar to that for Im G,,(E) given in eq 14:
The validity of eq 15 can be readily seen by examination of eq 5 and 6, in conjunction with eq 14. The important feature to note regarding this expression is that it relates Re G, to the same quantities that can be extracted from Im G,, via the absorption spectrum. 2.2. Relating Im G,, to the Zero-Order Parameters. Our interest centers on the case where one state ICY)couples to a set of noninteracting states Ij31. In terms of this zero-order basis the Hamiltonian can be written as
Herefs denotes the coupling between ICY)and 18). The fact that the dark states are noninteracting is displayed through the zero ,,off-diagonal elements. This matrix may be written in block form as
The composition o f f and c is obvious from eq 16. The Green’s function matrix may be expressed in block form in the same manner as H;the (1,l) element of this matrix is the Green’s function of interest, Le., G,,(E): (18)
where x = G,,(E). We simply need now to relate x to the elements of H as given in eq 17. This can be done by making use of the fact that the product of a matrix with its inverse is the identify matrix, Z. Thus C(E)[ H - ( E iV)q = Z (19)
+
By substituting the block diagonal forms for C and H a n d expanding this expression we obtain
I (29) The term c,, comes from the expansion of la) in terms of 11). The expansion of 11) in terms of la) may therefore be couched with respect to coefficients c.,* that are the complex conjugates of the c,,.
A Re G,,(E) = A’ B2 B Im G,,(E) = A2 + B2
+
~
Equation 21b is the relation that we require: ima&ary part of the Green’s function for the zero-order state with oscillator strength explicitly in terms of the zero-order quantities. The definition of C(E) (see eq 2) demands that the quantity q in these equations approaches zero. However, as we have discussed, the nonzero line widths found in real spectra can in fact be mimicked here by leaving this term small and finite. 2.3. The Deconuolution: Extracting the Zero-Order States and Couplings. (i) The Dark States and Relative Coupling Strengths. Consider the term B ( E ) defined in eq 20:
Note that, since 7 is small, the sum over j3 dominates the expression. The sum over j3 is in fact a sum of Lorentzians. Each Lorentzian is centered at the zero-order energy of one of the dark states. Further, the height of each Lorentzian is proportional to the square of the coupling matrix element connecting that dark state to the light state. Thus the function B(E) contains explicitly the majority of the information we desire: the energy of the zero-order states of the background manifold, and their couplings to the (zero order) state carrying the absorption oscillator strength. Equations 21a and 21b may be solved simultaneously to give B(E):
The calculation of B(E) requires a knowledge of both Im G, and Re G,,. We show how these functions can be obtained directly from the absorption spectrum. Consider first the calculation of Im G,,. We have seen earlier that Im G,, is related directly to the absorption spectrum (see eq 13). The explicit connection that is useful for our purposes may be derived as follows. Direct integration of eq 14 together with the normalization condition on the coefficients ca1gives the result 1:Im
G,,(E) d E = A
(23)
We now examine the expression for u(E) given in eq 13. The factor E may be considered as being essentially constant over the small spectral region usually encountered in the near-resonance interactions associated with radiationless transitions. For example, the spectral region of interest may be a 1-cm-’ span in the vicinity of E = 30000 cm-’, for a typical S1-So absorption transition in a polyatomic. Hence we may approximate the product [ E in eq 13 as a constant f‘. Integration of eq 13 together with use of eq 23 then gives Im G,,(E) = A U ( E ) / ~ ~ Ud(EE )
(24)
In other words, Im C,,(E) may be obtained directly from the experimental absorption spectrum by performing the appropriate normalization.
Deconvolution of Molecular Eigenstate Spectra This means of extracting Im G,, is general. It may be used for both congested and uncongested spectra. In the case of uncongested spectra, however, a simpler means exists for extracting Im G,,. Consider the form of Im G,, given in eq 14. The unknowns in this expression can be extracted from the spectrum by inspection, provided that the absorption lines are sufficiently well separated to enable peak heights and positions to be obtained accurately. The lcall coefficients are given by the square root of the relative band intensities (normalized such that their sum is unity), and the band positions give the relative energies of the molecular eigenstates, cl. The remaining unknowns in eq 14 are the peak widths. These can be measured as the hwhm of the absorption bands, or they may be calculated from the lifetime. In practice, the peak widths are often determined by the Doppler width or by the experimental resolution. We recommend for practical purposes that the line widths be regarded as constant for all bands, and that they be used as a variable in the deconvolution calculations in order to obtain the solution with the required accuracy: the narrower the line width, the more accurately the zero-order energies and couplings can be specified. We turn now to consider how to calculate Re G,,, the other function needed to calculate B(E). As for Im G,,, we discuss first the general method, and then examine the simpler method possible for uncongested spectra. The real and imaginary parts of a function are related by the general expression
where P denotes that the principal value of the integral is to be taken, i.e., the pole at E’ = E is avoided. We outline in the Appendix a useful algorithm for calculating this principal value integral numerically. This principal value relationship between Re G,, and Im G,, is the means used by earlier worker^^^*^^ to obtain Re Gam. In the case of uncongested spectra, the need to perform this principal value integral may be obviated. Equation 15 shows an explicit expression for Re G,,. Comparison between this equation and the earlier expression for Im G,, (eq 14) shows that both relations are in terms of the same set of unknowns. Re G,,(E) can thus be constructed directly from the IC,,~ and e, measured from the absorption spectrum, in a manner analogous to that used for constructing Im G,,(E). Thus, except for situations in which the spectrum is heavily congested, preventing accurate measurements of band intensities and positions, both Re G,, and Im G,, may be obtained without recourse to the more cumbersome integral relations. Once Im G,, and Re G,, have been calculated, the energies and coupling strengths of the dark states may be calculated by using eq 22. The energies are obtained straightforwardly as the band positions in the B(E) function. Similarly the relative couplings are given by the square root of the relative intensities. (ii) Absolute Coupling Strenghts,fB. The absolute magnitude of the couplings is more difficult to extract. In the general situation of a congested spectrum, the relationship between the peak heights of B(E) and the absolute magnitude of the coupling strength is unclear. However, by integrating the function B(E) over the spectral region we see from eq 20 that
Le., the total coupling for the entire dark manifold is obtained by integrating the B(E) function. Since the relative couplings are known from the peak heights, the absolute magnitude of the coupling for each dark state can be assigned. If the spectrum is uncongested and Re G,, and Im G,, are generated from eq 14 and 15, then the line width (9 in eq 20) is a known input parameter. Thus, the absolute value of the couplings can be extracted directly from the peak heights of B(E): peak heights arefs2/s. Once again the case of uncongested spectra is much simpler and avoids an integration procedure.
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985 921 (iii)Energy of the Light State, e., There are a number of means available for calculating the energy of the light state. One relatively straightforward approach is to calculate the so-called “center of gravity” of the absorption spectrum.25 This method is independent of the calculation of the dark states discussed above and so removes any sources of cumulative error in the deconvolution procedure. Furthermore, as we discuss below, by calculating t, via both the “center of gravity” method and from the dark states already calculated, we have a means for establishing the relative accuracy of the calculations. In essence, the “center of gravity” method computes t, as the first moment of the intensity distribution, since e, is naturally the position where all of the intensity would be if it were concentrated at one energy rather than being spread over a number of bands. Thus ‘a=
s
I(E)E d E /
spectral region
1
I(E) dE
spectral region
(27)
A more formal proof of this relationship is given in ref 25. This expression is easily evaluated for both congested and uncongested spectra. In the latter case, however, it suffices to calculate the integral simply as the sum of the product of normalized peak heights with peak positions. (iv) Internal Check of the Accuracy of the Calculated Energies. At this point we have calculated all of the unknowns, namely, the energies of all of the zero-order states and the coupling matrix elements. The accuracy of this calculation can be checked here by calculating c, from the dark states calculated earlier, since this calculation for c, is independent from the one above. A general, yet simple means of calculating E, is as follows. A(E) can be calculated from Im G,, and Re G,, in the same manner as was B(E) (see eq 22):
Further, we note from the derivation of eq 20 that
A ( E ) = t, - E - -P x1
lmmz dE‘
(29)
Thus, t, can be calculated by equating these two expressions for A ( E ) at any particular energy, E . For uncongested spectra, an alternative method for calculating t, is to take advantage of the fact that the trace of a matrix remains invariant to transformations. Thus the sum of the energies of the zero-order states must equal the sum of the energies of the observed eigenstates. In other words, the sum of the energies of the dark states differs from the sum of the energies of the observed bands by the energy of the light state, e,. 2.4. Summary of the Theory: The Deconvolution Recipe. The procedure given in the preceding sections provides a method for taking an observed spectrum and calculating directly from it the energies and coupling matrix elements of the zero-order states that spawned it. In principle, the absolute accuracy of the zero-order quantities so extracted is limited only by the uncertainties associated with measuring the band positions and intensities experimentally. We reiterate that the deconvolution procedure is only valid when one of the zero-order states carries oscillator strength and the interaction occurs between this state and a number of states that do not interact among themselves. We note that these requirements demand a careful approach toward formulating the Hamiltonian associated with the problem. Given that the situation satisfies the conditions above, the zero-order states and their couplings may be obtained from the following procedures. We distinguish between the cases of congested and uncongested spectra. ( i ) Congested Spectra (General Formulae). Step 1: Im G,,(E) is calculated from the absorption spectrum, a(E), via eq 24. Step 2: Re G,,(E) is calculated from Im G,,(E) via eq 2 5 . Step 3: B(E) is calculated from Im G,, and Re G,, via eq 22. This provides the energies of the dark states (positions of peaks
922
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
in B ( E ) ) and their relative couplings to the light state (from the square root of the relative peak intensities). Step 4 The absolute magnitude of the coupling matrix elements is calculated from eq 26. Step 5 : The energy of the light state is calculated from eq 27. The accuracy of the calculations can be checked by also calculating t, from the results of steps 3 and 4, via eq 28 and 29. (ii). Uncongested Spectra. Step 1: The energies and intensities of the absorption spectrum are measured. The energies give the tI values and the intensities, once normalized to add to unity, give the IcalI2values. These tI and I C , ~ I ~ values are used to generate Re G,,(E) and Im G,,(E) from eq 15 and 14, respectively. The line width in these expressions is assigned an arbitrarily small value, sufficient to prevent any overlap of bands in either Im G,, or the B(E) function to be calculated. Step 2: B ( E ) is calculated from Im G,, and Re G,, via eq 22. The band positions in B(E) are the zero-order energies of the dark states. The squares of the coupling matrix elements are calculated from the band intensities by multiplying by the line width used in step 1 when formulating Im G,, and Re Gas, Step 3: The energy of the light state is calculated from eq 27. The sum of the zero-order energies calculated should equal the sum of the el measured. 2.5. Uniqueness of the Solution. Often, the analysis of spectra in terms of zero-order states and their interactions is tackled by an iterative trial-and-error procedure. In such cases suspicion is raised that the solution so obtained is not unique but is merely one of a number of possible solutions. The direct deconvolution procedure that we have illustrated here does not suffer from such ambiguity, since the zero-order parameters are obtained via algebraic relationships from the absorption spectrum. Indeed, Ziv and Rhodes25 have demonstrated rigorously that the Green’s function inversion procedure is unique, provided of course that the experimental absorption spectrum is free from ambiguities (such as noise). A devil’s advocate may ask, however, whether we have somehow “got something for nothing”.30 The following qualitative argument explains why all the zero-order parameters may be obtained directly from the absorption spectrum. Consider the total Hamiltonian matrix for this system as it is written in terms of the zero-order states (see eq 16). If there are N dark states, then the total number of nonzero, independent elements in the matrix H i s 2N + 1. These matrix elements are the energies of the dark states, their coupling strengths to the light state, and the energy of the light state. There are thus 2N + 1 unknowns that we wish to determine. By comparison, from our experimental measurement of the absorption spectrum we obtain the energies of the various molecular eigenstates ( N + 1 values), and the relative intensities of these states ( N independent values), giving a total of 2N + 1 independent pieces of data. Thus the number of unknowns matches the number of experimental observables. Since the inversion procedure is analytical, the elements of the Hamiltonian in the zero-order basis may be specified uniquely. In other words, the spectrum implicitly contains all of the information necessary to determine the zero-order energies and couplings. This relies of course on the Hamiltonian being of the form given in eq 16. If this were not so, for example if there was coupling among the dark states, the number of unknowns would exceed the number of observables and any solution based solely on the absorption spectrum would not be unique.
3. Application of the Deconvolution Procedure: Singlet-Triplet Interactions in *B3”Pyrazine As we discussed in the Introduction, Kommandeur and cow o r k e r ~have ~ ~ obtained the absorption spectrum of lBju pyrazine in the region of the 0-0 band a t very high resolution (Doppler limited: 30 MHz) in a molecular beam. The spectra show a number of discrete transitions where there would normally be a
-
~~~
Lawrance and Knight TABLE I: Relative Energies, f,, and Normalized Relative Intensities, Icmll*,of the P(1) Member of the 0-0 Transition in ‘Bh Pyrazine from Ref 19 q/10-2 cm-’ IC,~I~ c J ~ O - cm-’ ~ I.c,112 al/10-2 cm-’ IC,,I~ . -.,
.
-9.52 -8.39 -4.80 -1.79
0.012 0.007 0.040 0.042
-1.22 -1.08 -0.76 -0.19
0.122 0.037 0.248 0.042
0.17 2.03 2.46 2.77
0.129 0.017 0.248 0.055
TABLE II: Relative Energies and Coupling Matrix Elements for the Pure Triplet States Obtained from the Deconvolution of the MES Spectrum of Pyrazine’ triplet energyb S-T coupling strength our work our work ref 20, ref 20,
lO-*cm-’
MHz
MHz
cm-I
MHz
MHz
-9.41 15 -8.3238 -4.6047 -1.7303 -1.1 370 -1.0145 -0.3027 -0.0295 1.4566 2.0672 2.7230
-2821.5 -2495.4 -1380.5 -518.7 -340.9 -304.1 -90.7 -8.8 436.7 619.7 816.3
-2818 -2490 -1375 -515 -336 -300 -86 -3 441 626 823
0.96507 0.76194 0.95718 0.28022 0.17948 0.25302 0.4321 1 0.42526 1.41000 0.26366 0.18770
289.3 228.4 287.0 84.0 53.8 75.9 129.5 127.5 422.7 79.0 56.3
283 236 287 86 54 77 131 126 422 83 56
“The number of significant figures (our work) is indicative of the relative accuracy of the calculation. *Referred to zero-order singlet state.
single rotational band. The extra structure arises from coupling between the SI state and nearby levels in the triplet manifold. Pyrazine has been extensively studied in time-resolved experiments? and from some of these experiments average coupling matrix elements have been e ~ t r a c t e d . ~Time-resolved ~ studies have further revealed that the number of triplet states coupled to the singlet is much larger than can be accounted for by the vibrational state density of the triplet states, indicating the importance of rotations in the singlet-triplet c o ~ p l i n g . ~ ~ , ~ ~ , ~ ~ The high-resolution absorption spectra, however, enable much more detailed information to be extracted. It has already been shown, for example, that the previously observed fluorescence quantum beats can be calculated from the spectrum,19 at least for J’ = 0. We here demonstrate how the singlet-triplet couplings can be extracted from the deconvolution procedure outlined in the preceding sections. For the purposes of this demonstration we focus our attention on the P ( l ) transition. This transition terminates in the J = 0 level and so we are viewing the interaction in the absence of rotational contributions. While the spectrum of the P(1) band is only one of a number that have been reported,lg the spectra have been reduced in the publication sufficiently to prevent measurements of the intensities and positions with the required accuracy. In the case of P(l), we make use of the band intensities and positions that have been tab~1ated.I~ The spectrum shows quite discrete structure, thereby allowing us to make use of the simpler deconvolution procedure. We follow directly the procedure described in section 2,4(ii). The intensities of the bands give the Jcall*coefficients, while the positions give the el values. These values are shown in Table I. The Re G,,(E) and Im G,,(E) functions are calculated from these Ic,,12 and t, values by using a value of 3 X cm-’ for the line width 7 (see eq 14 and 15). Equation 22 is then used to calculate B ( E ) from Re G,, and Im G .,, From B ( E ) we extract the peak positions and intensities, giving the zero-order energies and coupling strengths of the triplet states shown in Table 11. The relative energy of the singlet, calculated from the “center of gravity” expression, is -1.25 X low4c d . In comparison, this value is
~
(30) An equivalent epigram, coined allegedly by the Rt. Hon. Malcom Fraser (Australian Prime Minister, 1975-1983), is that ”there is no such thing as a free lunch”.
(31) McDonald, D. B.; Fleming, G. R.; Rice, S. A. Chem. Phys. 1981.60. 335.
The Journal of Physical Chemistry, Vol. 89, No. 6, 1985 923
Deconvolution of Molecular Eigenstate Spectra
TABLE HI: Comparison between the Calculated and Observed Band . Positions and Intensities of the P( 1) Transition band position/ cm-’ band intensity’ diffb diffb calcd obsd calcd obsd -9.519993 -9.52 -8.390 033 -8.39 -4.800 024 -4.80 -1.789 997 -1.79 -1.220 093 -1.22 -1.079 998 -1.08 -0.759912 -0.76 -0.190 022 -0.19 0.170060 0.17 2.030007 2.03 2.459 976 2.46 2.770023 2.77
-0.000 007 0.050 008 0.000033 0.030 002 0.000 024 0.159997 -0.000 003 0.170061 0.000093 0.490 156 -0.000002 0.149761 -0.000 088 0.999 945 -0.000 022 0.170 021 -0.000 060 0.520 268 -0.000 007 0.070 097 0.000 024 1.ooo 000 -0.000 023 0.220 148
0.05 0.03 0.16 0.17 0.49 0.15 1.oo 0.17 0.52 0.07 1.oo 0.22
-0.000 008 -0.000 002 0.000003 -0.000 061 -0.000 156 0.000 239 0.000055 -0.000021 -0.000 268 -0.000 097 0.000 000 -0.000 148
‘The calculated relative band intensity refers to the square of the coefficient of the zero-order singlet state in each respective molecular eigenstate. The normalized eigenvectors are displayed in Table IV. Observed minus calculated band positions. calculated to be -1.28 X lo4 cm-’ from the difference between the calculated triplet energies and the observed energies. The difference between these two values (3 X lod cm-’) suggests that the relative energies of the zero-order states can be specified with five figure accuracy. For comparison purposes, the results of the de-diagonalization carried out by Kommandeur et a1.21are included in Table 11. Agreement with our results is satisfactory. It is interesting at this point to comment on the accuracy of the calculations as indicated by the comparison above. We have presented the energies and couplings to five figure accuracy, yet the experimental results suggest that no more than three figure accuracy is warranted. Clearly, we do not suggest that the results here are absolutely accurate to five significant figures: uncertainty in the experimental results precludes such accuracy. Rather, our aim is to focus on the relative precision of the calculation. If we take the experimental results as being absolutely accurate, how well can we specify the energies of the zero-order states and their coupling strengths? We shall demonstrate that five figure accuracy is easily achieved. Verification of the “Deconvolution”. To prove that the calculation has truly given us the zero-order parameters, and to demonstrate the accuracy of the procedure, we can use these derived zero-order values to calculate the absorption spectrum via a direct diagonalization of the Hamiltonian. The form of the Hamiltonian is given in eq 16. The positions of the absorption features are calculated as the eigenvalues of this matrix, and the relative intensities are given by the square of the coefficient of the singlet state in each eigenvector. The results of this calculation are given in Table 111. In this example, the absorption spectrum has been reproduced to within 0.01%. The deconvolution procedure is highly accurate, easily justifying the five significant figures given in Table 11, and is clearly superior to trial-and-error techniques. It comes, of course, as no surprise that we can reproduce the absorption spectrum with such accuracy, since the input parameters were obtained from it in the first place. Nevertheless, it is comforting to see the deconvolution procedure verified to such a degree. The accuracy with which the zero-order states and coupling strengths can be specified is ultimately governed by the uncertainty in the experimental measurements. The Size of the S-T Coupling Matrix Elements. It is interesting at this point to compare the size of the coupling matrix elements calculated here with those extracted from lifetime measurements. We stress that these time-resolved experiments provide only an average value for the coupling strength, designated ( VsT). For comparison we take the value obtained by McDonald, Fleming, and Rice3’ of ( VsT)= 2 X an-’.By averaging our results, we obtain a value for ( VsT) of 6 X cm-I. The close accord between these two estimates for ( VsT)may at first seem comforting. However, there are reasons why we do not expect the value of ( VsT)derived from the lifetime data to
provide a good match with that obtained from the deconvolution of Kommandeur et al.’s molecular eigenstate spectrum. The MES spectrum corresponds to the interaction between the J = 0 level of the S1 state and the pyrazine triplet manifold. In contrast, the experiments of McDonald et aL3I were performed at room temperature and thus their estimate for ( VsT)refers to coupling between a range of higher IJ,K) SI states and their counterpart triplet manifolds. The involvement of rotational states in singlet-triplet coupling seems to have been identified secure1y.32,33 The relevant matrix elements for coupling a specific (v,J,K) singlet state with the Iu’,J,K’) rovibronic levels of the T manifold may be evaluated, at least on a relative basis.” The general trend is for the coupling matrix elements to increase with J. For example, in the case where the coupling involves spin-orbit-coriolis coupling, the S-T coupling matrix elements are expected to scale with J for A,B axis coupling and with K for C axis c o ~ p l i n g . ~ ~ , ~ ~ Hence, at least qualitatively, we should expect McDonald et al.’s3’ estimate for ( VsT) to be substantially larger than that derived from Kommandeur et ale’sdata. There is an alternative view concerning aspects of McDonald et al.’s31 experiment that might assist in resolving this apparent dilemma. Kommandeur20 has directed attention toward the contribution of off-resonance Rayleigh-Raman scattering to the “fast component” in the fluorescence decay of intermediate case molecules when the frequency width of the excitation source exceeds the natural width of individuai absorption lines. The laser used in McDonald et al.’s31experiments had a reported bandwidth of 4 cm-I, hence it exceeds the ca. 200-kHz natural width of the transitions in pyrazine. Accordingly, we might expect that the measurements of Afast/Aslow, the ratio of amplitudes for the “fast” and “slow” decays (rate constants k+ and k- respectively), in McDonald et al.’s3’ experiments may be greater than they would otherwise be due to a contribution to Afastfrom off-resonance Rayleigh-Raman scattering. Thus, from the relationship between the ratio Afast/A,I,wand the average S-T coupling matrix elem e n t : ~ ~viz. ~
where AST = 27rpT( VsT2)and N = A S T T T , we see that an underestimate of ( VsT)is expected to be obtained in these circumstances. An increase in ( VsT)due to the involvement of higher IJ,K) states may therefore not be detected if off-resonance scattering Overall, we suggest confounds the measurement of Afast/ASlow. that the close accord between ( VsT)obtained from McDonald et al.’s3I experiments and our estimate of ( VsT)derived from the exact deconvolution of the MES spectrum may be coincidental rather than reassuring. Finally, we draw attention to the variation of the coupling strength across the eleven triplet states coupled to the SI origin, and the size of these couplings relative to the triplet spacing. The coupling strengths vary quite smoothly across a range spanning to 1.4 X cm-’. almost an order of magnitude, from 1.8 X This result should be contrasted with the usual assumption in radiationless transition theory that the coupling matrix elements are relatively constant.18 It may be that the case of pyrazine is not widely representative in this respect, although we believe it probably is. Our analysis of course treats the triplet manifold as being noninteracting, thus the dark states that emerge from our deconvolution include all triplet-triplet vibronic interactions. Since the S-T coupling is dependent on Franck-Condon overlap integrals connecting the singlet and triplet state^,^^^^^ we might (32) Howard, W. E.; Schlag, E. W. In “Radiationless Transitions”; Lin, Press: New York, 1980; p 81. (33) Novak, F. A,; Rice, S . A.; Morse, M. D.; Freed, K. F. In ‘Radiationless Transitions”; Lin, S. H., Ed.; Academic Press: New York, 1980; p 135. (34) Allen, H. C., Jr.; Crqss, P.C. “Molecular Vib-rotors”; Wiley: New York, 1963. Pyrazine is an oblate rotor. (35) van der Werf, R.; Kommandeur, J. Chem. Phys. 1976,16, 125. van der Werf, R.; Schutten, E.; Kommandeur, J. Chem. Phys. 1976, 16, 151. Spears, K. G.; El-Manguch, M. Chem. Phys. 1977, 24, 65. S . H., Ed.; Academic
924
The Journal of Physical Chemistry, Vol. 89, No. 6,1985
Lawrance and Knight
TABLE I V Eigenvalues and Eigenvectors for the Hamiltonian Expressed in Teras of the Zero-Order Basis0
Eigenvalues/ lo-* cm-' -9.52
-8.39
-0c
0.086
0.991 0.07 1 0.022 0.004 0.002 0.003 0.005 0.005 0.014 0.003 0.002
0.082 -0.993 -0.022 -0.004 -0.002 -0.003 -0.005 -0.004 -0.012 -0.002 -0.002
-4.80 - 0 0 -0.042 -0.043 0.976 0.018 0.010 0.013 0.019 0.018 0.045
0.008 0.005
-1.79
-1.22
-00
-02
-0.026 -0.024 -0.070 0.964 0.057 0.067 0.060 0.050 0.089 0.014 0.009
-0.041 -0.037 -0.099 -0.192 0.753 0.429 0.164 0.125 0.184 0.028 0.017
-1.08
&.g
-0.76
eigenvectors
0.022 0.020 0.052 0.083 0.607 -0.745 -0.107 -0.078 -0.107 -0.016 -0.0 10
-0.19
0.17
2.03
- 0 0
0.205
0.359
0.132
-0.057 -0.050 -0.124 -0.144 -0.237 -0.495 0.471 0.290 0.317 0.047 0.027
0.022 0.019 0.045 0.037 0.039 0.063 0.788 -0.544 -0.176 -0.024 -0.132
0.036 0.032 0.072 0.053 0.049 0.077 0.328 0.766 -0.394 -0.050 -0.026
0.011 0.010 0.019 0.010 0.008 0.01 1 0.024 0.027 0.324 -0.935 -0.036
2.46 -0-0.041 -0.035 -0.068 -0.033 -0.025 -0.036 -0.078 -0.085 -0.700 -0.334 0.356
2.77
0.234 0.019 0.016 0.030 0.015 0.01 1 0.016 0.033 0.036 0.251 0.088 0.933
'The elements of H were obtained by deconvoluting the absorption spectrum. The eigenvectors are listed vertically under each eigenvalue; the first element (underlined) is the coefficient of the singlet, and the following elements are the coefficients of the triplets, arranged in order of increasing energy. conclude that the order of magnitude variation reflects, in part, a naturally wide range of vibrational quanta comprising the triplet destination states. Based on current knowledge of level structure in singlet states of similar molecules a t medium vibrational energies, this is not so surprising. The Extent of Level Mixing. The triplet level spacings and the coupling strengths are both on the same order of magnitude. This necessarily contributes to a rather extensive mixing of the levels, especially those that are tightly bunched near the singlet. The extent of mixing can be seen in the lack of correlation between the coupling strength and the final absorption intensity. It will be recalled that if the couplings were small relative to the level spacings a first-order perturbation correction to the wave functions would be sufficient, and the amount of singlet mixed into each triplet would be related to the coupling strength for that triplet. In order to demonstrate the extent of the mixing between the zero-order states we have listed the eigenvectors for each eigenstate in Table IV. Examine, for example, the eigenvector for the state whose cm-l. This state in fact correlates relative energy is 0.17 X with the primary singlet state (zero-order relative energy t, = 0). We see that its own singlet character, initially unity, has been diluted by a factor of eight, i.e., (0.359)2 0.13. Its character is now dominated by the zero-order triplet state with relative cm-'. The most prominent singlet character energy -0.0295 X now resides in the two mixed states that lie at 2.46 X lo-* and -0.76 X cm-I relative energy, with singlet coefficients of 0.498 for both states. Accordingly, we see the transitions involving these states to be the brightest in the MES spectrum, and of equal cm-I, intensity (see Table 111). The latter state, at -0.76 X turns out coincidentally to be the one that is most extensively homogenized by the S-T coupling among the 12 zero-order states, as seen from its own eigenvector: the largest coefficient for any of its component zero-order triplet states is 0.495.
-
4. Conclusion We have demonstrated a Green's function inversion procedure for extracting directly from a tangled absorption spectrum the zero-order energies and interstate coupling strengths. Provided that the molecular characteristics conform with the model, the accuracy with which one can specify the zero-order values is limited only by experimental uncertainty. In essence, the method works because there are the same number of observables as there are unknowns. While we have given an account of the theoretical basis of the deconvolution procedure, a complete understanding of this is not required to apply it: section 2.4 gives a step-by-step guide to performing the deconvolution. We have demonstrated the power of the technique by deconvoluting the molecular eigenstate spectrum of pyrazine obtained by Kommandeur and co-workers. l9 The use of high-resolution measurements of absorption spectra in conjunction with highly accurate deconvolution procedures such as the one given here offers the key to a significant step forward in our understanding of radiationless transitions and
the composition of molecular eigenstates. Acknowledgment. Drs. A. J. O'Connor and J. Dobson contributed generously to the early development of this work. We are grateful to Prof. Bradley Moore for his hospitality during the completion of this work and for providing inspirational views of the Bay area in return for loving Nugget. Financial support from the Australian Research Grants Scheme and from the US.Australia Scheme for Scientific and Technological Cooperation (National Science Foundation and Department of Science) is gratefully acknowledged. Appendix: Computational Evaluation of Principal Value Integrals
We wish to evaluate numerically a principal value integral of the form
The principal value integral may be defined as Re G ( E ) = E'-E-*Im G(E? d E ' + " -,Er] Im G(E? E'- E E'-E E"E+6
i!$[Jm
For computational purposes 6 must be set at some small, but nonzero number. In doing this, however, we cannot avoid neglecting a small region of the integral. To overcome this limitation, we suggest a method based on the following analysis. We expand Im G(E? in a Taylor series about the point E'= E: Im G(E? = Im G(E'= E )
Im G(E? -E'- E
+ (E'-
E)
Im G(E? - c- d(") dE
