3298
J . Phys. Chem. 1986, 90, 3298-3301
Asslgnment of the Vlbronic Level Structure of Cu, Ground State Josef W. Zwanziger,t Robert L. Whetten,* and Edward R. Grantt Department of Chemistry, Baker Laboratory, Cornel1 University, Zthaca, New York 14853 and Department of Chemistry and Biochemistry, University of California. Los Angeles, Los Angeles, California 90024 (Received: May 23, 1986)
A Jahn-Teller Hamiltonian successfully accounts for all resolved levels of Cupelectronic ground state observed in the emission spectra of Rohlfing and Valentini. The parameters of this model describe an energy surface moderately stabilized by distortion (D= 305 cm-I) with barriers to pseudorotation (1 11 an-') slightly less than the zero-point energy. The symmetric stretching frequency is assigned to be up= 269.5 cm-I. Calculated intensities agree roughly with experiment, and deviations point to additional excited-state complexity resulting from multistate interactions.
Introduction Metal cluster science has emerged as a fast-growing new subdiscipline, spanning the fields of chemistry, physics, and materials research. Among the many factors propelling this expanding interest is the question of the elementary properties of these materials.'-3 With high permutational symmetry and strong electron correlation effects, metal clusters challenge conventional notions about the theory of electronic structure and b ~ n d i n g . ~ . ~ Their ground and excited electronic states characteristically exhibit high term degeneracies or near-degeneracies. Vibronic coupling under such circumstances produces a vibrational level structure that appears irregular from the point of view of separable normal Moreover, the commensurate nature of electronic and vibrational frequencies requires that any consideration of the overall heavy particle dynamics also include the dynamics of the electronic degrees of freedom.*-'O At an elementary level the observable properties, dynamic and otherwise, of an isolated metal cluster are most precisely described in terms of its stationary states. Some experimental and theoretical progress toward resolving quantum level structure has been made, notably for alkali11-17 and copper Ionization thresholds have also been systematically determined as a function of cluster size for a variety of transition and main-group ~lusters.2~ Experimentally, the acquisition and labeling of cluster rovibronic states presents a formidable challenge.26 Low internal temperatures are required to resolve distinct transitions and reduce hot-band congestion, and in a cluster distribution one must record the spectrum specific to the n cluster of interest. Free-jet cluster technology coupled with resonant laser photoionization mass spectrometry appears to offer one solution to these problems, with the above cited successes for Nap and CU,, where in each case a simple vibronic model has been found to fit the absorption spectrum and predict properties of electronically excited states. We are concerned here with a cluster ground state, that of Cu,. This species has had an extensive history of study from both e ~ p e r i m e n t a l ~ ~and - ~theoretica12132 ~ ~ ~ ~ * * ~ perspectives. Elementary molecular orbital considerations predict the ground state of Cup to be degenerate, and effort has focussed on the question of whether the zero-point level of this molecule is made fluxional or localized by Jahn-Teller distortion. The precise frequency of the symmetric stretch has also emerged as an issue in experimental and theoretical interpretations. Very recent emission spectra of Rohlfing and Valentini2, have resolved the ground-state vibronic level structure of Cup. In the present work, we find that this structure has an accurate and complete interpretation in terms of an elementary two-parameter Jahn-Teller Hamiltonian. The parameters of this Hamiltonian resolve the question of localization vs. fluxionality for the isolated molecule; with a pseudorotation bamer of 111 cm-', the zero-point level of CU,is fluxional. The completeness of the fit to 19 bands 'Department of Chemistry, Cornell University. *Address correspondence to the author at Department of Chemistry and Biochemistry, University of California, Los Angeles.
0022-3654/86/2090-3298$01 .50/0
to within an average deviation of less than 7.5 cm-' clearly identifies, by its omission, the first excited level of symmetric stretch at w, = 269.5 cm-'. In these spectra, the intensities carry information on the upper state vibronic wave functions. This more complete perspective suggests a rather more complex nature for vibronic coupling in the electronically excited state.
Spectra and Assignment Rohlfing and Valentini report emission spectra of jet-cooled Cup excited to either of two levels of the lowest excited electronic state first characterized by Morse et al.*O These spectra, which display the levels of the ground state in a series of bands displaced to the red of the exciting frequency, are reproduced in Figure 1. The most distinctive feature of the level system evidenced by these data is the number and apparent irregularity of low-frequency vibrational states. (1) Jortner, J. Ber. Bunsenges. Phys. Chem. 1984, 88, 188. (2) Weltner, W., Jr.; VanZee, R. J. Annu. Rev. Phys. Chem. 1984,35,291. (3) "Diatomic Metals and Metallic Clusters", Faraday Symp. SOC.1980, 14. (4) Advanced Theories and Computational Approaches to the Electronic Srructure of Molecules, Dykstra, C . E., Ed.; Reidel: Dordrecht, 1983. (5) Comparison of Ab Initio Quantum Theory with Experiment, Bartlett, R. J., Ed.; Reidel: Dordrecht, 1986. (6) Bersuker, I. B. The John-Teller Effect and Vibronic Interactions in Modern Chemfstry;Plenum: New York, 1984. (7) KBppel, H.; Domcke, W.; Cederbaum, L. S.Ado. Chem. Phys. 1984, 57, 59. (8) Whetten, R. L.; Ezra, G. S.; Grant, E. R. Annu. Reu. Phys. Chem. 1985, 36, 277. (9) Zwanziger, J. W.; Whetten, R. L.; Ezra, G. S.; Grant, E. R. Chem. Phys. Lett. 1985, 120, 106. (10) Zwanziger, J. W.; Grant, E. R.;Ezra, G. S.J. Chem. Phys., in press. (11) Gerber, W. H.; Schumacher, E. J. Chem. Phys. 1978, 69, 1692. (12) Martin, R. L.; Davidson, E. R. Mol. Phys. 1978, 35, 1713. (13) Herrman, A.; Hoffmann, M.; Leutwyler, S.;Schumacher, E.; Wikte, L . Chem. Phys. 1979,62, 216. (14) Lindsay, D. M.; Thompson, G. A. J. Chem. Phys. 1982, 77, 1114. (15) Martins, J. L.; Car, R.; Buttet, J. J. Chem. Phys. 1983, 78, 5646. (16) Delacrttaz, G.; Wikte, L. Surf. Sci. 1985, 156, 770. (17) Delacrbtaz, G.; Grant, E. R.; Whetten, R. L.; WBste, L.; Zwanziger, J. W. Phys. Rev. Lett. 1986, 56, 2598. (18) DiLella, D. P.; Taylor, K. V.; Moskovits, M. J. Phys. Chem. 1983, 87, 524. Moskovits, M. Chem. Phys. Lett. 1985, 118, 111. (19) Howard, J. A.; Preston, K. F.; Sutcliffe, R.; Mile, B. J. Phys. Chem. 1983, 87, 536. (20) Morse, M. D.; Hopkins, J. B.; Langridge-Smith, P. R. R.; Smalley, R. E. J. Chem. Phys. 1983, 79, 5316. (21) Thompson, T. C.; Truhlar, D. G.; Mead, C. A. J. Chem. Phys. 1985, 82, 2392. (22) Walch, S. P.; Laskowski, 8. C. J. Chem. Phys. 1986, 84, 2734. (23) Rohlfing, E. A.; Valentini, J. J. Chem. Phys. Lett. 1986, 126, 113. (24) Crumley, W. H.; Hayden, J. S.; Gole, J. L., to be sumitted for publication. (25) Kappes, M. M.; SchBr, M.; Radi, P.; Schumacher, E. J . Chem. Phys. 1986, 84, 1863. (26) Ezra, G. S.;Berry, R. S.J. Chem. Phys. 1982, 76, 3679. Amar, G. Kellman, M. E.; Berry, R . S . J. Chem. Phys. 1980, 73, 2387. 1979. 70, 1973. Kellman, M. E.; Berry, R. S. Chem. Phys. Lett. 1976, 42, 327.
0 1986 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 90, No. 15, 1986 3299
/I
A
I;
I I
00
1000
I
x5
I x10 I
II I
II
2000 3000 4000 5000 cm-1, relative t o excitation frequency
6000
Figure 1. Comparison of experimental and theoretical spectra. The experimental traces are redrawn from Rohlfing and Valentini‘s results. The top trace is their spectrum A; the lower is spectrum B. The upper solid-line stick spectrum is derived from our calculation of the electronic ground-state eigenvalues and eigenstates coupled with an eigenstate derived for the origin of the *E” excited electronic state from the assignment of Thompson et al. These bands represent transitions to states with E’ vibronic symmetry and should be compared with spectrum A. The lower stick spectrum mixes calculated transition intensities from the first excited E” level of the upper state (solid lines), with a separate set of calculated relative intensities, for hypothetical transitions from the first A i level of the upper state (dashed lines). Note that the correspondence between theoretical and experimental intensities for this latter pair of upper states degrades substantially for transitions to the red of -300 cm-’ (cf. Table I). The theoretical intensities in all cases are normalized to the strongest band in each respective symmetry block and scaled as indicated.
The irregular character of the vibrational level system of Cu, is qualitatively explained by considering the electronic structure of the ground state. The bonding in this state arises from interaction of the 4s electrons of the 4s’3d1° ground-state configuration of the copper atoms. In D3hsymmetry the 4s electrons yield a ground-state a: e’‘ configuration, i.e. a single 2E’ term.27 This degenerate term is characterized by irregular vibrational structure and low-frequency fluxionality as the electronic states are mixed by displacements in the e’ nuclear coordinates (JahnTeller effect).6,28 Quantitative account of the effect of such coupling on the spectrum of eigenstates can be made by solving the dynamical Jahn-Teller problem variationally, diagonalizing the fully coupled nuclear and electronic Hamiltonian.** This is computationally simplest in a two-state diabatic basis. The resulting Hamiltonian complete to second order is
( g) +
fi = PN +
+
J+)(kpe-’@ 1/2gp2e2i@)(-l+ Hermitian conjugate (1)
where PN is the nuclear kinetic energy operator, I+) and I-) are the degenerate diabatic electronic wave functions, k and g are the linear (distortion) and quadratic (localization) Jahn-Teller parameters, and p and d the amplitude and phase of the distortion coordinate. This Hamiltonian gives energies in units of oo, the frequency of the degenerate mode in D3hsymmetry. Symmetric displacement fails to lift the degeneracy and is factored out. Energy levels and vibronic wave functions are obtained as eigenvalues and eigenvectors by diagonalizing (1) in a suitable basis (27) Inclusion of correlation of the 3d’* electrons produces some contraction of the bond lengths but does not alter the simple MO prediction of the lowest D,, electronic term. (28) Longuet-Higgins, H. C. Adu. Spectrosc. 1961, 2, 429.
--.. -.. -..--...*-*..
2 0.0
0.4
0.8
1.2
1.8
2.0
2.4
2.8
3.2
k g = 0.2 Figure 2. Eigenvalues of Hamiltonian (1) relative to the ground state for g = 0.2 and a range of k . Solid lines are of vibronic symmetry E, dashed lines of symmetry A*, and light dashed lines of symmetry A , .
of isotropic harmonic oscillator wave functions. By this procedure, coupling linear in p mixes the vibrational and electronic angular momenta to produce a conserved vibronic angular momentum j . j is half-odd-integral; we note that this is an example of the effect of a “diabolical point” in the parameter space of the associated adiabatic wave functions, as discussed recently by Berry29and others.3w34 Vibronic levels have symmetry species E, AI, and At, where AI and A2 are accidentally degenerate. The coupling term quadratic in p causes mixing of the vibronic angular momentum states such that j mod(3) remains conserved. In the process, the AI/A2 accidental degeneracy is lifted. A systematic view of the predicted level structure of such calculations for a range of k and g is conveniently obtained by means of correlation diagrams. Such a diagram showing the evolution of the level structure for a single g and range of k pertinent to the present problem is given in Figure 2. A reasonable fit to the spectrum of Rohlfing and Valentini is found by inspection at k = 1.9 in the diagram (assuming w0 = 137 cm-’). A three-dimensional nonlinear least-squares search for optimum parameters in this range yields a simultaneous fit of 19 levels of the spectrum to an average deviation of less than 7.5 cm-’ for best values k = 1.86 and g = 0.223. The fit to eigenvalues is summarized in Table I. Certain details of this fit and interpretation are worth discussing. To begin, we adopt the Morse et ai. assignment of the term (29) Berry, M. V. Proc. R . SOC.London, Ser. A 1984, A392, 45. Berry, M. V. J . Pkys. A 1985, 18, 15. (30) Simon, B. Phys. Reu. Lett. 1983, 51, 2164. (31) Wilczek, F.; Zee, A. Phys. Rev. Lett. 1984, 52, 2111. (32) Hannay, J. H. J . Phys. A 1985, 18, 221. (33) Kuratsuji, H.; Iida, S. Prog. Theor. Phys. 1985, 74, 439. (34) Moody, J.; Shapeve, A.; Wilczek, F. Phys. Rev. Lett. 1986, 56, 893.
3300 The Journal of Physical Chemistry, Vol. 90, No. 15, 1986
Letters 0
TABLE I: Fit to Experimental Positions and Intensities in the Emission Spectrum of Cu3 (Rohlfing and Valentini Spectra A and B) for Ground-State Linear + Quadratic Jahn-Teller Parameters k = 1.86. g = 0.223. uo = 137.0”
experimental intensity band origin
u,
A
B
100 154 220 235 304 352 378 402 462 486 519 556 58 1 625
intensity u,
cm-I
A
B
1.000 0.163 0.001 0.683 0.045 0.190 0.049 0.098 0.030 0.040 0.100 0.017 0.056 0.094
1.000 0.001 0.0003 0.092 0.105 0.449 0.177 0.450 0.240 0.263 0.930 0.229 0.770 1.595
gx
0
0
A1 A2 A3 A4,A5 A7 A8 A9 A10 A 11 A12 A13b A14 A15
cm-’
N 0
theoretical
1.000 0.067 0.18 0.70 0.018 0.27 0.002 0.13 0.13 0.067 0.018 0.072 0.033 0.039
1.00 0.06 0.04 0.24 0.00 0.03 0.16 0.15
0.00 0.027 0.036 0.020 0.013 0.030
101 143 223 234 296 361 382 404 462 481 514 553 573 628
States of the 3 / 2 Block as Observed in Spectrum B B1 B3 B7
B8
1.oo 0.64 0.18 0.17
16 144 274 324
12 148 281 300
1.000 1.530 0.545 0.828
‘us = 269.5 cm-l; we take band 13a of spectrum A at 546 cm” as the first overtone of the symmetric stretch.
symmetry (2E’’) and lowest vibronic level structure of the excited electronic states. Accordingly both the origin and the band shifted to the blue by 146 cm-l are of E’’ vibronic symmetry and thus can only fluoresce to E’ vibronic states. However, only a fraction of a wavenumber away from the latter E” state is found a state of A I ” vibronic symmetry, which can radiatively couple to A i levels in the ground state. We note that Coriolis interactions will mix these nearby vibronic states by in-plane rotation. Thus, Rohlfing and Valentini’s spectrum A (emission from the origin) should exhibit states entirely of E’ vibronic symmetry, while spectrum B (from Too 146 cm-I) should show E‘ and possibly A i states. The appearance of extra bands in the experimental spectrum B confirms this expectation. Consequently all bands in Rohlfing and Valentini’s spectrum A are assigned as E’. We also assign bands 4 and 5 of spectrum A as a single peak, since these features appear to coalesce into a single level as band 6 in spectrum B. We interpret as real the very small peak between bands 8 and 9 in spectrum A; this corresponds to the prominent band 9 of spectrum B. Our intensity calculation (see below) predicts this feature to be small in A. Bands in spectrum B not present in A are assigned as having A,‘ vibronic symmetry. These are bands 1, 3, 7, and 8. Theoretical spectra are fit, by a nonlinear least-squares search through the three-dimensional parameter space of wo, k , and g , to bands 1-5 and 7-15 of spectrum A and the above-mentioned 1, 3, 7, and 8 of spectrum B. Band 6 of A is excluded because it fails to fit the pattern of levels predicted by any reasonable calculation. It is therefore assigned as the symmetric stretch mode. At 269.5 cm-I it is considerably lower than the value of 352 cm-l suggested by previous experiments.’*
+
Adiabatic Surfaces: Fluxionality vs. Localization
The parameters of this spectroscopic fit can be applied to generate a Born-Oppenheimer picture of the lower surface in the ground electronic state. Setting TNto zero, or equivalently mass to infinity in the Hamiltonian (l), yields an expression for the surfaces:
For our values of k = 1.86 and g = 0.223 the usual simplification of the above equation by treating g p / k as “small” is not very
0
0
7 -4.0
-2.0
0.0
i.0
40
Qx
Figure 3. Born-Oppenheimerlower surface for 2E’Cu3 as predicted by fit to experimental spectra. Dimensionless units are used; contour spacing corresponds to 27.4 cm-I. Dashed contours give the classical turning points of the zero-point level.
accurate. Hence the saddle points on the lower surface fall at a value of p quite different from its value at the three equivalent minima. Thus, also the oft-used simple expression for the barrier height in terms of k and g is inappropriate here as well. With our parameters, we find numerically a Jahn-Teller stabilization energy of 305 cm-’, and a pseudorotation barrier height of 11 1 cm-I. Zero-point energy in this mode is 118 cm-l, indicating that no levels of the ground state are localized. We therefore infer from the spectrum that ground-state *E’ Cu3 is a fluxional molecule, undergoing continual hindered pseudorotation. In other words, it has no single well-defined geometrical structure. A contour map of the experimentally derived lower surface which indicates the amplitudes of the classical turning points of the vibrational ground state is given in Figure 3. Intensities and the Nature of Vibronic Coupling in the Excited State
Solutions to (1) include vibrational wave functions, which, in combination with excited state wave functions, predict the emission spectrum intensities. We have carried out such calculations for transitions from the excited states calculated by Thompson et aL2l to our model for the lower state. These intensity determinations for degenerate-to-degenerate transitions are somewhat more involved than standard calculations of nondegenerate to degenerate intensities. As mentioned above, for linear plus quadratic couplingj mod(3) remains conserved so that we have three type of states, labeled j’ = - I f 2, and j‘ = 3/2. The first of these mixes states by j’ = ‘I2, of vibronic angular momentum j = ’I2,etc. Each j state in the mix is itself a linear combination of - 5 / 2 , isotropic harmonic oscillator states, denoted I u , ~ ) , where only states with 1 = j - II2(A) are included in the sum. A = f l labels the electronic component. Hence a state of j’ = is written as
Similarly, states with j ’ = are denoted by IE-) and are constructed the same way. Finally, states with j ’ = 3/2 are written IA) and include both A, and A2 vibronic symmetry. In the case of Cu, we have E” E’ transitions, effected by A2’ the component of p which transforms as A,; and A,” transitions also effected by the A; dipole operator. In both cases the electronic component of the transition can to a good approximation be considered separately from the vibrational part. Thus matrix elements of the form
-
-
( f”IFA24f’)
must be evaluated, where
If”)
transforms as ( x z f iyz) in D3,,,
J . Phys. Chem. 1986, 90, 3301-3313 and
I*’)
as (x f iy). Elementary considerations then show that (4)
and the other matrix elements are zero. Hence both types of transitions have intensities given by
I(E”,AI”IILA~,,~E’,A~~)IZ )(+”IA2/’1+’)12(FC++ + FC--)’
(5)
FC is the Franck-Condon factor, that is, the overlap of the lv,l) basis functions associated with the appropriate electronic components I*) of the two eigenstates in question. Relative intensities calculated from eq 5 (for arbitrarily scaled electronic matrix elements) are listed in Table I and compared graphically with experiment in Figure 1. While it is difficult to gauge the accuracy of experimental intensities, it is clear that theory fails badly for a couple of bands, even though more qualitative broader trends are reasonably reproduced. One possible explanation is that the excited-state assignment of Thompson et al. needs refining. We have investigated this possibility but have yet to find a set of upper state coupling parameters that improves the intensity agreement, while retaining a fit to observed excited-state energy levels. More likely to be of greater importance than further optimization within the simple picture of the two-state isolated degeneracy demanded by Hamiltonian (1) is the inclusion of coupling to additional electronic states. Redissociation is evident in the experimental data of Morse et aL20 and Crumley et Configuration interaction involving excited states of 3d manifold can be expected to be important. In fact, Walch and Laskowski2* suggest on the basis of their calculations that the lowest allowed excited state is of electronic term symmetry ’Al’. Additional intensities from a greater number of the excited-state levels will be necessary to more fully resolve the vibronic character of the excited state.
3301
Conclusions To summarize, we have constructed a Jahn-Teller vibronic Hamiltonian that successfully accounts for all of the resolved level structure of the Cu3 electronic ground state. The parameters of this fit ( k = 1.86, g = 0.223 in reduced units of the zeroth order frequency wo = 137 cm-I) describe a surface moderately stabilized by distortion ( D = 305 cm-’) with barriers to pseudorotation (1 11 cm-l) just less than the zero-point energy (1 18 cm-’). The symmetric stretching frequency is assigned to be ws = 269.5 cm-I. Theoretical intensities, using Thompson et a1.k parameters to generate eigenfunctions for excited state levels, agree roughly with experiment, tending to confirm the published fit to the excited state structure. Deviations, however, point to possible added complexity in the optical spectrum due to multistate interactions in the excited state. Since completion of this work we have learned of a parallel effort by T r ~ h l a and r ~ ~co-workers to assign the Cu3 emission spectrum on the basis of a surface derived from the ab initio calculations of Walch and Laskowski. Their slightly optimized coupling parameters of k = 1.56 and g = 0.247 &firm the present more comprehensive assignment and show that the ab initio results give a reasonably correct description of vibronic coupling in Cu3.
Acknowledgment. We are grateful to E. Rohlfing and J. Valentini for providing their spectra in advance of publication, and to K. S. Haber for stimulating discussions. We thank D. Truhlar for communication of his results. This work was supported by the National Science Foundation under Grant No. CHE-8213162. (35) Truhlar, D. G.; Thompson, T. C.; Mead, C. A. to be submitted for publication.
FEATURE ARTICLE Infrared Laser Photodissociation and Spectroscopy of van der Waals Molecules R. E. Miller Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 2751 4 (Received: February 14, 1986; In Final Form: April 10, 1986)
Infrared-laser-inducedvibrational predissociationof weakly bound van der Waals molecules has become a topic of considerable interest, both experimentally and theoretically. In particular, the partitioning of the excess energy in the fragments and the vibrational relaxation lifetimes have attracted much attention. The spectroscopy of these systems is also of interest since it can provide valuable molecular structure information which is important in obtaining a detailed understanding of the dynamics. This article is intended to provide a concise account of our present understanding of these processes as well as to point out where future research in this area might be directed.
Introduction Weakly bound van der Waals molecules have fascinated scientists for many years, in part as a result of the promise they hold for bridging the gap, in a more or less continuous fashion, between the gas and solid (or liquid) states of matter.’ As pointed out by Levine,2 the idea of detecting these molecular aggregates by using spectroscopic methods dates back into the early l i t e r a t ~ r e . ~ (1) G. D. Stein, Phys. Teach., 17, 506 (1979). (2) H. B. Levine, J. Chem. Phys., 56, 2455 (1971).
0022-3654/86/2090-3301$01.50/0
Indeed, as so beautifully demonstrated for the Ar, the spectroscopy of atomic and molecular clusters holds much promise in obtaining accurate intermolecular potential surfaces. In larger systems, where the dimensionality of the potential becomes so large ~~
~~
~
~
(3) J. 0. Hirschfelder, F. T. McClure, and I. F. Weeks, J . Chem. Phys., 10, 201 (1942). (4) Y. Yanaka and K. Yoshino, J . Chem. Phys., 53, 2012 (1970); Y. Yanaka, K. Yoshino, and D. E. Freeman, J . Chem. Phys., 59, 5160 (1973). ( 5 ) E. A. Colburn and A. E. douglas, J . Chem. Phys., 65, 1741 (1976). (6) G. C. Maitland and E. B. Smith, Mol. Phys., 22, 861 (1971). (7) R. A. Aziz and H. H. Chen, J . Chem. Phys., 67, 5719 (1977).
0 1986 American Chemical Society
