J. Phys. Chem. 1992, 96, 5325-5331 It should be noticed that the lifetime of the luminescence in the spectral range of the Y-emission is about 38 ns in the a-perylene crystaP* and about 2.5 11s in the LB-Pe sample. The luminescence lifetime of the monomeric Bperylene crystal is about 5 115,’ similar to that of perylenc monomers?’ The 2.5-115 lifetime in the LB-Pe sample is in agreement with Y-type emission as an allowed transition from the u state12that might make a slight excimer-like configurational change. In contrast, Y-type emission with about a 38-ns lifetime in a-perylene could originate as a forbidden transition from the second excited g state in the a-perylene crystal.’* Our experiments on the luminescence of perylene dimers aligned in a Langmuir-Blodgett film have brought out three new results: first, Y emission with a short 2.5-11s lifetime; second, E emission that persists down to 1.5 K; third, generation of E emission via excitation of the lower excited g dimer state. These results agree with predictions of theoretical models for excimer formation in perylene sandwich dimers2J but are very different from results obtained with the a-perylene crystal. Further experimental work is necessary to clarify the nature of Y-type emission4J’J2with perylene dimers incorporated into different environments. It appears also necessary to further test and develop theoretical models of excimer formation. Our present results suggest that the a-perylene crystal represents a special case, rather than a general model case, for excimer formation dynamics in sandwich dimers.
Acknowledgment. We are grateful to the Deutsche Forschungsgemeinschaft, SFB 337, for financial support. F.W. and D.W. thank Dr. David Brown for an inspiring discussion of excimer formation in perylene dimers. Registry No. Perylene, 198-55-0; arachidic acid, 506-30-9.
5325
References and Notes (1) FBrster, Th. Pure Appl. Chem. 1962, 4 , 121; 1963, 7, 73. (2) McGlynn, S. P.; Armstrong, A. T.; Azumi, T. Modern Quantum Chemistry. Part I I I Action of light and Organic Crystals; Sinanoglu, O., Ed.; Academic Press: New York, 1965; p. 203. (3) Warshel, A.; Huler, E. Chem. Phys. 1974, 6, 463. (4) Cohcn, M. D.; Haberkorn, R.; Huler, E.; Ludmcr, Z.; Michel-Beyerle, M. E.; Rabinovich, D.; Sharon, R.; Warshel, A.; Yakhot, V. Chem. Phys. 1978, 27, 21 1. (5) Tanaka, J. Bull. Chem. SOC.Jpn. 1963, 36, 1237. (6) Donaldson, D. M.; Robertson, J. M.; White, J. G.Proc. R. Soc. London 1953, A220, 31 1. (7) von Freydorf, E.;Kinder, J.; Michel-Beyerle, M. E . Chem. Phys. 1978, 27, 199. (8) Walker, 9.; Port, H.; Wolf, H. C. Chem. Phys. 1985, 92, 177. (9) Hochstrasser, R. M. J . Chem. Phys. 1964, 40, 2559. (10) Hochstrasser, R. M.; Nyi, C. A. J. Chem. Phys. 1980, 72, 2591. (11) Sumi, H. Chem. Phys. 1989, 130,433. (12) Wu, T.-M.; Brown, D. W.; Lindenberg, K. J . Lumin. 1990,45,245. (13) Durfee, W. S.;Storck, W.; Willig, F.; von Frieling, M. J . Am. Chem. SOC.1987, 109, 1297. Durfee, W. S.;Tiishaus, M.; Schweizer, E.;Storck. W.; Willig, F.;von Fricling, M. Thin Solid Films 1988, 159, 331. (14) Storck, W. Unpublished results. (15) Gaines Jr., G. L. Insoluble monolayers at liquid-gas interfaces; Interscience Publishers: New York, 1966; p 338. (16) Van der Auweraer, M.; Willig, F. Isr. J . Chem. 1985, 25, 274. (1 7) OConnor, D. V.; Phillip, D. Time-correlated single photon counting Academic Press: New York, 1984. (18) von Frieling, M.; Bradaczek, H.; Durfee, W. S. Thin Solid Films 1988, 159,451. (19) Matsui, A.; Mizuno, K.; Iemura, M. J. Phys. SOC.Jpn. 1982, 51, 1871. (20) Kietzmann, R.; Willig, F. Unpublished results. (21) Mataga, N.; Torihashi, Y.; Ota, Y. Chem. Phys. Lett 1967, 1, 385. (22) Kasha, M.; Rawls, H. R.; El-Bayoumi, M. A. Pure Appl. Chem. 1965, 11, 371. (23) Weiss, D.; Kietzmann, R.; Storck, W.; Lehnert, J.; Willig, F. Makromol. Chem. 1991, 46, 65. (24) Ferguson, J. J . Chem. Phys. 1966, 44, 2677.
I s AIAr,, Icosahedral? Dario A. Estrin, Li Liu, and Sherwin J. Singer* Department of Chemistry, The Ohio State University, Columbus, Ohio 4321 0- 11 73 (Received: January 18, 1992)
The energy of AlAr12with a central aluminum atom is significantlylowered when the cluster distorts from icosahedral symmetry. A range of scenarios is possible, including loss of all symmetry. The Born-Oppenheimer surface for AIArI2is estimated using combined Hartree-Fock simulated annealing techniques. Also, the stability of high symmetry configurations and the possibility of nonadiabatic effects are explored in an approximate model based on a linear expansion in just three electronic states that are degenerate in icosahedral symmetry.
I. Introduction
AlAr12appears as a “magic number” cluster in these experiments. The distortion of AIArlz can be viewed in a simple physical Our answer to the question posed in the title is that the lowmanner as the response of the cluster to the egg-shaped electron energy isomers of AlAr12 are derived from the icosahedral distribution of the Al(2P) atom in its ground state. Where there structure, but that the energy is significantly lowered when the is just a single low-energy icosahedral isomer of A1Arl2 when cluster distorts from icosahedral symmetry. The distortion orialuminum is modeled as a spherical atom, in a more realistic ginates along two sets of Jahn-Teller active H, modes of the triply treatment we find many “electronic” isomers that represent difdegenerate Al(*P) atom in an icosahedral environment. The ferent packings of argon atoms around the egg-shaped (2P) aluenergy lowering upon distortion is quite large (-600-700 cm-l), minum. While Al solvated in argon may be regarded as somewhat perhaps about 30kT a t typical temperatures of van der Waals exotic, common aqueous ions like Fez+ have nonspherical ground clusters formed under experimental conditions and still several states as well.5 kT at r c ” temperature. We are motivated to understand electronic spectra of clusters with simple atomic chromophores, In the calculations reported here we treat the three aluminum like metal atoms,’V2since their simplicity offers hope of detailed valence electrons explicitly and incorporate the electrons of the comparison of theory and experiment. Some of the best experA13+core and argon atoms with pseudopotentials. Finding the imental data on purely atomic van der Waals clusters has been ground state of the valence electrons as a function of nuclear obtained for AlAr,, 1 I n I 28, by Whetton and c o - w o r k e r ~ . ~ * ~ configuration gives us the ground Born-Oppenheimer surface for 0022-365419212096-5325%03.00/0 0 1992 American Chemical Society
Estrin et al.
5326 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992
nuclear motion. The present work is mainly concerned with local minima on that surface. We search for local minima using classical mechanics as a tool, quenching classical trajectories until the particles settle into a locally stable arrangement. Simultaneous classical motion in concert with optimization of a Hartree-Fock wave function for the aluminum valence electrons is accomplished using simulated annealing techniques. These calculations provide the geometry and relative energy of many local minima found for AlAr12.Combined Hartree-Fock, simulated annealing molecular dynamics has been performed for isolated water and formamide molecules by Fielde6 Also, Dutta and Bhattacharyya have annealed electronic wave functions using Metropolis Monte Carlo technique^.^^ We also solve a different and more approximate set of dynamical equations for AlArI2based on a linear expansion of the wave function in three electronic states that transform as TI, in icafahedral symmetry. The more approximate model allows us to gauge the likelihood of nonadiabatic transitions among the three electronic states that correlate with the (2P) levels of isolated aluminum. We can precisely analyze the stability of high-symmetry configurations within the linear expansion model. Whetton et aL4 have obtained electronic spectra of AlAr, clusters by resonant two-photon, one-color ionization via the A1 3p 4s transition. The spectra are mass-resolved to the extent that fragmentation of AlAr,+ clusters with -0.6 eV exenergy can be ignored. Since the evolution of the spectrum with cluster number n is gradual, fragmentation will not alter the qualitative conclusions. Since we expect that the two-photon ionization cross section is largely determined by the resonant transition to the (?3) A1 intermediate state, we discuss the experimental spectrum as a simple absorption spectrum here. The experimental electronic spectrum4 for AlAr12exhibits a broad (300 cm-I fwhm) main peak centered near 26 000 cm-I. The transition frequencies from the two spin-orbit components of isolated (zP) A1 to the (%)-state lie at 25 236 and 25 348 ~ m - l so , ~the main absorption peak of AlAr12lies -730 cm-I to the blue of the atomic absorption line. In addition, there is a weak absorption peak further to the blue of the main peak at 26 600 cm-I, and an even weaker absorption to the red of the main peak a t 25 300 cm-I. This work suggests possible mechanisms for the broadened and shifted AlAr,, electronic spectrum. Section I1 reviews the construction of pseudopotentials by which the three aluminum valence electrons interact with the A13+and argon cores, and the wave function used to describe the valence electrons. In section I11 we present simulated annealing Hartree-Fock results for high-symmetry configurations of AIAr12. The more approximate model used to gauge nonadiabatic effects and precisely analyze the stability of the high-symmetry configurations is given in section IV.
-
11. Methods
To make calculations for AlAr12tractable we only treat the three valence electrons of the aluminum atom as active electronic degrees of freedom. The electrons of the A13+core and the Ar solvent atoms are incorporated in the form of nonlocal pseudopotentials. This reduces the problem to finding two R H F orbitals as a function of nuclear position, which is treated by simulated annealing methods. In this section we discuss the nonlocal solvent pseudopotentials in a restricted Hartree-Fock (RHF) theory framework. A. Nonlocal Argon Pseudopotential. Our pseudopotential is designed to mimic R H F calculations. Semilocal (radially local, angularly nonlocal) effective core potentials designed to mimic atomic cores like A13+ have been developed and tested against all-electron calculations.I0 Our A13+core pseudopotential is of the form VAi-ECp = C l l m ) W ( W (1) Im
and is taken from the work of Stevens, Basch, and Krauss.” We have found from our experience with NaAr, clusters that the semilocal ansatz does not work as well for the Ar solvent atoms.‘* Channel potentials ul(r) obtained by fitting electronargon elastic scattering data gave poor potential surfaces for the
NaAr dimer. Treating the argon atom in the first approximation as a frozen HartreeFock core gave much better results. The small response of the argon electrons to the solute is described by a polarization term. Czuchaj et al. have previously noted similar effects for other alkali metal-rare gas dimersI3J4 and the improvement with a frozen HartreeFock rare gas core.I5 A rather complicated nonlocal potential is generated by the frozen argon orbitals and projection operators needed to maintain orthogonality of the solute electrons to the argon core. We found that this potential could be fit to a much simpler form, still nonlocal, and thus the whole procedure could be tractable for statistical mechanical simulations.12 Our procedure could be described as a tractable but accurate implementation of the Phillips-Kleinman pseudopotential.I6 While we hope that this general strategy wil be useful in a variety of systems, the particular form of the fit is tailored to each system for reasons of efficiency. Here we construct a solvent pseudopotential that takes advantage of the physical characteristics of AlAr12. There are two R H F orbitals for the three A1 valence electrons, one for the two spin-paired electrons that becomes the 3s-orbitals in isolated aluminum and another that becomes a 3p-orbital for the isolated atom. Of course,these orbitals will distort in a cluster or condensed phase environment. The R H F equations for the orbitals in the condensed system are
where Fi is a Fock operator for orbital i (3) ~ ~ ) , , t eis the potential of the A13+core (treated with an effective core potential) and valence shell electrons,
is the potential of solvent atom 1 treated as a frozen Hartree-Fock core
4’) and K‘,‘) stand for the usual Coulomb and exchange operators associated with core orbital c of solvent atom 1. The index i labels the two aluminum valence orbitals, with i = 1 the s-orbital and i = 2 the porbital of isolated aluminum. The index c ranges over the nine R H F argon orbitals. Equation 2 must be solved subject to the constraint that 4l and c $are ~ orthogonal to all of the solvent orbitals. We will eventually solve for the orbitals variationally, and it is convenient to optimize pseudoorbitals x i from which any overlap with core orbitals q5L’ are removed by subsequent projection. +i
= (1 - C P / ) x i
(7)
mJ!Q) (4Pl
(8)
I
PI =
c
xi satisfy the modified Fock equations (1 - C P / ) F i ( l - CP,)xi = ei(l - CP/)xi, i = 1, 2
The pseudoorbitals /
/
/
(9)
Assuming that overlap between orbitals of different solvent molecules can be neglected (P,FiPI,= 0 for I # 19, the above equation reduces to IFi
+ c [ - P , F i - FiP/ + P,FjP/ + P/t,]JXj= /
-j/2V2 +
fl&\ute
+ C[q2,l + 6’AcJ~= i tixi I
(10) (11)
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5321
Is AlAr12Icosahedral? TABLE I: Argon PseudopotentialParameters" i= 1 i=2 i=3
function for the three aluminum valence electrons with respect to the Hamiltonian
~~
C,
a, C,
a,
R,
6.587216 3.710697
5.486964 4.118423
0.668297 0.655356
11.129240 0.902738 0.055387
0.149875 0.201785 0.247789
-3.219584 -0,787563 -0.063487
i=4
3
m=l
"The pseudopotential has exactly the same form as in our previous work on the Na-Ar system.'* All parameters are unchanged except for the ones given here, which replace Table 2 in our earlier work.12
The effective potential from solvent atom L has been broken into a local term,
and nonlocal solvent potential, V$$cl
= -E@' - P/Fj - FjPI C
+ P/FjP/ + P/ej
(13)
The local and nonlocal solvent potentials were fit to the same local and nonlocal ansatz used in our earlier work on the NaAr, system12 (see Table I). Argon Hartree-Fock orbitals were taken from the work of Roos and Siegbahn." The local part is spherically symmetric and presents no special problems. We have made further approximations in the nonlocal part. In particular, projections of core orbitals onto the A13+ effective core potential were neglected.
I4k1
P/vAl-ECP
e
O,
vAl-ECPP/
(14)
We have verified that this is a very good approximation.I2 Cross terms in P/F,and F,P, involving overlaps between orbitals of different solvent molecules are also neglected.
The above approximation would not be justified for an ionic solvent. However, the Coulomb operators cancel the nuclear attraction for the neutral argon solvent a t long range, and we expect that this is also an excellent approximation. Under these two approximations core projections onto the valence electron Fock operators reduce to P/Fl
e r
with similar results for FjP,. The nonlocal solvent potential E#clreven with the above approximations, is still state-dependent. The state dependence is removed (i.e., V$$cl V(nlnlcl) upon fitting to a computationally convenient nonlccal operator. (See Table I and our earlier work.'*) The key point is that matrix elements of ( x ~ ~ ~ can ~ ~bel c reproduced by matrix elements of a nonlocal operator that is independent of the orbital index i. This is possible because E!$!lcl acts on a different orthogonal subspaces for each i. The major state dependence contained in Fiis still explicitly retained in the pseudo-Fock equations ((9)-(11)); only the nonlocal part of the solvent pseudopotential arising from core projections onto F, is further modified in this manner. The advantage is that the solvent pseudopotential can now be written as a one-body (nonlocal) external potential. Under these conditions, the Fock equations are equivalent to variational minimization of an R H F wave
-
1 2
H o =E h ( m ) + -
-1.881033 2.126277
5
m.m'=l (m+"J
1 Irm - rmtl
(18)
It is desirable for us to return to the variational formulation to make use of simulated annealing techniques in molecular dynamics simulations. The fit to the nonlocal ansatz (Table I) is optimized to reproduce matrix elements of fi'& between valence electron orbitals. It is not meant to be a accurate globally, but only over the Hilbert space typical of the aluminum valence orbitals.'* These valence orbitals do not drastically change when the aluminum atom is solvated by argon atoms. Therefore, it is appropriate to develop a nonlocal operator that reproduces matrix elements of V$!$cl between the 3s and 3p orbitals of the isolated aluminum atom held at various distances from the solvent argon atom. The orbital energy ei appearing in eq 13 was also held at the isolated aluminum value. We took the isolated aluminum pseudoorbitals and energies from the work of Stevens, Basch, and Krauss." B. Solvent Polarization. Dispersion forces are not included at the RHF level, yet they are needed to describe the weak van der Waals attraction between the metal and rare gas atoms. We follow the common practice of adding the asymptotic polarization response of each solvent atom I to the electric field E,. produced _ by A13+ and the three valence electrons. up0l=
aAr
--l-CIE/12 I
The functions fAr and f, compensate for the fact that the asymptotic polarization interaction in eq 20 is not sensible at short range and would cause divergences if it were not cut off. The polarization response of the A13+ core, which is much less polarizable than an argon atom, is neglected. The polarization interaction given above is that of solvent atoms that respond instantaneously to the motion of the valence electrons. The adiabatic limit should be appropriate for AlAr, since electronic excitation energies for aluminum are well below that of argon. Interactions between induced dipoles on argon atoms are neglected. This should be an excellent approximation in high-symmetry clusters. However, low-symmetry environments lead to an asymmetric electron distribution on the aluminum, and hence to a static dipole moment. This moment induces a stronger polarization response from the solvent atoms. We suspect that our model overbinds these low-symmetry clusters, a t least in part because interactions between induced moments on solvent atoms ~ are ~ neglected. x i ~ The cutoff parameters Roand ro (Table I) were chosen in an empirical manner so that the dimer potential curves would match experimental data. Spectroscopic constants for the AlAr dimer have been determined by Gardner and Lester19 and by Callender, Mitchell, and Hackett.20 The experiments are consistent with each other, but considerable uncertainty in the well depths remain since only a limited number of vibrational states were accessed in these experiments. The ground Born-Oppenheimer surface has a well depth in the neighborhood of 170 cm-1.19920 The equilibrium bond distance has been estimated to be 6.6 ao. We
Estrin et al.
5328 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 chose cutoff parameters (Table I) that yield a ground-state well depth of 167 cm-l and an equilibrium bond distance of 7.05 ao. The choice of &, and ro are, within reasonable bounds, determined by comparison with experimental data, and would be revised if more refined experimental information would become available. After the electric field in eq 21 is inserted into eq 22, cross terms lead to two-electron operators, one-electron operators, and terms that only depend on nuclear coordinates. Since the two-electron operators factor, we can assemble the two-electron terms from products of one-electron matrix elements. C. C o r d o r e Interactions. Interactions between the closedshell argon atoms are described by a Lennard-Jones potential with standard radius and well depth.21 The interaction between argon atoms and the A13+core is estimated using a Gombas-typedensity functional f ~ r m u l a , ~ ~ . ~ ~
where pN3+ and pAr are aluminum core and argon densities taken from the original work of G o m b a ~ .The ~ ~ results for AlAr,, are less sensitive to the interaction between the metal ion core and argon than in our previous work on alkali metal-rare gas clusters1*2 because of the larger number of valence electrons in the present case. D. HartreeFock Simulated Anneabg. To search for locally stable isomers of the A1Ar12cluster we solve classical equations for nuclear motion on the ground Bom-Oppenheimer surface. The Born-Oppenheimer surface is obtained by minimizing
(+lrr+)/(+I+)
(25)
at each nuclear configuration, where H is the sum of Ho in eq 18 and the polarization potential of eq 20. Optimization of the electronic wave function in concert with nuclear dynamics is accomplished using simulated annealing24as implemented for molecular dynamics simulations by Car and Parrinel10.~~The total Born-Oppenheimer energy of the cluster also includes A13+-Ar core-core interactions from eq 24 and Lennard-Jones interactions between argon atoms, but these are independent of the valence electronic wave function The wave function for the three A1 valence electrons is the antisymmetrized RHF product wave function
+.
= (1/JV)AI41$1421
(26)
where JV is a normalization constant and A is the antisymmetrizer. The Hartree-Fock orbitals 41and 42were expanded as a sum over the same eight floating Gaussian basis functions. The floating Gaussian basis26was introduced into simulated annealing calculations by Sprik and Klein,27-29 and by Pederson et al.30 We optimize exponents and positions of the eight Gaussians, plus a set of linear coefficients for each bf the two orbitals. Although we assumed orthogonal, canonical R H F orbitals to derive the pseudo-Hamiltonian in eq 18, we found it convenient in this application not to constrain the orbitals to be orthogonal during the variational calculation. Full energy expressions for nonorthogonal orbitals were evaluated. For example, the expectation value of a symmetric two-electron operator for the aluminum valence electrons becomes
($1
c dm,m?l+) =
mSm’
(1/JV2)[(41411~4141)(42142)+ 2(41421A4142)(41l41) 2(41411gJ4241)(42141) - (4142lgl4241)(41141)I (27)
((4,4jlgl4&4d= (4,(1)4j(2)Ig(l,2)14&(1)4~(2))), malization integral is
and the nor-
(+I+)= (1/JV2)[(41141)2(4*142) - (41142)2(41141)1
(28)
Real orbitals are assumed in the above expressions. For an isolated aluminum atom we obtain a minimum energy in which d2was a sum of Gaussians lined up along a particular axis. The Gaussians were in pairs about the origin with coefficients
Figure 1. Apparent minimum-energy configuration of AlAr12with Dz,, symmetry obtained by simulated annealing. The spheres represent positions of argon atoms (radius is arbitrary). A contour surface at 20% of the maximum aluminum valence electron density is shown.
of opposite sign so that 42has a nodal plane. Gaussians near the origin had very small coefficients in $2. The orbital $1 consisted of Gaussians near the origin, plus a mixture of the paired Gaussians with the same sign. Therefore, 41for isolated A1 was a slightly prolate ellipsoid in which the major axis was perpendicular to the reflection plane of 42. The overall symmetry of the full wave function was presumably close to the desired P symmetry. It is quite possible that low-energy solutions with improper symmetry can be obtained from a pure variational approach, as we experienced in test simulated annealing calculations for twoelectron systems.31 A sufficient number of basis functions is essential. In our calculations we occasionally experienced problems with variational collapse of 42to dl. Therefore, we included a penalty function of the form 1.796
I(61142)12 (41141)(42142)
the prefactor based on numerical experience. We have recently described the methods by which we optimize the wave function in conjunction with molecular dynamics simulations2and do not repeat the details here. The general procedure is that of Car and Parrinel10.~~Following Sprik, we use a canonical molecular dynamics a l g ~ r i t h mto~ thermostat ~~~~ the electronic degrees of freedom at a very low temperature.
III. High-Symmetry Configurations of AIArI2 We first allowed the A1Ar12cluster to seek low-energy configurations by damping classical trajectories until the atoms came to rest. A perfectly icosahedral arrangement of argon atoms about the aluminum was clearly unstable, as it should be according to group theoretical a n a l y ~ i s . ~ “Two ~ ~ sets of H, vibrational modes in the icosahedral cluster are Jahn-Teller active. We obtained apparent minima of Dzhand Dsdsymmetry in this manner (Figures 1 and 2). We are careful to call these apparent minima. Using an approximate wave function in section IV we can carefully calculate the Hessian matrix of the potential in positions of D2h and DSdsymmetry and find saddle points in that case. Unfortunately, because electronic parameters “rattle” around the Bom-Oppenheimer minimum, we camiot numerically differentiate the force with enough accuracy in our simulated annealing calculations to obtain a Hessian matrix suitable for stability analysis. While damped classical trajectories head toward DZhand D5d configurations, we cannot be absolutely sure these are not saddle points as well. The aluminumargon distances break into three sets in the Du cluster. The four argons closest to the major axis of the ellipsoidal electron density are 7.97 a. from the aluminum, the four argons
Is AlAr12Icosahedral?
The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5329 E(cm-')
0
-200
-400
Figure 2. Apparent local minimum of AlAr12 with Djd symmetry obtained by simulated annealing. The spheres represent positions of argon atoms (radius is arbitrary). A contour surface at 20% of the maximum aluminum valence electron density is shown.
-600
ltl
Dfid
DPh
Figure 4. Comparison of the energy of AlAr12 in the 4 , Dzh, and DSd configurations using (a) valence electron wave functions obtained from full simulated annealing at that configuration (solid line), (b) the wave function obtained for the minimum-energy icosahedral configuration in all three structures (dashed line), and (c) the wave function obtained at the D2h configuration in all three structures (dot-dash line). The ground-state wave function is transferable between the DZhand DSd configurations, but not with the z h structure. The zero of energy is taken to be the minimum icosahedral cluster energy.
cluster is near a symmetric energy minimum which is explored in the following section.
Figure 3. Minimum-energy structure of AlAr12 constrained to have icosahedral symmetry as obtained from simulated annealing. The spheres represent positions of argon atoms (radius is arbitrary). A contour surface at 20% of the maximum aluminum valence electron density is shown.
lying along the equatorial plane (Figure 1) are 6.73 a,from the aluminum, and the remaining four are at intermediate distance 7.45 a,,. In DSdA1Ar12the argons break into two groups (Figure 2), two along the axis at 7.90 from the central metal atom, and two rings of five at 7.24 a. from the aluminum. To gauge the Jahn-Teller stabilization energy, we found the minimum energy of AlAr12constrained to have icosahedral symmetry (Figure 3). That is, we let the cluster relax along a one-dimensional breathing mode of Ih A1Ar12. The equilibrium distance was found to be 7.42 and the major axis of the electron density was oriented toward the midpoint of an argonargon bond (Figure 3). The D2* and DSdconfigurations had energies which lay 593 and 328 cm-l, respectively, below the minimum icosahedral energy. For typical beam cluster temperatures (-20-30 K) these Jahn-Teller stabilization energies are considerable, in the neighborhood of 30kT. We would expect AlAr12in a beam to rarely approach the icosahedral configuration on energetic considerations,and even more so when entropic effects are considered. The wave functions of the D2hand DSdconfigurations are quite similar, and both are different from the minimum energy Ih which is "squashed" at its extremities. This is apparent from the Gaussian parameters of the annealed wave function, but is barely visible in the figures. The DZhand DShwave functions can be interchanged with little effect on the energy, as shown in Figure 4. This suggests an approximation scheme for AlAr12when the
IV. Low-Symmetry AlAr12: Approximate Linear Expansion In this section we develop and study a model of AlAr12that is more approximate than the full simulated annealing calculation. The advantage is that the electronic wave function is fairly well described by only three basis function coefficients, so that we can easily calculate second derivatives of the Born-Oppenheimer potential surface and analyze the stability of structures of various symmetries. We observed in the preceding section that the ground valence electron wave function in DZhand DSdAlAr12configurations is nearly the same, but the calculated energy is noticeably different from the annealed electronic wave function at the minimum-energy icosahedral geometry. A common procedure in Jahn-Teller calculations is to adopt a crude Born-Oppenheimer basis set of degenerate states at the high-symmetry p ~ i n t . ~ ~ *The ~ Oabove observations suggest that this procedure will not be very accurate for MATl2. Because the distortion and Jahn-Teller stabilization energy is so large, the electronic ground state at the DZhand DSd configurations would be poorly described as a linear combination of the members of the ground-state triplet. In effect there is strong mixing of different angular momentum components going from Ih to either or DSd. This mixing is captured in the full simulated annealing calculations, but not in a small linear expansion. The data in Figure 4 suggests that a basis of three states formed from the ground state at either the Dur and Dsdconfiguration (and not the icosahedral point) would perform much better at Zowenergy configurations awayfrom the icosahedral point. Therefore, we adopt the following somewhat unorthodox procedure for constructing approximate wave functions for a Jahn-Teller distorted system: We form the total wave function as a linear combination of three states that are triply degenerate members of the T1, representation at icosahedral configurations.
However, the basis states are the ground state at the DZh configuration (say, with the major axis of the electron density along the z-axis), and this state rotated 90° to both the x-axis and y-axis. This basis provides an improved description of low-energy con-
5330 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992
Estrin et al. E(cm-‘)
a#@
3000 1
I
b
-e-
C
\ OI
I
-1000
, 0
10
20
30
40
50
time (ps)
Figure 5. Three unstable modes of the &, structure in the linear expansion model (section IV). In each of the panels the length of the “stick” emerging from the atomic positions indicates the relative magnitude of atomic displacements along each of the unstable modes. The position of argon atoms are represented by spheres of arbitrary radius. The position of the aluminum and the orientation of the electron density (as specified by the coefficients c,, c,,, and c, in eq 32) is indicated by two overlapping spheres near the center of the cluster. The most unstable mode (a) takes the symmetry down to C, as the aluminum departs from the center of the cluster. The second most unstable mode (b) is primarily a twist of the cluster about the major axis of the electronic density. The least unstable mode (c) can be pictured as a tilt of the anisotropic electron cloud away from the midpoint of a bond, where it is located at the DZh point.
figurations like DZhand DSd,but the energy of structures like Ih are overestimated. Since we are most interested in characterizing stable configurations and least interested in rarely sampled structures, this is acceptable. It would presumably make very little difference if we formed our basis states from the DSdground state instead of the DZhground state. We emphasize that the linear expansion does not have the same flexibility of the full simulated annealing calculation. In particular, distortion of the electron cloud which breaks P-like symmetry is not allowed in the linear expansion over just three states. Therefore, the results from the two models may be different in some respects. Both are approximations and must be interpreted with some caution. Since we can analyze the stability within the linear expansion model, we exploit this aspect to learn what possible scenarios might describe the departure of AlAr12from icosahedral symmetry. A. DZhand Dsd Revisited. We obtain a 3 X 3 Hamiltonian matrix using the wave function in eq 32. This matrix can be diagonalized to obtain ground and excited Born-Oppenheimer surfaces within the linear expansion approximation. In both the DZhand D5hconfigurations the ground state is nondegenerate. Therefore, we can take derivatives of the lowest energy surface near these points without concern about cusps. By numerically differentiating the force, we obtain the Hessian in mass-weighted coordinates. This analysis is performed at minimum-energy structures with DZhand DSdsymmetry within the linear approximation. We obtain normal modes by diagonalization of the Hessian. From this analysis, we find that both the D2,, and DSdconfigurations are unstable. There are three unstable modes in DZh symmetry, and seven unstable modes at the DSdpoint (in addition to six modes of zero frequency in each case). These unstable modes are very difficult to pick up without examining second derivatives of the potential. In our experience, these structures could appear to be “apparently” stable from examining damped
Figure 6. Evolution of the three energy eigenvalues given by the linear expansion model (section IV) for AIArlz during a molecular dynamics simulation. The aluminum atom remained in the center of the cluster during the entire run. The average kinetic energy corresponds to a temperature of 17.4 K. The zero of energy is that of the minimum-energy icosahedral structure.
classical trajectories. The three unstable modes at the DZhpoint are explained in Figure 5. They involve displacement of the aluminum away from the center of the cluster, or a tilt or twist of the cluster so that the argons do not directly face the “long” end of the electron density. and We verified that local minima of lower energy than the D5dconfigurations can be found by displacement of the cluster along the unstable modes followed by conjugate gradient minimization. In each case, a point was reached where the cluster had no discernible symmetry. By choosing different combinations of displacements from the high-symmetry (DZhor DSd)structures, various local minima could be found. More local minima were found by quenching points taken at regular intervals from a molecular dynamics trajectory on the surface defined by the linear expansion model (see below). The most stable structure found in this manner had a Jahn-Teller stabilization energy of 758 an-’. We wish to stress our qualitative result that a scenario is possible for distortion of A1Ar12from icosahedral symmetry through a sequence of symmetries, possibly all the way to no symmetry. Details of the electronic structurecalculation will affect the delicate symmetry breaking, and we do not claim that either the full simulated annealing or the linac expansion in three basis functions is sufficiently accurate to determine the ultimate structure. B. Molecular Dynamics of AlAr12in the Linear Expansion Model. The ground state in the linear expansion model (see above) provides an approximate potential surface for classical trajectories of A1Ar12. We simulated a cluster at 17 K to estimate the importance of nonadiabatic effects and as a means to search for other low-energy structures. A 3 X 3 matrix was diagonalized at each. molecular dynamics step. The derivatives of the lowest eigenvalue with respect to nuclear position determined forces for the classical trajectories. The evolution of the three energy eigenvalues obtained from the linear expansion approximation during a molecular dynamics simulation of AlAr12is shown in Figure 6. The average separation of the ground state from the first excited level is 1531 cm-l over this run, indicating that nonadiabatic effects for a cluster that has found its way to the ground state are minimal. Higher symmetry points are unfavored by both energetic and entropic considerations in this model. We took quenched 50 configurations at regular intervals from the run in Figure 6 and obtained many different minimum-energy configurations. The lowest energy configuration so obtained had a Jahn-Teller stabilization energy
The Journal of Physical Chemistry, V O ~96, . NO. 13, 1992 5331
Is AIArlz Icosahedral? E(cm”)
a real or imaginary frequency; that is decided theoretically by electronic structure calculations. While our pseudopotentials and techniques for calculating the electronic wave function should give a good indication of the magnitude of the distortion, we do not want to overinterpret the results because the final twists and turns as the cluster seeks a minimum-energy configuration are quite delicate. Rather, our results portray possible scenarios in the lowering of the symmetry from the icosahedral starting point.
300011 2ooo
Acknowledgment. The calculations reported here were made possible by a grant from the Ohio Supercomputer Center. This research was supported in part by the Camille and Henry Dreyfus Foundation in the form of a Distinguished New Faculty Grant, and by N S F grant CHE-9115615.
References and Notes
,1000 0
10
20
30
40
50
t i m e (ps)
Figure 7. Evolution of the three energy eigenvalues given by the linear expansion model (section IV) for AIAr,2 during a molecular dynamics simulation. The run was initialized with the aluminum on the surface of the cluster and an argon atom occupying the central position. The aluminum was found on the surface during the entire run. The average kinetic energy corresponded to a temperature of 17.3 K. The zero of energy is that of the minimum-energy icosahedral structure.
of 758 cm-I. The other minima were distributed within 7 0 cm-’ of the lowest one. We also considered a similar molecular dynamics trajectory initialized with the aluminum on the surface of a cluster. The temperature was again close to 17 K. The evolution of the three energy eigenvalues obtained from the linear expansion model is shown in Figure 7. In this case the average separation of the ground and first excited state was only 647 cm-I, far less than for aluminum in the interior of the cluster. (Figures 6 and 7 are plotted on the same scale.) At several points the two lowest electronic states for surface aluminum come within -250 cm-’ of each other. Nonadiabatic transitions will be more likely for aluminum on the surface, although nonadiabatic effects will not be dominant in this case either. We note that the ground state of surface aluminum is not much higher than aluminum in the interior. The lowest energy surface configuration we found by quenching 50 configurations taken at regular intervals from the run in Figure 7 was 532 cm-l lower than the minimum energy icosahedral structure. It is likely that beam clusters of AlAr,, contain significant amounts of both interior and surface aluminum atoms.
V. Conclusion Our central result is that the distortion of AIArl, with a central aluminum atom from icosahedral symmetry is strongly favored. The distortion can proceed through possible sequences determined by removing symmetry elements from high-symmetry structures. We have illustrated this effect using simulated annealing HartreeFock calculations, and a more approximate model based on a linear expansion of the electronic wave function. It is possible for the cluster to be left with no symmetry. Whether this actually happens and what the ultimate configuration will be is a question left as a challenge to experimentalists and more sophisticated electronic structure calculations. The initial distortion from icosahedral symmetry is predicted as a linear Jahn-Teller effect and is certain. In other cases we observe symmetry breaking in a configuration where the ground state is nondegenerate. Both the D5! and DZhstructures are examples of the latter case. While this is common, there is no theorem in this case that tells us whether a particular mode has
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