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Density functional calculations of beryllium clusters Ben, n=2–8
Chemical Physics 262 (2000) 15±23
www.elsevier.nl/locate/chemphys
Density functional calculations of beryllium clusters Ben, n 2±8 Martin K. Beyer a,b, Leonid A. Kaledin c, Alexey L. Kaledin c, Michael C. Heaven c, Vladimir E. Bondybey a,* a
Institut fur Physikalische und Theoretische Chemie, Technische Universit at M unchen, Lichtenbergstraûe 4, 85747 Garching, Germany b Department of Chemistry, University of California, Berkeley, CA 94720-1460, USA c Department of Chemistry, Emory University, Atlanta, GA 30322, USA Received 22 May 2000
Abstract Neutral beryllium clusters Ben , n 2±8, were investigated by density functional techniques. To minimize errors, geometry optimization, frequency and energy calculations were all carried out on the same level of theory, employing a large 6-311++G(3df) basis set. The method reproduces well the experimentally known bond length and vibrational frequency of the dimer, but its binding energy is still signi®cantly overestimated. The computed trends of the vibrational frequencies, bond lengths and binding energies of the clusters as a function of the number of atoms are discussed. The binding energies are found to increase rapidly as a function of size, and approach the binding energy of the bulk metal, 54.1 kJ per bond. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The ground state beryllium atom with its 1s2 2s2 con®guration has a closed 2s shell and no unpaired electrons, and the ground state of the dimer should in the simplest MO theory have a bond order of zero [1±3]; bonding arises only due to the contribution of the excited 2p orbitals. In contrast to the dimer, beryllium is a high melting, high boiling metal, with a rather high enthalpy of atomization [4]. Due to these unusual properties, beryllium clusters have been extensively investigated by
*
Corresponding author. Tel.: +49-89-2891-3421; fax: +4989-2891-3416. E-mail address: [email protected] (V.E. Bondybey).
theoreticians, and in particular, the dimer is a benchmark problem for quantum mechanical computations [5±10]. Quite surprisingly, the only sources of experimental information available appear to be a few studies of the dimer [9,11,12]. Ab initio calculations had considerable diculties in obtaining a reasonable description of Be2 , and in fact, the ®rst reasonable estimates of its properties came not from post-Hartree±Fock methods, but from density functional calculations by Jones [13]. Even though the density functional has the tendency to overestimate signi®cantly the binding energy of the dimer, this calculation yielded reasonable values of its bond length and vibrational frequency a few years before the more extensive computations of Liu and McLean [14] or of Harrison and Handy [15].
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 2 5 6 - 1
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
While in the recent years, more than 40 theoretical studies dedicated to the dimer have appeared, much less attention was given to the larger Ben clusters. Also, despite the initial success of the density functional techniques on the dimer, there seem to be almost no computational studies extending this work to larger beryllium clusters. Among the numerous studies employing post-HF methods [16±23], only two report vibrational frequencies [21,22], both on the Hartree±Fock level. In these works, the geometry optimization had to be done at a considerably lower level of theory than the ®nal energy calculation, introducing a considerable uncertainty into the results. With the availability of faster computers and parameter®tted density functional methods like B3LYP, it now becomes feasible to perform geometry optimization, frequency and energy calculations at the same high level of theory. The question of how rapidly the cluster binding energies converge from the tenuously bound dimer to the very stable bulk metal appears to be of considerable interest. We have therefore undertaken such a study of beryllium clusters Ben , n 2±8, whose results are reported in this paper.
2. Computational methods The calculations were carried out on either an SGI Power Challenge, a DEC Alpha Station 500 or a Cray Y-MP EL using the G A U S S I A N 9 4 program package [24]. The B3LYP hybrid functional [25] as implemented in this package was used. All electrons were explicitly treated using the wellbalanced 6-311++G(3df) basis set. The basis set superposition error (BSSE) is expected to be small and was therefore ignored in these calculations. In order to enhance the probability that the global minimum will be found, we have typically carried out, for each cluster, several computations departing from dierent initial geometries. For instance, calculations were started from con®gurations occurring in the metal lattice or from structures obtained by adding an atom to the converged structure of a smaller cluster.
3. Results and discussion In order to experimentally check the reliability of the method, we have also carried out computations on the Be atom and the Be2 dimer. The present level of theory yielded energy values of ÿ14.6713143 and ÿ14.5811151 hartrees for the 2s2 1 S ground state, and the lowest triplet state (2p 3 P0 ) state of atomic beryllium, respectively. The dierence of 19 796.4 cmÿ1 is in acceptable, yet not spectacular agreement with the known 21 974.4 cmÿ1 experimental excitation energy of the triplet state [26]. Even though the beryllium dimer is generally considered to be one of the most demanding problems in computational chemistry, as previously shown [13,26], standard DFT methods appear capable of yielding reasonable values for its bond length and vibrational frequency. This is con®rmed by our results, with the calculated re comparing favorably with the value of 2.48 A [11]. When experimental bond length of 2.45 A the usual ``scaling factor'' of 0.95 is applied to the calculated vibrational frequency of 289 cmÿ1 , the agreement with the experimental value of 276 cmÿ1 is also satisfactory. As is usually the case with DFT studies, our calculation substantially overestimated the binding energy, which in the speci®c case of Be2 , appears to be the most dicult property to reproduce. Still, even though the result of 18 kJ molÿ1 obtained with the B3LYP hybrid technique and relatively large basis set used throughout our computations overestimates the De value by a factor of 1.5, this is considerably closer to the best current estimates of the binding energy, which range from 10 to 12.6 kJ molÿ1 [5± 10], than that in the previous BLYP study [27]. It should be noted that while a factor of 1.5 in the computed energy may at ®rst sight seem quite unsatisfactory, the result appears much more favorable when one realizes that the absolute error is only some 6±7 kJ molÿ1 , a value which would in most cases be considered quite acceptable from the chemical point of view. The remaining error may in part be due to the BSSE, which, even though as mentioned above is expected to be small, may affect the weakly bound ground state of Be2 . In contrast with the singlet, low lying triplet states of Be2 are strongly bound. Triplet compu-
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
17
tation, starting with an initial re value of 2 A ÿ1 converged to a state located 5530 cm above the X 1 Rg ground state, probably the lowest 3 Pg [9], with an re 2:0015, and a harmonic frequency 676 cmÿ1 . The relatively reasonable agreement obtained for the Be atom, and even for the Be2 ground state which is a notoriously dicult problem to compute, encourages us to believe that the chosen level of theory, namely, B3LYP in combination with a large basis including diuse and polarization functions, should be capable of providing satisfactory results for the larger, neutral beryllium clusters. These are expected to be much more strongly bound so that the problems associated with the ®ne balance between bonding and antibonding interactions which plague the ground state computations of the dimer should be absent. 3.1. The singlet states of beryllium clusters The optimized geometries of the singlet state Ben clusters are shown in Fig. 1, with the computed total energies being listed in Table 1. Previous MP2 results [22] are also shown for comparison. The calculated harmonic vibrational frequencies, their degeneracies and infrared intensities are given in Table 2. Regardless of the starting geometry, the n 3±5 clusters converge smoothly to an equilateral triangle, tetrahedron, and trigonal bipyramid, respectively, with our calculated DFT bond lengths being systematically shorter than those obtained in the about 0.05 A MP2 study [22]. In contrast with the symmetric small clusters, the optimized geometries of the n 6 and 7 species are signi®cantly distorted, with their structures deviating signi®cantly from the structure of solid metal, and also exhibiting differences compared with the MP2 results. Thus n 6 is neither an octahedron, the most compact Be6 con®guration occurring in the metal lattice, nor a Be5 trigonal bipyramid with one of the faces capped. Starting from a slightly distorted octahedron, the optimization converges to strongly distorted C2v structure which can be described as a vertical Be2 dimer molecule (bond length 2.481 A) ``solvated'' in the horizontal plane by two Be2 di The mers with very short bond lengths (1.854 A).
Fig. 1. Three-dimensional structures of Ben clusters, n 2±8. Geometries have been optimized on the B3LYP/6-311++G(3df) level of theory.
Be7 cluster converges to a distorted pentagonal bipyramid, again resembling a vertical beryllium dimer, solvated in the horizontal plane this time by ®ve additional beryllium atoms. Unlike the symmetric smaller clusters, the distorted n 6 and 7 species have geometries which do not appear in solid HCP beryllium metal. This tendency towards distorted geometries was pointed out previously by Bauschlicher and Pettersson [17] in their calculations on Be13 . The largest cluster considered in this work, Be8 , reverts to a symmetric and rather compact con®guration, basically a doubly capped octahedron with a D3d symmetry. Earlier studies, conducted at the HF or MP2 level, sometimes employed for the geometry optimization smaller basis sets, and then used a larger
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
Table 1 and point groups of Ben , n 2±8, B3LYP/6-311++G(3df) level of theory comparison with earlier MP2 results [22] Bond lengths (A) n
Singlet
MP2a
Symmetry
Triplet
Symmetry
Be2 Be3
2.481 3 2:158
2.218
D1h D3h
D1h C2v
Be4
6 2:027
2.083
Td
Be5
3 1:982
2.031 2.085
D3h
2 2:182 2.482 2.562 4 2:029 4 2:039
C2v
2.0015 2 1:961 1 2:211 4 1:985 2 2:162 2 2:008 1 2:333 4 2:000 2 2:007 6 2:117 6 1:956
16 1:866 )2.469
C1
6 2:041 Be6 b
Be7 Be8
2 1:854 2.324 2.959 4 2:010 4 2:058 1 2:453 16 1:865 ÿ2.469c 6 1:964 6 2:081 6 2:108
D2d C2v
D3d
D3d
a
MP2 results taken from Ref. [22] are given for comparison. Quintet Be6 is computed and has an Oh symmetry with bond lengths 12 2:040 A. c Singlet Be7 is a distorted pentagonal bipyramid, the bond lengths between atoms 1±5 in the ``equatorial plane'' are 2.143, 2.079, 2.171, 2.084, 2.149; bond lengths to atom 6 above the plane: 1.865, 1.986, 2.320, 2.341, 2.002; atom 7 below: 2.469, 2.165, 1.899, 1.891, 2.143 distance 6±7: 2.448 A. A; b
Table 2 Harmonic vibrational frequencies (cmÿ1 ) with their degree of degeneracy, and, in parentheses, infrared intensities (km molÿ1 ) of Ben , n 2±8, calculated at the B3LYP/6-311++G(3df) level of theory Be2 Be3 Be4 Be5 Be6 Be7 Be8
289(0) 2 466(1.7) 2 495(0) 2 344(1.5) 91(12) 598(0.1) 26(2.3) 387(54) 687(0.3) 2 164(1.2) 2 525(2.5)
534(0) 3 595(17) 2 474(0) 292(6.9) 614(16) 146(14) 429(3.0) 749(4.5) 345(0.0) 2 622(0.0)
700(0) 482(0) 324(0) 615(1) 229(2.3) 519(10) 756(0.8) 2 359(0.0) 2 629(53)
basis set to compute the ®nal energies at the optimized geometry. Since the geometry often depends sensitively upon the theoretical model and the size of the basis set, such energy computations at a level dierent from that used in the optimization process may be done at geometries diering from the true minimum, and may, in particular for the larger clusters, lead to signi®cant errors in the binding energies. To avoid such problems, the
2 590(14) 422(5.0) 621(37) 286(2.0) 529(3.5)
621(37) 495(0.1) 782(4.4) 369(5.4) 617(0.0)
743(0) 528(100) 794(0) 383(14) 636(0.0)
394(0.0) 635(0.1)
2 397(0.2) 734(223)
401(117) 779(0.0)
same basis sets were used throughout the optimization and energy computation in the present study. The atomization energies computed here which are considerably higher than those obtained in previous works may be the consequence of this self-consistent approach. As noted in Section 1, of particular interest are trends of properties as functions of size, and speci®cally, the question of how quickly do the
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
properties of the discrete small clusters converge to those of the metallic beryllium. The most dramatic change is, of course, the increase in the atomization energy per atom from the dimer (5 kJ molÿ1 ) to that of the bulk solid (324.3 kJ molÿ1 ). As seen in the third column of Table 3, the computed values rapidly increase as a function of n, but the increase is far from smooth, and is not monotonic. Even for the largest clusters studied here, the computed energies are less than half of the 324.3 kJ molÿ1 bulk value. This is mainly due to the fact that while in the bulk solid, every atom has 12 nearest neighbors, in the small clusters, the coordination number of the atoms is much lower. Even in the largest cluster studied here, Be8 , two atoms are bound to three neighbors, while six are pentacoordinated, yielding an average coordination number of 4.5. If one considers this problem in more detail, it becomes clear that those properties that are strongly aected by atoms being on the surface will converge towards the bulk solid value rather slowly. It is instructive to remember that even for an n 1000 cluster, nearly half of the atoms will be located on its surface. In view of this eect of coordination, it is perhaps more meaningful and instructive to look at energy per bond, obtained by dividing the total atomization energy by the number of bonds. This of course may introduce in some cases a degree of arbitrariness in deciding whether two atoms are bonded or not. In the fourth column of Table 3,
19
one sees that there is a more than a factor of two increase of the bond energy for Be3 . As noted above, the DFT computations substantially overestimate the binding energy of Be2 ; if one considers the known experimental De value for Be2 , the 45 kJ molÿ1 binding energy per bond computed for Be3 in fact amounts to an increase of more than a factor of four. Already for Be4 , however, the energy per bond reaches a maximum of 67.7 kJ molÿ1 . For still larger clusters, the energy ¯uctuates somewhat, but seems to have an overall decreasing tendency, approaching the value deduced from the known 54.1 kJ molÿ1 atomization enthalpy of the closely packed bulk hexagonal beryllium. The computed trends are perhaps not dicult to understand qualitatively. The simplest MO theory predicts a zero order bond for the dimer with two bonding and two antibonding electrons. This explains the experimentally observed weak bond, and also is the underlying reason behind the diculties of theoretical computations: the bonding arises mainly from the mixing of excited con®gurations into the ground state. When a third atom is added, three Be±Be bonds can be formed; this makes an s±p rehybridization of the beryllium atoms, in spite of the high 2s±2p promotion energy, much more favorable than in the dimer. This trend is continued in the tetramer, where the addition and rehybridization of just one more atom makes it possible to form three additional
Table 3 Total equilibrium binding energies De , De per atom and per bond, incremental binding energies DDe n ! n ÿ 1 at the B3LYP/6311++G(3df) level of theorya
a
Ben
De
De /atom
De /bond
DDe n ! n ÿ 1
De b
Be2 Be3 Be4 Be5 Be6 Be7 Be8 Bulke
18.4 134.9 407.7 561.8 691.3 899.8 1105.2
9.2 45.0 101.9 112.4 115.2 128.5 138.2 324.3
18.4 45.0 67.9 62.4 53.2 56.2 61.4 54.1
18.4 116.6 272.8 154.1 129.5 208.5 205.4 324.3
12.6d 100 347 460
De values from two earlier studies are given for comparison. All energies are given in kJ molÿ1 : Best estimates of Lee et al. [20]. c MP4/6-311+G level of theory [22]. d Ref. [9]. e DH298 value from Ref. [28]. b
De c 90.3 348.6 510.3 571.0 770.4
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
metal±metal bonds, for a total of six. This is re¯ected in the large atomization energy and Dat H per bond value of Be4 . Even though the same argument could be used when proceeding to the trigonal bipyramidal Be5 , where addition of the ®fth atom again allows three more bonds to form, the trend is now reversed. The increase in atomization energy is much smaller, and the energy/bond actually decreases. This undoubtedly re¯ects another property of beryllium, well known from its chemistry: electron de®ciency. The Be atom has only two valence electrons, and thus in Be3 , there will be six electrons for three bonds or two electrons per bond available. As the clusters get larger, the number of possible metal±metal bonds increases, and naturally, the number of electrons available per bonding interaction must decrease. Thus eight electrons per six bonds or 1.33 per bond are available for Be4 , and 10 electrons per nine bonds or 1.11 per bond in Be5 . If one tries to extrapolate this trend to the bulk metal, in the HCP solid beryllium, each atom interacts with 12 nearest neighbors so that only one-third electron will be available per Be±Be interaction. The same trends discussed above in connection with the energy per bond should also be apparent when one examines the bond lengths and vibrational frequencies, and this is con®rmed by the re of the Be2 dimer, even calculations. The 2.49 A though considerably shorter than that of Li2 , is still considerably longer than the nearest neighbor in metallic beryllium. In the distance of 2.12 A triangular D3h trimer, the bond lengths are sig and this trend is ni®cantly reduced to 2.158 A, continued for n 4 and 5 clusters. Here, the bonds are further shortened, and in fact ``undershoot'' the bulk value, becoming considerably shorter nearest neighbor distance in bulk than the 2.12 A beryllium metal. For still larger clusters, this trend is again reversed; while in the large clusters, not all bonds are equivalent, and the individual lengths vary, the average bond length rapidly approaches the nearest neighbor distance of the bulk metal. The behavior of the computed harmonic frequencies and force constants can be seen in Table 2, but in view of the mixing of stretching and deformation vibrations, quantitative comparisons
are more dicult. Still, one can see that compared to the xe 289 cmÿ1 in the dimer, the larger clusters exhibit much higher frequencies, with the highest frequency stretching modes approaching 800 cmÿ1 . A consideration of the vibrational frequencies also provides some interesting insights into the cluster structures and dynamics. The structures of the n 3±5 clusters as well as that of n 8 are highly symmetric. They are quite robust and well de®ned, and the geometry optimizations smoothly converged to the global minimum. Quite dierent are the Be6 and Be7 clusters, which have low symmetries, and for which the computation convergence was found to be very laborious and slow. This behavior is also re¯ected in the computed vibrational frequencies. While all the vibrational frequencies of the former, symmetric clusters are well de®ned and rather high, each of the latter clusters, Be6 and Be7 , exhibits one very low frequency mode. As noted above, the structures of both clusters can be described as a ``vertical'' Be2 molecule, ``solvated'' in the horizontal symmetry plane by 4 and 5 additional Be atoms, respectively. In each case, the lowest computed frequency, 26 cmÿ1 in Be7 and 85 cmÿ1 in Be6 , corresponds to tilting of this ``vertical'' Be2 axis with respect to the atoms in the equatorial plane. The potential for this vibration is extremely ¯at so that in some computations, the optimization stopped before reaching the global minimum, resulting in imaginary values for this frequency. Of interest are also the incremental adiabatic dissociation energies of the beryllium clusters given in the ®fth column of Table 3. One can see huge increases of ± when one takes the experimental dimer value ± more than a factor of 10 when going from n 2 to 3, and an additional factor of more than two from Be3 to Be4 . The reasons for this were, in fact, already discussed above. In the trimer, the promotion energy needed to rehybridize the three atoms can be ``paid for'' by forming three strong metal±metal bonds, and this trend is continued in the very stable tetramer, where rehybridization of one additional atom makes possible three more metal±metal bonds. The trend then reverses sharply for Be5 and in particular for Be6 ; both clusters have geometries which do not occur in metal lattice, and are apparently unfavorable.
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
A most compact hexamer taken out of a Be lattice would be a regular octahedron, but as seen from the strong distortion, such a con®guration is not energetically favored when not stabilized by the remainder of the lattice. Furthermore, for these larger clusters, the above mentioned de®ciency of electrons may already be taking eect. Even though the n 7 cluster also has a distorted geometry quite unlike the solid, the relatively compact bipyramidal structure is apparently quite favorable, as re¯ected by the rather large DDe value. The n 8 optimized cluster again has a favorable geometry very close to the bulk, and exhibits a rather large dissociation energy. It shows that while an unsupported octahedral structure alone is not a stable minimum, already two additional atoms are sucient to make an octahedron with two faces capped the global minimum on the Be8 potential energy surface. If the atomization energy of the metal is taken as the limiting DDe value, then it is clear that even for the most stable clusters studied, the computed dissociation energies are considerably smaller. This must again be attributed to the fact that in the case of small clusters, one will be removing an apex or edge atom of relatively low coordination. 3.2. States of higher multiplicity: triplets and quintets As discussed above, the Be2 dimer is predicted to have a zero bond order by a simple MO theory, which is in agreement with the low dissociation energy found both experimentally and by quantum mechanical computations. In contrast with the
21
singlet ground state, the theory predicts numerous triplets, including the lowest one, to be strongly bound, and this is again con®rmed both by our present calculations as well as by more extensive computations elsewhere. The obvious conclusion is that the lowest triplet states of the dimer must lie signi®cantly lower than the 3 P atomic state at 21 974.4 cmÿ1 , (236.8 kJ molÿ1 ), and indeed, our computation predicts the triplet ground state to lie mere 52.5 kJ molÿ1 above the singlet ground state. In view of this, one should consider the possibility that the ground states for some of the larger clusters might in fact be states of higher multiplicity. We have therefore also carried out calculations on triplet and quintet states of clusters Ben up to n 8, with the results being summarized in Table 4. Our computations indeed yield also for the n 3±5 very low lying triplet states, located 47.4, 104.1 and mere 21.3 kJ molÿ1 , respectively, above the corresponding singlet ground states. For the hexamer, we ®nd the triplet to lie 7.6 kJ molÿ1 below the singlet, so Be6 is predicted to have a triplet ground state. The decreasing separation between the ground state and the states of higher multiplicities with increasing cluster sizes may be viewed as an indication of decreasing band gap, and approach to the metallic properties of bulk beryllium. It is interesting to note that the higher multiplicity states have not only dierent geometries but also dierent symmetries than the singlets. Thus, the lowest triplet state of Be3 at 47.4 kJ molÿ1 is not an equilateral triangle, but has a lower C2v symmetry. Similarly the Be4 and Be5 triplet states are no longer a regular tetrahedron or
Table 4 Total energies (hartrees) and symmetries of Ben clustersa
a b
n
Singlet
Triplet
1 2 3 4 5 6 7 8
ÿ14.671314 ÿ29.349621 ÿ44.065328 ÿ58.840551 ÿ73.570534 ÿ88.291168 ÿ103.041917 ÿ117.791473
ÿ14.581115 ÿ29.329624 ÿ44.047287 ÿ58.800889 ÿ73.562422 ÿ88.293949 ÿ103.031665 ÿ117.786126
D3h Td D3h C2v C1 D3d
Bold letters indicate ground state values. Single-point calculation with singlet geometry.
Quintet
C2v D2d C2v D3d C1 D3d
ÿ44.027951 ÿ58.721531 ÿ73.500583 ÿ88.289124 ÿ103.036859 ÿ117.739286b
C2v C2v C2v Oh C2v D3d b
22
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
a trigonal bipyramid, but have lower symmetries, D2d and C2v , respectively. Interestingly, unlike the C2v singlet, the triplet hexamer has a higher, D3d symmetry. The quintet state of Be6 has a still higher, Oh symmetry, but lies 5.4 kJ molÿ1 above the singlet, and 12.7 kJ molÿ1 above the computed triplet ground state. Up to now, with the exception of the Be dimer, very little is known about the larger clusters experimentally. The harmonic frequencies computed here might provide some help in identifying and characterizing these species spectroscopically. Unfortunately, the vibrational frequencies lie relatively far in the infrared with their intensities being in most cases low, and this may be one of the reasons why they have not yet been detected in matrix experiments. Perhaps, the most favorable cases are the frequency near 528 cmÿ1 of the pentamer (intensity 100 km molÿ1 ) or the 734 cmÿ1 asymmetric stretch of the octamer (intensity 223 km molÿ1 ). In searching for these absorptions, the above mentioned factor of 0.95 derived from the beryllium dimer values could be used to correct the listed frequencies, but clearly applying the same scaling to every mode without worrying about the nature of the motion involved is at best a very crude approximation. Even if the vibrations should be detected, their assignment to a cluster of a particular size would remain a problem; the best opportunity would seem to oer an experiment with size selected clusters.
4. Conclusions While the Be2 dimer ground state is only weakly bound, the binding energies of larger clusters rapidly increase, with the energy per bond actually reaching a maximum for the tetramer. The relative energy of excited higher multiplicity states decreases rapidly with n, with Be6 being computed to have a triplet ground state. This may perhaps be viewed as re¯ection of the decreasing energy gap and approach to metallic properties. Some cluster sizes have stable and rigid high symmetry structures, others, for instance, singlet Be6 and Be7 are characterized by distorted geometries with a ¯at
potential surface and very low frequency vibrational modes.
Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. M.K.B. thankfully acknowledges a Feodor Lynen Research Fellowship from the Alexander von Humboldt-Foundation.
References [1] G. Herzberg, Zeitschrift f ur Physik 86 (1929) 601. [2] G. Herzberg, Molecular Spectra and Molecular Structure I, second ed., van Nostrand-Reinhold, New York, 1950. [3] R.S. Berry, S.A. Rice, J. Ross, Physical Chemistry, Wiley, New York 1980. [4] CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1992. [5] I. Rùeggen, J. Alml of, Intern. J. Quant. Chem. 60 (1996) 453 and references therein. [6] J. St arck, W. Meyer, Chem. Phys. Lett. 258 (1996) 421. [7] L. F usti-Molnar, P.G. Szalay, Chem. Phys. Lett. 258 (1996) 400. [8] J.M.L. Martin, Chem. Phys. Lett. 303 (1999) 399. [9] L.A. Kaledin, A.L. Kaledin, M.C. Heaven, V.E. Bondybey, J. Mol. Struct. (Theochem) 461±462 (1999) 177 and references therein. [10] R.J. Gdanitz, Chem. Phys. Lett. 312 (1999) 578. [11] V.E. Bondybey, Chem. Phys. Lett. 109 (1984) 436. [12] V.E. Bondybey, Science 227 (1985) 125. [13] R.O. Jones, J. Chem. Phys. 71 (1979) 1300. [14] B. Liu, A.D. McLean, J. Chem. Phys. 72 (1980) 3418. [15] R.J. Harrison, N.C. Handy, Chem. Phys. Lett. 98 (1983) 97. [16] G. Pacchioni, D. Plavsic, J. Kouteck y, Ber. Bunsenges. Phys. Chem. 87 (1983) 503. [17] C.W. Bauschlicher Jr., L.G.M. Pettersson, J. Chem. Phys. 84 (1986) 2226. [18] R.J. Harrison, N.C. Handy, Chem. Phys. Lett. 123 (1986) 321. [19] M.M. Marino, W.C. Ermler, J. Chem. Phys. 86 (1987) 6283. [20] T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 92 (1990) 489. [21] H. Kato, E. Tanaka, J. Comp. Chem. 12 (1991) 1097. [22] P.V. Sudhakar, K. Lammertsma, J. Chem. Phys. 99 (1993) 7929. [23] W. Klopper, J. Alml of, J. Chem. Phys. 99 (1993) 5167.
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23 [24] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople,
GAUSSIAN 94,
[25] [26] [27] [28]
23
Revision D.4, Gaussian, Pittsburgh, PA, 1995. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. C. Moore, Atomic Energy Levels, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. US, 35/V.I, 1971. C.W. Murray, N.C. Handy, R.D. Amos, J. Chem. Phys. 98 (1993) 7145. M.W. Chase Jr., NIST-JANAF Thermochemical tables, Fourth ed., J. Phys. Chem. Ref. Data, Monograph 9, 1998.
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Density functional calculations of beryllium clusters Ben, n 2±8 Martin K. Beyer a,b, Leonid A. Kaledin c, Alexey L. Kaledin c, Michael C. Heaven c, Vladimir E. Bondybey a,* a
Institut fur Physikalische und Theoretische Chemie, Technische Universit at M unchen, Lichtenbergstraûe 4, 85747 Garching, Germany b Department of Chemistry, University of California, Berkeley, CA 94720-1460, USA c Department of Chemistry, Emory University, Atlanta, GA 30322, USA Received 22 May 2000
Abstract Neutral beryllium clusters Ben , n 2±8, were investigated by density functional techniques. To minimize errors, geometry optimization, frequency and energy calculations were all carried out on the same level of theory, employing a large 6-311++G(3df) basis set. The method reproduces well the experimentally known bond length and vibrational frequency of the dimer, but its binding energy is still signi®cantly overestimated. The computed trends of the vibrational frequencies, bond lengths and binding energies of the clusters as a function of the number of atoms are discussed. The binding energies are found to increase rapidly as a function of size, and approach the binding energy of the bulk metal, 54.1 kJ per bond. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The ground state beryllium atom with its 1s2 2s2 con®guration has a closed 2s shell and no unpaired electrons, and the ground state of the dimer should in the simplest MO theory have a bond order of zero [1±3]; bonding arises only due to the contribution of the excited 2p orbitals. In contrast to the dimer, beryllium is a high melting, high boiling metal, with a rather high enthalpy of atomization [4]. Due to these unusual properties, beryllium clusters have been extensively investigated by
*
Corresponding author. Tel.: +49-89-2891-3421; fax: +4989-2891-3416. E-mail address: [email protected] (V.E. Bondybey).
theoreticians, and in particular, the dimer is a benchmark problem for quantum mechanical computations [5±10]. Quite surprisingly, the only sources of experimental information available appear to be a few studies of the dimer [9,11,12]. Ab initio calculations had considerable diculties in obtaining a reasonable description of Be2 , and in fact, the ®rst reasonable estimates of its properties came not from post-Hartree±Fock methods, but from density functional calculations by Jones [13]. Even though the density functional has the tendency to overestimate signi®cantly the binding energy of the dimer, this calculation yielded reasonable values of its bond length and vibrational frequency a few years before the more extensive computations of Liu and McLean [14] or of Harrison and Handy [15].
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 2 5 6 - 1
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
While in the recent years, more than 40 theoretical studies dedicated to the dimer have appeared, much less attention was given to the larger Ben clusters. Also, despite the initial success of the density functional techniques on the dimer, there seem to be almost no computational studies extending this work to larger beryllium clusters. Among the numerous studies employing post-HF methods [16±23], only two report vibrational frequencies [21,22], both on the Hartree±Fock level. In these works, the geometry optimization had to be done at a considerably lower level of theory than the ®nal energy calculation, introducing a considerable uncertainty into the results. With the availability of faster computers and parameter®tted density functional methods like B3LYP, it now becomes feasible to perform geometry optimization, frequency and energy calculations at the same high level of theory. The question of how rapidly the cluster binding energies converge from the tenuously bound dimer to the very stable bulk metal appears to be of considerable interest. We have therefore undertaken such a study of beryllium clusters Ben , n 2±8, whose results are reported in this paper.
2. Computational methods The calculations were carried out on either an SGI Power Challenge, a DEC Alpha Station 500 or a Cray Y-MP EL using the G A U S S I A N 9 4 program package [24]. The B3LYP hybrid functional [25] as implemented in this package was used. All electrons were explicitly treated using the wellbalanced 6-311++G(3df) basis set. The basis set superposition error (BSSE) is expected to be small and was therefore ignored in these calculations. In order to enhance the probability that the global minimum will be found, we have typically carried out, for each cluster, several computations departing from dierent initial geometries. For instance, calculations were started from con®gurations occurring in the metal lattice or from structures obtained by adding an atom to the converged structure of a smaller cluster.
3. Results and discussion In order to experimentally check the reliability of the method, we have also carried out computations on the Be atom and the Be2 dimer. The present level of theory yielded energy values of ÿ14.6713143 and ÿ14.5811151 hartrees for the 2s2 1 S ground state, and the lowest triplet state (2p 3 P0 ) state of atomic beryllium, respectively. The dierence of 19 796.4 cmÿ1 is in acceptable, yet not spectacular agreement with the known 21 974.4 cmÿ1 experimental excitation energy of the triplet state [26]. Even though the beryllium dimer is generally considered to be one of the most demanding problems in computational chemistry, as previously shown [13,26], standard DFT methods appear capable of yielding reasonable values for its bond length and vibrational frequency. This is con®rmed by our results, with the calculated re comparing favorably with the value of 2.48 A [11]. When experimental bond length of 2.45 A the usual ``scaling factor'' of 0.95 is applied to the calculated vibrational frequency of 289 cmÿ1 , the agreement with the experimental value of 276 cmÿ1 is also satisfactory. As is usually the case with DFT studies, our calculation substantially overestimated the binding energy, which in the speci®c case of Be2 , appears to be the most dicult property to reproduce. Still, even though the result of 18 kJ molÿ1 obtained with the B3LYP hybrid technique and relatively large basis set used throughout our computations overestimates the De value by a factor of 1.5, this is considerably closer to the best current estimates of the binding energy, which range from 10 to 12.6 kJ molÿ1 [5± 10], than that in the previous BLYP study [27]. It should be noted that while a factor of 1.5 in the computed energy may at ®rst sight seem quite unsatisfactory, the result appears much more favorable when one realizes that the absolute error is only some 6±7 kJ molÿ1 , a value which would in most cases be considered quite acceptable from the chemical point of view. The remaining error may in part be due to the BSSE, which, even though as mentioned above is expected to be small, may affect the weakly bound ground state of Be2 . In contrast with the singlet, low lying triplet states of Be2 are strongly bound. Triplet compu-
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
17
tation, starting with an initial re value of 2 A ÿ1 converged to a state located 5530 cm above the X 1 Rg ground state, probably the lowest 3 Pg [9], with an re 2:0015, and a harmonic frequency 676 cmÿ1 . The relatively reasonable agreement obtained for the Be atom, and even for the Be2 ground state which is a notoriously dicult problem to compute, encourages us to believe that the chosen level of theory, namely, B3LYP in combination with a large basis including diuse and polarization functions, should be capable of providing satisfactory results for the larger, neutral beryllium clusters. These are expected to be much more strongly bound so that the problems associated with the ®ne balance between bonding and antibonding interactions which plague the ground state computations of the dimer should be absent. 3.1. The singlet states of beryllium clusters The optimized geometries of the singlet state Ben clusters are shown in Fig. 1, with the computed total energies being listed in Table 1. Previous MP2 results [22] are also shown for comparison. The calculated harmonic vibrational frequencies, their degeneracies and infrared intensities are given in Table 2. Regardless of the starting geometry, the n 3±5 clusters converge smoothly to an equilateral triangle, tetrahedron, and trigonal bipyramid, respectively, with our calculated DFT bond lengths being systematically shorter than those obtained in the about 0.05 A MP2 study [22]. In contrast with the symmetric small clusters, the optimized geometries of the n 6 and 7 species are signi®cantly distorted, with their structures deviating signi®cantly from the structure of solid metal, and also exhibiting differences compared with the MP2 results. Thus n 6 is neither an octahedron, the most compact Be6 con®guration occurring in the metal lattice, nor a Be5 trigonal bipyramid with one of the faces capped. Starting from a slightly distorted octahedron, the optimization converges to strongly distorted C2v structure which can be described as a vertical Be2 dimer molecule (bond length 2.481 A) ``solvated'' in the horizontal plane by two Be2 di The mers with very short bond lengths (1.854 A).
Fig. 1. Three-dimensional structures of Ben clusters, n 2±8. Geometries have been optimized on the B3LYP/6-311++G(3df) level of theory.
Be7 cluster converges to a distorted pentagonal bipyramid, again resembling a vertical beryllium dimer, solvated in the horizontal plane this time by ®ve additional beryllium atoms. Unlike the symmetric smaller clusters, the distorted n 6 and 7 species have geometries which do not appear in solid HCP beryllium metal. This tendency towards distorted geometries was pointed out previously by Bauschlicher and Pettersson [17] in their calculations on Be13 . The largest cluster considered in this work, Be8 , reverts to a symmetric and rather compact con®guration, basically a doubly capped octahedron with a D3d symmetry. Earlier studies, conducted at the HF or MP2 level, sometimes employed for the geometry optimization smaller basis sets, and then used a larger
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
Table 1 and point groups of Ben , n 2±8, B3LYP/6-311++G(3df) level of theory comparison with earlier MP2 results [22] Bond lengths (A) n
Singlet
MP2a
Symmetry
Triplet
Symmetry
Be2 Be3
2.481 3 2:158
2.218
D1h D3h
D1h C2v
Be4
6 2:027
2.083
Td
Be5
3 1:982
2.031 2.085
D3h
2 2:182 2.482 2.562 4 2:029 4 2:039
C2v
2.0015 2 1:961 1 2:211 4 1:985 2 2:162 2 2:008 1 2:333 4 2:000 2 2:007 6 2:117 6 1:956
16 1:866 )2.469
C1
6 2:041 Be6 b
Be7 Be8
2 1:854 2.324 2.959 4 2:010 4 2:058 1 2:453 16 1:865 ÿ2.469c 6 1:964 6 2:081 6 2:108
D2d C2v
D3d
D3d
a
MP2 results taken from Ref. [22] are given for comparison. Quintet Be6 is computed and has an Oh symmetry with bond lengths 12 2:040 A. c Singlet Be7 is a distorted pentagonal bipyramid, the bond lengths between atoms 1±5 in the ``equatorial plane'' are 2.143, 2.079, 2.171, 2.084, 2.149; bond lengths to atom 6 above the plane: 1.865, 1.986, 2.320, 2.341, 2.002; atom 7 below: 2.469, 2.165, 1.899, 1.891, 2.143 distance 6±7: 2.448 A. A; b
Table 2 Harmonic vibrational frequencies (cmÿ1 ) with their degree of degeneracy, and, in parentheses, infrared intensities (km molÿ1 ) of Ben , n 2±8, calculated at the B3LYP/6-311++G(3df) level of theory Be2 Be3 Be4 Be5 Be6 Be7 Be8
289(0) 2 466(1.7) 2 495(0) 2 344(1.5) 91(12) 598(0.1) 26(2.3) 387(54) 687(0.3) 2 164(1.2) 2 525(2.5)
534(0) 3 595(17) 2 474(0) 292(6.9) 614(16) 146(14) 429(3.0) 749(4.5) 345(0.0) 2 622(0.0)
700(0) 482(0) 324(0) 615(1) 229(2.3) 519(10) 756(0.8) 2 359(0.0) 2 629(53)
basis set to compute the ®nal energies at the optimized geometry. Since the geometry often depends sensitively upon the theoretical model and the size of the basis set, such energy computations at a level dierent from that used in the optimization process may be done at geometries diering from the true minimum, and may, in particular for the larger clusters, lead to signi®cant errors in the binding energies. To avoid such problems, the
2 590(14) 422(5.0) 621(37) 286(2.0) 529(3.5)
621(37) 495(0.1) 782(4.4) 369(5.4) 617(0.0)
743(0) 528(100) 794(0) 383(14) 636(0.0)
394(0.0) 635(0.1)
2 397(0.2) 734(223)
401(117) 779(0.0)
same basis sets were used throughout the optimization and energy computation in the present study. The atomization energies computed here which are considerably higher than those obtained in previous works may be the consequence of this self-consistent approach. As noted in Section 1, of particular interest are trends of properties as functions of size, and speci®cally, the question of how quickly do the
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
properties of the discrete small clusters converge to those of the metallic beryllium. The most dramatic change is, of course, the increase in the atomization energy per atom from the dimer (5 kJ molÿ1 ) to that of the bulk solid (324.3 kJ molÿ1 ). As seen in the third column of Table 3, the computed values rapidly increase as a function of n, but the increase is far from smooth, and is not monotonic. Even for the largest clusters studied here, the computed energies are less than half of the 324.3 kJ molÿ1 bulk value. This is mainly due to the fact that while in the bulk solid, every atom has 12 nearest neighbors, in the small clusters, the coordination number of the atoms is much lower. Even in the largest cluster studied here, Be8 , two atoms are bound to three neighbors, while six are pentacoordinated, yielding an average coordination number of 4.5. If one considers this problem in more detail, it becomes clear that those properties that are strongly aected by atoms being on the surface will converge towards the bulk solid value rather slowly. It is instructive to remember that even for an n 1000 cluster, nearly half of the atoms will be located on its surface. In view of this eect of coordination, it is perhaps more meaningful and instructive to look at energy per bond, obtained by dividing the total atomization energy by the number of bonds. This of course may introduce in some cases a degree of arbitrariness in deciding whether two atoms are bonded or not. In the fourth column of Table 3,
19
one sees that there is a more than a factor of two increase of the bond energy for Be3 . As noted above, the DFT computations substantially overestimate the binding energy of Be2 ; if one considers the known experimental De value for Be2 , the 45 kJ molÿ1 binding energy per bond computed for Be3 in fact amounts to an increase of more than a factor of four. Already for Be4 , however, the energy per bond reaches a maximum of 67.7 kJ molÿ1 . For still larger clusters, the energy ¯uctuates somewhat, but seems to have an overall decreasing tendency, approaching the value deduced from the known 54.1 kJ molÿ1 atomization enthalpy of the closely packed bulk hexagonal beryllium. The computed trends are perhaps not dicult to understand qualitatively. The simplest MO theory predicts a zero order bond for the dimer with two bonding and two antibonding electrons. This explains the experimentally observed weak bond, and also is the underlying reason behind the diculties of theoretical computations: the bonding arises mainly from the mixing of excited con®gurations into the ground state. When a third atom is added, three Be±Be bonds can be formed; this makes an s±p rehybridization of the beryllium atoms, in spite of the high 2s±2p promotion energy, much more favorable than in the dimer. This trend is continued in the tetramer, where the addition and rehybridization of just one more atom makes it possible to form three additional
Table 3 Total equilibrium binding energies De , De per atom and per bond, incremental binding energies DDe n ! n ÿ 1 at the B3LYP/6311++G(3df) level of theorya
a
Ben
De
De /atom
De /bond
DDe n ! n ÿ 1
De b
Be2 Be3 Be4 Be5 Be6 Be7 Be8 Bulke
18.4 134.9 407.7 561.8 691.3 899.8 1105.2
9.2 45.0 101.9 112.4 115.2 128.5 138.2 324.3
18.4 45.0 67.9 62.4 53.2 56.2 61.4 54.1
18.4 116.6 272.8 154.1 129.5 208.5 205.4 324.3
12.6d 100 347 460
De values from two earlier studies are given for comparison. All energies are given in kJ molÿ1 : Best estimates of Lee et al. [20]. c MP4/6-311+G level of theory [22]. d Ref. [9]. e DH298 value from Ref. [28]. b
De c 90.3 348.6 510.3 571.0 770.4
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
metal±metal bonds, for a total of six. This is re¯ected in the large atomization energy and Dat H per bond value of Be4 . Even though the same argument could be used when proceeding to the trigonal bipyramidal Be5 , where addition of the ®fth atom again allows three more bonds to form, the trend is now reversed. The increase in atomization energy is much smaller, and the energy/bond actually decreases. This undoubtedly re¯ects another property of beryllium, well known from its chemistry: electron de®ciency. The Be atom has only two valence electrons, and thus in Be3 , there will be six electrons for three bonds or two electrons per bond available. As the clusters get larger, the number of possible metal±metal bonds increases, and naturally, the number of electrons available per bonding interaction must decrease. Thus eight electrons per six bonds or 1.33 per bond are available for Be4 , and 10 electrons per nine bonds or 1.11 per bond in Be5 . If one tries to extrapolate this trend to the bulk metal, in the HCP solid beryllium, each atom interacts with 12 nearest neighbors so that only one-third electron will be available per Be±Be interaction. The same trends discussed above in connection with the energy per bond should also be apparent when one examines the bond lengths and vibrational frequencies, and this is con®rmed by the re of the Be2 dimer, even calculations. The 2.49 A though considerably shorter than that of Li2 , is still considerably longer than the nearest neighbor in metallic beryllium. In the distance of 2.12 A triangular D3h trimer, the bond lengths are sig and this trend is ni®cantly reduced to 2.158 A, continued for n 4 and 5 clusters. Here, the bonds are further shortened, and in fact ``undershoot'' the bulk value, becoming considerably shorter nearest neighbor distance in bulk than the 2.12 A beryllium metal. For still larger clusters, this trend is again reversed; while in the large clusters, not all bonds are equivalent, and the individual lengths vary, the average bond length rapidly approaches the nearest neighbor distance of the bulk metal. The behavior of the computed harmonic frequencies and force constants can be seen in Table 2, but in view of the mixing of stretching and deformation vibrations, quantitative comparisons
are more dicult. Still, one can see that compared to the xe 289 cmÿ1 in the dimer, the larger clusters exhibit much higher frequencies, with the highest frequency stretching modes approaching 800 cmÿ1 . A consideration of the vibrational frequencies also provides some interesting insights into the cluster structures and dynamics. The structures of the n 3±5 clusters as well as that of n 8 are highly symmetric. They are quite robust and well de®ned, and the geometry optimizations smoothly converged to the global minimum. Quite dierent are the Be6 and Be7 clusters, which have low symmetries, and for which the computation convergence was found to be very laborious and slow. This behavior is also re¯ected in the computed vibrational frequencies. While all the vibrational frequencies of the former, symmetric clusters are well de®ned and rather high, each of the latter clusters, Be6 and Be7 , exhibits one very low frequency mode. As noted above, the structures of both clusters can be described as a ``vertical'' Be2 molecule, ``solvated'' in the horizontal symmetry plane by 4 and 5 additional Be atoms, respectively. In each case, the lowest computed frequency, 26 cmÿ1 in Be7 and 85 cmÿ1 in Be6 , corresponds to tilting of this ``vertical'' Be2 axis with respect to the atoms in the equatorial plane. The potential for this vibration is extremely ¯at so that in some computations, the optimization stopped before reaching the global minimum, resulting in imaginary values for this frequency. Of interest are also the incremental adiabatic dissociation energies of the beryllium clusters given in the ®fth column of Table 3. One can see huge increases of ± when one takes the experimental dimer value ± more than a factor of 10 when going from n 2 to 3, and an additional factor of more than two from Be3 to Be4 . The reasons for this were, in fact, already discussed above. In the trimer, the promotion energy needed to rehybridize the three atoms can be ``paid for'' by forming three strong metal±metal bonds, and this trend is continued in the very stable tetramer, where rehybridization of one additional atom makes possible three more metal±metal bonds. The trend then reverses sharply for Be5 and in particular for Be6 ; both clusters have geometries which do not occur in metal lattice, and are apparently unfavorable.
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
A most compact hexamer taken out of a Be lattice would be a regular octahedron, but as seen from the strong distortion, such a con®guration is not energetically favored when not stabilized by the remainder of the lattice. Furthermore, for these larger clusters, the above mentioned de®ciency of electrons may already be taking eect. Even though the n 7 cluster also has a distorted geometry quite unlike the solid, the relatively compact bipyramidal structure is apparently quite favorable, as re¯ected by the rather large DDe value. The n 8 optimized cluster again has a favorable geometry very close to the bulk, and exhibits a rather large dissociation energy. It shows that while an unsupported octahedral structure alone is not a stable minimum, already two additional atoms are sucient to make an octahedron with two faces capped the global minimum on the Be8 potential energy surface. If the atomization energy of the metal is taken as the limiting DDe value, then it is clear that even for the most stable clusters studied, the computed dissociation energies are considerably smaller. This must again be attributed to the fact that in the case of small clusters, one will be removing an apex or edge atom of relatively low coordination. 3.2. States of higher multiplicity: triplets and quintets As discussed above, the Be2 dimer is predicted to have a zero bond order by a simple MO theory, which is in agreement with the low dissociation energy found both experimentally and by quantum mechanical computations. In contrast with the
21
singlet ground state, the theory predicts numerous triplets, including the lowest one, to be strongly bound, and this is again con®rmed both by our present calculations as well as by more extensive computations elsewhere. The obvious conclusion is that the lowest triplet states of the dimer must lie signi®cantly lower than the 3 P atomic state at 21 974.4 cmÿ1 , (236.8 kJ molÿ1 ), and indeed, our computation predicts the triplet ground state to lie mere 52.5 kJ molÿ1 above the singlet ground state. In view of this, one should consider the possibility that the ground states for some of the larger clusters might in fact be states of higher multiplicity. We have therefore also carried out calculations on triplet and quintet states of clusters Ben up to n 8, with the results being summarized in Table 4. Our computations indeed yield also for the n 3±5 very low lying triplet states, located 47.4, 104.1 and mere 21.3 kJ molÿ1 , respectively, above the corresponding singlet ground states. For the hexamer, we ®nd the triplet to lie 7.6 kJ molÿ1 below the singlet, so Be6 is predicted to have a triplet ground state. The decreasing separation between the ground state and the states of higher multiplicities with increasing cluster sizes may be viewed as an indication of decreasing band gap, and approach to the metallic properties of bulk beryllium. It is interesting to note that the higher multiplicity states have not only dierent geometries but also dierent symmetries than the singlets. Thus, the lowest triplet state of Be3 at 47.4 kJ molÿ1 is not an equilateral triangle, but has a lower C2v symmetry. Similarly the Be4 and Be5 triplet states are no longer a regular tetrahedron or
Table 4 Total energies (hartrees) and symmetries of Ben clustersa
a b
n
Singlet
Triplet
1 2 3 4 5 6 7 8
ÿ14.671314 ÿ29.349621 ÿ44.065328 ÿ58.840551 ÿ73.570534 ÿ88.291168 ÿ103.041917 ÿ117.791473
ÿ14.581115 ÿ29.329624 ÿ44.047287 ÿ58.800889 ÿ73.562422 ÿ88.293949 ÿ103.031665 ÿ117.786126
D3h Td D3h C2v C1 D3d
Bold letters indicate ground state values. Single-point calculation with singlet geometry.
Quintet
C2v D2d C2v D3d C1 D3d
ÿ44.027951 ÿ58.721531 ÿ73.500583 ÿ88.289124 ÿ103.036859 ÿ117.739286b
C2v C2v C2v Oh C2v D3d b
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M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23
a trigonal bipyramid, but have lower symmetries, D2d and C2v , respectively. Interestingly, unlike the C2v singlet, the triplet hexamer has a higher, D3d symmetry. The quintet state of Be6 has a still higher, Oh symmetry, but lies 5.4 kJ molÿ1 above the singlet, and 12.7 kJ molÿ1 above the computed triplet ground state. Up to now, with the exception of the Be dimer, very little is known about the larger clusters experimentally. The harmonic frequencies computed here might provide some help in identifying and characterizing these species spectroscopically. Unfortunately, the vibrational frequencies lie relatively far in the infrared with their intensities being in most cases low, and this may be one of the reasons why they have not yet been detected in matrix experiments. Perhaps, the most favorable cases are the frequency near 528 cmÿ1 of the pentamer (intensity 100 km molÿ1 ) or the 734 cmÿ1 asymmetric stretch of the octamer (intensity 223 km molÿ1 ). In searching for these absorptions, the above mentioned factor of 0.95 derived from the beryllium dimer values could be used to correct the listed frequencies, but clearly applying the same scaling to every mode without worrying about the nature of the motion involved is at best a very crude approximation. Even if the vibrations should be detected, their assignment to a cluster of a particular size would remain a problem; the best opportunity would seem to oer an experiment with size selected clusters.
4. Conclusions While the Be2 dimer ground state is only weakly bound, the binding energies of larger clusters rapidly increase, with the energy per bond actually reaching a maximum for the tetramer. The relative energy of excited higher multiplicity states decreases rapidly with n, with Be6 being computed to have a triplet ground state. This may perhaps be viewed as re¯ection of the decreasing energy gap and approach to metallic properties. Some cluster sizes have stable and rigid high symmetry structures, others, for instance, singlet Be6 and Be7 are characterized by distorted geometries with a ¯at
potential surface and very low frequency vibrational modes.
Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. M.K.B. thankfully acknowledges a Feodor Lynen Research Fellowship from the Alexander von Humboldt-Foundation.
References [1] G. Herzberg, Zeitschrift f ur Physik 86 (1929) 601. [2] G. Herzberg, Molecular Spectra and Molecular Structure I, second ed., van Nostrand-Reinhold, New York, 1950. [3] R.S. Berry, S.A. Rice, J. Ross, Physical Chemistry, Wiley, New York 1980. [4] CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1992. [5] I. Rùeggen, J. Alml of, Intern. J. Quant. Chem. 60 (1996) 453 and references therein. [6] J. St arck, W. Meyer, Chem. Phys. Lett. 258 (1996) 421. [7] L. F usti-Molnar, P.G. Szalay, Chem. Phys. Lett. 258 (1996) 400. [8] J.M.L. Martin, Chem. Phys. Lett. 303 (1999) 399. [9] L.A. Kaledin, A.L. Kaledin, M.C. Heaven, V.E. Bondybey, J. Mol. Struct. (Theochem) 461±462 (1999) 177 and references therein. [10] R.J. Gdanitz, Chem. Phys. Lett. 312 (1999) 578. [11] V.E. Bondybey, Chem. Phys. Lett. 109 (1984) 436. [12] V.E. Bondybey, Science 227 (1985) 125. [13] R.O. Jones, J. Chem. Phys. 71 (1979) 1300. [14] B. Liu, A.D. McLean, J. Chem. Phys. 72 (1980) 3418. [15] R.J. Harrison, N.C. Handy, Chem. Phys. Lett. 98 (1983) 97. [16] G. Pacchioni, D. Plavsic, J. Kouteck y, Ber. Bunsenges. Phys. Chem. 87 (1983) 503. [17] C.W. Bauschlicher Jr., L.G.M. Pettersson, J. Chem. Phys. 84 (1986) 2226. [18] R.J. Harrison, N.C. Handy, Chem. Phys. Lett. 123 (1986) 321. [19] M.M. Marino, W.C. Ermler, J. Chem. Phys. 86 (1987) 6283. [20] T.J. Lee, A.P. Rendell, P.R. Taylor, J. Chem. Phys. 92 (1990) 489. [21] H. Kato, E. Tanaka, J. Comp. Chem. 12 (1991) 1097. [22] P.V. Sudhakar, K. Lammertsma, J. Chem. Phys. 99 (1993) 7929. [23] W. Klopper, J. Alml of, J. Chem. Phys. 99 (1993) 5167.
M.K. Beyer et al. / Chemical Physics 262 (2000) 15±23 [24] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople,
GAUSSIAN 94,
[25] [26] [27] [28]
23
Revision D.4, Gaussian, Pittsburgh, PA, 1995. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. C. Moore, Atomic Energy Levels, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. US, 35/V.I, 1971. C.W. Murray, N.C. Handy, R.D. Amos, J. Chem. Phys. 98 (1993) 7145. M.W. Chase Jr., NIST-JANAF Thermochemical tables, Fourth ed., J. Phys. Chem. Ref. Data, Monograph 9, 1998.