3042
J . Phys. Chem. 1988, 92, 3042-3046
compounds 5a and 5b, the small energy difference (2 kcal/mol) between 3f and 3g precludes the certain identification of 3g as the most stable two-silicon isomer. For the three-silicon structures (4a-d), isomer 4d is the most stable, since this isomer avoids the high strain energies of the bicyclobutanes 4a and 4b. The Si-C double bond is also much stronger than the Si-Si double bond.48 However, the trade-off of a weak Si-Si bond in 4d for a stronger Si-C bond results in a net 4c - 4d energy difference of less than 2 kcal/mol. Recent ab initio calculations by Schleyer et aI.*O have predicted the presence of a bond-stretch isomer of 5a; Le., an isomer that differs from the one discussed above primarily in the much greater length of the bridge bond. Generalized valence bond (GVB)49 calculations from this laboratorySo on 5a are in qualitative agreement with their results. We also have preliminary evidence for the existence of a similar isomer for compound 3a, and the possibility of ”bond-stretch” isomers for all of the bicyclobutane systems is under investigation. A detailed analysis will be presented in a forthcoming paper.s0 However, as noted by Schleyer et aI.,*O the “bond-stretch” isomers are not likely to be observed experimentally if the hydrogens at the bridging positions are replaced by larger substituents. This is, in reality, most likely to be the case.
IV. Conclusions The conclusions to be drawn from the present study are as follows: (1) The bicyclobutane structures are more highly strained than the cyclobutene isomers. Within a given subset of bicyclobutane isomers, the most highly strained systems are those containing Si-Si bridge bonds. The strain in the cyclobutene structures (49) Bobrowicz, F. W.; Goddard, W. A., 111 In Methods of Electronic Structure Theory; Schaefer, H. F., 111, Ed.; Plenum: New York, 1977. (50) Boatz, J. A.; Gordon, M. S . , manuscript in preparation.
decreases as the number of silicon atoms increases. (2) With the exception of the all-silicon molecules 5a and 5b, the cyclobutene with the fewest possible number of silicon atoms in the double bond appears to be the thermodynamically favored compound within a subset of isomers. However, this is not a definitive observation due to the presence of several isomers having energies within a few kilocalories per mole of the lowest energy structure. (3) The bicyclobutanes have highly bent Si-Si and Si-C bonds. The displacement of the MED path away from the internuclear axis can be as large as 0.34 A, and the difference between the bent bond length and the companion internuclear distance can exceed 0.15 8, (as in 3a). Indeed, some bonds with very short (apparently double-bond-like) internuclear distances have bent bond lengths that are more like normal single bonds. This emphasizes the utility of the bent-bond analysis for interpreting the electronic structure of highly strained compounds.
Acknowledgment. This work was supported by grants from the National Science Foundation (NSF Grant CHE-86-4077 1 and the Air Force Office of Scientific Research (AFOSR Grant 87-0049). The calculations were performed on the North Dakota State University IBM 3081/D computer, on a VAX 11/750 purchased by AFOSR Grant 84-0428, on a microVAX I1 purchased with the aid of NSF Grant CHE-8511697, and on the San Diego Supercomputer Center Cray X-MP (time provided by the NSF). We are most grateful to P. J. MacDougall and R. F. W. Bader for providing their AIMPAC program for computing bent bond lengths and to Professor P. v. R. Schleyer, who pointed out the presence of bond-stretch isomers in tetrasilabicyclobutane. Registry No. Za, 112021-18-8; 2b, 89418-57-5; Zc, 112021-19-9; 2d, 112021-20-2; 3a, 110797-75-6; 3b, 112021-21-3; 3c, 79647-93-1; 3d, 112021-22-4; 3e, 112021-23-5; 3f, 112021-24-6; 3g, 102839-40-7; 4a, 112021-25-7; 4b, 112021-26-8; 4c, 112021-27-9; 4d, 112021-28-0; Sa, 98721-26-7; Sb, 109672-91-5.
Selected Properties of Be Clusters in ab Initio Model Approximations W. C. Ermler,* R. B. Ross, Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
C . W. Kern, R. M. Pitzer, Department of Chemistry, The Ohio State University, Columbus, Ohio 43210
and N. W. Winter Physics Department, Lawrence Livermore National Laboratory, Livermore, California 94550 (Received: August 17, 1987; In Final Form: November 1 1 , 1987) Results of ab initio self-consistent field calculations are reported for electronic states of beryllium clusters comprised of 13, 19, 21, 33, 39, 51, 57, and 63 atoms. The clusters correspond to the second through ninth coordination shells of a central Be atom with internuclear separations taken from the lattice constants of the bulk hcp metal. Effective core potentials have been employed to replace the 1s core electrons, thereby reducing the complexity of the calculations. In addition, the use of the full D,,point group symmetry of each cluster results in substantial reductions of the numbers of two-electron integrals that must be computed and processed. Properties are calculated for selected electronic states and include binding energy, orbital energies, electric field gradient, nuclear-electron potential, diamagnetic shielding constant, second moments, and Mulliken populations. Overall net charges, second moments, and nuclear potentials become approximately constant in the larger clusters, indicating bulk behavior.
Strategy Particularly stringent tests of “supermolecule” approaches to condensed-phase properties arise for metals, where the electrons are highly delocalized compared to, say, an ionic solid. It is now feasible to carry out calculations on systems with sufficient numbers of atoms to investigate the interface between finite clusters and the bulk. The inverse r dependence of such properties 0022-3654/88/2092-3042$01.50/0
as the diamagnetic shielding constant and the electric field gradient suggests that these Properties of a solid may be. well approximated by finite clusters of representative atoms or molecules.’ That is, the nearest neighbors are responsible for the most important (1) (a) Gornostansky, S . D.; Kern, C. W. J . Chem. Phys. 1971,55, 3 2 5 3 . (b) Sudicura, X.;Davidson, E. R. Chem. Phys. Lett. 1984, 26, 54.
0 1988 American Chemical Society
The Journal of Physical Chemistry. Val. 92. No. 11. 1988 3043
Selected Properties of Be Clusters
Q
Be
13
Be 19 Be 2 1
Be 33
.
Be 39 Be 51
Be 57
Be 63 Figure 1. Clusters corresponding to 1-9 coordination shells of B central Be atom. Actual geometries were dcrivd from the lattice constants of Be metal. (Wyckoff,R. G.'C r p r o l Sl~ucrures;Interscience: New Yark. 1974.)
effects, the second nearest neighbors provide a less important contribution, and so on. In the present study, the HartreeFock-Roothaan (HFR) approximation' and ab initio effective core potentials' (EP) are employed to explore this supermolecule model from a purely nonempirical standpoint. The major limitation in applying ab initio approaches to the calculation of wave functions for very large clusters of atoms bas been the necessity to deal with the fourth power dependence of the number of two-electron integrals on the number of basis functions. Finite cluster approximations to periodic systems contain many symmetry elements, however, which leads to a substantial reduction in the number of unique multicenter integrals over basis functions! A further simplification results from the incorporation of symmetry-adapted linear combinations of basis functions into the self-consistent field (SCF) step of the HFR calculations. The resulting block diagonalization of Fock matrices transforms the problem from one of finding the eigensolutions of a large matrix to that of solving several matrices of lower order. In addition to making use of the inherent point group symmetry of clusters to reduce the number of integrals that must be processed, the explicit treatment of electrons that are 'corelike" may be eliminated by making use of EP's. If it is assumed that all bonding processes involve only the valence electrons, whereby the wre electrons remain independent of their chemical environment, then it should be possible to account for the atomic cores in any molecular or crystalline system by using a quantity derived f r o m and uniquely characteristic of, the constituent atoms. The E P is a one-electron operator which replaces the twoelectron operators corresponding to the repulsion between inner-shell electrons and valence electrons in a given atom. EP's reduce the number of basis functions centered on each atom by as many functions as are required to describe the core electrons and replace them by the EP operator, leading to only as many additional integrals as are (2) Roothaan, C. C. J. Reo. Mod. Phys. 1960, 33, 179. (3) Krauss. M.; Stevens. W. J. Annu. Reo. Phys. Chem. 1984, 35, 357. (4) Pitzer, R. M. J. Chem. Phys. 1973, 58, 311 1.
due to either the kinetic energy or nuclear attraction operators. Thus, corelike basis functions which, in the electronic energy, lead to a fourth-power dependence on their number, are replaced by EP's which involve only a second-power dependence. The availability of experimental results is an important criterion to consider when choosing a model system for an exploratory theoretical study. In addition to the fact that many of its bulk properties as well as its electric field gradient are known, beryllium metal is of special theoretical interest for other reasons. For example, the Be atom possesses a closed-shell ground-state configuration and forms a dimer which is only weakly bound. Cohesive energies are expected to be too small a t the Hartree-Fock level of approximation, where Be, is unbound. Nonetheless, since HartreeFock wave functions generally lead to reasonably accurate values of one-electron properties of atoms and molecules,' it is probable that the wave functions for large enough Be clusters should produce charge densities which are representative of the bulk metal, given comparable boundary conditions for both the supermolecular cluster and solid-state representations. This work is a continuation of a study of Be clusters embedded in the bulk metal. Electronic states and properties of Be,,, the smallest cluster with a central atom coordinated by I 2 nearest neighbors, have been discussed.6 Calculations on larger clusters of 51 and 57 atoms have also been presented.' These are complementary to studies by Bagus, Bauschlicher, and co-workers on the properties of Be clusters? To investigate the effects of electron ( 5 ) (a) Fraga. S.; Malli, G. Many Elecrron Sysrems: Properries ond (b) Neumann. D.; M a -
hrerocrions; Saunders: Philadelphia, PA, 1968. kowitz, J. J. Chem. Phys. 1968, 49. 2056.
w.
(6) Ermler. W.C.: Kern, C. W.: Pitzer, R. M.: Winter, N. W. 3. Chem. Phys. 1986.81. 3937. (7) Ross,R. B.: Ermler, W. C.; Kern, C. W.; Pitzer, R. M. Chem. Phys. Lerl. 1987, 134, 115. (8) (a) Bagus, P. S.;Schaefer, H. F.; Bauschlicher, C. W. J. Chem. Phys. 1983.78, 1390. (b) Bagus, P. S.:Nelin, C. J.: Baurhlicher, C. W. Surf Sei. 1985, 156,615. (c) Bausehlicher, C. W.Chem. Phys. Lett. 1985. 117. 33. (d) Pettenon. L. G. M.; Bauschlichcr. C. W. Chem. Phys. Lett. 1986, 130, 111.
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
3044
Ermler et al.
TABLE I: Be Clusters by Coordination Shell and hcp’ Layer shell R,b 8, z coord‘ 3c/2
c c/2 0 -c/2
0 0.00
2 2.29 = a
1 2.23
3 3.19
4 3.58
5 3.93
6 3.96
7 4.25
8 4.58
1 12 7 12 1
1 12 13 12 1
7 12 13 12 7
7 12 19 12 7
12 33
6 39
12 51
6 57
1
3 1 3
3 7 3
6 7 6
1 6 7 6 1
1 1
6 7
6 13
6 19
2 21
-c
-3c/2 added total
9 5.53
10 5.97
3 7 12 19 12 7 3 6 63
3 7 12 31 12 7 3 12 75
‘With undistorted hcp structure, shells 1 and 2 would be combined, as would shells 5 and 6, 7 and 8, and 9 and 10. (i.e. c / a = 81/2/3= 1.633). bDistance from central Be atom. c c = 3.58 A; c / a = 1.567. TABLE 11: Restricted Hartree-Fock Energies of States of Be,, MO configuration“ 1. 2. 3. 4. 5. 6. 7. 8.
state
9al’
loal’
sE“ ’E’
2 2 2 2
1 1 1 1
2
0 0 0 1
4 elec‘ 6 elece IA,’
3E‘
1
2 elec‘ 3 eled
1 2
4a2/ 0 1 0 1 0 2 0 0
8aF 2 2 2 2 2 2 2 2
9a2/1 2 1 2 1 2 2 2 2
lle’ 4 4 4 4 4 4 4 4
12e’ 2 2 2 2 0 1 1 2
9e” 1 1 1 1 4 4 4 0
2aI” 2 2 2 2 2 2 2 2
valence energy* -62.384 32 -62.37241 -62.371 53 -62.34944 -62.32970 -62.26746 -62.265 59 -62.202 18
binding energye
excitn energyd
24.72 24.60 24.59 24.37 24.18 23.56 23.54 22.91
0.32 0.35 0.95 1.49 3.18 3.23 4.618
‘(8)ai’, (3)a;, (7)a,”, (9)e’, @)e”, and ( l ) a l ” valence MO’s fully occupied. bHartrees. ‘Per atom in kcal/mol. EJcf= -0.95083 hartree for Be. eV relative to state no. 1. eAverage of configuration. /Positive ion, average of configuration. ZIonization potential relative to four-electron average state.
correlation and geometry optimization, the work of Whiteside et al? has-been extended and enhanced in configuration interaction and Maller-Plesset perturbation theory calculations on Be clusters containing three to seven atoms.I0
Calculations All of the Be clusters that can be formed by considering successive nearest neighbors of a central Be atom possess D3hpoint group symmetry. Table I shows the buildup of clusters to Be7, in terms of the number of atoms lying on successive coordination spheres around the central Be. Perspective vibws of the clusters are shown in Figure 1. In the present study, we have chosen to constrain the clusters to geometries corresponding to their positions in the hcp metal lattice. An EP replaces the 1s electrons, and a valence basis set of contracted Gaussian-type functions (GTF) describes the 2s electrons. The representation of the EP in the form of an expansion in G T F s and the (3s2p)/[2slp] valence basis set are given in ref 6 . The reduction of the basis set size through the use of E P s and the incorporation of the full point-group symmetry is dramatic. For each successive cluster, the required number of two-electron integrals is reduced by 4 orders of magnitude in comparison to a (9s 2p) all-electron basis set of GTF‘s.~ In the case of Be63,the reduction is from 2 X 1Ol1to 1 X lo7 two-electron integrals. To maintain consistency, in addition to the lowest energy state, the lowest closed shell (lAi’) state has been investigated for each cluster. This procedure is similar to that followed by Bagus et al. in their studies on the nature of the Be metal suiface and its catalytic properties.* Although it is known9J0 that the smaller Be clusters of 3-7 atoms and even the smallest cluster that is a subset of the pure hcp metal, Bel3: possess low-lying states of high-spin multiplicity, it is expected that properties localized at the central Be may become approximately constant once the cluster has increased beyond some threshold size. A purpose of these (9) Whiteside, R. A.; Krishnan, R.; Pople, J. A.; Krogh-Jespersen, M.-B.; Schleyer, P. R.; Wenke, G.J . Compur. Chem. 1980, 1, 307. (10) Marino, M. M.; Ermler, W. C . ‘ J . Chem. Phys. 1987, 86, 6283. ( 1 1) %e Lucken, E. A. C. Nuclear Quadrupole Coupling Constants; Academic: New York, 1969.
TABLE 111: One-Electron Properties of Bes$ state
5E,! ’E’ 4 elecb 6 elecb ’Ai’
’EE’ 2 elecb 3 elect
u 0.0166 0.0064 0.0131 0.0183 0.0112 0.0147 0.0144 0.0059
o -1.049 -1.034 -1.045 -1.035 -1.042 -1.045 -1.045 -0.952
ad -19.62 -19.60 -19.62 -19.61 -19.61 -19.62 -19.62 -19.52
(x2)1/*
(Z*)I/~ (?)i/2
46.24 46.03 46.24 46.34 46.17 46.17 46.17 45.87
57.39 51.87 57.38 57.34 57.59 57.58 57.58 56.90
e
87.00 50.00 87.1 1 -24.99 87.01 50.19 87.09 64.65 87.06 19.74 87.05 21.69 87.05 21.48 86.29 71.99
“All in au. For explicit definitions see ref 5b. bAverage of configuration (see Table 11). ‘Positive ion, average of configuration (see Table 11).
investigations on the larger clusters is to test this conjecture. The calculations on Bes3 were carried out on a Cray X-MP supercomputer. Integrals were evaluated and stored for use in subsequent S C F calculations on a number of low-lying neutral and ion states. S C F convergence was achieved for each state in 50-1 50 iterations by using Hartree-damping, level shifting, and DIIS procedures, as was required’ for Besl and Bes7. Results for Be63 are given in Tables 11-IV. Total valence energies, binding energies per atom, and excitation energies for six states and the lowest positive ion average state are listed in Table 11. One-electron properties for these states are reported in Table I11 and results of Mulliken population analyses are given in Table IV. Properties having origin at the central Be include the electric field gradient (q), nuclear-electron potential (a), and square roots of second moments of charge (( x 2 ) and ( z 2 ) Results of S C F calculations on the 13-63-atom clusters are given in Table V for comparable closed-shell states. Total binding energies, binding energies per atom, and ionization potentials are presented, as are one-electron properties and population analyses.
Discussion As shown in Table I and in Figure 1, the Be,, cluster incorporates atoms on the third layer along the c axis, lying at 3 c / 2 . This ninth coordination shell is the next logical extension of cluster size beyond the 51- and 57-atom systems. The tenth coordination shell, corresponding to the addition of 12 atoms in the xy plane contiining the central Be, appears to give somewhat more of an
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3045
Selected Properties of Be Clusters TABLE I V Population Analysis for Electronic States of Be, state Be(0) Be(A) Be(B) Be(C) Be(D) 5Ef’ 0.40 -0.94 0.24 -1.20 -0.66 0.23 -1.20 -0.65 -0.93 ’E’ 0.40 0.24 -1.21 -0.65 -0.94 4 elecd 0.39 0.23 -1.20 -0.65 -0.94 6 elecd 0.39 0.25 -1.13 -0.65 -0.96 IA,’ 0.39 -1.14 -0.96 0.24 E’ 0.39 -0.65 -0.65 -0.96 0.24 -1.14 2elec“ 0.39 0.24 -1.23 -0.66 -0.92 3 elec‘ 0.38
Be(E)
Be(F)b
Be(G)b
Be(H)
Be(1)
Be(J)
0.30 0.31 0.31 0.32 0.29 0.29 0.29 0.31
0.17 0.13 0.16 0.13 0.12 0.14 0.14 0.14
0.32 0.31 0.32 0.31 0.25 0.26 0.26 0.30
0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29
0.12 0.13 0.12 0.13 0.15 0.14 0.14 0.04
0.13 0.12 0.13 0.12 0.18 0.17 0.17 0.06
N ( ~ S ) ~N(2p)‘ 0.94 0.95 0.94 0.95 0.96 0.94 0.94 0.95
1.46 1.44 1.45 1.44 1.44 1.45 1.45 1.43
‘See Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833 for definitions. Be(0) is the net charge on the central Be atom. Entries (A) through (J) are the net charges on a single Be atom in coordination shells 1 through 9, respectively. bAtoms of types F and G all lie in coordination shell number 6 (see Table I, Figure 1 and ref 7). CGrossatomic orbital population on the central Be. dAverage of configuration (see Table 11). ‘Positive ion, average of configuration (see Table 11). TABLE V Properties of Singlet Be Clusters ab aC ( x x ) ’ / ~(~z z ) I l Z d eC cluster BE‘ 14.0 14.3 -17.8 10.0 0.191 -1.028 19.4 18.6 6.3 13.9 -0.063 -1.058 19.5 23.8 -30.4 15.6 -0.030 -1.068 31.4 29.1 49.0 12.1 0.016 -1.013 36.2 30.2 -19.3 19.3 -0.030 -1.029 39.6 45.4 41.0 28.3 0.045 -1.071 45.4 45.2 -19.3 23.0 0.007 -1.086 46.2 57.6 19.7 24.2 0.011 -1.042 76.5* 0.0049‘
IPf IPS 4.15 5.21 5.08 5.10 3.68 4.62 4.08 4.35 4.34 4.39 4.76 4.5 1 3.921
“Binding energy per atom in kcal/mol (see Table 11). *Electric field gradient along the D3 axis in au. cNuclear electron potential at the central Be in au. dSquare root of electronic expectation value in au. ‘Molecular quadrupole moment in au. ’Ionization potential (ev). (Calculated as total valence energy difference.) SIonization potential (ev). (Estimates based on Koopmans’ theorem.) *Taken from Kittel, C. Introduction to Solid State Physics; Wiley: New York, 1976; p 74. ‘References 12 and 13. ’Tompa, G. S.;Seidl, M.; Ermler, W. C.; Carr, W. E. Surf. Sci. 1987, 185, L453.
overall sphericity to the cluster. However, calculations on Be75 are considerably more difficult than on Be63 and require the extension of the basis set from 315 to 375 functions. In addition, rather acute difficulties have been encountered in achieving S C F convergence for the clusters in which atoms have been added to the x y plane. (Be5, exhibited more anomalous behavior than did Be51; but Be,, displayed a monotonic convergence pattern.) A strategy for carrying out calculations on the 75-atom cluster is being planned. The alternate clusters 13, 21, 39, 57, and 75 are seen to be more spherical and possess similar characteristics. This is clearly exhibited in the molecular quadrupole moment, 8,which is defined as the difference between the second moments R,,and R,, where the latter include both electronic and nuclear contributions (Table V). On the other hand, it appears that certain properties are less sensitive to increased cluster size by the time the number of atoms reaches 63 (note BE, q, in Table V). However, whereas in the case of BeS7we observed a range of atomic orbital populations 2s:2p at the central Be from 1:1.3 to 1:1.5, it remains close to 2s12pi,sfor all of the states of Bes3, including the positive ion. This is reminiscent of metallic p-band electrons. The population analysis for Be6, shown in Table IV also gives indications of bulk behavior. The net charges on the central Be atom vary by less than 0.02 for all of the states studied here. Other shells of Be atoms show a similar consistency. While the net charges are not exactly zero, as is true for the bulk, the lack of preference for electronic state is a characteristic of bulk systems. It is not unexpected that small net charges are seen because the finite cluster approximation leads to alternating net charges by shell in the context of Mulliken’s definitions. In contrast to BeSl and Bes7, where closed-shell ground states were found, the lowest state of Be,, is a quintet (Table 11). There is also a low-lying septet state and the 4- and 6-electron average energy states lie within 0.95 eV of the quintet. This behavior is indicative of the onset of bulk characteristics, Le., the presence of a partially filled band in which spin couplings are not important energetically. Binding energies of all of the states are within 1.8 kcal/mol. They are also less than 3 kcal/mol different from those calculated for Be,, and Be,, which is suggestive of convergence
for this property. The residual difference of -50 kcal/mol from the bulk cohesive energy (Table V) may be attributed to a combination of basis set deficiency problems, the need for the inclusion of electron correlation, and the finite cluster approximation. The use of a larger basis set may be expected to increase the binding energy by only about 4 kcal/mol.s‘ The ionization potential of Be6,, obtained as the difference between total valence energies of the ground and ionic states, is 0.2 eV higher than that calculated for Be,,, but is still within about 0.5 eV of the work function of the bulk. The bulk value given in Table V was measured relative to a polycrystalline sample and, thus, is an average of work functions from different faces. The highest occupied orbital energies of Bes3 are quite constant as a function of state (-0.16 hartree) for the seven states of Table 11, with nearly 10 orbitals having values between 4 . 1 6 and -0.18 hartree for all of the states, including the cation. One-electron properties for the eight states of Be63(Table 111) show a high degree of constancy, with the exception of the quadrupole moment, which is a sensitive measure of the asphericity of the cluster. The electric field gradient varies by less than a factor of 3, a clear improvement over the factor of 50 seen for Be,, and The electric field gradient of beryllium metal has been of considerable experimental interest. From several measurements of the quadrupole coupling constant ( e q Q / h )of the metal,12 together with an independent determination of the quadrupole moment (Q) of the 9Be nucleus,13an accurate value of the electric field gradient ( q ) of the metal can be extracted. Taking (eqQ/h) = 57 f 1 kHz12 and Q = 0.049 f 0.003613the resulting value of q is (1.60 f 0.07) X lo1, esu/cm3 or 0.00494 f 0.00022 au. Taking beryllium metal as composed of S Be2+ ions and 2 s conduction electrons, Pomerantz and Das14were able to separate the field gradient calculation into the determination of an ionic lattice term, q(lat), and a conduction electron term, q(ce). The former was computed by summing over the lattice with all 2, = 2, a procedure which is slowly convergent. They included in their result for q(1at) a factor, which was designed to account for the Sternheimer effect,’l which takes into account the polarization of the core (1s) electrons of the Be atom and was obtained from a calculation on the Be2+ ion. The effect of this factor was to decrease the value of q(1at) by 18.5%. Their value of q(ce) was computed by averaging the field gradient operator over a metal wave function constructed from orthogonalized plane waves. The final results for q(lat), q(ce), and q reported by Pomerantz and Das are, respectively, 0.005 95, 0.000 46, and 0.006 42 a.u. A subsequent study of Mohapatra et al.” showed that an important contribution to q(ce) involves the nonuniform electron density in the Wigner-Seitz cell around the Be nucleus at which the electric field gradient is measured. They recomputed q(ce) as a sum of this local contribution and a “distant” contribution arising from (12) Barnaal, D. E.; Barnes, R. G.: McCart, B. R. Mohn, L. W.;Torgeson,
D.R.Phys. Rev. 1967, 157, 510. (13) Blackman, A. G . ; Lurio, A. Bull. Am. Phys. SOC.1966, 1 1 , 343. (14) Pomerantz, M.; Das, T. P. Phys. Reu. 1960, 119, 70. ( 15) Pomerantz, M . ; Das, T . P. Phys. Rev. 1961, 123, 2070. (16) Mohapatra, N . C.; Singal, C. M.; Das, T.P.;Jena, P. Phys. Rev. Lett. 1972, 29, 456.
3046
J . Phys. Chem. 1988, 92, 3046-3048
the surrounding Wigner-Seitz cells. Using a revised result for q(1at) of 0.005 92 au, they obtained q(ce,distant) = 0.0017, q(m,local) = 4 . 0 0 1 51, and q = 0.004 58 au. The inclusion of this local contribution leads to a result for q(ce) that is consistent with the prediction of Barnaal et al.'* The values of q computed for the successively larger Be clusters, while showing a general trend toward the experimental value, do not converge in a monotonic fashion (Table V). This is due to the sensitivity of this property to electronic state, especially for the smaller clusters. In Bel3,q ranged from 0.19 to -0.17 au. for the 22 states investigated.6 This wide variation is not observed for the larger clusters, e.g., 0.076 > q > 0.007 for Be,,' and 0.016 > q > 0.006 for Bea (Table 111). Thus, q has converged to a value of 0.01 1 f 0.005 au for Be63. The correction for the Sternheimer effect6 amounts to a 16-18% increase in magnitude in q which, although in the wrong direction, still results in a range for q that is close to experiment. This favorable result for the field gradient supports the premise that localized properties of bulk systems can be calculated by using cluster models, provided the number of
shells of nearest neighbors included in the cluster is sufficiently large. For the ab initio model employed here for Be clusters, it appears that about 10 shells beyond the central atom create a prolate spheroid of dimensions 5.53 by 5.97 A which is sufficient to simulate bulk metal behavior.
Acknowledgment. We are grateful to M. M. Marino for her assistance with the calculations on Be63and for helpful discussions. This research was partially supported by the National Science Foundation under Grants CHE-8214689 and CHE-83 12286. The NSF Office of Advanced Scientific Computing is acknowledged for an allocation of time at the Pittsburgh Supercomputing Center. This work was performed under the auspices of the Division of Material Sciences of the Office of Basic Energy Sciences, U S . Department of Energy, and Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-8. Registry No. Be,,, 66019-92-9; Bel9, 113723-64-1; Be2,, 113723-65-2; Be3,, 113703-90-5; Be3,, 113703-91-6; Be,,, 113703-92-7; Be5,, 113703-93-8; Be,,, 113703-94-9.
Ab Initio versus Molecular Mechanics Calculations for Conformational Energies of 2-Propanol and Cyclohexanol William E. Palke and Bernard Kirtman* Department of Chemistry, University of California, Santa Barbara, California 931 06 (Received: August 19, 1987; In Final Form: November 23, 1987) Ab initio calculations are carried out and compared with molecular mechanics (MM2 and MOLMEC) for determination of conformational energies in cyclohexanol and 2-propanol. In 2-propanol the ab initio OH torsional potential is fairly well reproduced by MOLMEC but not by MM2. The same is true of equatorial cyclohexanol for which 2-propanol is a good model. However, both molecular mechanics methods underestimate steric repulsion in the 180' (HCOH dihedral angle) conformation of axial cyclohexanol. They also fail to reproduce the global minimum predicted by the ab initio treatment to occur for axial cyclohexanol at -60'. On the other hand, the current experimental AGO favors the equatorial conformer in solution.
Introduction This work addresses several related topics: (1) What is the most stable conformation of cyclohexanol? The essential comparison is among four configurations-equatorial and axial hydroxyl, each with -60' and 180' HOCH dihedral angles. A chair framework is assumed throughout. (2) How helpful is knowledge of the relative energies of the conformers of 2-propanol in predicting relative energies of the analogous conformers in cyclohexanol? Inspection of molecular models indicates that in the equatorial conformer of cyclohexanol the OH torsional potential should be nearly the same for both molecules. In the axial conformation, steric interactions are expected to raise the energy at the 180' dihedral angle geometry. (3) How do the predictions of molecular mechanics compare to ab initio calculations? Our original intent in this work was to select a molecular mechanics method that would yield accurate conformational energies for the cyclohexanol entity which occurs in many biomolecules of interest. Since experimental data to test the accuracy of conformational energy predictions are sparse, we deemed it worthwhile to compare with a b initio calculations. A 3-21G basis set was used in Hartree-Fock calculations on cyclohexanol.' For 2-propanol the treatment was extended to include polarization functions and some correlation.2 The two molecular mechanics methods designed for the purpose at hand ,
