J. Phys. Chem. 1990, 94, 7711-7174 TABLE 111: Coefficients of Fitting Equation E , or E = xA,(ln m)" (Eq 3) for E,ImV for EImV for EJmV for EImV -10.446 -0.631 -0.819
A, A, A,
A4 A5
-15.287 0.486 -0.721
a/mV
-0.351 -0.065 0.017
-0.249 -0.030 0.027
TABLE I V Cation Transference Numbers t + O
m
present
Hittorf
0.2 0.3 0.45 0.6 0.8
0.425 0.410 0.387 0.367 0.342 0.317 0.186 0.067 -0.064 -0.153
0.420 0.410 0.390 0.370 0.346 0.325 0.196 0.08 1 -0.05 I -0.15
1 .o
2.0 3.0 4.5 6.0
McOuillanl' 6 emf 0.005 0.004 0.003 0.005 0.004 0.004 0.007 0.010 0.015 0.03
0.425 0.414 0.396 0.378 0.347 0.321 0.205 0.099 -0.055 -0.153
6 0.006 0.003 0.006 0.006 0.004 0.005 0.005 0.017 0.017 0.046
"The uncertainties quoted for the Hittorf values are assessed from McQuillan's analytical estimates and the variations between anode and cathode compartment results; those for his emf values are as given by him. For uncertainty of present results see discussion. certainties below 4 m. The differentiation required was not easy given the rapid change (to negative values) of the transference number of the cation constituent. The Helmholtz calculation has been done with the present data using t(Cd2+) = 1 - (dE,/d In m)/(dE/d In m )
(2)
For this calculation the E values were taken from ref 15 via eq I ; it was thought better to use a single set of data for the differentiation, rather than to accept the derivatives given in ref 13 which were a best fit to a number of different sets. The E, values were referred to the 1 m half-cell as shown in Table 11. A polynomial in In m was found to give much better fits than one including powers of m; the coefficients are given in Table 111. In both cases the fit with five terms was chosen, even though a further halving of the already small standard deviation of the E, data occurred with six terms. The differentials were evaluated at round molalities so that the transference numbers (Table IV) could be
7771
compared with those of McQuillan. The differentiation cannot be trusted at the ends of the fitted range so Table IV runs from 0.2 to 6 m.
Error Assessments and Comparisons with Published Work It is not easy to assess by statistical theory the reliability of the ratio of two differential Coefficients derived from two separate polynomial curve fits. The excellence of the five-term fits in Table I l l is encouraging, especially since the successive coefficients rapidly decrease in magnitude. It is often observed that differentials from such cases are much more stable than if the coefficients are all of similar magnitude and oscillate in sign. The following empirical method of error assessment was tried: To each of the experimental E, values an artificial error was added, ranging randomly in sign and magnitude between 0.1 and -0.1 mV. These erroneous values were fitted to eq 3, giving a slightly different set of coefficients, and the standard deviation increased 4-fold to 0.067 mV. The resulting transference numbers differed from those in column 2 of Table IV by amounts ranging between 0.005 and -0.007. It is reasonable to suggest that the uncertainty in the transference numbers is proportional to the standard deviation of the polynomial fit. If so, the values given in Table IV should be reliable within 0.002. Agreement with McQuillan's emf-derived values is quite good except for the region 2-5 m, where an inconsistency between the data for the two types of transport cells (anion or cation reversible) and the activity coefficients has been noted above. One might prefer the highly stable and reproducible amalgam electrode results in this region. Agreement with the Hittorf results is good in view of the severe demands the Hittorf method makes on analytical accuracy at high concentrations. The work of McQuillan]' was followed up by very careful indirect moving boundary measurements16 from 0.0038 to 0.0873 m. These show a rather flat maximum of 0.430 from 0.03 m up, joining smoothly with the present results. Acknowledgment. Much of the experimental work reported here was done in 1982 at the Diffusion Research Unit of the Research School of Physics, Australian National University, Canberra. I also thank a referee for drawing my attention to some recent work. Registry No. CdCl,, 10108-64-2; CdHg, 12787-26-7; Cd2+,744043-9.
Ab Initio Properties of Electronic States of Be,, R. B. Ross, C. W. Kern, R. M. Pitzer,* Department of Chemistry, The Ohio State University, Columbus, Ohio 43210
W. C . Ermler, Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030
and N. W. Winter Physics Department, Lawrence Livermore National Laboratory, Livermore. California 94550 (Received: October 23, 1989; In Final Form: February 12, 1990)
Results of ab initio Hartree-Fock calculations are presented for the nine-coordination-shell cluster Be69. Binding energy, ionization potential, electric field gradient, nuclear-electrostatic potential, diamagnetic shielding constant, second moments of charge, and Mulliken populations are reported for several electronic states. Properties centered at the innermost Be atom exhibit convergence toward bulk Be metal values.
Ab initio self-consistent field (SCF) calculations are reported here for low-lying electronic states of Be6.+ This work is an 0022-3654/90/2094-777 1$02.50/0
extension of a series of studies on smaller Be clusters.'-3 Electronic states and properties of Bel3,the smallest cluster with a central 0 1990 American Chemical Society
Ross et al.
7772 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 TABLE I: Be Clusters by Coordination Shell and Hcp Layef shell
R,b %, z-coordC 3c/2
0
1
2
3
4
5
6
7
8
9
0.00
2.22
2.28
3.19
3.58
3.92
3.96
4.25
4.57
5.08
5.34
1 12 7 12 1
1 12 13 12 1
7 12 13 12 7
7 12 19 12 7
7 18 19 18 7
13 18 19 18 13
2 33
6 39
2 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
c/2 0 -c/ 2 -C
-3c/2 added total
1
1
1
2 69
1 0 1 1 1 2 1 3 1 4 1 5 1 6
1
2 81
5.53 3 13 18 19 18 13 3 6 87
5.57
5.80
5.98
3 3 13 19 21 21 19 19 21 21 13 19 3 3 6 1 2 93 105
6 19 21 19 21 19 6 6 111
6.02
6.04
6 6 19 19 27 27 19 31 27 27 19 19 6 6 1 2 1 2 123 135
"With undistorted hcp structure, shells 1 and 2 would be combined, as would shells 5 and 6 and 15 and 16 (Le., c / a = 8'/2/31/2 = 1.633). bDistance from central Be atom. E~ = 3.58 A; c / a = 1.567."
TABLE 11: Properties of Be,9 for Average Energy of Configuration States state 4" 4b ad (x2) 1/2d 2 elec 0.0 159 -1.06 -20.98 55.3 4 elec 0.0 164 -1.06 -20.99 55.3 closed shell 0.021 1 -1.07 -20.99 55.3 6 elec 0.01 63 -1.06 -20.98 55.3 exptl f0.0046' f0.0003
(z2) 1/2d 48.8 48.8 48.8 48.9
8c
IPf
BEg
-12.6 -1 8.4 -18.2 -23.5
3.62 3.48 2.58' 2.79 3.92k
28.2 28.0 27.6 26.3 75.9
AE~ 0.00 0.65 1.89 5.60
" Electric field gradient (au). Nuclear-electrostatic potential (au). e Diamagnetic shielding constant (au). dSecond moment of charge (au). 'Quadrupole moment (au). f AE(SCF) ionization potential (ev). BBinding energy per atom (kcal/mol). hExcitation energy in eV relative to the 2-electron state having total valence energy E = -68.705962 hartrees. 'Koopmans' theorem IP = 2.94 eV. 'Reference 8. kReference 10. atom coordinated by nearest neighbor atoms, have been discussed in detail2 Calculations on the larger clusters of 51,, 57,3and 63l atoms have also been reported. A perspective view of the Be,, cluster is shown in Figure 1. This cluster corresponds to the first through ninth coordination shells of a central Be atom with internuclear separations corresponding to the lattice constants of the bulk hcp metal.4 Table I is a corrected and extended version of Table I of ref 1. The purpose of these tables is to show the relationship between number of coordination shells and number of atoms for Be clusters that are increasingly larger parts of the bulk metal. The former Table I' correctly represents clusters comprised of successive nearest neighbors through shell 8. Table I here reproduces that information but goes further to provide the proper extension from shell 9 through 16 with the number and positioning of atoms in each case. Ab initio effective core potentials (EP), expressed as an expansion in Gaussian functions, have been employed to replace the effects of the Be Is core electron^.^ The valence electrons are described with a (3s2p)/[2sl p] set of contracted Gaussian-type atomic orbitals. The EP and basis set were reported in our previous study2 on Be,,. The overall advantage of using EPs, contracted Gaussians, the full D3,,.symmetry, and supermatrices can be obtained in steps. An equivalent basis set including functions to describe the inner shells is (9s2p) or 15 functions per atom, 1035 functions total, ( I ) Ermler, W. C.; Ross, R. B.; Kern, C. W.;Pitzer, R. M.; Winter, N. W,. J. Phys. Chem. 1988.92, 3042. (The columns headed "shell" 9 and 10 in Table I, which showed a total of 6 atoms at *3c/2 and 19 and 31 atoms at 0,now appear as shells 11 and 16 in the present Table I. Thus, the Be63 cluster discussed in this reference, while being a reasonably chosen system to represent the first set of Be atoms that occur at f3c/2 along the c direction, is not a member of the series of clusters derived from successive concentric shells around a central Be atom. The binding energy per atom for the lowest energy singlet state of BeSl in Table V should have been listed as 25.8 and not 28.3 kcal/mol as shown.) (2) Ermler, W. C.; Kern, C. W.; Pitzer, R. M.; Winter, N. W. J. Chem. Phys. 1986,84, 3937. (3) Ross, R. B.; Ermler, W. C.; Pitzer, R. M.; Kern, C. W. Chem. Phys. Lett. 1987, 134, 115. (4) Wyckoff, R. G . Crystal Structures; Interscience: New York, 1974. ( 5 ) Ermler, W. C.; Ross, R. B.; Christiansen, P. A. Adu. Quantum Chem. 1988, 19, 139.
Figure 1. Be69 cluster corresponding to nine coordination shells of a central Be atom. The actual geometry was derived from the lattice constants of Be metal as given in ref 4.
giving 1.4 X 10'' two-electron integrals. Using EPs and reducing the basis to (3s2p) yields 9 functions per atom, 621 functions total, giving 1.9 X 1O'O integrals. Contraction of the basis set to [2slp] yields 5 functions per atom, 345 functions total, 1.8 X lo9 integrals. The use of symmetry6 reduces the number of independent integrals by approximately a factor equal to the order of the point group (12) to yield 1.5 X lo8 integrals. The supermatrix formulation for SCF calculations7 reduces the number approximately by a further factor of the order of the group (see Appendix) to a final estimated number of 1.2 X lo7 integrals. The actual number found in the calculation was 1.4 X lo7. The overall reduction is thus a factor of IO4. A second supermatrix is needed to carry out open-shell calculations, although, depending on the number of symmetries with open-shell molecular orbitals (MO), only part of this supermatrix is needed during the SCF iterations. Closed-shell and open-shell states represented as average energies of configuration, of two, four, and six unpaired electrons were studied for Be69. We chose to complete calculations on average energies of MO configurations because our previous results (6) Pitzer, R. M. J. Chem. Phys. 1973, 58, 3111. (7) Roothaan, C. C. J. Reu. Mod. Phys. 1960,32, 179.
Ab Initio Properties of Electronic States of Be69
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7773
TABLE 111: Population Analysis for Electronic States of Be69(Electron Units)” state Be(0) Be(A) Be(B) Be(C) Be(D) Be(E) Be(F)b 0.23 2 elec 0.28 -0.69 0.31 -0.54 0.11 -0.51 0.27 -0.69 0.28 4 elec 0.27 -0.51 0.13 -0.53 closed shell 0.29 -0.71 -0.51 -0.52 0.11 0.28 0.38 0.26 6 elec 0.29 -0.68 0.15 0.26 -0.51 -0.56
Be(G)b -0.38 -0.41 -0.32 -0.41
Be(H) 0.27 0.28 0.28 0.31
Be(1) 0.18 0.19 0.16 0.19
Be(J) 0.20 0.21 0.18 0.21
N(2s)e 0.85 0.84 0.86 0.86
N(~P)~ 1.43 1.42 1.43 1.43
“See ref 1 1 for definitions. Be(0) is the net charge on the central Be atom. Entries A-J are the net charges on a single Be atom in coordination shells 1-9 (Table I). bAtoms of types F and G all lie in coordination shell number 6 (see Table I and Figure 1). cGross atomic orbital population on the central Be.
on large Be clusters showed only small differences attributable to different couplings of the M O S . ’ ~ ~ The low-lying states of Be clusters consisting of 13, 19, 21, 33, 39, 5 1, 57, and 63 atoms have been characterized previously.’-3 Those clusters were selected because they form a series of clusters bounded by coordination shells of a central Be atom. In addition to exhibiting monotonic convergence of many properties toward bulk values, spherically balanced clusters are more likely to have charge distributions that are representative of the bulk. The lowest lying state of Be69 is an open-shell average-ofconfiguration state3 with the two unpaired electrons occupying the 14e’ shell, (1 la1’)0(4a2’)2(14e’)2(”l’’)o(9a2/l)o”e’’)4. In addition, a four-electron average-of-configurationstate, (1 1a1’)O(4a;)’( 1 4e’)2(3al’’)0(9”/’)1(9e’’)4,a closed-shell state, (1 (4a2’)2(14e’)O( 3al”)o(9a2/l)o(9e”)4,and a six-electron averageof-configuration state, (1 1al’)O(4ai)I( 14e’)2(”l’’)2(9a2/l)1(9e11)2, were found at 0.65, 1.89, and 5.60 eV, respectively, above the ground state. As shown in Table 11, the calculated binding energies per Be atom in these Be69states range from 28.2 to 26.3 kcal/mol as a function of electronic state. SCF binding energies for the same EP and basis set in the ground states of Bel3, BeS1,BeS7,and Be63 are 12.04,* 25.8,3 25.4,3 and 24.7’ kcal/mol, respectively. In comparison to the smaller clusters, the Be69 studies show an increase in binding energy of 2-3 kcal/mol in the direction of the accepted experimental value of 75.3 kcal/mol.8 The difference of 47 kcal/mol is largely attributable to the inherent neglect of electron correlation in the Hartree-Fock treatment of the cluster. One-electron properties calculated with respect to the central atom of the Be69cluster are also shown in Table 11. The (constant) contribution of the central-atom core has been omitted even when it is not zero. The cores of other atoms were approximated by reducing the nuclear charges. All of the charge distribution properties: except the diamagnetic shielding and second moments, contain nuclear contributions. The nuclear-electrostaticpotential 4 and second moments of charge (x2) and ( z 2 ) are essentially invariant to changes in state and electron configuration. The molecular quadrupole moment 8 shows a variation of 42% as a function of state (Table 11). The quadrupole moment, being a sensitive measure of asphericity, typically shows large differences with state arising from the large contribution made by the most diffuse molecular orbitals.2 Extension to even a larger and more spherical cluster such as Bel35(Table I) is expected to reveal greater constancy with state for such properties. The spreads of values for the second moments and the quadrupole moment as a function of state are substantially smaller in Be69than those calculated in the earlier studies on Be!3,2 Be51,3 and BeS7.3As can be seen in Figure 2, with the exception of the quadrupole moment for Bel3,there is a systematic decrease in the spread as the cluster size increases. This is indicative of the onset of convergence (at the SCF level) for these properties. The small spread of values for the quadrupole moment of Bel3is an artifact of the moiety itself, which exhibits more the properties of a molecule than a piece of the solid. Ionization potentials, as shown in Table 11, range from 2.58 to 3.69 eV. This range of 1.1 1 eV is somewhat larger than that of other properties, indicative of the change in MO stability as a function of electron configuration. It is noteworthy that the (8)Blaha, P.;Schwarz, K. J . Phys. F: Met. Phys. 1987, 17, 899. (9) Neumann, D.; Moskowitz, J. W. J . Chem. Phys. 1968, 49, 2056.
Spread of Second a n d Quadrupole Moments (a.u.) 800
0 8-r VI
1I
t
c
E,
07-
Legend 70.0
2
i
quadrupole moment 0,
06-
600
05-
50 0
04-
40 0
03-
300
0 c
In
3
0
L
,o
5 ?
‘0 L
m v)
2 0
>
;0 2 -
-
0
U
H
-0
200
0‘-
100
00
00
2 E
a v)
5,)
3c,,
$+e6,
Figure 2. Spread of calculated values of the quadrupole moment (1.344 91 X esu.cm2/au) and square roots of second moments (5.291 77 X cm/au) for Bel3, Be5,, Bes,, and Be69 for states lying within 5.6 eV of the ground state. Both nuclear and electronic contributions are included in the calculated values of the quadrupole moment.
value for the lowest state is within 0.3 eV of the experimental work function of Be metal.’O A Mulliken population analysis for each of the electron configurations studied is given in Table 1II.I’ The variation in overall net charges with MO configuration for the 11 sets of symmetry distinct atoms (Figure 1, Table I) is 0.05e or less, except for the atoms lying in shell 6, where the range is 0.09 and 0.15e. The 2s:2p ratio of gross atomic orbital populations on the central Be is 1.O: 1.7 for all four MO configurations, which is in accord with the observed behavior of the 2s and 2p bands in bulk Be.8 For the six atoms added to BeS7to form Be63,the maximum variation in net charge was 0.15 electron units.’ This proves to be quite parallel to that due to the 12 atoms added here from BeS7to form Be69. One might expect that the interior atoms would be close to neutral. It is not clear whether their actual computed charges are due to insufficient cluster size or to some other difficulty. The diamagnetic shielding constant ad (Table 11) varies by less than 0.01 au in comparison to ranges of 0.04,0.07, and 0.02 au for Be51,3Be57,3and Be63,’ respectively. This constancy as a function of electron configuration is strongly indicative of bulk behavior since, by contrast, in a stable gas-phase molecule such configuration changes are generally accompanied by large differences in these properties. Calculated values of electric field gradient q, shown in Table 11, vary from 0.016 to 0.021 au. The ranges of values calculated in earlier studies on Be13: Bes,: and Be5? are 0.17 to -0.19,0.035 to 0.076, and 0.007 to 0.079 au, respectively, for states lying within 5.6 eV of the ground state. A comparative plot of these ranges, along with the range of values from various experiments, can be seen in Figure 3. In comparing the smaller clusters to Be6,), the observed decrease in spread indicates convergence toward the bulk SCF value for this environment-sensitive propertye8 As can also be seen in the figure, the respective midpoints of the calculated ranges approach that of the bulk experimental range as the cluster (10)Tompa, G.S.;Seidl, M.;Ermler, W. C.; Carr, W. E. Surf. Sci. 1987, 185, L453. (11) Mulliken, R. S. J . Chem. Phys. 1955, 23, 1833.
J . Phys. Chem. 1990, 94, 7774-7780
7774
Electric Field G r a d i e n t of Be ( a . u . )
-0 2
'
Be
I
1
1
Be,,
Bes7
Bes3
r
Bulk
Figure 3. Range of values of electric field gradient (3.241 23 X I O L 5 es~.cm-~/au) for Be,3,Be5,, Be5,, Be69and experiment as given in ref 8 (bulk value assumed to be positive).
size increases. As we proceed to clusters beyond Be69in future studies, we may expect some oscillations in convergence patterns that depend on cluster shapes. Acknowledgment. This research was partially supported by the National Science Foundation under Grants CHE-8214689 and CHE-8912674 and by a grant from Cray Research, Inc. The Ohio Supercomputer Center and the Pittsburgh Supercomputing Center are acknowledged for grants of supercomputer time. We thank Dr. Victor Luana for helpful discussions. This work was performed under the auspices of the Division of Material Sciences
of the Office of Basic Energy Sciences, US. Department of Energy, and Lawrence Livermore National Laboratory under contract No. W-7405-Eng-8. Appendix The group theoretical arguments for approximating the reduction in the number of integrals arising from the use of point-group symmetry depend on the number of pairs of symmetry orbital sets for each irreducible representation (irrep), as arises in the supermatrix definition.' If there are nA atoms, none of which are located on symmetry elements, and each has a basis set of nF functions, then the A 0 representation consists of m (=nqnF) regular representations,I2 and the total number of basis functions is n (=gm), where g is the order of the group. A regular representation contains each irrep (A) a number of times equal to its dimension (d,).l2 Assuming that m is a large enough number that only its leading power need be kept, the total number of pairs of symmetry orbital sets from the same irreps are C(mdA)(mdA)/2= (m2/2)Cdx2 = m2g/2 h
A
and the total number of supermatrix elements is (m2g/2)(m2g/2)/2 = m 4 2 / 8 = (n4/8)/g? Since (n4/8) is the total number of A 0 integrals, the overall reduction factor is g?. Further analysis shows that approximately (n4/8)/g A 0 integrals are independent and need to be computed but that only (n4/8)/$ linear (supermatrix) combinations of these integrals need to be used in S C F calculations. Registry No. Be, 7440-41-7; Be69,127685-42-1. (12) Slater, J. C. Quantum Theory of Molecules and Solids; McGrawHill: New York, 1963; Vol. I .
Rotational Spectrum and Structure of the HCN-( CO,), Trimer H. S. Gutowsky,* Jane Chen, P. J. Hajduk, and R. S. Ruoff Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois 61801 (Received: iovember 0, 1989)
Rotational spectra have been observed and rotational constants determined for HCN-(C02)2,. H'3CN-(C02)2. HCIJN-(C02)2, and HCN-l C02C02by using the Fourier transform, Flygare/Balle Mark I1 spectrometer with a pulsed nozzle. Less extensive observations were made of DCN-(C02)2, HCN-( 13C02)2, and two '*O-substituted isotopic species. The rotational constants found for the parent asymmetric top are 1852.844, 1446.159, and 98 1.48 MHz for A, B, and C and -0.1085, -0.0297, -0.0241, -0.0190, and -0.0066 MHz for T ~ T, ~ T,,, Tbbbb, and T, respectively. The isotopic substitution reveals a ground-state geometry with the C2 symmetry of the slipped parallel (CO,), subunit and having the HCN along the C2 axis, the N end closest to the (CO,),. The C, symmetry is confirmed by the absence of eo and oe states as predicted for 2-fold symmetry with only equivalent bosons off-axis. The two carbons of the (CO,), lie in a plane R = 3.098 A below the center of mass of the HCN. The C-C distance in this subunit is 3.522 A, which is 0.077 A shorter than reported for the free (CO,), dimer. Also, an inertial analysis shows the individual C02's to be counterrotated by y = 20.3' out of the ac plane containing the carbons, the inner oxygens rotated away from the HCN. The OCC "slip" angle fi is 60.8' in the (CO,),. The torsional oscillations of the HCN are anisotropic, with an average displacement of 12.4', as determined from isotopic substitution and the I4N hyperfine structure (hfs). Virtually all of the hf components are doublets separated by 10-200 kHz. We attribute the doubling to an inversion of the clusters by a 140' counterrotation of the C02's. The inversion does not affect the dipole moment of the cluster, so the observed doubling is the difference in tunneling splittings of the rotational states for each transition.
( I ) Gutowsky, H. S.; Hajduk, P. J.; Chuang, C.; Ruoff, R. S. J . Chem.
Phys. 1990, 92, 862.
0022-3654/90/2094-7774$02.50/0
(2) Fraser, G . T.; Pine, A. S.; Lafferty, W. J.; Miller, R. E. J . Chem. Phys.
1987, 87, 1502.
0 1990 American Chemical Society
