4834
J . Phys. Chem. 1992, 96, 4834-4840
1991, 176, 1. (c) Martin, J. M. L.; Francois, J. P.; Gijbels, R. J. Chem. Phys. 1991,94, 3753. (d) Bernholdt, D. E.; Magers, D. H.; Bartlett, R. J. J. Chem. Phys. 1988,89, 3612. (e) Pacchioni, G.; Koutechy, J. Ibid. 1988,88, 1066.
(20) Moller, C.; Plesset, M. S. Phys. Reu. 1934, 46, 618. Pople, J. A.; Binkley, J. S.; Seeger, R. Int. J . Quantum Chem. Symp. 1976, SIO, 1. Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72, 4244. Bartlett, R. J.; Purvis, G. D. Int. J . Quantum Chem. 1978, 24, 561. (21) Pople, J. A.; Head-Gordon, M.; Fox, D. J.; Raghavachari, K.; Curtis, L. A. J. Chem. Phys. 1989, 90, 5622. (22) Pople, J. A,; Head-Gordon, M.; Raghavachari, K. J . Chem. Phys.
(f) Michalska, D.; Chojnacki, H.; Hem, B. A,, Jr.; Schaad, L. J . Chem. Phys. Lett. 1987, 141, 376. (g) Magers, D. H.; Harrison, R. J.; Bartlett, R. J. J. Chem. Phys. 1986, 84, 3284. (h) Ritchie, J. P.; King, H. F.; Young, W. S. Ibid. 1986, 85, 5175. (i) Lammertsma, K.; Pople, J. A,; Schleyer, P. v. R. J . Am. Chem. SOC.1986, 108, 7. (j) Whiteside, R. A.; Raghavachari, K.; DeFrees, D. J.; Pople, J. A,; Schleyer, P. v. R. Chem. Phys. Lett. 1981, 78, 538. (k) Slanina, Z.; Zahradnik, R. J . Phys. Chem. 1977, 81, 2252. (14) Raghavachari, K. J . Chem. Phys. 1986,84,5672. Rachavachari, K.; Logovinsky, V. Phys. Rev. Lett. 1985, 55, 2853. (15) GAUSSIAN 90: Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M.; Binkley, J. S.; Gonzales, C.; &Frees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Steward, J. J. P.; Fluder, E. M.; Topiol, S.;Pople, J. A. Gaussian, Inc., Pittsburgh, PA. (16) For an introduction to the methods employed see: Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (17) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (18) Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72,
authoritative review on C4,see: Weltner, W. Jr.; Van Zee, R. J. Chem. Reu.
4244. (19) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. J. Comput. Chem. 1983, 4, 294.
1989, 89, 1713. (28) (a) Dewar, M. J. S.; Ford, G. P. J . Am. Chem. SOC.1979, 101,783. (b) Cremer, D.; Kraka, E. J . Am. Chem. SOC.1985, 107, 3800.
1987,87, 5968. (23) DeFrees, D. J.; McLean, A. D. J. Chem. Phys. 1985.82, 333. (24) (a) Goddard, W. A. 111; Dunning, T. H.; Hunt, W. J.; Hay, P. J. Acc. Chem. Res. 1973,6, 368. (b) Bobrowicz, F. W.; Goddard, W. A. I11 Modern Theoretical Chemistry; Schaefer, H. F. 111, Ed.; Plenum Press: New York, 1977; Vol. 3. (25) (a) Bader, R. F. W. Atoms in Molecules-a Quantum Theory; Oxford University Press: New York, 1990. (b) Bader, R. F. W. Acc. Chem. Res. 1985,18,9. (c) Bader, R. F. W.; Nguyen-Dang, T. T. Ado. Quantum Chem. 1981,14,63. (d) Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y. Rep. Prog. Phys. 1981, 44, 893. (26) Cremer, D.; Kraka, E. Croat. Chim. Acta 1984, 57, 1259. (27) Algranati, M.; Feldman, H.; Kella, D.; Malkin, E.; Miklazky, E.; Naaman, R.; Vager, Z.; Zajfman, J. J . Chem. Phys. 1989,90,4617. For an
The Laplaclan of the Electron Density and the Electrostatic Potential of Danburlte, CaB,SI,O, James W.Downs*?+and R.Jeffrey Swopel Department of Geological Sciences, The Ohio State University, Columbus, Ohio 4321 0 (Received: August 20, 1991)
The crystal structure, electrostatic potential, and total electron density and its Laplacian distribution, V2p(r), in danburite (CaBzSi208),a framework borosilicate, have been retrieved from single-crystal X-ray diffraction intensities measured to The topology of the Laplacian distributionand the fact that V2p > 0 at the bond critical points indicates sin O/A = 1.1 18 the Ca-0, B-O, and Si-0 interactions to be mainly closed shell in nature. The pattern of oxygen valence shell charge depletion as revealed in maps of V2p(r) indicates that the oxygen valence shell is less locally depleted near bond paths and more depleted between bonds. This pattern of local charge concentration and depletion appears consistent with features in the electrostatic potential around bridging oxygens.
Introduction Danburite (CaB2Si208)is a naturally occurring borosilicate with charge-compensating Ca atoms embedded in a framework of corner-sharing BO4 and S i 0 4 tetrahedra.'-3 Topologically, the tetrahedral framework of danburite differs from that of the feldspars anorthite (CaAlzSiz08), albite (NaA1Si308), and reedmergnerite (NaBSi308). Furthermore, it has a unique ordering scheme with three of the five crystallographically unique bridging oxygens in SiB07 groups, one in a B2O7 group, and one in an Si207 This is in contrast to the aluminum analogue of danbunte, anorthite, which exhibits a perfect alteration of A104 and S i 0 4 tetrahedra consistent with Lowenstein's rules5 Lowenistein's rule therefore appears unique to aluminosilicate frameworks. We find the borosilicate framework of danburite to be of interest because of the possibility of tetrahedral boron in boron-substituted zeolites.6 In particular, we are interested in the detailed electron density distribution, p(r), and electrostatic potential, $(r), about the bridging oxygen of the SiOB group as a possible Bransted acid site.6 In this paper we present an analysis of an electron density distribution derived from X-ray diffraction data where we focus on the pattern of local charge depletion in the valence shell of oxygen as revealed through the Laplacian of p(r). This property is then related to features in the electrostatic potential distribution. 'Chemical Phvsics Graduate Facultv. *Current addiess: Department of -Geological Sciences, University of Colorado, Boulder, CO 80309.
TABLE I: Experimental Conditions crystal size, mm space group cell dimensions, A radiation temperature, K scan type step size, deg; time, s scan range range in sin 8 / A linear absorption coeff p, cm-' transmission factor range
0.07 X 0.14 X 0.09
Pnam a = 8.0456 (7)" b = 8.7629 (4) c = 7.7341 (7) Zr-filtered Mo KCY(A = 0.71069 %.) 300 w-28 step scans 0.04 w ; 4.0 Aw = 2.0 0.7 tan 8 0.084-1.118 A-'
+
15.49 0.84-0.97
Esd's in parentheses refer to last decimal place.
Experimental Section Table I contains information relevant to data collected and reduction. A large colorless crystal of danburite from Toroku Mine, Miyazaki, Japan, was obtained from the National Museum of Natural History (USNM No. 105586). The crystal was broken into equant fragments, approximately 5 mm in diameter, that were heated and then thermally shocked by rapid immersion in liquid nitrogen in an attempt to decrease crystallite size, increase mosaic spread, and thereby reduce secondary extinction. A fragment, bounded by six semiplanar faces, with approximate dimensions 0.07 X 0.14 X 0.09 mm was selected and mounted on a glass fiber. Preparatory to computing path lengths for absorption and ex-
0022-365419212096-4834%03.00/0@ 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 12, I992 4835
Crystal Structure of Danburite TABLE II: Multipole Population Parameters
EPo/K K
D1 02 03 Q1 Q2 Q3 Q4 Q5 0 1 02 03 04 05 06
07
Ca
B
Si
18.200 (3) 1 .o
4.61 (17) 1.02 13) 0.01 (15) -0.45 (14) -0.37 (14) 4 (3) 1 (6) 9 (7) -3 (6) 0 (5) -0.16 (4) -0.21 (4) -0.01 (4) 0.13 (4) -0.22 (5) -0.01 (4) -0.17 (5)
13.59 (23) 0.90 (2) 0.21 (15) -0.33 (20) 0.17 (15) -8 ( 5 ) 1 (11) 3 (11) -7 (12) 1 1 (10) -0.27 (3) -0.12 (3) 0.05 (3) -0.06 (4) -0.20 (4) -0.09 (4) -0.24 (4)
01 8.44 (6) 0.981 (4) 0.09 (8)‘ 0.24 (9) 0.03 (9) -1.4 (7) -1.9 (13) 0.5 (14) 0.8 (14) -0.7 (1 1) 0.00 (2) 0.02 (2) 0.00 (2) 0.05 (2) 0.02 (2) 0.01 (2) 0.05 (2)
tinction corrections, the perpendicular distance of each face from an appropriate origin was accurately measured using a spindle stage and image splitting ocular mounted on a polarizing microscope. The Miller index of each face was obtained from the spindle-stage coordinates and diffractometer orientation matrix by direct linear tran~formation.~ The crystal was mounted on a Picker four-circle diffractometer automated by a Kreisel control system. An orientation matrix and unit cell parameters were obtained by least squares from the corrected angles of 19 automatically centered reflections in the range 52O I29 I66O. Profiles for reflections consistent with space group Pnam were measured. The full sphere of reflection out to 28 41° was scanned, and a half-sphere 28 from 41° to 105O. Three standard reflections were monitored every 8 h. Reflection profiles were corrected step by step for Lorentz and polarization effects, reduced to integrated intensities, and corrected for time-dependent fluctuations by a polynomial fit to standard reflections using the methods of Bles~ing.~Analytical absorption corrections were completed using program ABSORB.^^ After rejection of outliers, 16 243 intensities were averaged in h u e class mmm to 33 15 observations with and internal agreement factor R = 4.3 % ( R = C(llFo12- IFo12(mean)l)/CIFo12).The rather large agreement factor was partly due a preponderance of weak high-angle reflections. Because of these weak data we decided to only use 2197 observations with I > 2 c ~ ( l F ~ during 1 ~ ) leastsquares refinements even though this practice tends to bias the high-angle data by rejecting underestimated intensities and retaining overestimated ones.” The variance for each symmetrically unique observation was taken to be the weighted mean of the counting statistical variance for each measurement used in the average.
CSF Model In addition to positional, anisotropic vibrational, and a type I Lorentzian isotropic extinction parameter,12 generalized scattering factors (GSF’s) to the octupole level were included in the model in order to probe the details of electron density distribution, p(r), within the rigid pseudoatom approximation.I3J4 During refinement 6 = Cw(lFOl2- IFc(2)2 was minimized with w = 1/ u2(IFOl2). Real anomalous dispersion corrections from Cromer and LibermanIs were applied as augmented by the Z-dependent corrections of Kissel and Pratt.I6 Since our major interest was the electron distribution of the tetrahedral framework Ca was described with a monopole form factor only computed from the SCF wave function for Ca2+of Clementi and Roetti.17 Prior to computing radial form factors the Clementi and Roetti SCF wave functions for B, Si, and 0 were density localized using the method of Stewart.’” The K-shell radial functions for these atoms were formed using is X is orbital products. During refinement the K-shell electron populations of B, Si, and 0 were constrained to equal the CaZ+K-shell population. Valence monopoles for B and 0 computed from localized Lshell
02 8.39 (6) 0.981 -0.08 (8) 0.08 (9) 0.04 (8) -1.0 (6) 0.8 (13) -2.2 (14) -2.4 (14) -1.5 (11) 0.04 (2) 0.16 (2) 0.05 (2) 0.05 (2) 0.04 (2) 0.01 (2) 0.01 (2)
03 8.45 (6) 0.981 0.22 (9) 0.08 (9) -0.11 (8) -0.2 (6) 3.4 (14) -0.3 (13) -0.7 (13) -1.2 (11) 0.03 (2) 0.04 (2) -0.02 (2) 0.01 (2) 0.07 (2) -0.01 (2) -0.01 (2)
04 8.40 (7) 0.981 -0.16 (13) 0.17 (13)
05 8.43 (8) 0.981 -0.12 (12) -0.06 (14)
-2.6 (9) 0.4 (20)
-1.9 (9) -0.7 (20)
2.2 (16) -0.02 (3)
-2.3 (17) 0.04 (3) 0.05 (3)
-0.02 (3) -0.01 (3)
0.08 (3) 0.02 (3)
-0.05 (3)
atomic orbital (AO) products consisted of 31 Slater type functions (STFs) each, and the localized M-shell of Si of 72 STFs. These valence monopoles were scaled with a variable K parameterlg with each of the five crystallographicallyunique 0 atoms sharing the same K . Dipole radial form factors were taken from localized valence shell s X p A 0 products consisting of 24 STF’s for B and 0 and 64 STF’s for Si. Quadrupole radial form factors were formed similarly from localized p X p A 0 products consisting of 10 STFs for B and 0, with 36 S T F s for Si. Dipole and quadrupole form factors were not K scaled. The octupole radial functions were of single-exponentialtype Pe-OLrwith n = 3, 3, and 4 and standard molecularZoradial scalars LY = 2.52,3.06,and 4.47 bohr-’ for B, 0, and Si, respectively. Tesseral harmonics of all multipoles were described relative to the same Cartesian coordinate system.” During preliminary conventional refinements it was noticed that the weighted residuals of the 040 and 080 observations were far larger than for any other reflection, even with an extinction correction. These residuals were therefore not included in the computation of the least-squares shifts. The full model consisted of 168 variables (nv) fit to 2195 observations (no). Of the 168 variables, 104 were electronic parameters whose refined values and associated esd’s are given in Table 11. The final cycles were performed with the full Hessian matrix including second derivatives for all parameters and refinement was considered converged when the projection of the least-squares error vector onto the vector of shifts was 9.88 X lo-”. The figures of merit are R(1q2) = 0.0346,R,(1q2) = 0.058,gof = [t/(no - n ~ ) ] =~ 0.873. / ~ The above model is on a relative scale and may be put on an absolute scale by dividing the electron population parameters by an effective scale factor taken as the sum of the monopole populations divided by the number of electrons in the unit cell giving Kdmted = 1.026 (6). The isotropic extinction parameter refined to g = 0.087 (4) with the largest extinction correction for the 130 observation of y = 0.76,or an estimated 24% attenuation on lFOl2.The fact that the intensities were measured on a relative scale under nonkinematic conditions was probably the most serious limiting factor in this analysis. Refinements were carried out using the VALRAY system of programs.I4
Crystal Structure The refined positional and vibrational parameters are given in Table 111. An ORTEP drawing of danburite is shown in Figure 1. As described by early workers, the framework is based on four-membered rings of tetrahedra that are linked together to form the “double crankshaft” chains running parallel to the 2 crystallographic axis which are in turn cross-linked through the 0 2 oxygens to form a continuous three-dimensional framework. All oxygens save 0 4 are considered bonded to Ca. Ca, 04,and 0 5 are located on a mirror planes Z = 1/4, 3/4. The mean square amplitudes of vibration along the principal directions of the vibrational ellipsoids are deposited in Table SI (see paragraph at the end of this article).
Downs and Swope
4836 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 TABLE 111: Atomic Positions and Vibrational Parameters Darameter Ca B Si 0.25898 (9) 0.05330 (2) X 0.38555 (3) 0.41935 (10) 0.19256 (2) Y 0.07653 (2) 0.42074 (10) -0.05584 (3) Z 0.25 0.0043 (2) 0.00399 (6) Ull, A2 0.00749 (6) 0.00384 (6) 0.0053 (2) U22,A2 0.00648 (6) 0.00363 (6) U3,, A2 0.00675 (6) 0.0052 (2) -0.00006 (5) UI2,A2 -0.00018 (6) -0.0002 (2) -0.0000 (2) 0.00001 (5) u13,A’ 0 -0.0006 (6) 0.0001 (2) u23,A2 0
01 0.19282 (10) 0.06781 (10) -0.00332 (1 1) 0.0075 (2) 0.0049 (2) 0.0092 (2) 0.0025 (2) -0.0022 (2) -0.0015 (2)
TABLE IV: Selected Interatomic Distances (A) and Angles (deg) 05-B-01 115.02 (9) B-014 1.4792 (12) 05-B-02 106.96 (7) 1.5003 (11) B-0212 I 1 1.OS (8) 1.4631 (1 1) 05-B-03 B-0312 102.54 (7) 1.4541 (9) 01-B42 B-05 110.65 (6) 01-B-03 mean 1.4742 02-B-03 110.18 (7) mean 109.40
02 0.12650 (10) 0.36484 (9) -0.04248 (10) 0.0066 (2) 0.0044 (2) 0.0072 (2) -0.0017 (2) -0.0020 (2) 0.0005 (2)
Si-01 Si42 Si-039 Si-04’ mean
03 0.39967 (9) 0.31355 (10) 0.07837 (9) 0.0050 (2) 0.0067 (2) 0.0057 (2) 0.0011 (2) 0.0013 (2) 0.0010 (2)
1.6187 (8) 1.6238 (8) 1.6150 (8) 1.6158 (8) 1.6183
04 0.51380 (16) 0.66360 (16) 0.25 0.0096 (3) 0.0096 (4) 0.0031 (3) 0.0023 (3) 0 0
01-Si42 01-Si44 01-Si43 02-Si44 02-Si43 04-Si43 mean
05 0.18391 (14) 0.42812 (17) 0.25 0.0051 (3) 0.0120 (4) 0.0042 (3) 0.0017 (3) 0 0
111.12 (4) 111.03 (6) 110.31 (4) 109.05 (6) 105.52 (4) 109.66 (5) 109.45
T-0-T Angles Ca-01I2 X 2* Ca4Px2 Ca-03I2 X 2 Ca-0s2 mean
2.4998 (9) 2.4549 (8) 2.4675 (8) 2.4008 (1 1) 2.4636
Si-01-B7 Si-02-Bi2 Si8-03-BiZ Si4-04-Si1O B-05-BIZ
132.55 (7) 126.42 (6) 128.03 (6) 136.67 (9) 130.49 (9)
Bond distances and angles are given in Table IV. A table (Table SII) containing lFal, lFJ,u(IFal), and the absorption weighed path length for each observation is given in the supplementary material.
Electron Density Analysis Elastic X-ray diffraction intensities are the Fourier transform of the mean-thermal one-electron density function in the crystallographic unit Even though electron population parameters are refined in the presence of anisotropic vibrational parameters, and the mean square amplitudes of motion are relatively small in danburite, a full deconvolution of the electron density from the vibrational motion is not expected. Although model densities of the type shown in this paper are traditionally referred to as “static”, in fact the electron density distribution presented here is something between the full thermally smeared distribution and that sought after by theorists who solve the equations of electron motion in the field of fixed nuclei. It is not clear how best to compare experimentally derived densities of this type with those computed from theoretical wave functions for crystals or model molecular clusters. Traditionally, electron density results have been displayed as the deformation density, Ap(r), the difference between p(r) and the electron density distribution obtained from a superposition of spherically averaged atomic densities fixed at the atomic positions of the crystal or molecule. Here, we depart from this tradition for the following reasons: (1) density functional theory2* tells us that the total energy is a functional of p(r) not Ap(r); (2) although &(r) is purported to reveal the electron density changes due to chemical binding, the analogous energy term, the binding energy, is defined relative to atoms separated at infinity, not relative to a hypothetical model of spherical atoms forced into a nonquantum state in a crystal; (3) finally, only by analysis of p(r) and its functionals can we hope to probe the underlying mechanics which underly this functi01-1.~~ We restrict our electron density analysis therefore to our pseudoatom model for the total electron density distribution. There is no constraint in the GSF model that forces positivity in the resulting electron density distribution. It is therefore necessary to test the final model for positivity throughout the asymmetric portion of the crystallographic unit cell. The total electron density of the GSF model and its variance (as derived
Figure 1. ORTEP drawing of danburite structure viewed down Ycrystallographic axis; Z is to the right. Atoms are plotted as 98% probability vibrational ellipsoids.
from the inverse Hessian matrix) were sampled over the entire asymmetric unit with a 40 X 8 1 X 20 grid and 0.1 A between grid points. The GSF model density was found to be positive definite at all points sampled. Figure 2 shows p(r) through the bonding planes of bridging oxygens 0 3 , 0 4 , and 0 5 , located at the center of each map. The maps are truncated at 15 e/A3. For danburite p(r) only exhibits local maxima at the positions of nuclei, shown by the large truncated peaks in the maps. The small subsidiary maxima are not true maxima in three dimensions, whereas the truncated maxima at the nuclei are. Maps for the 0 1 a n 8 0 2 oxygens are not shown since they are quite similar to 0 3 . Figure 2d is a map of the esd in p(r) for the Si-03-B plane. The map was computed using the elements of the inverse Hessian of the electronic parameters and is truncated at 0.5 e/A3. The actual esd at the Si nucleus is 5.1 e/A3, whereas p(r) = 415.56
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 4837
Crystal Structure of Danburite Si
1
Si
03
04
B
Si
TABLE V:
n
Ca-0 1 Ca-02 Ca-03 Ca-05 B-01 B-02 B-03 B-05 Si-0 1 Si42 Si-03 si-04
p
(e/A3) and V2p (e/As) at (3,-1) Critical Points in p(r)
0.142 (3) 0.166 (3) 0.161 (3) 0.184 ( 5 ) 0.93 (6) 0.99 (3) 1.13 ( 5 ) 1.15 ( 5 ) 1.01 (4) 0.95 (3) 0.95 (4) 0.94 (4j
3.23 3.73 3.57 4.14 11.85 6.04 7.63 6.95 17.56 17.16 19.46 18.30
0.2880 0.3791 0.3927 0.5368 0.2205 0.21 39 0.3052 0.2342 0.1094 0.0856 0.4883 0.4725
0.0733 -0.029 1 0.1949 0.0736 -0.0325 0.3989 0.3850 0.4202 0.1385 0.2639 0.3129 0.6802
0.1228 0.3553 0.1622 0.25 -0.0594 0.0438 0.0784 0.1358 -0.0334 -0.0485 -0.0009 0.3617
e/A3 at this point. At the minimum in p(r) between 0 3 and B, p(r) = 1.13 e/A3 with an esd of 0.05 e/A3. The electron density
B
05
03
i
B
B
Figure 2. GSF electron density distribution truncated at 15 e/A3, from top to bottom: (a) Si-03-B, (b) Si-04-Si, (c) B-05-B; (d) esd for Si-03-B truncated at 0.5 e/A3. (e) Residual Fourier map for Si-03-B plane based on F, - F,, contour interval 0.1 e/A3. Maps are 6.0 X 6.0
A.
is thus most precisely determined at the nuclear positions. In the valence region the esd map is dominated by oxygen with a value of 0.05 e/A3 in the cols between the atoms rising to 0.18 e/A3 at the ledge adjacent to the central peak. Esd maps for the 0 4 and 0 5 planes are similar. Figure 2e is a residual Fourier map based on F, - F, using the GSF model for the Si-03-B plane. Although the features are not small, ranging from -0.54 to 0.45 e/A3 in magnitude, they are essentially randomly distributed. Residual maps calculated throughout the unit cell are similar with features ranging from -0.64 to 0.58 e/A3. The residual densities indicate a rather noisy data set, as does the poor agreement among symmetry equivalent reflections. In such a case the GSF model can be viewed as a filter through which the significant electronic features can be clearly seen. The topology of p(r) may be analyzed by locating critical points where the gradient of p(r) vanishes. These critical points (cp’s) are classified as (r,s)where r, the rank of the critical point, is the number of nonzero eigenvalues in the Hessian matrix of p(r), and s, the signature, is the sum of the signs of the eigenvalue^.^^ The nuclear positions are expressed in p(r) as (3,-3) cp’s since the electron density falls off in all directions heading away from a nuclear position. Locating the (3,-3), cp’s in p(r) has been the task of crystallographersfor over 75 years. However, a complete description of a crystal structure in terms of p(r) demands the location of cage (3,+3), ring (3,+1), and bond (3,-1) cp’s as well. We will limit ourselves here to (3,-3) cp’s, given in Table 111, and (3,-1) cp’s given in Table V. The saddle points in p(r) along the bonds in Figure 2 show where the gradient of p(r) goes to zero. The 3 X 3 matrix of second derivatives of p(r) (Le., the Hessian of p(r)) has three nonzero eigenvalues corresponding to the three principal curvatures of p(r) calculated at the critical point. Close to but not necemrily on the nearest-neighbor internuclear vector is a line where p(r) is a maximum relative to any neighboring line. This is the bond path.24 Along the bond path the curvature of p(r) is positive, whereas the curvatures are negative perpendicular to the bond path. The sum of the signs of the curvatures (-1-1+1) yields the signature, -1, of the (3,-1) or bond critical point. The locations of the bond cp’s of danburite and the value of p(r) at these points, p(rq), are given in Table V. All Ca-0 bonds show a very low value of p(r) at the bond critical point whereas B 4 and S i 4 values are much larger, with the average B 4 value slightly greater than the average Si-0 value. The atomic coordinates of Table I11 and the (3,-1) cp coordinates of Table V may be used in comparisons of critical point locations obtained from theoretical charge density functions. The electron density function of an isolated atom is a monotonically decreasing function of the radial coordinate and does not reveal the shell structure of the atom.25 It is seen in Figure 2 that this characteristic is retained in the solid if the “atom” encloses only those trajectories in the gradient vector field of p(r) that terminate at tis nucleus.26 The radial density function of the atom, D(r) = 4 d p ( r ) , where p(r) is the spherically averaged atom, is often invoked to show an atom’s shell structure, but this function loses its meaning for the “atom” in a molecule or crystal.27
4838 The Journal of Physical Chemistry, Vol. 96, No. 12, I992 Si
03
si
03
B
Downs and Swope Ca
B
B
Figure 3. (a, top) Vzp(r) and (b, bottom) -V2p(r) for Si-03-B plane truncated at r50 e/A5. Figure 5. V2p(r) for B-05-B plane truncated at 750 e/AS from two perspectives.
Si
04
Figure 4. a top) and (b, bottom) VZp(r)for Si-04-Si plane truncated at 750 elk;; (c) for plane bisecting Si-04-Si linkage.
The quantum shell structure of the atom, be it isolated or bound in a crystal, is revaled through the Laplcian of the electron density, V2p(r).If V2p(r)< 0 then p(r) is locally concentrated; if V2p(r) > 0 then p(r) is locally depleted.23 Each quantum shell of the atom will exhibit a region of + and -V2p(r). Maps of V2p(r)are shown in Figures 3-5. Values that plot above the horizontal plane show where p(r) is locally depleted, and values that plot below the plane show where p(r) is locally concentrated. These maps have been arbitrarily truncated at r 5 0 e/A5. Figure 3a shows V2p(r)for the Si-03-B plane. The spike at the center of Si is from the K-shell, and the large truncated peak from the L-shell. The diffuse valence M-shell of an isolated Si atom is missing. Boron exhibits only a central K-shell charge
depletion. The most interesting features are associated with oxygen which dominates the electron density in the “valence region” (see Figure 2). Like B, the spike near the 0 3 nucleus shows the charge depletion associated with the K-shell. The undulating ridge around oxygen corresponds to its valence shell charge depletion (VSCD). The most curious result of our analysis are the undulations that occur in the VSCD’s of these bridging oxygens. An isolated, spherically averaged oxygen would have a circular ridge of the same height in V2p(r),like the circular Ca2+M-shell evident in Figure 5b. However, we find that the VSCD’s of oxygen in danburite exhibit local maxima and minima, with the minima always near the bond paths. This means that, although the valence electron distributions of these oxygens always exhibit regions of local depletion, the least amount of local valence charge depletion is found along the bonds. The Laplacian of p(r) is intimately associated with the energetics that govern the electron distribution. If V(r) is the local potential energy density and G(r) is the positive definite local kinetic energy density then the following relation obtains. (h2/4m)V2p(r)= 2G(r)
+ V(r)
(1)
The competition between the local kinetic and potential energies in lowering the energy of the system is thus revealed through V2p(r)?3Since this particular definition of the local kinetic energy density, G(r), is positive definite, regions where V2p(r)< 0 indicate where the potential energy dominates. The kinetic energy contribution is dominant over the potential energy term when V2p(r) > 0. The topology of V2p(r) and the value of V2p(r)at a bond cp exhibit particular properties for the limiting cases of shared and closed-shell interactions. For homonuclear diatomics, where valence electrons are shared between the two atoms, it is found that V2(r,) < 0 and the region of -V2p(r) is contiguous between the two atoms.23 For interactions between closed-shell species, as in essentially pure ionic bonds, the region of -V2p(r) is localized about the anion, the region of VSCD associated with the cation is missing, and V2p(r)< 0 at the bond critical point. However, for interactions of a mixed nature, as between Si-0 and B-0, the distinction is not as clear cut and we cannot merely evaluate V2p(r)at the (3,-1) cp’s in order to classify the bond as “ionic” or “covalent”.27 The values of V2p(r)at the bond cp’s are given in Table V for each bond path. It is seen that they are uniformly positive, suggesting atomic interactions of a closed-shell nature. Furthermore, the regions of VSCD of the B and Si are missing. However, the Laplacian distributionsof Figures 3-5 are of a form unlike those of a pure closed-shell or shared interactions. The
Crystal Structure of Danburite
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 4839
Figure 6. Mean thermally averaged electrostatic potential with mean inner potential removed for (a) Si-03-B, (c) Si-O4-Si, (e) B-05-B, electronegative contours dashed, contour interval 0.1 e/A; (b), (d), (f) esd maps for same planes, add 0.05 to get true esd; small dashes negative, solid positive, large dashes zero. Maps are 6.0 X 6.0 A.
large values of V2p(rcp)for Si-0 relative to B - 0 bonds are due to the much larger positive curvature for Si-0 while the negative curvatures for B-O and Si-O are nearly identical. This should, however, probably not be interpreted in terms of relative ionicities of the B-0 and Si-0 bonds in danburite. The Laplacian distributions of Figure 3-5 indicate therefore that the Ca-0, B-O, and Si-0 bonds of danburite are largely closed shell in nature and that the valence shell charge depletions of oxygen are a minimum along the bonds. From (1) it is seen that the lower VSCD's near the bond paths may be due to less dominance of the kinetic energy density in these regions. No maps of the esd of V2p(r) have been presented since an algorithm for their computation has yet to be incorporated into our codes. It is estimated that the values of V2p(r) in Table V and the features in Figures 3-5 may be in error by more than 30%
of the value of the property. Furthermore, these values are highly dependent on the details of the GSF model, which was chosen to be rather restricted because of the modest quqlity of the X-ray data. Nonetheless, we believe that even these modest results yield a more complete picture of the nature of the electron density distribution and bonding in danburite than would be obtained from a superior set of X-ray diffraction intensities by relying on the deformation density alone. Electrostatic Potential The first inner moment of p(r) is the electrostatic potential, t$(r), which may be computed from a theoretical electron density function or may be derived from X-ray diffraction data.28-30 The electrostatic potential is not calculated directly from the GSF model but is computed through a combination of Fourier sum-
4840 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
mation and direct space lattice sum techniques. It is given as $(I) = A$(r)
+ ~ I A M+ 40
(2)
where A$(r) is the mean thermal deformation electrostatic POtential given by A$(r) = -(1/4aV)C(F0 - FIAM)~-~*~".'(S~~ e/x)-2 (3) hkl
where v i s the unit cell volume, Fais the observed structure factor with Dhase anale from the GSF model, F U is~ the structure factor comiuted frGm a superposition of spbeically averaged Hartree-Fock atoms" (Le., the independent atom model or IAM), and H is the Bragg vector with magnitude IHI = 2 sin e/A. This expression is very similar to the Fourier series represenation of the deformation density save for the constant term and the fact that in (3) each Fourier coefficient is weighted by lHl-2 which means that the deformation potential is much better resolved than the deformation density when computed from the same data set. The second term in (3), is the total potential of the IAM and is computed as a lattice sum in direct space. This effectively adds back in the contribution of the IAM substracted out in the first term. The reason for this contrivance is to overcome the singularity caused if the Fooois weighted by IHI-2 in a calculation of the total potential directly as a Fourier series over F0.29 The final term, &, is an estimate of the mean inner potential of the crystal as encountered in electron microscopy. For most solids c $ ~ is about 10 eVa3' With (2) we are therefore mapping the mean thermally averaged electrostatic potential with the mean inner potential removed. The addition of do= -1.0419 e/A for danburite guarantees that the integral of 4(r) over the unit cell vanishes so that the unit cell remains electrostatically neutral (1 e/A = 14.4 eV/unit charge). Maps of 4(r) and u[(4(r)] for the same planes as Figure 2 are shown in Figure 6. In order to differentiate between minima and maxima in the esd maps, a constant of 0.05 e/A was subtracted before contouring. Addition of 0.05 e/.& will yield the true esd. The questionable values of IFoI for the 040 and 080 observations were replaced with lFcl computed from the GSF model prior to computing these maps. In the maps the electropositive equipotential contours are solid, the small dashes are electronegative, and the large dashes electroneutral. Near the nuclei 4(r) is dominant4 by the electropositive nuclear potential so that most of the electropositive contours have been omitted. 4(r) is of most interest in regions far removed from the nuclei. The esd maps show that 4(r) tends to have the largest error in these extranuclear regions with a maximum estimated error of about 0.06 e/A, and therefore the contour interval for the 4(r) maps was chosen to be 0.1 e/A. The electronegative equipotential lines show where a noninteracting positive test charge would be attracted by the crystal potential. Moving up gradient from a local electronegative minimum one invariably approaches an oxygen. The Ca-05 bond in Figure 6e and the Ca-03 bond in Figure 6a seem particularly prone to electrophilicattack. The electronegative minima in Figure 6, a, c, and e, are -14.2 (7), -15.8 (7), and -17.0 (7) eV, respectively. There is an interesting relationship between features in 4(r) about the bridging oxygens and the local maxima and minima in the oxygen VSCD's encountered in the Laplacian maps.
Downs and Swope Starting from a bridging oxygen nucleus in +(r) and heading by steepest descent down gradient one passes over the regions of largest VSCD in V2p(r). For example, compare the broad electronegative minimum evident on the back side of 04 in Figure 6c with the broad, undeformed VSCD of Figure 4b. The steepest gradients in #(r) about 0 3 and 0 5 correlate well with the maxima in the VSCD's of these atoms shown in Figures 3a and 5b, respectively. Therefore, the regions about the oxygens most prone to nucleophilic attack are where the oxygen valence density is not locally depleted. Acknowledgment. We thank G. V. Gibbs, Department of Geological Sciences, Virginia Tech, Blacksburg, VA, who graciously allowed us substantial instrument time on the four-circle diffractometer in his laboratory. This work was supported by National Science Foundation Grant EAR-8618834 awarded to J.W.D. Supplementary Material Available: Table S I listing eigenvalues and eigenvectors of vibrational ellipsoides (1 page); Table SI1 listing X-ray structure factor moduli for danburite (17 pages). Ordering information is given on any current masthead page. References and Notes (1) (2) (3) 79. (4) (5) (6)
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