6849
J. Phys. Chem. 1995,99, 6849-6856
Electron Density and Electrostatic Potential of Coesite James W. Downs? Department of Geological Sciences, The Ohio State University, Columbus, Ohio 4321 0 Received: December 21, 1994; In Final Form: February 27, 1995@
Coesite is a framework silicate containing Si-0-Si groups similar to the potential Br#nsted acid sites of zeolite catalysts. In this paper it is demonstrated that the details of the electrostatic potential and electron density near these bridging oxygens can be determined from single-crystal X-ray diffraction intensities. The static deformation electron density exhibits bond maxima near each 0 of about 0.6(1) e A-3 in the Si-0 bonds, which are strongly polarized toward Si. The total electron density (e) reaches a minimum of, on average, 1.05(2) e A-3 in the Si-0 bonds, with VZe at this point +20.3(5) e A-5, indicating bonds of an intermediate ionickovalent nature. On average, bond critical points are located 0.68 A from Si and 0.94 A from 0. Maps of the deformation density, deformation electrostatic potential, and -VZg show the influence of a local concentration of electron density in the nonbonding region of one bridging oxygen with an Si0-Si angle of 137". Maps of the electrostatic potential near two representative bridging oxygens are quite different, supporting the commonly held conclusion that theoretical calculations on zeolite catalysts must properly account for long-range interactions which may significantly perturb local electron density features.
Introduction It is generally accepted that the catalytic activity of zeolites is due to the release of protons from Br#nsted acid sites. These sites are formed when a Si in an Si-0-Si group is replaced by Al, permitting the bridging oxygen to accept a proton. A major problem is to locate the active Bransted acid sites in a zeolite and to predict their relative acidity. It has long been recognized that the protonation energy of a Br#nsted acid site will depend on the details of the charge distribution at the bridging oxygen. These details have been probed theoretically by selecting appropriate aluminosilicate clusters where the cleaved Si-0 or A1-0 bonds of the solid are terminated with H or OH. However, the intermediate ioniclcovalent nature of the Si-0 bond indicates that calculations on very large clusters or, ultimately, on the crystal itself are required in order to account for the effect of the long-range Coulomb field on the local electric fields at the Bransted sites.',2 For example, Greatbanks et al. recently performed SCF calculations on aluminosilicate clusters embedded in a zeolite l a t t i ~ e . The ~ electrostatic potential from the surrounding structure was modeled by nuclear centered point charges fitted to a potential generated by a periodic Hartree-Fock calculation at the minimal basis level. The purpose of this paper is to demonstrate that there is an alternative experimental approach to observing the detailed electrostatic potential and electron density about potential Br#nsted acid sites in aluminosilicates. Accurate X-ray structure factors derived from single-crystal diffraction data contain a wealth of information on the vibrationally averaged electron density distribution! Using such data to examine the electron distribution is attractive for three reasons. First, the electron distribution being probed may be viewed as nature's solution to the electronic structure problem, with no approximations. Second, the effects of lattice dynamics are naturally included in the scattered X-ray intensities. Third, the experimental approach may be applied just as easily to large, complex structures as to simple ones, the major factor being the availability of accurate diffraction data. This experimental + @
Chemical Physics Program. Abstract published in Advance ACS Abstracts, April 15, 1995.
0022-365419512099-6849$09.0010
approach may be viewed as complimentary to using the methods of computational chemistry to probe electronic structure. Preliminary to examining the detailed electron distribution of an actual zeolite we have chosen to study the Si-0-Si groups in the mineral coesite, a high-pressure polymorph of Si02. Coesite has often been chosen to study bonding in silicates, since there are eight unique Si-0 bonds and five unique Si-0-Si groups in the unit cell. A study of the deformation electron density of coesite has been reported by Geisinger, Spackman, and G i b b ~ .The ~ reader is referred to their paper for a discussion of the crystal structure and a review of previous analyses of this material. Using the same set of diffraction data, their study is herein extended to include the electrostatic potential and the gradient, curvature, and Laplacian of the electron density.
Electron Density Model The X-ray diffraction data of Geisinger et al.5 consists of 1716 structure factor moduli measured using Mo K a radiation to a maximum sin 8/Aof 0.91A-'. Although these structure amplitudes are related to the Fourier components of the electron density, the resolution is not sufficient for a direct mapping of the electron density by a Fourier series. Using the generalized X-ray scattering factor (GSF) model of Stewart,6 as coded in program VALRAY? a direct space, analytical representation of the total electron density was fit to the squares of the observed structure factor moduli IFOl2. The figures of merit for several least-squares refinements are given in Table 1. Independent atom (IA) scattering factors were computed from the Hartree-Fock atomic wave functions of Clementi and Roetti8 and used in a conventional structure refinement usin only high-order (HO) data with the minimum sin 81A = 0.75 - I . Scattering from core electrons is believed to dominate in this angular region, and the IA model is assumed to accurately account for core electron density features that are within the resolution of the data. The purpose of the highorder refinement was to obtain an unbiased estimate of the scale factor to serve as a point of comparison during refinement of the GSF model. In all refinements the anomalous dispersion corrections of Cromer and Liberman9 modified by the Z-
K
0 1995 American Chemical Society
Downs
6850 J. Phys. Chem., Vol. 99, No. 18, 1995
Table 1. The major difference between these models is that Geisinger et aL5 fixed used neutral, spherical atoms (the IA model) as fixed monopoles and used single-exponential radial functions with variable radial parameters on all atoms up to 2.17 1.02 937 57 924 2.32 IA-HO 1.014(3) the hexadecapole level. Although the agreement factors for the 1.31 1.09 1.17 IAM+g 1.011(1) 0.26(1) 1716 211 2565 1.35 1.23 GSF 1.005(7) 0.223(6) 1716 161 2843 1.25 I A M f model are somewhat lower than those of the present study, the GSF model requires only 161 parameters compared Isotropic extinction parameter x lo4 (radians-I). no = number to 21 1 for IAMf. The vibrationally averaged atomic positions of observations. nv = number of least-squares variables. R(F) = XI IFo12 - l ~ c l 2 l 2 l X I ~ o le2R4F) . = [ X ~ ( l F ol ~I ~ c l Z ) 2 ~ C ~ I ~ 0 1 4 1 1 ~ 2and f g ~ vibrational f displacement parameters are given in Table 2, = goodness of fit = [€/(no - nv)]'/*. g Geisinger, Spackman, and interatomic distances and angles in Table 3, mean square Gibb~.~ displacements along the principal directions of the vibrational dependent relativistic corrections of Kissel and Prattle were tensors in Table 4, and radial parameters and multipole applied. Refinements were based on minimization of e = populations in Table 5 . Multipole populations are expressed - lFc12)2 with the least-squares weight w = l/a2(IFo12). as momentsI2 such that the local dipole of a pseudoatom may The GSF model consisted of a multipole expansion about be obtained by evaluating (D12 D22 D3*)'". In all tables, the nucleus of each pseudoatom up to the octupole level. numbers in parentheses are estimated standard deviations and Valence monopole scattering factors for Si and 0 were refer to the last decimal place. constructed from linear combinations of Jacobi functions fitted Since it is a probability density distribution, the electron to the density-localized valence shells derived from the wave density must be positive everywhere. However, there is nothing functions of Clementi and Roetti.s The electron populations inherent in a GSF model that forces the resulting total density and K scaler of the valence monopoles for each unique to be positive. The electron density and its estimated standard pseudoatom were independently varied. Dipole and quadrupole deviation from the GSF model were computed at 0.1 A intervals radial functions were constructed from linear combinations of over the entire asymmetric portion of the unit cell. It was found Slater type functions taken from the density-localized atomic to be negative at several points, all far removed from the nuclei. orbitals of Clementi and Roetti.* The dipole radial functions The most negative density found was -0.033(7) e A-3. were formed from localized 2s 2p orbital products, and the Although the total electron density at such a point is strictly quadrupole radial functions, from localized 2p 2p products. The nonphysical, it is believed that the values are small enough so dipole and quadrupole functions on Si shared a common K scaler, as to not disqualify the GSF model from further examination. similar to the case for 0. Octupole radial functions for Si and To test the quality of fit, residual density maps were computed 0 were taken as with n = 4 for Si and n = 3 for 0. The as a Fourier series based on FobS - FGSF. Although all radial parameter, a,was fixed at 2.5 boh-' = 4.72 A-'. observations were included in least-squares modeling, only the Several low-order reflections exhibited significant departures 1520 observations with lFol > 3a(lF,I) were included in from kinematic scattering conditions. Extinction was treated calculating the residual maps. This cutoff eliminated 196 of according to the Becker-Coppens' model as isotropic Type-I the weakest high-order observations. Residual maps for the with a Lorentzian mosaic distribution. The largest extinction Si2-02-Si2 and Sil-05-Si2 planes are shown in Figures correction was for the 040 observation and amounted to l a and 2a, respectively. The features in these maps range from increasing lFol by 10%. Six out of the 1716 observations -0.34 to 0.31 e A-3 and appear to be caused by randomly received extinction corrections larger than 3%. distributed noise in the data. Importantly, there does not appear The figures of merit for the GSF model of this study are to be any accumulation of scattering density in the binding compared with those of model IAM+ of Geisinger et aL5 in
TABLE 1: Scale Factors, Extinction Parameters, and Figures of Merit from Least-Squares Refinements scale ga nob nvc 6 %R(F)d%RW(F)'goP
+
+
TABLE 2: Positional and Vibrational Parameters for GSF Modela Si1 0.14034(2) 0.10832(1) 0.07233(2) 0.00505(5) 0.00412(5) 0.00469(5) -0.00102(3) 0.00251(4) -O.O0069( 3)
X
Y Z
U11,@ U22, A2 U33, u12, A 2
A*
U13, A2 u23, A2
Si2
01
02
03
04
05
0.50669(2) 0.15801(1) 0.54064(2) 0.0053 l(5) 0.00476(5) 0.00397(5) -0.00035(4) 0.00257(3) -0.00032(3)
0 0 0 O.OOSO(2) 0.0050(2) 0.0097(2) -0.0036(2) 0.0036(2) -0.0014(2)
112 0.1 1622(4) 314 0.0 107(2) 0.0082(2) 0.0051(2) 0 0.0058(2) 0
0.26628(5) 0.12309(3) 0.9401O(5) O.OOSl(2) 0.0118(2) 0.0099(1) -0.0023( 1) 0.0070(1) -0.0008( 1)
0.31097(5) 0.10374(3) 0.32799(5) 0.0093(2) 0.0 107(1) 0.0041(1) -0.0028( 1) 0.0006(1) -0.00 17(1)
0.01772(6) 0.21176(3) 0.47854(6) 0.0091(2) O.O42( 1) 0.0 125(2) -O.O006( 1) 0.0060( 1) 0.0004( 1)
a Space group C2/c, second setting, origin at i on c-glide plane. Unit cell dimensions 7.1367(4) A, 7.1742(5) A, p = 120.337(3)' from Geisinger U&*2 2U12hku*b* 2U13hla*c* 2U23klb*c*)]. et al.5 T = e x p [ - 2 ~ r ~ ( U , l h ~ u * ~
+
+
+
+
+
TABLE 3: Mean Thermal Interatomic Distances (A) and Angles (deg) for GSF Model Sil-01 -03b -04
1.595(1) 1.613(2) 1.61l(4) 1.6208(9)
Si2-02 -03' -04 -05d
1.6118(3) 1.6140(6) 1.605(4) 1.6186(7)
01-Sil-04 -03'
109.29(2) 110.41(1) 109.80(2) 110.39(2) 108.91(2) 108.02(2)
02-Si2-04 -05d -03' 04-Si2-05d -03' 05d-Si2-03e
109.30(2) 110.24(3) 109.59(2) 109.3l(2) 108.87(2) 109.51(2)
-0s -09
04-Si1 -03b
-0s
03b-Sii - O S OX,
y, z
+ 1.0.
b
~ y ,,
z - 1.0.
-x, -y,
-2.
d
-
+ 0.5, -y + 0.5, -z + 1.0. ~
e
-x
Si1 -Ol-Silc Si2-02-Si2' Si 1"-03 -Si2' Sil-04-Si2 Silf-05-Si2d
+ 1.0, y , -z + 1.5.
f-X,
y,
180.0(4) 142.59(3) 144.43(2) 149.56(3) 137.20(3)
-Z
+ 0.5.
. I Phys. . Chem., Vol. 99, No. 18, 1995 6851
X-ray Diffraction Studies of Coesite
TABLE 4: Eigenvalues and Eigenvectors of Vibrational Tensors for GSF Model atom
u (A2 x
Si 1
01 02 03 04 05
r = xa
Y 0.067 27 -0.008 22 -0.044 08 -0.014 04 0.071 58 -0.034 86 0.067 16 0.036 41 0.026 44 0.000 00 0.080 84 0.000 00 0.017 07 0.042 38 -0.066 69 0.022 76 0.067 25 0.038 67 0.079 10 0.016 11 0.004 42
0.068 15 -0.022 74 0.135 71 -0.017 82 0.067 82 0.146 43 0.090 32 -0.112 26 -0.074 82 0.010 44 0.000 00 0.162 02 0.104 28 0.091 20 0.084 65 0.108 22 0.030 69 -0.117 07 0.025 64 -0.144 79 0.068 82
0.343(5) 0.465(6) 0.57 l(7) 0.362(5) 0.464(5) 0.548(7) 0.26(2) 0.99(2) 1.13(3) 0.26(3) 0.82(2) 1.07(3) 0.35(3) 0.98(2) 1.3l(2) 0.32(2) 1.03(2) 1.45(2) 0.40(2) 0.84(3) 1.26(2)
Si2
xa
100)
Z
0.065 78 0.125 86 0.076 91 -0.145 37 0.005 64 0.070 14 0.042 78 -0.135 13 0.077 40 -0.133 85 0.000 00 0.090 37 -0.050 30 0.135 03 0.072 93 0.150 58 -0.057 32 0.01 1 04 -0.005 69 -0.016 15 0.160 59
+
+ yb + zc, where (a, b, c> is the direct crystal basis.
regions that has not been accounted for by the GSF model. Due to the amount of noise evident in the residual maps, it was decided to use F G ~ F rather than Fobs when computing the electrostatic potential.
Electron Density Analysis Static Deformation Density. The static deformation density A@is computed by evaluating a direct space lattice summation over pseudoatoms of the GSF model and subtracting the IA model. All pseudoatoms within 5 of a particular point are included in the calculation. Although the Si and 0 atoms of the IA model are derived from ground state atomic wave functions (both 3P states), they have been spherically averaged. For diatomic molecules, one may choose to use properly
a
oriented ground state reference atoms or spherically averaged ones, resulting in strikingly different A@ maps in those cases where the total angular momentum of the ground state is greater than zero. This implies an unavoidable arbitrariness in A@maps that limits their usefulness for rigorous chemical interpretation. Notwithstanding this difficulty, A@ may be used to compare similar bonds in a given structure. Furthermore, since it has become traditional in both experimental and theoretical electron density studies to display A@maps, we continue that tradition here. Figure l b is a map of A@ for the plane Si2-02-Si2 with Si-0-Si angle 142.59(3)"; the 0 2 nucleus is at the center. Solid contours show where the GSF model has more electron density than the IA model; dashed contours show where it has less. The obvious accumulations of electron density are around oxygen nuclei, with well defined maxima in each Si-0 bond. The other two oxygens near the top of the map are both 0 5 atoms with nuclei 0.15 8, out of the plane. The other signs in Figure Ib nearest the Si2 positions indicate the projections of the 0 3 and 0 4 nuclei that are -1.38 and 4-1.22 A, respectively, out of the plane. The Si2-02 bond exhibits a Although this deformation density maximum of 0.6( 1) e experimental result is not strictly on an absolute scale, the fact that the scale factor for the GSF model of 1.005(7) is close to that from the high-order refinement of 1.014(3) suggests that the A@maps reported here are nearly on an absolute scale. Using the periodic Hartree-Fock model with a 6-21G** basis set, Nada et a l . I 3 have estimated the maximum A@for the Si-0 . A@ maps from bond of a-quartz to be about 0.5 e k 3 The SCF cluster calculations reported by Geisinger et aL5 exhibit bond maxima of about 0.35 e A-3 using a 6-31G* basis. It has long been known that the most general feature of the deformation density is an accumulation of electron density in the binding regions and a depletion in the extranuclear regions. Most notably, in regions far removed from the nuclei, A@is usually fairly constant and slightly negative. This means that pseudoatom densities in condensed matter are generally more con-
TABLE 5: Radial Parameters and Multipole Moments for GSF Modela Si1 l.OO(1) 9.98(1) 2.74(23)
K
core valence K
D1 (e A) D2 D3 K
Q1 (e A2) Q2 43 Q4 Q5
0 1 (e 02 03 04 05 06 07
3,
Si2 1.05(2) 9.98 2.34(21)
0.87(3) - 1.9(9) -0.9(8) -2(1)
0.87 0.1(9) 2(1) -0.7(8)
0.87 1.5(4) 3.2(9) -5(1) 0.7(8) -5.3 (9)
0.87 1.3(5) - 1.8(8) - 1.1(7) - 1.6(8) -1.8(7)
4.72 0.81(8) -0.01(8) -0.9(2) 0.1(4) 0.2(6) -0.10(7) 0.54(5)
4.72 0.51(9) -0.52(8) O.lO(2) 0.82(4) -0.41(6) -0.36(6) -0.10(5)
01
02
Monopoles 0.960(6) 0.962(6) 2.00 2.00 6.70(9) 6.73(9) Dipoles 0.46(2) 1.8(4)
0.46 0.07(8) 1.0(2) 0.0(1) 0.1(2) -0.7(1)
Quadrupoles 0.46 0.11(7) - 1.1(2) 0.2(1) Octupoles 4.72 0.08(4) -0.4(2) 0.12(3)
03 0.967(5) 2.00 6.54(7) 0.46 0.6(6) 0.4(3) 1.2(3)
04
05
0.949(4) 2.00 6.87(7)
0.95 l(4) 2.00 6.85(7)
0.46 1.1(4) 0.6(3) - 1.0(5)
0.46 - 1.7(4) 0.8(6) -0.6(3)
0.46 0.29(6) 0.0(1) -0.5(1) -0.26(9) -0.64(9)
0.46 0.05(4) O.Ol(9) 0.4(1) 0.2(1) 0.4(1)
0.46 -0.09(6) 0.1(1) 0.12(9) 0.0(1) -0.4 l(8)
4.72 -0.08(3) 0.02(3) 0.09(7) 0.1(2) -0.09(2) -0.01(2) -0.07(2)
4.72 -0.0 l(3) -0.02(3) 0.37(7) 0.3(2) 0.06(3) 0.08(3) -0.16(2)
4.72 0.25(4) -0.20(4) 0.24(8) -0.3(2) 0.05(2) -0.07(3) 0.06(2)
Moments for higher multipoles have been multiplied by 10. Estimated standard deviations are shown in parentheses unless value fixed or constrained.
6852 J. Phys. Chem., Vol. 99, No. IS, 1995
Downs
Figure 1. Electrostatic properties in the Si2-02-si2 plane: (A) residual density based on Fogs - FGSF,observations with IFob$l < 3a(lF&,l) excluded, contour interval 0.1 e A-3; (B) static deformation density A@(r),contour interva! 0.05 e A-3; ( C )total electrostatic potential d(r), contour (D) deformation electrostatic potential A+(r), contour interval 0.1 e A-I. interval 0.1 e kl; tracted than those of free atoms. The low values of he from cluster calculations when compared to the periodic HartreeFock results appear to reflect this trend. Assuming that our scaling is nearly correct, the fact that coesite is about 10% more dense than a-quartz may account for the slightly larger bond maxima in he observed for coesite when compared to those of a-quartz. Physically, as the average distance between positively charged nuclei decreases, the nuclear charges are less effectively shielded from one another. A contraction of electron density about at least one nucleus of a bonded pair may serve to accomplish this shielding. It is the shape of the electron density that is of most importance here, not absolute magnitudes. The major accumulation in he is about 0, but this density is strongly polarized toward the Si nuclei. Although there is a protrusion of deformation density away from 0 2 toward the exterior of the Si-0-Si angle, there is no distinct accumulation of deformation density in this nonbonding region. The major difference between Figure 2b and he maps from both cluster and periodic SCF calculations is that the experimental results apparently exhibit somewhat larger polarization of 0 toward Si. The deformation densities about 0 in the theoretical maps
are somewhat more spherical in shape. This could result from inadequacies in basis sets or from the fact that Hartree-Fock approximation itself usually leads to an overestimation of ionic character. The he map of Nada et al.I3 for a-quartz also exhibits an obvious accumulation of deformation density in the nonbonding region, a feature notably absent in Figure lb. Figure 2b shows the static deformation density in the plane of the Sil-05-Si2 group, which has the smallest Si-0-Si angle in the structure at 137.20(3)'. 0 5 is at the center with a clear accumulation of deformation density associated with each Si-0 bond. The 01 atom at the left edge of the map is 0.08 A out of the plane. The signs above Si1 represent the projection of the 0 4 and 0 3 nuclear positions at +1.02 and f1.27 A, respectively; those below Si2 indicate 0 2 and 0 4 positions at - 1.48 and +1.02 A, respectively. The most striking difference between the deformation densities about 0 5 and 0 2 is that there is a clear accumulation of deformation density in the nonbonding region of 05. The bond maximum is displaced toward the interior of the Si-0-Si angle by 9" for the Sil0 5 bond and the 14" for Si2-0 bond. The angle between the bond maxima is 114", and the average angle between the nonbonding maximum and the bond nearest the bond maximum
+
X-ray Diffraction Studies of Coesite
J. Phys. Chem., Vol. 99, No. I S , 1995 6853
IC
.. ..............."
I
".+
.-' *.____-
I-'
I - - -
I.-'
'-I
+ '\ / -.
',\
\
'
/
I
/
/
*---------'I
I
-. \
I I
/
/
Figure 2. Electrostatic properti$s in the Sil-05-Si2
plane: (A) residual density based on Fobs - FGSF, observations with < 3a(lF&sl) excluded, contour interval 0.1 e A-3; (B)static deformation density A&), contour interval 0.05 e A-3; (C) total electrostatic potential +(r), contour interval 0.1 e A-l; (D) deformation electrostatic potential A+(r), contour interval 0.1 e A-l. is 123". This appears consistent with the valence shell electron pair repulsion model14 and localized orbital ab initio studies where the repulsion between bonding and lone-pair hybrid orbital densities is greater than between bonding hybrids.15 A calculation of A@perpendicular to this plane, bisecting the Si105-Si2 angle, reveals only a single maximum in the nonbonding region corresponding to that shown in Figure 2b. Assuming a simple model based on the hybridization of atomic orbitals centered on oxygen, the deformation density at 0 5 would therefore appear consistent with sp2rather than sp3 hybridization. However, caution must be exercised in such interpretations because most of the electron density of the oxygen has been removed when forming the deformation density, and the reference oxygen atom lacks the quadrupolar component to the density of the true ground state atom. Furthermore, one cannot extract a unique set of orbitals, localized or otherwise, from an electron density. Therefore, it is tenuous even to speak of orbitals while gazing at an electron density map, particularly one of the deformation density. The other unique bridging oxygens of coesite exhibit deformation density features similar to those of 0 2 , with 0 5 being the only oxygen to exhibit a distinct concentration in the
nonbonding region. It is not clear whether the nonbonding feature at 0 5 is a purely local feature resulting directly from the narrow Si-0-Si angle or if Coulomb interactions with second neighbor Si's may play a significant role. The theoretical he maps of Geisinger et aL5 show the nonbonding feature growing more pronounced as the Si-0-Si angle narrows. However the lack of any smooth correlation of this deformation density feature with Si-0-Si angle in the present study indicates that long-range interactions may play a significant role in determining the shape of such detailed deformation density features. In zeolites therefore, we may anticipate that electronic structure calculations using robust basis sets on crystals or properly embedded clusters will be required in order to characterize potential BrQnsted acid sites. Topology of the Total Electron Density. The ambiguities of choosing a reference model for constructing he are removed if we limit our analysis to the total electron density e. Here we follow a procedure based upon a topological analysis of the electron density.I6 Figure 3 shows a surface plot of the total electron density of the GSF model truncated at &15 e A-3 for the Sil-05-Si2 plane, with oxygen at the center. A minimum in e associated with each Si-0 bond is clearly seen. A
6854 J. Phys. Chem., Vol. 99, No. 18, 1995
Downs
si 1
si2
students, to actually view the size of atoms in terms of the total electron density. The mean radii for coesite are R(Si) = 0.68 8, and R ( 0 ) = 0.94 These are quite different than the traditional ionic radii of, say, Shannon and Prewitt19 with R(IVSi4+)= 0.26 A and R("02-) = 1.35 A. The power of ionic radii such as these is in the prediction of bond lengths, not in giving a physical picture of where one atom ends and another begins. The total electron density at the bcp's averages 1.05 e A-3. Note that the bcp's do not'coincide with the location of the bond maxima in A@, the latter being located much closer to oxygen. The average value of the Laplacian of Q at the bcp's is +20.3 e A-5, clearly placing the Si-0 bond in either the intermediate or ionic camp. An examination of the individual principal curvatures shows that the large positive value of vQ(r,) is due mainly to the fact that A3 is so large compared to A1 and 1 2 . The A3 curvature, and one of the negative curvatures along the Si2-05 bond is clearly seen in Figure 3. The art of a flexible pseudoatom refinement is largely in the choice of radial functions. The more accurate and extensive the data, the more one is able to allow the radial functions to be determined by the data. In this study, the radial functions may be considered as somewhat restricted. This is not unlike the common procedure in computational chemistry of selecting a basis, such as 6-31G*, and performing a routine ab initio SCF calculation. In a normal calculation, the orbital exponents are not optimized for the molecule in question, only the electron populations are adjusted. The curvatures which sum to give vQ(rc) are particularly sensitive to the choice of radial functions. A preliminary model fit to this same data resulted in an average value of v Q ( r c )of 12 e A-5, much lower than that of the current mode1.23>24 This value was the result of both an average reduction of A3 and an average increase in A, and A2. At least in this case therefore, the curvatures of appear to be highly dependent on the radial functions chosen. A goal of current charge density research is to obtain more reproducible values for these curvatures from experimental data, which are a very sensitive measure of the shape of the electron distribution. When one considers that VQis related to the fourth derivative of the electrostatic potential, it is clear that only an extremely accurate potential will yield a reliable Laplacian distribution. It would be very desirable to quantitatively compare theoretical and experimental electron densities for crystals on the basis of results such as those shown in Table 6. A map of - v g through the Sil-05-Si2 plane is shown in Figure 4. 0 5 is at the center, and the view is toward the nonbonding region. For clarity, the map is truncated at f 1 5 0 e A-5 and the horizontal plane is where VQ(r) = 0. Points which plot above the plane show where V'-Q -= 0, where the total electron density is locally concentrated. Each quantum shell of at least light atoms exhibits a distinct pair of positive and negative regions in V 2g.20The spike at the center of 0 5 is the K-shell electron concentration, and the sharp, circular ridge surrounding the central spike is the L-shell electron concentra-
A.
Figure 3. Total electron density Q(r) in the Sil-05-Si2 truncated at 15 e A-3.
plane
chemical bond may be defined in terms of the topology of Q as a line joining two nuclei along which Q is a maximum relative to any neighboring line.I6 Along this topological bond path Q reaches a minimum at a bond critical point (bcp). The bcp's for the two Si-0 bonds are clearly seen in Figure 3. At a bcp, e(rc) has a positive curvature (A3) along the bond path and negative curvatures in both principal directions normal to the bond path (A, and A,). It has been shown that in covalent bonds the contractions of toward the bond midpoint dominate over contractions toward the nuclei such that the vQ(rc)= (1, 1 2 A,) -= 0 at the bond critical point. In bonds of ionic or intermediate type, VQ(rc)> O.I7 In Table 6, the total electron density of the GSF model is analyzed at each of the eight unique bond critical points of coesite. The total GSF density, g(rc), its Laplacian Vg(rc), the three principal curvatures of e at the bcp (A,, 22, and A,), and the fractional coordinates (xc, y,, and z,) giving the location of each bcp are given. The distance from the respective nuclei to the bond critical point is given by R(Si) and R(0). Note that if R(Si) R ( 0 ) is greater than the conventional bond length (measured between thermally averaged nuclear positions and given in Table 3), then the bcp necessarily cannot lie along the internuclear vector. In the case of coesite the bond paths and internuclear vectors very nearly coincide. R(Si) and R ( 0 ) are sometimes referred to as bonded radii,'* one of the many possible measures of the radius of an atom. Bonded radii imply the definition of what constitutes an atom in a molecule or solid to be that defined by BaderI6 in the Quantum Theory of Atoms in Molecules where atoms are separated by distinct surfaces along which VQ(r)= 0. Notwithstanding the rigor of Bader's theory, the definition of an atom in a molecule is certainly an arguable point which we will not belabor here. However, the idea of atomic or ionic radius is so entrenched in the psyche of inorganic and solidstate chemists that it can be illuminating, particularly for
+
+
+
TABLE 6: Analysis of Bond Critical Points for GSF Model Si1-01
Si2-02
Si1-03
Si2-03
1.11(2) 22.1(5) -7.5 (2) -7.0( 1) 36.7(4) -0.0798(8) -0.0634(2) -0.0432(8) 0.669 0.926
1.03(2) 20.3(5) -7.2( 1) -7.0( 1) 34.4(4) 0.4973(7) 0.140 1(4) 0.8723(5) 0.677 0.935
1.16(2) 1835) -8.1(1) -7.3(1) 33.9(5) 0.1964(5) 0.1 137(4) 1.0206(7) 0.672 0.942
1.08(2) 19.0(5) -7.1( 1) -6.9( 2) 33.0(5) 0.3983(5) 0.1448(4) 0.9536(8) 0.678 0.937
Si1 -04
Si2-04
Sil-05
Si2-05
0.93(2) 22.0(5) -6.9( 1) -6.5(2) 35.3(4) 0.21 17(8) 0.1073(4) 0.1801(3) 0.678 0.933
0.95(2) 21.4(5) -6.7( 1) -6.2(1) 34.2(5) 0.4260(8) 0.1366(5) 0.4470(5 ) 0.680 0.927
1.04(2) 19.6(5) -7.2( 1) -6.8(1) 33.7(6) -0.0773(8) 0.1525(3) 0.4467(8) 0.676 0.945
1.07(2) 19.7(5) -7.2(1) -7.0(2) 33.8(5) 0.0020(8) 0.2874( 1) 0.4649(8) 0.677 0.942
X-ray Diffraction Studies of Coesite
Si2
Figure 4. Laplacian of the electron density -Ve(r) in the Sil-05Si2 plane truncated at &150 e k5.
tion or the valence shell charge concentration (VSCC). Each Si pseudoatom exhibits similar features. The large, truncated columns are L-shell concentrations, and the tiny spikes within the columns are K-shell concentrations. The M-shell concentration of even an isolated Si atom at this scale is so small as to not really be visible. In the valence region, the map is clearly dominated by the VSCC of oxygen. This is in accord with the electron density of states calculated by Nada et al. for a-quartz, which shows the valence band to be nearly totally due to oxygen 2s and 2p states. Referring to Figure 3, note that when moving from the 0 5 nucleus toward the viewer is strictly a monotonically decreasing function. There are no quantum shells evident in e. However, the quantum shells become obvious by merely taking the Laplacian of the total electron density, without comparison with any arbitrary reference model. The map of - v e shows small peaks in the VSCC of oxygen near each Si-0 bond, indicating increased local concentration in the bond. There is also a very small, broad peak in the nonbonding region. Therefore, the nonbonding feature so clearly evident in A@ persists in -v@ and is not simply an artifact of subtracting out the IA model when computing A@. 0 5 is the only oxygen of coesite that exhibits three maxima in its VSCC; all others only exhibit two, one associated with each Si-0 bond. Threedimensional sections through 0 5 also reveal that the VSCC maxima evident in Figure 4 are in fact the true maxima in this three-dimensional function.
Electrostatic Potential Analysis At least in regions removed from the immediate vicinity of the nuclei, the electrostatic potential is well resolved given accurate X-ray diffraction data within a sphere of sin 0/d = 1.0 A-'. Although the present data are limited to a maximum sin 0/d of 0.91 A-1, we nonetheless attempted a mapping of this property using a combination of reciprocal space and direct space methods. The method used has been outlined by Stewart.21 The mean thermal deformation electrostatic potential, A$, is first evaluated by Fourier summation as in
J. Phys. Chem., Vol. 99, No. 18, 1995 6855 potential is formed in reciprocal space, and then the independent atoms are added back in by a direct space lattice summation with static point nuclei included. The final term is the mean inner potential due to the IA model and is a sum over the unit cell. This term ensures that the electrostatic potential integrates to zero over the unit cell. As discussed by O'Keeffe and Spence,22there are several possible strategies for computing the mean inner potential and defining the zero of the potential. In order to facilitate comparisons between theoretical electrostatic potential calculations, this author encourages the use of a standard definition of the zero of the potential for such studies. Choosing zero such that the potential integrates to zero over the unit cell seems an obvious choice for crystalline systems. Maps of the total electrostatic potential for the Si2-02-Si2 and Si1-05-Si2 planes are shown in Figures IC and 2c, respectively. Electronegative contours are dashed, and the contour interval is 0.1 e A-' = 33.21 kcaymole of charge. The line of zero equipotential is shown by the large dashes. Since the zero of the potential chosen here is quite different than that chosen by Greatbanks et al.3 or by White and Hem2 in their theoretical studies, a direct comparison of absolute values is unfortunately not possible. The most striking difference between these two maps is that a well defined minimum in $ occurs near 0 5 in the nonbonding region whereas a broad electronegative region exists in the same region of 0 2 . It is not immediately clear how much of this difference is simply due to the distribution of electropositive nuclei and how much is due to the local electron density accumulations in the nonbonding regions of 0 5 and also 0 1 in Figure 2c. Figures 2d and 3d show the deformation electrostatic potential A$(r) for the 0 2 and 0 5 planes, respectively. The deformation potentials about 0 2 and 0 5 are clearly different and mimic in a subdued manner the deformation density features. In particular, the nonbonding region of 0 5 is distinctly electronegative in comparison to that of 0 2 . The localized minimum in $ near 0 5 therefore appears consistent with local features in A@,A$, and -vQ. Electrostatic potential maps for these same planes were computed using Si4+and 02-ions and a monopoles-only fit to the X-ray data. Neither of these models agree with the GSF model. The major point is that regardless of the local vs nonlocal origin of these features, the electrostatic potentials about 0 2 and 0 5 are rather different. Since the Si-0-Si angles about 0 2 and 0 5 are similar, it is clear that if these observations were to be repeated theoretically, a crystal calculation or a calculation on a properly embedded cluster would be required. These results suggest that the Brgnsted acidity of an Si-OH-A1 site in a zeolite would be strongly affected by longrange electrostatic interactions.
Acknowledgment. The author wishes to thank Dr. Mark A. Spackman for providing the coesite data and for reviewing an earlier version of the manuscript. References and Notes
where V is the unit cell volume and H is the Bragg vector with magnitude 2n sin 0/d. F(H),b, is scaled, corrected for extinction, and phased by the GSF model. The total electrostatic potential is formed by evaluating
where $(r)IAM is the electrostatic potential due to the static IA model, including nuclei. In other words, the deformation
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