J . Phys. Chem. 1989, 93, 7979-7984
7979
FEATURE ARTICLE Experimental Charge Densities in Chemistry: What is Next? Philip Coppens Chemistry Department, State University of New York at Buffalo, Buffalo, New York 14214 (Received: February 17, 1989)
After extensive development of experimental and calculational methods and concepts, experimental charge densities from diffraction measurements are now being more widely applied to chemical and physical problems. Examples are the calculation of the Coulombic contribution to molecular interactions, the study of bonding-induced changes in transition-metal ligands, the establishment of the electronic ground state of transition-metal complexes, and the calibration of theoretical calculations. Further experimental advances may be expected from the availability of the new synchrotron sources, which allow the use of smaller crystals, harder radiation, and shorter data collection times.
Introduction
The possibility of measuring the charge density in a crystal from its X-ray diffraction pattern was envisioned many years ago when Debye and Scherrer discussed the possibility that the halos on a powder photograph were images of the electron orbits around the atoms.' It took more theoretical developments, in particular the application of the theory of Fourier transforms, to establish the basic relationships between elastic X-ray scattering and the electron density distribution. Even then the preponderance of experimental errors in the measurements, and the success of the spherical atom description of the density, raised doubts about the feasibility of such studies. As a result, electron density studies in the 1960s and early 1970s were directed toward establishing the feasibility of the method and its limitations, given the sources and equipment available at that time. More recently the emphasis has shifted to applications to chemical and physical problems, including comparison with theoretical results, calculation of intermolecular interactions, and determination of the ground state in metal coordination complexes. The reproducibility of the diffraction method has been well established, in particular by parallel X-ray and neutron diffraction studies on oxalic acid dihydrate conducted in several laboratories.* But it is necessary to emphasize that charge densities cannot be obtained with data of routine quality, notwithstanding occasional claims to the contrary. The accuracy of the intensities must be monitored by measuring symmetry-equivalent reflections, data must be collected at reduced temperatures in particular in the case of molecular crystals, and systematic effects on the intensities, such as extinction, must be corrected for, or, better, minimized by the use of very small sample crystals or very high energy radiation. When these conditions are met, unique results can be obtained. Some examples and new developments are discussed in this article. Why Experimental Charge Densities?
Though electrons and electron probability distributions are essential concepts in a theory of chemical bonding, the electron density cannot be used to obtain the wave function of the system, except perhaps in the simplest cases.3 The use of density matrices to obtain the maximum amount of information from the diffraction (1) Debye, P.; Scherrer, P. Phys. Z . 1918, 19, 474. (2) Coppens, P.; Dam, J.; Harkema, S.; Feil, D.; Feld, R.; Lehmann, M.
S.; Gaidard, R.; Kruger, C.; Hellner, E.; Johansen, H.; Lamn, F. K.; Koetzle, T. F.; McMullan, R. K.; Maslen, E. N.; Stevens, E. D. Acta Crystallogr. 1984, A ~ O184. , (3) Massa, L.; Goldberg, M.; Frishberg, C.; Boehme, R. F.; LaPlaca, S. J. Phys. Rev. Lett. 1985, 55, 622.
0022-3654/89/2093-7979$01.50/0
experiment is an active field of investigation! The charge distribution as determined by X-rays is a ground-state property, whereas many processes of chemical interest involve electronically excited states. On the other hand, the charge density from a diffraction experiment is a detailed three-dimensional function, which successful theoretical methods must be able to reproduce. Thus it can be used for the calibration of theoretical methods and has, for example, confirmed the validity of density functional calculations on tetrafluoroterephthalonitrile (p-dicyanotetrafluorobenzene)s and oxalic acid dihydrate6 and demonstrated the deficiency of limited basis set theoretical calculations,' which do not properly predict the amount of delocalization of the valence electrons into the bonding regions. Recent advances in density functional theory8 have underlined the central role of the electron density as a function from which the Coulombic and the kinetic energy can be derived, the latter through the use of a density functional relating the density to the kinetic energy distribution. Though this article concerns the charge density, the possibility of combining experimental charge densities with spin densities from polarized neutron diffractiong and with momentum densities from Compton scattering measurements should not be ignored. Ultimately further advances in all three techniques may make experimental densities as routinely available as is now the case for atomic structure. Such an advance would provide an unusually powerful analytical technique for the characterization of solids. Some Definitions and Methods
The most commonly used methods of electron density analysis are the calculation of electron density maps and the least-squares fitting of parametrized analytical functions to the experimental structure factors. The electron density maps are obtained by the usual Fourier inversion of the structure factors and require knowledge of the phases of the observed structure factors. For noncentrosymmetric crystals, for which the phases are continuous variables, the model used to calculate the phases must be sufficiently sophisticated to include the deviations from spherical symmetry of the atomic densities.l0 This means that a nonspherical least-squares refinement must be used to determine the model parameters. For centrosymmetric crystals the spherical (4) Proceedings of the Conference on the Use of Density Matrices, Coimbra, 1988. (5) Delley, B.; Becker, P. J. Unpublished results. (6) Krijn, M. P. C. M.; Feil, D. J . Chem. Phys. 1988, 89, 4199. (7) Stevens, E. D.; Rys, J.; Coppens, P. J . Am. Chem. SOC.1978, 100, 2324.
(8) Dahl, J. P.; Avery, J. A. Local Density Approximations in Quantum Chemistry and Solid State Physics; Plenum: New York, 1984; p 851. (9) Coppens, P.; Koritsanszky, T. Chem. Scr. 1986, 26,463. (10) Coppens, P. Acta Crystallogr. 1974, 830, 255.
0 1989 American Chemical Society
7980 The Journal of Physical Chemistry, Vol. 93, No. 24, 1989
Figure 1. Deformation density map in the plane of the imidazole ring of the histidine molecule, showing differences between C-C bonds. Contours at O.le A-’. Zero and negative contours broken.35
atom treatment is quite adequate; such crystals are therefore inherently more suitable for the density analysis. Nevertheless, successful studies on acentric crystals have been reported; an example of such a study on iron(I1) tetraphenylporphyrin is discussed below. Since the total electron density is dominated by the core electrons and only very slightly affected by chemical bonding, difference densities are employed to illustrate bonding effects. The most widely used function is the “atom deformation density” or “standard deformation density” which is simply the difference between the total density and the density corresponding to superimposed spherical atoms A P ( ~ )= Ptotadr) -
C all
p(r-Ri)
(1)
atoms
where Ri is the nuclear position of atom i . The density obtained experimentally is the total density averaged over the vibrational modes of the crystal. In order to apply expression 1, the atomic densities subtracted must be thermally averaged, or the thermal motion must be deconvoluted from the total density. In the former case the thermally averaged deformation density ( Ap(r) ) is obtained. Either procedure requires a model for the thermal motion in the crystal, which is often based on the atomic thermal parameters from the X-ray experiment. In the case of a molecule the second term in (1) is referred to as the promolecule density; Le., it is the density prior to bond formation between the atoms. Such deformation densities can be excellent demonstrations of accumulation of density in the bond and lone-pair regions of a molecule (see, for example, Figure 1). But for elements with more than half-filled valence shells, bond peaks in the standard deformation density tend to be lower than expected or may even be absent.” The deformation density for hydrogen peroxide, for example, shows no bond accumulation in the 0-0bond, even though the oxygen atoms are clearly held together by a covalent interaction (Figure 2).12 This is because the spherically averaged oxygen atom contains more than one electron in the orbital which participates in bond formation. The effect has been very elegantly demonstrated in a study on a molecule that contains C-N, C-0, N-N, and 0-0 bonds, 1,2,7,8-tetraam-4,5,10,1 l-tetraoxatricyclo[6.4.1.1L7]tetradecane, in which, as expected, deformation density bond peaks decrease as the atoms participating in the bonds are more electron-rich.” Because of this, the standard deformation density has been criticized by theoretician^,^^*'^ and alternative definitions of the ~~
~
~~~
(11) Coppens, P. In Electron Distributions and the Chemical Bond Coppens, P., Hall, M. B., Eds. Plenum: New York, 1982; p 479. (12) Savariault, J. M.; Lehmann, M. S . J . Am. Chem. SOC.1980, 102, 1298. (13) Dunitz, J. D.; Seiler, P. J . Am. Chem. SOC.1983, 105, 7056. (14) Schwartz, W. H. E.; Valtazanos, P.; Ruedenberg, K. Theor. Chim. Acta 1985,68,471. Schwartz, W . H. E.; Mensching, L.; Valtazanos, P.; von Niessen, W. Int. J . Quantum Chem. 1986, 29, 909.
Coppen s
Figure 2. Deformation density map through the 0-0-H group of hydrogen peroxide. Contours at 0. l e A-’. Negative contours broken.I2
deformation density have been proposed in which either oriented atoms or hybridized atoms are subtracted, as was in fact done in much earlier work on the F2 molecule by Bader, Henneker, and Cade.16 While the alternative definitions are helpful, in particular in illustrating the bonding density, the spherically averaged atomic density is the only reference state that is unambiguously defined and does not depend on assumptions regarding orientation and hybridization of the atoms. An extension of the idea of subtraction of a prepared entity is to subtract a chemical fragment, or ligand, from a molecular density. This was first done in 1976 by Rees and Mitschler for chromium hexacarbonyl, Cr(C0)6, for which subtraction of a theoretical density of the carbon monoxide molecule clearly showed the decrease in u- and increase in *-density in the ligand due to bonding,” in accordance with the a-donation, ?r-back-donation concept of Dewar and Chatt.I8 The experimental work was very nicely extended in a theoretical density study of the dissociation of chromium hexacarbonyl by Sherwood and Ha11,I9 who concluded that *-orbitals contribute 25% of the Cr-C bond, except at large Cr-C distances where their effect becomes negligible. A similar study of the C-H ligand density in methylidyne tricobalt nonacarbonyl shows the ligand density to be intermediate between that of the 211 ground state and the 4E excited state of the C-H molecule.20 To obtain more quantitative information on the charge distribution than is possible from electron density maps, the refinement of charge density parameters such as valence shell populations, a valence shell ”shapen parameter, and populations of atomic dipole, quadrupole, and higher functions has been developed.21v22 They are usually refined together with the conventional structural parameters which include atomic positional and temperature parameters. The most widely applied charge density model of this kind consists of an expansion of atom-centered spherical harmonic functions The radial dependence or “shape” of the valence shell density in this “multipole model” is allowed to vary by the adjustment of a parameter K , which scales the radial coordinate rz2
(15) Kunze, K. L.; Hall, M. B. J . Am. Chem. SOC.1986, 108, 5122. (16) Bader, R. F. W., Henneker, W. H.; Cade, P. E. J . Chem. Phys. 1%7, 46, 3341. (17) Rees, J. J . Am. Chem. SOC.1976, 98, 7918. (18) Dewar, M. J. S . Bull. SOC.Chim. Fr. 1951, 18, C71. Chatt, J.; Ducanson, L. A. J . Chem. SOC.1953, 2939. (19) Sherwood, D. E., Jr.; Hall, M. B. Organometallics 1982, I , 1519. (20) Leung, P. Ph.D. Thesis, State University of New York at Buffalo, 1982. Coppens, P. Coord. Chem. Rev. 1985,65, 285. (21) Stewart, R. F. Acta Crystallogr. 1976, A32, 565. (22) Hansen, N . K.; Coppens, P. Acta Crystallogr. 1978, A34, 909.
The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 7981
Feature Article TABLE I: Comparison of X-ray and Spectroscopic Values of Fe Quadrupole Sptitting Constants (mm/s)‘ ref
compoundb Fe(II)S2 Fe( 1I)Pc
Fe(III)(OMe)TPP Fe(II)(py)2TPP Fe( I1)TPP
X-ray
Mossbauer
X-ray
0.8 (5) 2.2 (1.1) 0.6 (2) -1.9 (5) -4.8 (8)
0.634 (6) 2.62 (2) 0.56 (5) f1.15 f1.32 f1.52
60 30 63 53, 54
TABLE II: Comparison of Charges on COO- and NH3+ Groups in Different Compounds
DL-histidine
Mossbauer
group
61 62
COONH,’
64
“Using the shielding constants y = -10 and R = 0.22 and QFs = 0.2 cm2to convert the X-rays results to spectroscopic values. b P ~ = phthalocyanine, TPP = tetraphenylporphyrin, OMe = methoxy, py = pyridine.
X
where p is plus or minus for m I 1, P, is the valence shell population which multiplies the spherical valence density p(r), and the radial function R can have either a Slater type or a Hartree-Fock radial dependence. When this expansion is truncated after the second term a simple spherical atom model is obtained, which allows the determination of the net charge and the change in the radial dependence of the valence shell that must accompany any deviation from electron neutrality of an atom, because of the change in electron-electron repulsion. This “kappa refinement” is quite successful in giving net atomic charges of atoms, which can be used in molecular force field calculations, or in the calculation of molecular dipole moments for molecules in crystals. Examples of experimental solid state dipole moments found to be close to solutionor gas-phase values are those of f ~ r m a m i d e sulfamic ,~ acid,24 d alanine,^' and the water molecules in oxalic acid dihydrate,25 and MgS04.6H20?7 The agreement may be somewhat fortuitous, however, as dipolar interactions in solids are expected to enhance the dipole moment through polarization. On the other hand, the experimental dipole moment is reduced by truncation of the tails of the molecular distribution in the space-partitioning process.28 Use of the full multipole expansion; including the higher order terms, leads to the determination of dipole and higher moments of the “pseudoatoms”, so labeled because of the ambiguity of any partition of a continuous charge distribution. Quadrupole coupling constants obtained in this way are often in agreement with spectroscopic values, as in and in iron(I1) phthalocyanine?” (but not in iron(I1) tetraphenylporphyrins3qs4)(Table I). Though they are much less accurate than the spectroscopic values, they give insight in the origin of the moment and include determination of its sign. In such work the multipole least-squares analysis is also used to deconvolute the effect of thermal motion in the crystal from the electron density, as the thermal motion is represented by the usual anisotropic harmonic temperature factor (in some studies anharmonic temperature factors have been use4i31). Within the limits of the validity of the thermal motion model a static density is obtained when the multipolar functions are plotted. It is often directly compared with theoretical results (for example, tetrafl~oroterephthalonitrile~).On the other hand, it is possible to thermally smear the theoretical density using the thermal motion parameters obtained experimentally. Such an analysis was applied (23) Coppens, P.; Gum Row, T. N.; Leung, P.; Baker, P. J., Yang, Y. W.; Stevens, E. D. Acta Crystallogr. 1979, A35, 63. (24) Bats, J. W.; Coppens, P.; Koetzle, T. F.Acta Crystallogr. 1988,833, 35. (25) Destro, R.; Marsh, R. E.; Bianchi, R.J . Phys. Chem. 1988, 92, 966. (26) Destro, R.; Bianchi, R.; Morosi, G. J . Phys. Chem. 1989, 93,4447. (27) Bats, J. W.; Fuess, H.; Elerman, Y. Acta Crystallogr. 1986,842, 552. (28) Moss, G.; Coppens, P. In Molecular Electrostatic Potenfials in Chemistry and Biochemistry; Politzer, P., Truhlar, D., Eds.; Plenum: New York, 1981. (29) Stevens, E. D.; DeLucia, M. L.; Coppens, P. Inorg. Chem. 1980, 19, 813. (30) Coppens, P.; Li, L. J . Chem. Phys. 1984, 81, 1983. (31) Mallinson, P. R.; Koritsanszky, T.; Elkaim, E., Li, N.; Coppens, P., Acfa Crystallogr. 1988, A44, 235.
ref 35 -0.68 (8) 0.57 (8)
L-alanine ref 35 ref 25 -0.63 (7) 0.38 (6)
-0.59 (2) 0.42 (4)
a-glycylglycine ref 36 -0.5 0.43
to f~rmamide.~ It clearly demonstrates the disappearance of sharp features, in particular near the nuclei, due to the vibrations of the molecules. In a covalently bonded molecule the overlap density in the bonds between atoms is effectively projected into the atom-centered functions of the multipole expansion. But when the overlap populations are very small the atom-centered functions of the model directly represent the population of the atomic orbitals. To a good approximation this situation exists in transition-metal complexes, for which the population of the d-orbitals can be derived from the multipole parameters within the validity of the approximation. The relation between d-orbital populations and the multipole coefficients can be derived when the products of the spherical harmonic d-orbital functions are written as linear combinations of spherical harmonic^.^^-^^
For a transition-metal atom the d-orbital density is also described by the summation in the multipole formalism defined by (2). The second term in (2) then describes the non-d (i.e., mainly 4s) part of the spherical valence density. It follows that the multipole populations PIw and the orbital populations are related through a matrix equation PImp= MP,
(4)
where P!w is a vector containing the coefficients of the 15 spherical harmonic functions with 1 = 0, 2, or 4, which are generated by the products of d-orbitals. Thus, the d-orbital populations P can be obtained from the refinement results, the Pimp, througg application of the inverse of (4). The appropriate elements of M have been r e ~ o r t e d . ~ ~ , ~ ~ Experimental Charge Densities and Intermolecular Interactions
Experimental charge densities are making significant contributions to the study of intermolecular interactions. In molecular mechanics and lattice energy calculations the electrostatic interaction is often approximated by the interaction between atomic monopoles, using charges from theoretical treatments of the isolated molecules or fragments. Net atomic charges from the experimental diffraction data obtained by least-squares refinement, or with other formalism^?^ apply to a molecule in a crystalline environment and therefore may be more appropriate than isolated molecule values. To examine the reproducibility of such charges we have applied the kappa refinement to a set of 125 K data on histidine^^ and to 23 K data on L-alanine by Destro et al.25 The agreement between the charges on the carboxylic and NH3+ groups in these two compounds and those in gly~ylglycine~~ in quite good (Table II), as are the individual charges on comparable atoms in the two molecules.35The same kappa method was applied to the nucleotide 2’-deoxycytidine 5’-monophosphate by Pearlman and Kim;7 who (32) Holladay, A.; Leung, P. C.; Coppens, P. Acfa Crystallog. 1983, A39, 311. (33) Coppens, P. International Tables for X-ray Crystallography; Vol. B, in press. (34) H,irshfeld, F. L. Theor. Chim. Acta 1977, 44, 129. (35) Li, N. Ph.D. Thesis, State University of New York at Buffalo, 1989. Li, N.; Coppens, P. To be published. (36) Griffin, J. F.; Coppens, P. J . Am. Chem. SOC.1975, 95, 3496. (37) Pearlman, D. A,; Kim, S.-H. Biopolymers 1985, 24, 327.
7982 The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 found that the net atomic charges to be in reasonable agreement with theoretical calculations for most atoms, but much lower than predicted by theory for phosphorus, for which a net charge of +0.57e was obtained. The difference appears mainly a matter of partitioning the charge among the atoms of the phosphate group, however, as the total charge for this group (about -1.3e) is in much better agreement with the theoretical results. The electrostatic interaction between two charge densities pA and pB is the interaction of the charge distribution B with the electrostatic potential of A at B (or vice versa)
Coppen s
orbital dxz-g 4 2
dxz dY2 dXY
total
exptl
ext Huckel
0.35 (4.8%) 1.05 (14.4%) 1.93 (26.5%) 1.93 (26.5%) 2.02 (27.7%) 7.29
0.81 (11.0%) 0.724 (9.8%) 1.93 (26.1%) 1.93 (26.1%) 1.99 (27.0%)
1.2 1.2 1.2 1.2 1.2
spherical
7.40
6.00
(20%) (20%) (20%) (20%) (20%)
dx2.gz
-
which with
leads to
where pA and pB include the nuclear charges. The interaction contains contributions of atomic dipole-dipole, quadrupolequadrupole, and higher pole interactions, which are neglected in most calculations, plus a penetration term which contributes when the charge distributions o ~ e r l a p . ~A~ full ,~~ treatment using experimental higher moments was made by Leiserowitz and co-workers in their analysis of the solid-state packing motifs of amidesa and carboxylic acids.41 In this work experimental amide and carboxyl group atomic moments were transferred to a much larger set of molecules of which the packing was analyzed and explained by the combination of electrostatic and atom-atom interactions. Features such as the catemer (Le., head to tail) motif of formic acid and a-oxalic acid, as opposed to the common centrosymmetric dimer arrangement, are satisfactorily explained by the analysis. Recognizing a need for a consistent set of neutral atom-atom potentials (representing dispersive and exchange interactions) to be combined with theoretical or experimental charges, S p a ~ k m a n ~ ~ derived a set of coefficients for an exp-6 potential, using the Gordon-Kim density functional model. By combining these functions with electrostatic potentials from atomic multipole interactions up to the octapole (i.e., I = 3) level, Spackman obtained very good agreement with calorimetric and quantum-mechanical values for the lattice energies of a number of molecules, including imidazole, urea, and g-meth~ladenine.~~ Furthermore, optimization of the energy of two molecules with the experimental charge distribution “lifted” out of the crystal gave geometries in close agreement with experiment. It appears that much is to be gained from more widespread application of experimental charge densities in molecular conformation and lattice energy calculations, and more such work may be expected. The electrostatic potential within a crystal can also be obtained by direct summation over the structure factors, analogous to the Fourier summation of the structure factors F(h) to give the charge density p
where Vis the volume of the unit cell and F’(h) differs from the (38)Buckingham, A. D.Q.Rev. 1959, 183. (39)Spackman, M.A. J. Chem. Phys. 1986, 85, 6587. Leiserowitz, L. J. Am. Chem. SOC.1980, 102, (40) Berkovitch-Yellin, Z.; 7677. (41) Berkovitch-Yellin, Z.; Leiserowitz, L. J. Am. Chem. SOC.1982, 104, 4052. (42) Spackman, M. A. J. Chem. Phys. 1986.85, 6579. (43) Spackman, M.A,; Weber, H. P.; Craven, B. M. J . Am. Chem. Soc. 1988, 110, 715.
.-
....,... t2g
.’......‘:. .. dxz,d,,
Octahedral Square Planar Figure 3. d-Orbital energy level diagram for the iron porphyrin complexes.
X-ray structure factor by the addition of a nuclear contribution. To obtain the electrostatic potential each structure factor must be divided by the square of the magnitude of h, which is the scattering vector:“ 1 V(r) = -xF(h)/(h12 TV
exp(-2~ih.r)
(7)
This expression has been used in a series of studies on the interpretation of crystalline proper tie^.^^ It has a singularity at h = 0 which must be carefully considered as the zero term gives a constant contribution equal to the average potential over the unit cell.& This term cannot be neglected when the electrostatic potential is different crystals is compared. Experimental Charge Densities and the Ground State of Transition-Metal Complexes About 6 years ago we realized that states which may be very closely spaced energetically may be more easily distinguishable from their experimental densities. Prime examples, of biological interest, are the four-coordinate iron(I1) tetraphenylporphyrins and phthalocyanines, for which several different ground states have been proposed on the basis of spectroscopic and magnetic data and theoretical calculations. They are unusually large molecules for a charge density determination, especially when they occupy sites of low symmetry, but they are fairly rigid and often have centrosymmetric crystal structures. Because of the existence several low-lying states, configuration interaction is essential in the theoretical treatment of the intermediate spin iron complexes, leading to varying conclusion^!^ But as the different states differ in the distribution of the electrons over the energy levels, their charge density will be radically (44) Stewart, R. F. Chem. Phys. Lett. 1979, 65, 335. (45) Spackman, M.A,; Stewart, R. F. In Chemical Applications of Atomic and Molecular Potentials; Politzer, P.,Truhlar, D.G., Eds.; Plenum: New York; p 407. (46) Becker, P.;Coppens, P. To be published. (47) Edwards, W. D.; Weiner, B.; Zemer, M. C. J. Am. Chem. SOC.1986, 108, 2196.
The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 7983
Feature Article TABLE I V d-ElectronOrbital Population in FePc and FeTPP
experimental term symbol
3Eg
IAze
FePc
%g
dXxyz dxrIvr
dXY
1(17%) 3 (49%) 2 (33%)
2 (33%) 2 (33%) 2 (33%)
0.70 0.93 2.12 1.68
1(17%) 4 (67%) 1(17%)
t Figure 4. The iron(I1) meso-tetraphenylporphyrinmolecule.
different, in particular in the region around ‘the metal atom. In order to test the applicability of the X-ray method, we first showed that the experimental electron density in the high-spin, six-coordinate, bis(tetrahydrofuran)(meso-tetraphenylporphinato)iron(II) differs as expected from the six-coordinate low-spin complex bis(pyridine)(meso-tetraphenylporphinat0)iron(II), in which the two axial tetrahydrofuran ligands have been replaced by pyridine m o l e c ~ l e s . For ~ ~ the ~ ~ ~former compound an extended Huckel (EH) calculation is available.% Though the experimental and theoretical methods are based on very different premises, and electron populations are calculated in different ways (a Mulliken population analysis in the case of the Huckel results), the agreement between the d-orbital populations obtained with the two methods is rather remarkable (Table 111). Both predict preferential occupancy of the t2gtype orbitals d + z and d, ,with a slightly larger population of the latter (see Figure 3 for the energy level diagram). Reasonable agreement is also obtained for the d9 orbital. But the d+g orbital shows a discrepancy. The occupancy of this orbital is due to the covalent interaction with the ligand s- and p-orbitals, which is clearly overestimated by the EH method. Nevertheless, the general agreement is better than might be expected given the approximate nature of the one-electron Hamiltonian used in the EH method and the inadequacies of the Mulliken population analysis. The deviations from the spherical symmetry assumed in standard crystallographic treatments (last column of Table 111) are large. The nature of the ground state of the intermediate-spin fourcoordinate Fe( 11) complexes is more controversial. From theoretical and spectroscopic results the main contenders for leading contributor to the ground state are the 3A2, and the 3Egconfigurations, though the 3B2s state has also been proposed.s1 The main difference between the first two states is a shift of one electron from the dxryr orbitals in 3E, to the dzz orbital in ’A2,. Such a shift gives a large difference in the electron deformation density map, as confirmed by theoretical calculations.52 We have performed low-temperature charge density analyses of both iron(I1) phthalocyanine (FePc) and iron(I1) (mesotetraphenylporphyrin (FeTPP) (Figure 4). A surprising result of our work is that the electron deficiency in the axial direction predicted for the 3E state is clearly observed for FePq30 but not for FeTPP, which shows a large excess density above and below the iron atom.53ss4 The d-orbital populations for the two complexes, given in Table IV,lead to the same conclusion: while the (48) Li, N.; Coppens, P.; Landrum, J. Inorg. Chem. 1988, 27, 482. (49) Lecomte, C.; Blessing, R. H.; Coppens, P.;Tabard, A. J. Am. Chem. SOC.1986, 108, 6942. (50) Scheidt, W. R. Private communication. (51) Barraclough, C. G.; Martin, R. L.; Mitra, S.;Sherwood, R. C. J. Chem. Phys. 1970, 63, 1643. (52) Rohmer. M. M. Chem. Phvs. Lett. 1985. 116. 44. (53j Tanaka,’K.; Elkaim, E.; L i h g , L.; Jue, Z. N.; Coppens, P.; Landrum, J. J . Chem. Phys. 1986, 84, 6969. (54) Li, N.; Coppens, P.; Landrum, J. To be published.
(7) (12.9%) (6) (17.1%) (7) (39.1%) (10) (30.9%)
FeTPP 0.24 (15) 2.10 (14) 2.28 (18) 1.53 (15)
(3.9%) (33.7%) (36.5%) (25.9%)
FePc distribution shows the population ratio’s typical for the E-type state, FeTPP has the more even electron distribution over the dzz, dxzyx,and d, orbitals typical for the more symmetric A state. Agreement with spectroscopic values for the electric field gradient splitting is quite reasonable (Table I). As may be expected, there are deviations from the idealed ionic states, the populations of which are listed in Table IV. But the conclusion is inescapable that in the crystal the two four-coordinate complexes have different ground states. This different may very well be due to the effect of the intermolecular interactions, which are quite different in the two crystals. In monoclinic FePc, nitrogen atoms in the bridging (meso) position of a neighboring molecule are located at 3.42 A above and below the iron atoms, thus providing an axial “pseudo”-ligand. In tetragonal FeTPP the molecules are aligned perpendicular to the 4 axis of the space group 142d, and no such approach exists. The sensitivity of the ground state of the porphyrins to axial ligation has been discussed by several authors, including Mispelter, Momenteau, and L h o ~ t e , ~ ~ who show that small axial perturbations can induce a reversal of the ground state in capped ferrous porphyrin derivatives. A similar effect can be caused by interactions in the crystal, and probably also by solvent interactions in solutions. Thus, the X-ray measurements give information on a molecule in a particular environment, which is not the isolated state commonly treated in theoretical work. We conclude that the absence of strong interactions in the FeTPP crystal supports the view that the ground state of this molecule in its isolated state has a pronounced 3A2gcharacter.
Further Developments To obtain meaningful results great care is needed in collecting and processing of the data. In each of the porphyrin studies, for example, more than 20 000 reflections were collected at liquid nitrogen temperatures over a time span of several months for each experiment. In one case4*the process had to be repeated completely because of an unfortunate mishap almost at the end of data collection. Data processing requires a careful analysis of each of the peak profiles, and an analysis of systematic effects which at best limit the accuracy of the experimental densities or at worst may render them useless. What can be done to reduce the effort and increase the accuracy of the results? For simpler cases very significant advances in accuracy have been reported. When perfect crystals are available, as in the case of silicon, special techniques allow determination of the structure factors with an accuracy of 0.1% or better (see, for example, SpackmanS6),which is at least a factor of 10 improvement over the accuracy achieved in the large data sets collected on less perfect crystals of more complicated structures. Other techniques are also being developed. For example, technical advances in electron diffraction of solids have made it possible to determine accurate structure factors of low-order reflections, as demonstrated in the study of GaAs5’ For the more complicated solids of general chemical interest the solution may be found in the brighter sources which are now becoming available. Present synchrotron radiation sources are a factor lo4 brighter than conventional X-ray tubes, while those under development, such as the Advanced Photon Source at Argonne National Laboratory and the European Synchrotron ~
(55) Mispelter, J.; Momenteau, M.; Lhoste, J. M. J. Chem. Phys. 1980, 72, 1003.
(56) Spackman, M. A. Acta Crystallogr. 1986, A42, 271. (57) Zuo, J. M.; Spence, J. C. H.; OKeefe, M. Phys. Rev. Lett. 1988,61, 353.
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Coppens commonly described as extinction and multiple reflection. These are much reduced when the path length of the beams in the crystal is smaller or the scattering power is reduced by the use of shorter wavelengths. The same is true for absorption, while the resolution of the diffraction experiment can be improved by the selection of shorter wavelengths available at synchrotron sources. Since the synchrotron source has a low divergence, the reflection peaks tend to be much narrower, which reduces the influence of thermal diffuse scattering on the net intensities obtained by peak integration. But synchrotron beams are often not very stable over longer time periods. To test the suitability of synchrotron radiation in charge density work, we performed a first charge density study at the A2 wiggler beamline at the Cornel1 High Energy Synchrotron Source (CHESS),58choosing the rather hard ionic crystal of chromium hexaammine hexacyanochromium and shortwavelength radiation with X equal to 0.302 A. Utmost care was taken to assure beam stability and to correct for beam decay, deadtime, and other effects. The resulting deformation density map, shown in Figure 5, has very low noise in areas away from the atoms and shows bond and lone-pair peaks which are much more clearly defined than in room-temperature maps of analogous compounds studied with conventional equipmenLs9 The study demonstrates that with proper precautions synchrotron radiation is eminently suitable for accurate studies. Much is to be expected from the very high intensity, third generation sources, now in the construction stage. However, the work will require stable sources and careful monitoring of any fluctuations in the beam conditions. Finally, the availability of very high intensity beams makes it possible to study the response of the charge distribution to an external perturbation. Very recently we have observed the effect of an external electric field on the X-ray scattering of the nonlinear optical material 2-methyl-4-nitroaniline. The first results indicate both a molecular reorientation and a polarization of the charge d i ~ t r i b u t i o n .Such ~ ~ studies, aimed at a detailed understanding of crystal polarizability at the atomic level, are a further extension of the charge density studies described in this article. Acknowledgment. Support of our work by the National Science Foundation (CHE8711736) and the National Institute of Health (5ROlHL2388408) is gratefully acknowledged. I thank Dr. M. D. Newton of Brookhaven National Laboratory for helpful comments.
(b)
Figure 5. Deformation density maps in the Cr(CN)4 plane in chromium hexaammine hexacyancchromium based on room-temperature synchrotron data. Contours a t O.OSe A-3, Negative contours broken (a) before averaging, and (b) after averaging over chemically equivalent regions.
Facility in Grenoble, France, promise an additional increase of at least 4 or 5 orders of magnitude. This is of crucial importance, because the factors limiting accuracy in the studies described above can all be reduced by a reduction of the size of the specimen crystal or the use of shorter wavelengths. For example, the rescattering of the diffracted beams within the crystal gives rise to what is
(58) Nielsen, F. S.; Lee, P.; Coppens, P.Acfa Crystallogr. 1986,842, 359. (59) Iwata, M. Acta Crysrallogr. 1977, 833, 59. (60) Stevens, E. D.; DeLucia, M. L.; Coppens, P. Inorg. Chem. 1980, 19, 813. (61) Finklea, S. L.; Cathey, L.; Amma, E. L. Acta Crystallogr. 1976, A32, 529. (62) Dale, B. W.; Williams, R. J. P.; Johnson, C. E.; Thorp, T.L. J . Chem. Phys. 1968, 49, 3441. Dale, B. W.; Williams, R. J. P.; Edwards, P. R.; Johnson, C. E. J . Chem. Phys. 1968, 49, 3445. (63) LeComte, C.; Chadwick, D. L.; Coppens, P.; Stevens, E. D. Inorg. Chem. 1983, 22, 2982. (64) Dolphin, D. H.; Sams, J. R.; Tsin, T. B.; Wong, K. L. J . Am. Chem. SOC.1978, ZOO, 1711. (65) Paturle, A,; Graafsma, H.; Boviatsis, J.; LeGrand, A.; Restori, R.; Coppens, P.; Kvick, A,; Wing, R. M. Acta Crysrallogr. Sect. A 1989, 45, FC25.
