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Supplemental Material for: Experimental electron densities from X-ray diffraction to derive optical properties: constrained wavefunction versus multipole refinement Daniel D. Hickstein1, Jacqueline M. Cole1,2,3, Michael J. Turner4, Dylan Jayatilaka4 1 Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE. UK Department of Physics, University of New Brunswick, P. O. Box 4400, Fredericton, E3B 5A3. Canada 3 Department of Chemistry, University of New Brunswick, P. O. Box 4400, Fredericton, E3B 5A3. Canada 4 School of Chemistry and Biochemistry, University of Western Australia, Nedlands 6009. Australia 2 Section 1: Theory A. Multipole Refinement The intensities of the diffracted beams (πΉβππ ) in an X-ray experiment are proportional to the electron density (ππ₯π¦π§ ) in the unit cell1: πΉβππ = ∫ ∫ ∫ πππ₯π¦π§ exp[2ππ(βπ₯ + ππ¦ + ππ§)]ππ₯ππ¦ππ§, (2) where π is the volume of the unit cell and β, π, π are the Miller indices of the diffracted beam. However, in the vast majority of X-ray diffraction experiments, the electron density of a crystal is simply modeled as a sum of spheres, one for each atom. Since the spheres do not interact with one another, this is known as the independent atom model (IAM). This is a very good approximation for heavy atoms, which contain a large number of tightly bound core electrons that account for the majority of X-ray scattering. Light atoms, however, have few core electrons and are thus not well described by a simple spherical model. Since the electron density is a continuous function over three-dimensional space, it is infinitely divisible and would take an infinite number of variables to represent perfectly. Consequently, several different models for the aspherical electron density have emerged. One, the maximum entropy method assumes little about the nature of the charge density and models the charge at every point on a three-dimensional grid. The maximum entropy method is praised for its lack of bias, but it is not often used owing to its lack of resolution. Other methods gain greater precision by incorporating assumptions about the nature of the charge density. The Hansen–Coppens2 multipole model has emerged as the most popular method of aspherical atom refinement. It uses multipole functions – which resemble atomic orbitals – to model the charge density, making the implicit (and generally effective) assumption that atomic orbitals are a good description of the electron density in a crystal. Using the multipole model, charge-density crystallography has “come of age”3, becoming an increasingly useful and popular tool for investigating crystal structure. In the Hansen–Coppens2 multipole formalism, the charge density π(π) at some point π in the crystal is given by a sum of atomic charge densities that have been suitably smeared for thermal motion of the atoms. The charge density for each atom (πatomic ) is equal to πatomic (π«) = ππ πcore + ππ£ π 3 πvalence (π π π) + π ′ ∑4π=1 π π (π ′ π) ∑ππ=π πππ± πππ± (π«/π), (3) where πcore and πvalence are spherical core and valence densities from Hartree-Fock calculations, πππ are the spherical harmonic angular functions, π π are Slater radial functions, π and π ′ are expansion-contraction coefficients for the monopole (π ) and higher order multipole functions (π ′), and ππ£ and πππ± are the population coefficients of each multipole. B. Wavefunction fitting (XCHF) The wavefunction fitting method4–8 is fundamentally different from the multipole model. Most importantly, instead of simply obtaining the charge density of the material, the wavefunction fitting method generates an “experimental wavefunction”, from which most molecular and material properties can easily be calculated. Additionally, the wavefunction fitting method does not simply maximize the agreement to the experimental X-ray structure factors. Instead, the wavefunction is modified to fit the experimental structure factors while still keeping the Hartree-Fock energy low. Thus, the wavefunction fitting method represents a hybrid of experimental and theoretical techniques A full description of the theory involved in the fitting of a wavefunction to a charge-density-quality X-ray dataset is given by Jayatilaka and Grimwood4 and is summarized here. A basic principle of crystallography is the relationship between the observed structure factors, πΉ(π), at each Miller index (π = (β, π, π) ) and the electron density adjusted for thermal motion, π(π), at each position π in the unit cell: πΉ(π) = ∫cell π(π) exp(2πππ ⋅ π©π) ππ (4) where π© is the reciprocal lattice matrix, which depends only on the shape of the unit cell, and the integral is completed in real space over the three-dimensional unit cell. Otherwise stated, the structure factors are the Fourier transform of the electron density. In the wavefunction fitting model, the unit cell density is constructed from molecules calculated using a standard self-consistent field (SCF) method on a noninteracting molecular fragment. Consequently, it can be shown that Eq. 4 can be written in terms of π·, the half-closed density matrix, which is the central property in a Hartree-Fock type calculation. Then, the calculated structure factors (πΉ π (π)) at each Miller index are given by: πΉ π (π) = trπ·π°(π) (5) where π° is a function of only the basis set and the unit cell geometry. These calculated structure factors then need to be corrected for the smearing of the electron density due to the thermal motion of the nuclei and a secondary extinction factor can optionally be applied. Now that the structure factors can be calculated from the wavefunction, they can be compared with the experimental structure factors. The π 2 statistic is defined in Section II C and is used to assess the agreement between the calculated and experimental structure factors since it is quite sensitive to small differences in fit and because it takes into account the experimentally measured error in each structure factor. π 2 is equal to the square of the un-weighted goodness of fit (GOF) statistic used in crystallography. However, it should be noted that many IAM refinement packages, SHELX9 in particular, apply weighting schemes to the GOF statistic, making significant deviations from one unusual. Here, π 2 is un-weighted, so deviations from unity should not always be interpreted as an indicator of particularly poor fit. In order to modify the wavefunction to fit the experimental data, a Lagrange multiplier, λ, is introduced. Then, instead of simply minimizing the Hartree-Fock energy (E(c, ε)), the Lagrange equation is minimized: πΏ(π, π, π) = πΈ(π, π) − π[π 2 (π) − π₯] (7) where π is a matrix of molecular-orbital coefficients, β is the desired error in π 2 , and π are the Lagrange multipliers related to the orthogonality of the orbitals. Though the implementation is somewhat more complicated than has been presented here, the basic scheme is simple: we are minimizing the Hartree-Fock energy subject to a penalty function that is based on the error statistic π 2 . Since this is essentially just a Hartree-Fock calculation that is constrained to fit the experimental structure factors, the wavefunction fitting method is also referred to as the X-ray Constrained Hartree-Fock (XCHF) method. C. Hirshfeld Refinement Like the IAM, the Hirshfeld refinement is a method for obtaining the positions and ADPs of atoms in a crystal. In contrast to the IAM, the Hirshfeld refinement achieves far more precise atomic positions and ADPs by using aspherical (instead of spherical) electron densities for each atom. These electron densities are obtained by performing a Hartree-Fock SCF calculation on the entire molecule and breaking this density into individual “Hirshfeld atoms” that can have their positions refined according to a least-squares refinement method10. The Hirshfeld partitioning scheme breaks up the molecular electron density according to the original electron density of each isolated atom, where, “like partners in a stockholder’s corporation, each atom thus partakes in the gain or loss in direct proportion to its share in the capital investment” 11. In this scheme, there is a sharing function at every point π ππat (π) π€π (π) = ∑ at π ππ (π) , (8) where ππat (π) is the density of isolated atom π. Then, the density of the molecule πmol (π) is simply split into Hirshfeld atoms ππH.A. (π) according to this weighting scheme ππH.A. (π) = π€π (π)πmol (π) . (9) Once the Hartree–Fock electron density is broken into Hirshfeld atoms, the atomic positions and ADPs are then refined to minimize the π 2 statistic introduced above. This procedure (SCF, Hirshfeld partitioning, and refinement) is then repeated until the positions, ADPs, and π 2 values converge. Section 2. Hartree-Fock Energies for the XCHF refinements The XCHF method generated wavefunctions that presented a modest increase in the Hartree-Fock energy of the molecule, relative to gas-phase HF calculations. In a previous study, Grimwood and co-workers5 showed that the increase in the HF energy owing to wavefunction fitting process for oxalic acid dihydrate, OAD (the de facto reference compound for X-ray diffraction studies) was 1130 kJ/mol, or 0.063% of the total HF energy on the compound. In this study, coumarin has similar C=O functional groups and a roughly comparable HF energy to OAD. For coumarin, the total HF energy increased from -494.13 Hartrees at π = 0 to -493.85 Hartrees at π = 0.18, a difference of 750 kJ/mol or 0.058%, in good agreement with the OAD study. This energy increase is much higher than energies that are associated with intermolecular interactions which one might intuitively expect as the primary difference, cf. ~20 kJ/mol for a hydrogen bond. Rather, is on the same order as the energy of a carbon–carbon bond (~600 kJ/mol). From this we conclude that the wavefunction fitting calculation is making a significant difference in the molecular wavefunction, but that the difference is of similar magnitude to previous studies. All of the increases in the HF energy scale roughly in proportion with the unconstrained HF energy (Table 1). Strikingly, this energy change is much smaller when coordinates and ADPs from the Hirshfeld atom refinement are used rather than coordinates and ADPs from the multipole refinement. On average, the HF energy increased by 0.044% when multipole coordinates were used and 0.019% when Hirshfeld coordinates were used. From this difference, one can draw two conclusions: (1) a large portion of the changes to the wavefunction incurred by the XCHF method result from small deviations from the ideal atomic coordinates and ADPs and (2) when seeded with coordinates and ADPs from the Hirshfeld atom refinement, the wavefunction fitting calculation makes significantly smaller changes to the wavefunction. Table S1. The effect of the wavefunction fitting refinement on the Hartree-Fock (HF) energy of the compound. The increase in the HF energy caused by the wavefunction fitting method is a roughly constant fraction of the total HF energy on the compound, an average of 0.044% when coordinates and ADPs from the multipole refinement are used and 0.019% when atomic coordinates and ADPs from the Hirshfeld atom refinement are used. OAD represents the oxalic acid dihydrate wavefunction fitting study completed by Grimwood and Jayatilaka5 and is included as an external comparison. Compound Coumarin DED MBADNP ZTS OAD Coordinates HF Energy XCHF Energy Difference Difference Difference (Hartrees) (Hartrees) (Hartrees) (kJ/mol) (percent) Multipole -494.13 -493.85 0.28 747 0.058% Hirshfeld -494.13 -494.04 0.09 244 0.019% Multipole -913.49 -913.34 0.14 371 0.015% Hirshfeld -913.48 -913.42 0.06 154 0.006% Multipole -1016.48 -1016.08 0.40 1048 0.039% Hirshfeld -1016.47 -1016.31 0.16 420 0.016% Multipole -4114.30 -4111.75 2.55 6701 0.062% Hirshfeld -4114.16 -4112.78 1.39 3648 0.034% Multipole -680.61 -680.18 0.43 1129 0.063% Section 3. Three-dimensional views of ZTS electron density As mentioned in the main text, ZTS is not a planar molecule like the rest of the molecules in the study. As a consequence, it is valuable to look at three-dimensional views of the electron density. The most striking feature seen in these three-dimensional plots occurs near the zinc center. When the Hartree-Fock (HF) calculation is completed without fitting to X-ray data, there is very little diffuse electron density around the zinc atom compared to the spherical atom model (Fig. S1a). However, when the wavefunction is constrained by the experimental X-ray structure factors (Fig. S1b), considerable diffuse electron density disappears on one side of the zinc atom and appears on the other side. When the two calculations are subtracted (Fig. S1c), the diffuse electron density near the zinc atom indeed appears as the largest discrepancy. FIG S1. Effect of wavefunction fitting on ZTS. These deformation density plots show the electron density minus the electron density from the spherical atom model. a) The deformation density from the Hartree-Fock (HF) calculation without wavefunction fitting. b) The deformation density from the X-ray constrained wavefunction (XCHF) calculation. c) The difference between the XCHF and the HF charge densities. Blue isosurfaces correspond to areas of electron density increase while red contours represent areas of electron density depletion. Isosurfaces are drawn at 0.1 e/Å3. By examining the electron densities generated by the wavefunction fitting method (X-ray constrained Hartree-Fock, XCHF) with different starting coordinates and atomic displacement parameters (ADPs), it is clear that the differences in the electron density near the zinc center arise from the use of coordinates and ADPs derived from the multipole model. If we compare the XCHF calculation with the HF calculation when using the coordinates and ADPs from the multipole refinement (“multipole coords”), we observe a large change in the diffuse electron density near the zinc atom (Fig. S2a). However, if we use the atomic coordinates and ADPs from the Hirshfeld atom refinement (“Hirshfeld coords") and compare the electron densities (Fig. S2b), we observe that the change in the electron density near the zinc atom is much smaller and more comparable to the small changes that appear close to the nuclei for the other atoms in ZTS. By comparing the electron densities generated by the XCHF method using the different starting coordinates, we see that there are systematic and significant differences with the difference near the zinc atom being the most obvious. Thus, in these three-dimensional visualization of the electron density of the ZTS molecule, we arrive at a similar conclusion to that reached in the main text: the choice of starting coordinates and ADPs has a significant effect on the electron density generated by the XCHF method and that it appears that the coordinates and ADPs from the Hirshfeld atom refinement represent a more suitable starting point than coordinates obtained from a multipole refinement. FIG S2. Effect of the Hirshfeld refinement coordinates on the electron density generated from the X-ray constrained Hartree-Fock (XCHF) method. a) The effect of the wavefunction fitting calculation (XCHF minus HF) when the multipole coordinates are used. b) The effect of the wavefunction fitting calculation (XCHF minus HF) when coordinates and ADPs from the Hirshfeld atom refinement are used. To a first approximation, the electron densities are very similar. c) The difference between the electron densities generated by the wavefunction fitting calculation using the different coordinates (Hirshfeld minus multipole). There are significant differences near the zinc center and near most of the nuclei. Isosurfaces are drawn at 0.1 e/Å3. References 1 2 3 D. Watkin, J. Appl. Cryst. 41, 491 (2008). N.K. Hansen and P. Coppens, Acta Cryst. A 34, 909 (1978). P. Coppens, Angew. Chem. Int. Ed. 44, 6810 (2005). 4 D. Jayatilaka and D.J. Grimwood, Acta Cryst. A 57, 76 (2001). D.J. Grimwood and D. Jayatilaka, Acta Cryst. A 57, 87 (2001). 6 I. Bytheway, D.J. Grimwood, and D. Jayatilaka, Acta Cryst. A 58, 232 (2002). 7 I. Bytheway, D.J. Grimwood, B.N. Figgis, G.S. Chandler, and D. Jayatilaka, Acta Cryst. A 58, 244 (2002). 8 D. Grimwood and I. Bytheway, J. Comp. Chem. 24, 470 (2003). 9 G.M. Sheldrick, Acta Cryst. A 64, 112 (2008). 10 D. Jayatilaka and B. Dittrich, Acta Cryst. A 64, 383 (2008). 11 F.L. Hirshfeld, Theor. Chim. Acta 44, 129 (1977). 5