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 (𝐹ℎ𝑘𝑙 ) in an X-ray experiment are proportional to the electron
density (𝜌𝑥𝑦𝑧 ) in the unit cell1:
𝐹ℎ𝑘𝑙 = ∫ ∫ ∫ 𝑉𝜌𝑥𝑦𝑧 exp[2𝜋𝑖(ℎ𝑥 + 𝑘𝑦 + 𝑙𝑧)]𝜕𝑥𝜕𝑦𝜕𝑧,
(2)
where 𝑉 is the volume of the unit cell and ℎ, 𝑘, 𝑙 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 𝜌(𝒓) at some point 𝒓 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 (𝜌atomic ) is equal to
𝜌atomic (𝐫) = 𝑃𝑐 𝜌core + 𝑃𝑣 𝜅 3 𝜌valence (𝜅𝑟 𝑟) + 𝜅′ ∑4𝑙=1 𝑅𝑙 (𝜅 ′ 𝑟) ∑𝑙𝑚=𝑙 𝑃𝑙𝑚± 𝑑𝑙𝑚± (𝐫/𝑟),
(3)
where 𝜌core and 𝜌valence are spherical core and valence densities from Hartree-Fock calculations, 𝑑𝑙𝑚 are
the spherical harmonic angular functions, 𝑅𝑙 are Slater radial functions, 𝜅 and 𝜅′ are expansion-contraction
coefficients for the monopole (𝜅) and higher order multipole functions (𝜅′), and 𝑃𝑣 and 𝑃𝑙𝑚± 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, 𝐹(𝒉), at each Miller index (𝒉 = (ℎ, 𝑘, 𝑙) ) and the
electron density adjusted for thermal motion, 𝜌(𝒓), at each position 𝒓 in the unit cell:
𝐹(𝒉) = ∫cell 𝜌(𝒓) exp(2𝜋𝑖𝒓 ⋅ 𝑩𝒉) 𝑑𝒓
(4)
where 𝑩 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 𝑷, the
half-closed density matrix, which is the central property in a Hartree-Fock type calculation. Then, the
calculated structure factors (𝐹 𝑐 (𝒉)) at each Miller index are given by:
𝐹 𝑐 (𝒉) = tr𝑷𝑰(𝒉)
(5)
where 𝑰 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 𝜒 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. 𝜒 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, 𝜒 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:
𝐿(𝒄, 𝜀, 𝜆) = 𝐸(𝒄, 𝜀) − 𝜆[𝜒 2 (𝒄) − 𝛥]
(7)
where 𝒄 is a matrix of molecular-orbital coefficients, ∆ is the desired error in 𝜒 2 , and 𝜀 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 𝜒 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 𝒓
𝜌𝑖at (𝒓)
𝑤𝑖 (𝒓) = ∑
at
𝑖 𝜌𝑖 (𝒓)
,
(8)
where 𝜌𝑖at (𝒓) is the density of isolated atom 𝑖. Then, the density of the molecule 𝜌mol (𝒓) is simply split
into Hirshfeld atoms 𝜌𝑖H.A. (𝒓) according to this weighting scheme
𝜌𝑖H.A. (𝒓) = 𝑤𝑖 (𝒓)𝜌mol (𝒓) .
(9)
Once the Hartree–Fock electron density is broken into Hirshfeld atoms, the atomic positions and ADPs are
then refined to minimize the 𝜒 2 statistic introduced above. This procedure (SCF, Hirshfeld partitioning, and
refinement) is then repeated until the positions, ADPs, and 𝜒 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 𝜆 =
0 to -493.85 Hartrees at 𝜆 = 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.
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2
3
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5