3. Crystals

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3. Crystals
What defines a crystal? Atoms, lattice points,
symmetry, space groups Diffraction B-factors R-fac
tors Resolution Refinement Modeling!
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Crystals
What defines a crystal? 3D periodicity anything
(atom/molecule/void) present at some point in
space, repeats at regular intervals, in three
dimensions. X-rays see electrons ? ?(r)
?(rX) ?(r) electron density at position
r X n1a n2b n3c n1, n2, n3
integers a, b, c vectors
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Crystals
What defines a crystal?
primary building block the unit cell
lattice set of points with identical environment
crystal
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Crystals
Which is the unit cell?
primitive vs. centered lattice
primitive cell smallest possible volume ? 1
lattice point
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Crystalsorganic versus inorganic
lattice points need not coincide with
atoms symmetry can be low unit cell
dimensions ca. 5-50Å, 200-5000Å3 NB 1 Å
10-10 m 0.1 nm
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Crystalssome terminology
solvates crystalline mixtures of a compound
plus solvent
c
a
b
- hydrate solvent aq - hemi-hydrate
0.5 aq per molecule polymorphs different
crystal packings of the same compound lattice
planes (h,k,l) series of planes that cut a, b, c
into h, k, l parts respectively, e.g (0 2 0),
(0 1 2), (0 1 2)
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Crystalscoordinate systems
Coordinates positions of the atoms in the unit
cell carthesian using Ångstrøms, and an
ortho-normal system of axes. Practical e.g. when
calculating distances. example (5.02, 9.21,
3.89) the middle of the unit cell of
estrone fractional in fractions of the unit
cell axes. Practical e.g. when calculating
symmetry-related positions. examples (½, ½, ½)
the middle of any unit cell (0.1, 0.2, 0.3) and
(-0.2, 0.1, 0.3) symmetry related positions via
axis of rotation along z-axis.
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Crystalssymmetry

• Why use it?
• efficiency (fewer numbers, faster computation
  etc.)
• less noise (averaging)
• finite objects crystals
• rotation axes (?) rotation axes (1,2,3,4,6)
• mirror planes mirror planes
• inversion centers inversion centers
• rotation-inversion axes rotation-inversion axes
• ----------------------------- screw axes
• point groups glide planes
• translations
• ---------------------------
• space groups

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Crystalssymmetry and space groups
symmetry elements translation vector
rotation axis screw axis mirror plane
glide plane inversion center
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Crystalssymmetry and space groups
symmetry elements translation vector
rotation axis screw axis mirror plane
glide plane inversion center
examples (x, y, z) ? (x½, y½, z) (x, y,
z) ? (-y, x, z) (x, y, z) ? (-y, x, z½) (x, y,
z) ? (x, y, -z) (x, y, z) ? (x½, y, -z) (x, y,
z) ? (-x, -y, -z)
equivalent positions
Set of symmetry elements present in a crystal
space group examples P1 P1 P21 P21/c
C2/c Asymmetric unit smallest part of the unit
cell from which the whole crystal can be
constructed, given the space group.
-
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CrystalsX-ray diffraction
diffraction scattering of X-rays by periodic
electron density diffraction
reflection against lattice planes,
if 2dhklsin? n?
X
? 0.5--2.0Å Cu 1.54Å
dhkl
?
Data set list of intensities I and angles ?
?path 2dhklsin?
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Crystalsinformation contained in diffraction data

• lattice parameters (a, b, c, ?, ?, ?) obtained
  from the directions of
• the diffracted X-ray beams.
• electron densities in the unit cell, obtained
  from the intensities of the
• diffracted X-ray beams.
• Electron densities ? atomic coordinates (x, y, z)
• Average over time and space
• Influence of movement due to temperature atoms
  appear smeared out
• compared to the static model ? ADPs
  (B-factors).
• Some atoms (e.g. solvent) not present in all
  cells ? occupancy factors.
• Molecular conformation/orientation may differ
  between cells
• ? disorder information.

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Crystalsinformation contained in diffraction data
How well does the proposed structure
correspond to the experimental data? ?
R-factor consider all (typically 5000)
reflections, and compare calculated structure
factors to observed ones. R ? Fhklobserved -
Fhklcalculated Fhkl ? Ihkl
? Fhklobserved OK if 0.02 lt R
lt 0.06 (small molecules)
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Crystals - doing calculations on a structure from
the CSD
We can search on e.g. compound name
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Crystals - doing calculations on a structure from
the CSD
We can specify filters!
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Crystals - doing calculations on a structure from
the CSD

• refcodes
• re-determinations
• polymorphs
• anthraquinone

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Crystals - doing calculations on a structure from
the CSD
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Crystals - doing calculations on a structure from
the CSD
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Crystals - doing calculations on a structure from
the CSD
Z molecules per cell
Z molecules per asymmetric unit
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Crystals - doing calculations on a structure from
the CSD
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Crystals - doing calculations on a structure from
the CSD
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Crystals - doing calculations on a structure from
the CSDexporting from ConQuest/importing into
Cerius
Cerius2
CSD
cif
cssr fdat pdb
Not all bond (-type) information in CSD data ?
add that first!
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Crystals - doing calculations on a structure from
the CSDChecking for close contacts and voids
how close is too close default 0.9xRVdW
minimal void size
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Crystals - doing calculations on a structure from
the CSDOptimizing the geometry
CSD optimized) a 7.86 7.76 b
3.94 4.36 c 15.75 15.12 ? 90
90 ? 102.6 107.4 ? 90 90
!
space-group symmetry imposed
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Crystals - doing calculations on a structure from
the CSDOptimizing the geometry
CSD opt/spgr opt) a 7.86 7.76
7.69 b 3.94 4.36 4.66 c 15.75
15.12 15.93 ? 90 90 90 ? 102.6
107.4 106.8 ? 90 90 90
space-group symmetry not imposed Is it
retained?
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Crystals - doing calculations on a structure from
the CSDOptimizing the geometry

• Application of constraints during optimization
• space-group symmetry -- if assumed to be known
• cell angles and/or axes -- e.g. from powder
  diffraction
• positions of individual atoms -- e.g non-H, from
  diffraction
• rigid bodies -- if molecule is rigid, or if it
  is too flexible...

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Crystalssingle crystal versus powder diffraction
Powder large collection of small single
crystals, in many
orientations Single crystal ? all reflections
(h,k,l) can be observed individually,
leading to thousands of data
points. Powder ? all reflections with
the same ? overlap,
leading to tens of data points.
Diffraction data can easily be computed ?
verification of proposed model, or
refinement (Rietveld refinement)
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Next week.
Modeling crystals how does it differ from small
systems? Applications predicting
morphology predicting crystal packing