Xray Charge Density Analysis and the XD programming package Buffalo NY,USA 1317 May 2003

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X-ray Charge Density Analysis and the XD
programming package Buffalo NY,USA13-17 May 2003

• Carlo Gatti
• CNR-ISTM, Milano, Italy

The topology of charge density in crystals
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Outline

• The topology of ED in crystals peculiarities

• Intermolecular and intramolecular interactions
  in crystals
• Does the ID contain information on
  intermolecular interactions ?
• ID and topological properties changes
• Hydrogen bond energetics from topological
  analysis
• Secondary interactions and BCPs in Ionic Crystals

• Scalar and vector fields used in QTAIM from exp.
  EDD
• ED topologies experiment vs theory

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Comprehensive Reviews

• Chemical Applications of X-ray Charge-Density
  Analysis
• T.S. Koritsanszky, P. Coppens, Chem. Rev.
  101, 1583-1627 (2001)
• Charge Densities from X-Ray Diffraction Data
• M. A. Spackman, Annual Reports, Section C,
  94, 177-207 (1997)

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Topology of the electron distributions in
perfectly periodic crystals

• W. Jones and N.H. March, Theoretical Solid
  State Physics (vol 1), Dover, New York 1985),
  Vol. 1, Appendix A 1.6
• C. K. Johnson, M.M. Burnett and W. D. Dunbar,
  Crystallographic Topology and Its Applications
  http//www.iucr.org/iucr-top/comm/ccom/School96/pd
  f/ci.pdf
• A. Martín Pendás, A. Costales, V. Luaña, PRB
  55, 4275 (1997)
• A. Costales, phD thesis, in Spanish, Oviedo
  University, 1998, http//www.uniovi.es/quimica.fi
  sica/qcg/qcghome.html

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The space domain of crystal structures
For an in vacuo molecule R3 Open domain without
boundaries and infinite in every direction
For a 3D lattice the space domain is homeomorphic
to S3, the 3-torus Closed domain, that is a
region without boundaries but finite in every
directions
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The 2-dimensional lattice and S2, the 2-torus

• Mapping is such that
• r and rR become the same point in the closed
  domain D, but all the values of r within the
  2-dimensional cell remain distinct when mapped
  into D
• it reflects the periodicity of some function
  f(r)f(rR)

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The 3-dimensional lattice and S3, the 3-torus
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The real crystal
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Critical points analysis for a single-valued
function defined over a closed domain (Morse
relations)
Rn is the number of distinct n-dimensional
regions (C1n , C2n, etc) in D which have no
boundaries and do not form boundaries of (n1)
dimensional regions of D Distinct regions are
those that are not deformable into one
another Deformation of a region of D implies
change of shape or size, or translation, with
deformation always taking place in D
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Betti numbers for the 2- and the 3-torus
3-Torus Ro 1 , C0 being any point on the
3-torus R3 1 , C3 being the domain itself R1
R2 3
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Betti numbers and CPs Morse relations
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Fulfillment of Morse relationship in urea crystal
P 21m
16 34 26 8 0
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The complete bond network in urea crystal P 21m
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Point-group symmetry and CPs
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Li hcp metal as an exampleHexagonal , P63/mmc
space group, Z2
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Critical Points in crystals some simple rules
Each fixed point Wyckoff position of the space
group must contain a critical point of the scalar
function
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Gradient paths in 3D Scalar field

• dr(s)/ds ?? r(s) X for a given
  r(0) ?r0
• r(s) r0 ? ?? r(t) X dt GRADIENT
  PATHS (GPs)

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Gradient Paths in any 3D Scalar FieldN.O.J.
Malcom, P.L.A. Popelier, J. Comp. Chem. 24,
437-442 (2003)
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The primary bundles(A.M. Pendás et al., PRB 55,
4275 (1997))
1c, 1n and m (r,b) here m2
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Atomic basins and Primary Bundles (PBs).
Proximity and Coordination polyedra
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Weighted Proximity Polyedra and Atomic Basins
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Primary bundles and the topological equivalent to
the Wigner Seitz cellPendas et al. PRB, 55, 4275
(1997) and Zou and Bader, Acta Cryst. A50, 714
(1994)
A bonded pair of C toms, defining the
Wigner-Seitz cell of diamond of D3d symmetry
C atom in diamond of Td symmetry
The C-C interatomic surface in diamond
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Intermolecular and intramolecular contacts
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Intermolecular and intramolecular contacts.
Examples 2
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Intermolecular and intramolecular contacts.
Examples 3
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Intermolecular and intramolecular contacts CP
data
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Intermolecular interactions and interaction
density
Are these effects likely to be measurable in an
experiment or observable in a multipole
refinement of the electron density?
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I. Influence of intermolecular interactions on
multipole-refined EDM.A. Spackman et al., Acta
Cryst A55, 30-47 (1999)
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II. Influence of intermolecular interactions on
multipole-refined EDin urea M.A. Spackman et
al., Acta Cryst A55, 30-47 (1999)
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Interaction densities M.A. Spackman et al., Acta
Cryst A55, 30-47 (1999)
Ice VIII
Acetylene
Formamide
Urea
?int at a resolution sin?/? 1.0Å-1 obtained by
Fourier summation of appropriate phased Ftheory
structure factors. Contours at 0.025e Å-3
intervals
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I. Interaction density and changes in BCP
properties of urea
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II. Interaction density and changes in BCP
properties of urea
Bulk -molecule
BCP displacements and ?N(?)
CG
1displacement
CG
OG
?? (bcp)
1 increase
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III. Interaction density and changes in BCP
properties of urea
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IV. Interaction density and changes in BCP
properties of urea. The dimer model
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Classification of atomic interactions
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Classification of Chemical interactions and
Interpretation of BCP changes
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Interaction density in urea
??MOLECULES
? crystal
ID, ?2,4,810-n n0-3, au
??ATOMS
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Interaction density and atomic basin changes
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Hydrogen bond energetics from topological
analysis of experimental electron densities.E.
Espinosa et al. , CPL 285, 170-173 (1998), Acta
Cryst B55, 563-572 (1999), CPL 300, 745-748 (1999)
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II. Hydrogen bond energetics from topological
analysis of experimental electron densities.
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III. Hydrogen bond energetics from topological
analysis of experimental electron densities.
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IV. Hydrogen bond energetics from topological
analysis of experimental electron densities.
Caveats!!
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V. HB energetics from topological analysis of
EDDs. Recognising the importance of the
promolecule (M.A. Spackman, CPL 301, 425-429
(1999))
Exp points are those of Espinosa et al. 26
contacts from more recent studies Solid lines
from the two-atom model
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VI. HB energetics from topological analysis of
EDDs. Recognising the importance of the
promolecule (M.A. Spackman, CPL 301, 425-429
(1999))
¼ ?2? (r) 2 G(r) V(r) Promol. underestimates
?2? and hence overestimates V
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I. Secondary Interactions and Bond Critical
Points in Ionic Crystals
Ionic interaction one in which atomic species
exhibit charges approaching the values for the
corresponding closed-shell ions. ?b and ?2?
typical of closed shell-interactions, G(?b)/ ?b
gt1
Are to be considered as repulsive interactions?
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II. Secondary Interactions and Bond Critical
Points in Ionic Crystals

• No account for any electrostatic contribution
  to the potential arising from distortions of the
  spherical ion densities
• Use of an electrostatic model to determine the
  cohesive energy of a crystal assumes the kinetic
  energy of the electrons to be the same for the
  crystal as for the isolated ions. V.Theorem not
  satisfied!

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III. Secondary Interactions and Bond Critical
Points in Ionic CrystalsR.F.W. Bader, A Bond
Path A Universal Indicator of Bonded
InteractionsJ. Phys. Chem. A102, 7314 (1998)

• The force determining whether a system is in
  electrostatic equilibrium is the Hellmann-Feynman
  force (F? -??E) acting on the nuclei. This
  force is zero for all of the nuclei in a crystal
  in an equilibrium geometry

• Attractive restoring forces are the only one
  operative in a bound state and these are present
  only when the nuclei are displaced from
  equilibrium

• If repulsive forces were present the crystal
  would be unstable and would either atomize or
  distort to an eq. geometry of lower energy

• Since ? is homeomorphic to the virial field V(r)
  a bond path is mapped on a line of maximally
  negative potential energy density, a virial path.

• Closed-shell interaction, weak or strong, and
  shared interaction both result from the local
  pairing of the densities of opposite spin and in
  the formation of a bond and a virial path. They
  only differ for the degree and the way of
  realization of such pairing.

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Scalar and vector quantities depending on ?1(r,r)
? ?(r)
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Scalar quantities depending on ?1(r,r)
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Scalar quantities depending on ? ? ? (r1,r2)
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I. Approximated kinetic and potential energy
densities in the (100) plane of LiFTsirelson
V.G. Acta Cryst B58, 632-639 (2002)
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II. Approximated kinetic and potential energy
densities in the (100) plane of LiFTsirelson
V.G. Acta Cryst B58, 632-639 (2002)
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I. Approximated ELFTsirelson V. and A. Stash,
CPL 351, 142-148 (2002)
?? (r) ELF? (r) 1?? ( r)-1 ??( r) ??( r)
/?? ( r)unif. el. gas ??( r) G?( r) - g?(r) G?
( r) ?i? ?Ñ?i(r) ?2 g? (r) 0.25 (Ñr?)2/ r?
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II. Appproximated ELF3D-Hydrogen Bonding network
in urea the ELF-DFT description from
experimental densities
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Hydrogen Bond the -?2? description
Hydrogen bonds may be discussed in terms of a
generalized Lewis acid and base interaction.
Generally the approach of the acidic hydrogen to
the base will be such as to align the (3,3)
minimum in the VSCC of the H with the most
suitable (3,-3) base maximum
(3,-3)
(3,3)
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3D-Hydrogen Bonding network in urea the -?2?
description
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I. Wavefunction and one-electron density matrix
from experiment
Jayatilaka D, PRL 80, 798-801 (1998) Jayatilaka D
and Grimwood DJD Acta Cryst A57, 76
(2001) Grimwood DJD. Jayatilaka et al., Acta
Cryst A57, 87 (2001) Bytheway I, Grimwood DJ, et
al., Acta Cryst A58, 244 (2002) Bytheway I,
Grimwood DJ, Jayatilaka D, Acta Cryst A58, 232
(2002) Grimwood DJ, Bytheway
I, Jayatilaka D, J. Comput. Chem. 24, 470-483
(2003) ELF, Kinetic energies , ESP, and RHO
TONTO (http//www.theochem.uwa.edu.au/tonto)
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II. Wavefunction from experiment

• Constrained variational method
• Choose an appropriate variational ansatz for
  the wf (e.g. HF calculations of isolated crystal
  fragments)