Quantitative EDSA for determining the electrostatic potential and chemical

bonding in crystalsThe modern state of method and some examples of

investigations

Anatoly Avilov

Institute of Crystallography of Russian Academy of sciences

• Introduction: Quantitative electron diffraction structure analysis (QEDSA)

• Electron diffractometry• The Fourier method of the reconstruction of the ESP• The reconstruction of the ESP by analytical methods• Some properties of the electrostatic potential • Investigation of chemical bonding• Examples

- Ionic crystals with the structure NaCl

- Сovalent type of bonding (Ge)• Concluding remarks

High energy electron diffractionpeculiarities

• Scattering on the inner electrostatic potential of crystal

• Strong interaction with substances, possibility of study small amount of material (thin films, surface slabs, micro- and nanocrystals, etc.)

• Very high diffraction intensities• More high in comparison X-ray diffraction sensiti-

vity to the effect redistribution of the valence electrons

Poisson equation Mott formula :

fe (s) = me2/(2h2) {[Z – fx (s)] / s2

Z - nuclei charge

s 0 , fx (s) Z,

fe (s) large changes !

Study of ESP is very important…!

ESP distribution - the crystal packing features

- peculiarities of the atomic and molecular

interactions

- at nuclear positions - the core -electron

binding energy

- gradient of the electric field at nuclei

(nuclear quadrupole resonance,

Mossbauer spectroscopy)

- diamagnetic susceptibility, refractive

index of electrons ….

The relation of ( r ) and ( r )with physical properties

Properties directly Properties indirectlydepending on depending on ( r ) and ( r ) ( r ) and ( r )_________________________________________________Diamagnetic Electron staticsusceptibility polarizability

Dipole, Nonlinearquadrupole and ( r ) opticalother momentum ( r ) characteristics

of nuclearCharacteristics of Intermolecularthe electrostatic interactions

fieldEnergy of

electrostatic interaction

ELECTROSTATIC (COULOMB) POTENTIAL (ESP)

• ESP is a scalar function :

r r rrdr• full charge density:

r = a rRa r

aand Ra - nuclear charge of atom “a” and its coordinates

r - electron density r includes 0 - mean inner potential, which depends

on crystal shape and surface structure

• {(h2 / 82 m ) 2b *cr Hb cr dv } b Eb b

= h2 m K2 b . K– modulus of wave vector in vacuum

r = (1/e) *cr Hb cr dv = (1/) h exp(- 2 i gr)

HEED TEM

wide primary beam; small electron beam;

electron diffraction micro- and nanodiffraction

structure analysis SAED

= EDSA or HRED;

specific diffraction

patterns

structure analysis of polycrystals and single microcrystals

Problems of development of the precise EDSA

• Development of the precise technics of measurements of electron DPs

• Elaboration of the methods of many beam calculations for the refinement of structure parameters

• Improvement of the means for the accounting for the inelastic scattering

• Working out the methods of modelling ESP on the base of experimental information and the estimate of its real accuracy

• Elaboration of the methods of the treatment of uninterrupted ESP distribution in terms of conception of physics and chemistry of solids

Scheme of the electron diffractometer

Package of program for mesaurements, treatments and structure calculations

1 – electron diffractometer,

2 – electron microscope,

3 – imaging plate,

4 – polycrystalline sample,

5 – textured sample,

6 – mosaic monocrystalline sample,

7 – refinement of the atomic structure,

8 – LSQ – method,

9 – direct methods,

10 – refinement of parame-ters of chemical bonding,

11 – Fourier maps

Statistical treatment of experimental intensities

accumulation mode for measurements, statistical treatment and «control point» - 1-2% statistical accuracy

How to reconstruct the electrostatic potential?

• Two methods will be considered:

• Summing of the Fourier series with using experimental structure amplitudes hkl

• Analytical reconstruction in the direct space on the parameters of the model, obtained from the experiment

The Fourier method of the reconstruction of ESP

φ (xyz) = 1/

hkl

hkl exp[-2 π i(hx + ky + lz)]

= | | expi [hkl] = VÅ3

Scaling : Parseval integral equation

H

H2obs = < φ2 > = )(celli

(1 / 22)

0

f2eTi (s) s2 ds

Wilson statistical method < 2obs(sin /) > = 2sin

1sin

f2eT(sin /) d(sin /)mean inner potential - 000 : < φcr > = 000 / = fe(0) / [V/Å

3]

potential projection (e.g. using reflexions hk0) :

’ (xy) = c 1

0

(xyz) dz = -(1/S) hk0 exp 2i(hx + ky)

[’ (xy)] = [Volt Å] = [mass1/2

lenght3/2

time-1

]

Form of peaks of the atomic potential distributionT (r) = (1/22)

0

feT(s) s2 (sin sr/sr) dsfeT(s) = 4

0

T (r) r2 (sin sr/sr) dr ;

total atomic potential : (r) dvr = 4

0

T (r) r2 dr = fe(0)

potential in the centre of the atom (at r=0) (0) = (1/22)

0

feT(s) s2 ds

for the projection of the atomic potential :’ (R) = (1/2)

0

feT(s) s Io(sr) ds; Io(sr) - Bessel function

in the atomic centre: ’ (0) = (1/2)

0

feT(s) s ds

for real conditions - approximated formula: (0) = KZ

q (K, , q are given by Vainshtein (1964))

Analysis of (0) gives useful information for EDSA

i (0) values provide information on the occupancies of

the atomic positions

• EDSA: (0) ~ Z 0.8 , XDSA: (0) ~ Z1.25

Localization of light atoms in the presence of heavy ones in EDSA is much better (e.g. hydrogen, lithium, oxygen, etc) than in X-ray diffraction

• difference Fourier syntheses is effective for the definition of (0)

Difference Fourier syntheses

Φdif (r) = 1/ hkl

(obs - ’calc ) exp(-2 π i rH)

obs = n

1

feTi exp (2 π i riH)’calc =

n'

1i

feTi exp (2 π i riH)

Fourier projection- cubic ice - (001) plane (with- (a) and without oxigen (b))

(obs - ’calc ) =

n'

1

n

i

feTi exp (2 π i riH)

Accuracies of determination of structure parameters and ESP by Fourier method (by Vainshtein)

• In determination of coordinate values:

xj= {2 h2expth2} [a d2 dxj2]

• Potential values:

a) from the exp. errors:

2exp = R (1/) hkl2

b) from the break of the Fourier-series:

2break= (1/) (1- q) hkl2

parameter q is determined by the experimental break of structure amplitudes (see Vainshtein)

Influence of the break of Fourier series for reconstruction potential’s maps

• Fourier maps for the ESP: (100) - left, and (110) - right.

• Two upper rows are experimental series up to (sin/)max 1,3 A-1.

• Two lower rows present theoretical Hartree-Fock calculations and experimental amplitudes with adding theoretical ones ( 15 A-1) .

• Appearence of false peaks (5-10 % from true peaks) and distortions of the forms of the ESP peaks GaAs examples is seen

Structural aspect...

• How to compare atomic potential of identical atoms in different structure?

• How to use ESP in the crystal-chemistry analysis for the decision more general questions on the crystal formation?

So: for the solving of the problems with quantitative investigations of the chemical bonding and electrostatic potential the analytical methods of the reconstruction are needed.

The reconstruction of the ESP by analytical methods

• Model’s parameters are founded by adjustment to experimental structure amplitudes

• Analytical methods are free from many errors:

- the break of Fourier series

- inaccuracies of intensity measurements and

transition to structure amplitudes

• Static ESP is calculated for the following analysis

• The calculation of ESP is realized in direct space by using Hartree-Fock wave functions

Chemical bonding in EDSA

• Multipole model Hansen-Coppens:

( r ) = Pcore core ( r ) + Pval val ( r) + Rl ( r) Plm ylm (r/r)

( r ) – electron density of each pseudoatom, core ( r ) and val ( r )

– core and spherical densities of valence electron shells

• Pval and Plm (multipoles) describe electron shell occupations

- and describe spherical and complex deformation in anysotropic cases

- y (r/r) is geometrical functions• for ionic bonding – spherical approximation (kappa –model):

( r ) = Pcore core ( r ) + Pval val ( r)

• electron structure amplitude, using Mott-formula:

(g) = ( g ) {Z – [ f core(g) + Pval fval (g/ )]}

• Rl ( r) Plm ylm (r/r) - nonspherical part, describing space anisotropy of the electron density

Experimental curve for polycrystalline film LiF

• Upper curve - experimental intensity distribution

• Lower one - the same after subtraction of background

Quantitative data for the ionic crystalsLiF, NaF, and MgOa

Structural - model Electron diffraction Hartree-Fock

Structure amplitudes structure amplitudescompound atom Pv R% Rw % Pv R% LiF Li 0.06(4) 1b 0.99 1.36 0.06(2) 1b 0.52 F 7.94(4) 1b 7.94(2) 1.01(1) NaF Na 0.08(4) 1b 1.65 2.92 0.10(2) 1b 0.20 F 7.92(4) 1.02(4) 7.90(2) 1.01(1) MgO Mg 0.41(7) 1b 1.40 1.66 0.16(6) 1b

O 7.59(7) 0.960(5) 7.84(6) 0.969(3)

a Structural - models were as followed- LiF: cation = 1s (r ) + Pval 3 2s ( r ),

anion = 1s (r ) + Pval 3 2s,2p ( r ); NaF and MgO: cation = 1s,2s,2p (r ) +

Pval 3 3s ( r ), anion = 1s (r ) + Pval 3 2s,2p ( r )b Parameters were not refined

Quality of the adjustment -model to exp. data for binary

compounds

F / F =

(Fexp-Fmod)/ Fmod

ELECTROSTATIC POTENTIALTOPOLOGICAL ANALYSIS

• Classical electrostatic field is characterized by the gradient field ( r ) and curvature 2 ( r ) (these characteristics do not depend on the mean inner potential 0) :

E ( r ) = - ( r ) • EP exhibits maxima, saddle pointes, and minima

(nuclear positions, internuclear lines, atomic rings, and cages).

• In “critical points”: ( r ) = 0

ELECTROSTATIC POTENTIALTOPOLOGICAL ANALYSIS (2)

• Theory Bader (analog for the electron density) was used for the ESP

• In “critical points”: ( r ) = 0 • Hessian matrix - H is composed from the second derivative ( r )

• For ESP in critical points 1 + 2 + 3 0, because ( r ) 0

1 + 2 3

• Nuclear of neighboring atoms and molecules and crystals are separated in the E ( r ) by “zero-flux” surfaces S ( r )

E ( r ) n ( r ) = - ( r ) n ( r ) = 0 , r S ( r )

These surfaces define the electrically neutral bonded pseudoatoms

Inside surfaces nuclear charge is fully screened by the electronic charge

ESP for binary compounds (analytical reconstruction)(110) - plane

• circle - (3,-1) - bonding lines - one-dim. minimum

• treangle - (3,+1) - two-dim. minimum

• square - (3,+3) - absolut minimum

ESP (left) and ED for (100) plane of LiF

• The location of CPs does not coincide, ESP does not fully determine the ED

• In ESP the main input in the distribution belongs to cations

ESP in (110) MgO

Characteristic peculiarities are seen along bonding lines

(ESP in maxima are given in logariphmic scale)

Analysis of the ESP along bonding in binary crystals LiF, NaF, MgO

• Distribution of ESP in binary compounds is along cation-cation (dotted),

anion-anion (solid) -left;

• The same one is along cation-anion - right side (ESP-values are in log of Volts)

ESP for bonded atoms in crystals

• Electrostatic potential as a function of the distance from the point of observation to the center of an atom for isolated ions in LiF and NaF crystals with the parameters of the -model obtained from the electron diffraction data

Laplacian of the ED for LiF and MgO, plane (110)

• Laplacian (-2 ( r )) allows one to analyse the overflow of the electronic charge at the bonding formation

• Inner electronic shells

are seen

Values of the Electrostatic potentials (V) at the nuclear positions in crystals and free

atoms and mean inner potentials (0 )

Compound atom electron Hartree-Fock (crystal) 0

diffraction free

- mode direct reciprocal ftoms

space space

LiF Li -158(2) -159.6 -158.1 -155.6 7.07

F -725(2) -726.1 -727.2 -721.6

NaF Na -968(3) -967.5 -967.4 -964.3 8.01

F -731(2) -726.8 -727.0 -721.6

MgO Mg -1089(3) -1090.5 -1088.7 -1086.7 11.47

O -609(2) -612.2 -615.9 -605.7

“Bonded radii” derived from the electrostatic potential and electron densitya

“bonded radii” (A)compound atom electrostatic potential electron density

LiF Li 1.084 0.779 F 0.928 1.233 NaF Na 1.355 1.064 F 0.964 1.255 MgO Mg 1.207 0.918 O 0.899 1.188

a “Bonded ionic radii” is defined as a distance from a nuclear position to the one-dimensional minimum in the electrostatic potential or electron density along the bond direction

Ge - covalent bonding

• Multipole model Hansen-Coppens:

( r ) = Pcore core ( r ) + Pval val ( r) + Rl ( r) Plm ylm (r/r)

( r ) – electron density of each pseudoatom, core ( r ) and val ( r )

– core and spherical densities of valence electron shells

• Pval and Plm (multipoles) describe electron shell occupations

- and describe spherical and complex deformation in anysotropic cases

- y (r/r) is geometrical functions

• for ionic bonding – spherical approximation (kappa –model):

( r ) = Pcore core ( r ) + Pval val ( r)

• it should be taken into account for nonspherical part :

Rl ( r) Plm ylm (r/r) , describing space anisotropy

of the electron density• radial functions Rl ( r) = r exp (- r),

• n = 4,4,4,4 ( l 4 ) and =2.1 a.u. are calculated theoretically

Results of the multipole model refinement of Ge crystalon the electron diffraction data

Electron Refinement with LAPW

Diffraction structure factors

' 0.922(47) 0.957

P32- 0.353(221) 0.307

P40 - 0.333(302) - 0.161

R(%) 1.60 0.28

Rw(%) 1.35 0.29

GOF 1.98 -

ED and ESP for (110) plane in Ge

Location of

critical points

is equivalent

• circle - (3,-1) - bonding

• square - (3,+3) - absolut minimum

• treangle - (3,+1) - two-dim. minimum

Laplacian of electron density for Ge

• -2 ( r )• fragment of structure of

Ge along plane (110), reconstructed from the ED-data

• The formation of Ge crystal is accompanied by the shift electron density to the Ge-Ge bonding line

• The inner electron shells are seen

Topological characteristics of the electron density in Ge at the bond, cage and ring critical points First row presents the ED results, second row presents the our calculations based on model parameters, obtained by LAPW dataCharacteristics of the CP (3,-1) for the procrystal: =0.357, 1=2=-0.65, 3= 1.85, 2= 0.55

critical point type (eÅ-3) 1(eÅ-5) 2 (eÅ-5) 3 (eÅ-5) 2and Wyckoff position

Bond critical point, 16c 0,575(8) -1.87 -1.87 2.04 - 1.70(5)

0.504 - 1.43 - 1.43 1.68 - 1.18

Ring critical point, 16d 0.027(5) - 0.02 0.013 0.013 0.25(5)

0.030 - 0.02 0.014 0.014 0.26

Cage critical point, 8 b 0.024(5) 0.05 0.05 0.05 0.15(5)

0.022 0.05 0.05 0.05 0.15

Direct calculation of some physical properties from ED data

• Diamagnetic susceptibility d

Spherical symmetry of atoms , ionic bonding

Classical Langeuven equation, with accounting for the symmetry

d = - (0 e2 NA a2 / 4m) [ N/96 + 1/(22 ) (-1)h/2 F (h00) / h2 ]

N – number of electrons in cubic cell, NA – Avogadro’s number, a – parameter of the elemental cell, 0 – magnetic constant value.

F (h00) – structure amplitude of h00 – reflection.

• Electron static (low frequency) polarizability

Kirkwood relation between number of electrons in molecules and mean square radius-vector of electrons in atom.

= 16 a4 /(a0 Ne) [ Ne /96 1/(2 2) (-1) h/2 F(h00) /(2 2 h2)]2

Ne – the number of electrons in molecular unit, a0 – Bohr-radius

Physical properties of crystals, calculated on the electron diffraction structure amplitudes

in comparison with other data

Determination method reference d 10-10 (m3 mole) (0) 10-30 (m-3)

electron diffraction 1, 2 2.06(4) 17.70(3) 3.82(3) 22.0(1) electron diffraction 3 2.0(1) -

X-ray diffraction 1,2 2.24(1) 20.80(2) - - X-ray diffraction 4 2.17(4) 9.71(2) 3.4(5) 25.82(4) Nonempirical Hartree-Fock 5 2.12(2) 18.71(2) calculation - - Independent experim.data 2.31 22.60 - 35.91 Red color – data for MgO Blue color – data for MnOREFERENCES: 1. J.Struct.Chemi. (1992) 70, 1996. 2. Acta Crystall. (1998) 54B, 1

3. Izv.ANSSSR. Ser.physich. (1984) 48, 1753. 4. Proc.Japn,Acad.Ser. (1979) B55, 43.

5.Crystall.Reports (1997) 42, 660.

Main conclusions

• The achieved level of the EDSA in the combination of the topological analysis of ESP and ED allow one to get a reliable quantitative information about chemical bonding and depending on it properties

• Electrostatic field in crystal is good structurized and depends from the type of structure

• The precise EDSA should be developed due to improvement: electron diffractometry, methods of the accounting for dynamic scattering and background, methods of the analysis of ESP and ED. EDSA has a good perspectives for this !