Maximum‐Entropy Methodin the analysis of Charge Density I‐ A Promising Tool for Charge Density Study ‐Masaki Takata
RIKEN SPring‐8 CenterOutline
1. Information Entropy and Charge Density Analysis
2. MEM Charge Density Study by Diffraction Data
3. Charge Density Refinement by MEM
MEM/Rietveld Analysis etc.
4. Nuclear Density Study by MEM
5. Application1

Information Theory (1948)
I just wondered how things were put together.

by Claude Shannon
American mathematician andelectronic engineer known as"the father of information theory"
Information Entropy :S   i nii

as a measure for the uncertaintyof information (data)

1916 - 2001

Jaynes’ Entropy (1968)

S   i ni

i
i

Innovation of the idea:
Prior information2

The Dawn ofMaxEnt Application toCharge Density Study

MEM application tocrystallography
MEM: Virtual Lens for
Diffraction Data Imaging

Electrostatic
PotentialImaging

Structure Model Refinement

High Brilliance
Synchrotron Radiation X-ray

Diffraction Data

Bonding Nature Visualization

N2FeN4

N1

Covalent Bonding Ionic Bonding

O2

C2H2Science, 298 (2002), 2358

Nature 436 (2005) 238

Gas Molecule Imaging Adsorbed in MOF

Charge Order Imaging

Metal

Insulator

Angew. Chem. Int’l Ed. 43 (2004) 3670

Charge Ordering Through M-I Transition of EDO-TTF

3

Basic Concept: MEM Analysis
The formalism of the constrained entropy is given
by using the method of Lagrange undetermined
multiplier in order to constrain the function C to beunity while maximizing the entropy.

Sir

SiFourier Synthesis

X-ray Data

Fcal (k)    (r ) exp[2ikr]

SiSi

Charge DensityQ     r  nr r   C  1 r  2Fcal k   Fobs k 1C N k
 2 k  (r )   F (k) exp[2ikr ]k2MEMTo maximize the entropyQ 0by r Extinction Free Precise Databy Pendellösung Method
Saka & Kato,
Acta Cryst. (1986)

MEM CD

MEM charge density formalism is obtained asr   exp  n r  N.Kato

F01
Fobs k   Fcal k exp - 2 ik  r N k  2 k 

Electrostatic Potential Formalism based on MEM
Phys. Rev. B, 74 (2006)172105

U (r )  4 ZetGt

H Tanaka

|G |2 / 2e  iGR t  FMEM (G ) iGrZte  erfc(η | r  l  Rt |)2
|G |t | r  l  Rt |lElectron charge contribution term

Nuclear charge contribution term
Ewald’s Technique

MEMAdvantages• Parameter free (in practice)• Direct method

MEM Electrostatic Potential Imaging
PbTiO3
Pb O

NaClO

ClNa+

Ti

0.86e/Å30.25e/Å3-0.2e/Å～0.0e/Å

-0.9e/Å～0.9e/Å

PbTiO34

MEM
SampleChargeDensitySynchrotron
Radiation

Electrostatic
Potential

Visualization ofAtoms/ MolecularInteraction
Data
Measurement
DataAnalysis

Problem: The Influence of the Data Completeness
Si MEM Charge Density by Takata & Sakata
M.Takata

5A way for reliable data measurement for MEM analysis- SR Large Debye-Scherrer Camera -MEM Charge Density

Intensity (Arb. Unit)Si

Si NIST

Rwp= 5.2%
RI = 2.5%(λ=0.4Å, Exposure Time: 20min.)

BL02B1@SPring‐82(degree)

A way to estimate net charge of atoms
Extended Mulliken Scheme for MEM CD Analysis

Net charge of A atom Q A   Qi  Z A ZA: Atomic Numberi A

QAB
Overlap Population

AB

Orbital Population

6

Ionic Valence investigation from MEM charge Density
BaTiO3

PbTiO3tetragonal

cubictetragonal

cubic

cubic

A: Pb, Ba+1.9(3)+1.8(3)+1.1(3)+2.0(3)+1.2(4)

B: Ti, Zr+1.9(4)+2.1(4)+2.4(4)+2.2(3)+2.7(5)O (1)-1.6(3)-1.3(3)-1.4(3)-1.4(3)-1.3(2)O (2)-1.1(3)--1.0(3)--a)b)

PbPbO1O2PsPbTiO2Pb

2.0 ÅxGas Adsorption in MOF
O NaNO

Cu2+ +O2O1z

S.Kitagawa

PbZrO30.1 ~ 5.5 e/Å3 , step = 0.2 e/Å3

[Cu2(pzdc)2(pyz)]n : CPL-1
NH2O

PbTiO3300K

Npyz

O
N

L
L

L
L

L
L

L
L

O Na

L= N

NPyrazine

Na2pzdc[Cu(pzdc)] n[Cu 2(pzdc)2(pyz)]nCu2(pzdc)2
sheet

CH4O2

HCNOCu7MEM/Rietveld Analysis

Final MEM Charge Density of CPL‐1 & CPL‐1/O2First Visualization of One Dimensional Arrayof Physisorbed O2 Molecules in MOF

Prof. Y.Kubota

Science, 298 (2002), 2358-2361
BeforeAdsorption

O2
AfterAdsorption

1.0e/Å 38

Refinement by Omit‐Difference MEM Mapping Method
Prof. E.Nishibori

4. Nuclear Density Study by MEM
Fundamental Equation of Neutron Diffraction
to treat negative scattering length

Information Entropy Equation of Nuclear Density
based on Neutron Diffraction

Two MEM equations for Nuclear Density

9MEM Nuclear Density of TiO2Ti：negative scattering length

MEM can just visualize
the information included in data.

Susie (original)
Poor
Counting Statistics

MEM Susie
Based on (b)BetterCounting Statistics

MEM Susie
Based on (d)10