Maximum‐Entropy Methodin the analysis of Charge Density I‐ A Promising Tool for Charge Density Study ‐Masaki Takata
RIKEN SPring‐8 Center

Outline1. 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 uncertainty
of information (data)

1916 - 2001

Jaynes’ Entropy (1968)

S   i n
i

i
i

Innovation of the idea:
Prior information

2

Application to Image Reconstruction
Maximum Entropy Method

Silver, R.N.,Sivia,D.S. and Gull,J.E.:Phys.Rev.B41 2380-2389(1990)

The Dawn of
MaxEnt Application toCharge Density Study

3

MEM application to
crystallography
MEM: Virtual Lens for
Diffraction Data Imaging

Electrostatic
Potential
Imaging

Structure Model Refinement

High Brilliance
Synchrotron Radiation X-ray

Bonding Nature Visualization

N4

C2H2
Nature 436 (2005) 238

Gas Molecule Imaging Adsorbed in MOF

Diffraction Data

N2
Fe

O2
Science, 298 (2002), 2358

Charge Order Imaging

Metal

N1

Covalent Bonding Ionic Bonding

Insulator

Angew. Chem. Int’l Ed. 43 (2004) 3670

Charge Ordering Through M-I Transition of EDO-TTF

2. MEM Charge Density by Diffraction Data
From Electron Density mapping
To Electrostatic Potential visualizing
X‐ray Scattering

Incident
X-ray

ScatteredX-ray

(r): Electron density
in material

Electrostatic Potential
Imaging

To visualize precise charge density & electrostatic potential
1.Reliable Diffraction Data: absorption, extinction, etc. data correction free
2.Sophisticated Data Analysis Technique
Maximum Entropy Method(MEM), Multipole refinment, etc.

4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.

Si

r

Si

Fourier Synthesis

X-ray Data

Fcal (k)    (r ) exp[2ikr]

Si

Si

Charge DensityQ     r  nr

 r  
 C  1 r  2

Fcal k   Fobs k 
1
C 
N k
 2 k  (r )   F (k) exp[2ikr ]
k

2

MEM
To maximize the entropyQ 
0by r 
Extinction Free Precise Data
by 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 The process of obtaining the MEM solution
Initial solution

After 2nd iteration

Charge Distribution(110) Plane

103

102

charge density[e/Å2]

1D Charge Density<111> Direction

charge density[e/Å2]

103

101
10

0

10-1
10-2

0.2

0.4

0.6

0.8

102
101
100
10-1
10-2

0.2

0.4

0.6

0.8

Coincidence ofAtomic Scattering Factor

5The iterative process of obtaining the MEM solution
After 10th iteration

Final results ( 775 iterations )Charge Distribution(110) Plane

103

10

charge density[e/Å2]

1D Charge Density<111> Direction

charge density[e/Å2]

103
2

101
100
10-1
10-2

0.2

0.4

0.6

102
101
100
10-1
10-2

0.8

0.2

0.4

0.6

0.8

Coincidence ofAtomic Scattering Factor

The Sketch of Electron Density&
Fourier Coefficient Distribution
Covalent Bond Model
Estimated Atomic Scattering Factors by MEM
Amplitude of atomic scattering factor

1210Best possible fit
with Observed Data8

observed
MEM

6

Unobserved Dataestimated by MEM: Super‐resolution

4

2

0
0

0.5

1

1.5

2

2.5

3

sinθ/λ [Å‐1]6Imaging of Diffraction Databy the MEM

Bonding Nature
Imaging

Real Space Image
Reciprocal Space
X-Ray
Diffraction Data

Imaging
Processor
Electron Density

MEM
Neutron
Diffraction Data

Nuclear Density

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

U (r )  4 

Ze
t

G

t

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

MEM
Advantages• Parameter free (in practice)• Direct method

MEM Electrostatic Potential Imaging
PbTiO3
Pb O

NaCl

O

ClNa+

Ti

0.86e/Å3
0.25e/Å3-0.2e/Å～0.0e/Å

-0.9e/Å～0.9e/Å

PbTiO37

MEM
SampleChargeDensity

Synchrotron
Radiation

Electrostatic
Potential

Visualization ofAtoms/ MolecularInteraction
Data
Measurement
Data
Analysis

Be Charge Density Study by Precise Single Crystal Data

Niels Hansen & Finn Larsen

8MEM Charge Density Study of Be
Prof. F.Larsen
Prof. Bo Iversen

M.Takata, B.Iversen, F.Larsen, et al. Acta Cryst. (1994) A50 330

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

9

Data Incompleteness gave an artifact in MEM density.

endellösung Method Data
Saka & Kato
Acta Cryst. (1986)

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)

10

Si(NIST) Powder Diffraction Data

Intensity (Arb. Unit)

complete data set with no absorption correction(λ=0.4Å, exposure time: 20min.)
Rwp= 5.2%
RI = 2.5%2(degree)

Si & Diamond MEM Charge Densities
Based on SR Powder Diffraction Data
(100K)

Si
0.55 e/Å3C

1.62 e/Å3

0.0～2.0 0.1[e/Å3]step11

Comparison with Theoretical Charge Density
Reliable data can give a meaningful precise charge density by MEM.

Si Valence Electron Density
Si

MEM Charge Density

First Principle Calculation
Wien 2000

The IP Camera Powder Method using High Energy SR
Whole Powder Pattern can be collected simultaneously.
Rapid data collection
No need of incident X‐ray beam monitoring
Same experimental conditions at each data point

High Resolution Data
=0.5Å~1.0Å

d= 0.4Å~0.8Å at 80º (2)Insignificant 2 dependence of absorption effects
even for heavy materials
For example PbTiO3 powder in 0.1 capillary
X-ray Energy (mm-1)

0º / 90º (%)

MoKÅ)

68.2

24.0

30KeV(Å)

17.2

1.7

 : Total linear absorption coefficient0º / 90º : The ratio of the coefficient at 0º and 90º(2)12PbTiO3 MEM Charge Densities
based on high energy powder data
=0.41Å : 30keV, Exposure Time : 60 min., d >0.47Å

Prof. Y.Kuroiwa

Cubic

800K

Tetragonal

300K

Y. Kuroiwa et al., Phys. Rev. Lett. 87 (2001) 217601

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 Number
i A

QAB
Overlap Population

A

B

Orbital Population

13

Ionic Valence investigation from MEM charge Density
BaTiO3

PbTiO3

PbZrO3tetragonal

cubic

tetragonal

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)

Pb

Pb

O1

O2

Ps

Pb

Ti

O2

Pb

O2

O1

z
2.0 Å

x

PbTiO3300K

0.1 ~ 5.5 e/Å3 , step = 0.2 e/Å3

High counting statistics data measurement for MEM
Overlaid Measurement Method
Prof. E.Nishibori

Lower Angle Data : D1

Higher Angle Data: D2

IP

IP

+

X-ray

X-ray
SampleSampleWith longer exposure
80000

1000000

40000

60000020000

4000000402000000

0

Improvement of statistics
in higher angle range

60000

8000002040502Deg. )

60

60

70

80

90

80

14

MEM valence density by overlaid measurement

With d > 0.34 Å
Diamond

E. Nishibori et al.,Acta. Cryst. A (2007)

Si

0.56eÅ-3

1.64eÅ-3

Si: 0.55～0.59eÅ-3

Diamond: 1.53～1.69eÅ-3Theoretical charge density values at the bond midpointYin M. T. And Cohen M. L., PRB26 (1982).
Van Camp P. E., et al., PRB34,(1986).,
Christensen N. E. et al., PRB36 (1987). , etc.

Single Crystal Diffraction Experiment
Large Cylindrical Imaging‐Plate Camera @ BL02B1/SPring‐8
Prof. H Sawa

Cylindrical Image-Plate

X-ray

1/4-goniometer- For
data completenessclose to 100%.

Single axis goniometer- for cryostat andspecial sample chamberon phi-axis for in-situ observation

Large IP camera and are avertable for a great variety of experiments.
- High- and low-temperature experiment- Photo-excited crystal structure- High pressure experimentetc.

Specification:
Dynamic range:
Camera radius:
Detector area:2 coverage:< 106 counts
191.3 mm
683(W) x 350(H) mm-60.0 ～ 144.0°

15Extremely high resolution in data collection
with high S/N ratio: DyMnO3 data
0.6 Å
0.2 Å

0.3 Å

0.8 Å

0.5 Å

DyMnO3
Crystal size (mm)Cell parameter:

35.24 keV (0.3517Å)High resolution data measurement
is available fora precise charge density study.

0.03 x 0.03 x 0.03
a = 5.82250(10) Åb = 7.3816(7) Åc = 5.27400(10) ÅV = 226.67(2) Å3
Z=4
Space group
Pnma (#62)
Wavelength:0.3517 Å ( 35.24 keV)
Temperature:
140 K
78.63  (0.27 Å)2 max.:Rmerge:4.00 %
R factor:
3.13 % (2(I)<I)
GOF:
1.125
Total measurement time: 6.5 hours

Vol. 2 (2010) 678-683

Li+

C60

MEM Charge
Density of Li@C60

Two Dimensional Head & Tail Layered Structure of[Li@C60] & (SbCl6)

Chemical & Engineering News 88 (2010) 38-39.

16Ultimate Single Crystal Structural Analysis
from 500 nm BaTiO3 single grain powder sample(Beam size 3.2W x 2.8H um)
ωtotal=225º, ∆ω=0.5º, 2θ=30º,
Exposure time 5sec/1 image

Structure Analysis
by Direct Method

0.0771(4350ref)
Rint
0.0436(2356ref)
R1
2θmax 65.6°
Completeness 0.925

450 images are superimposed

3. Charge Density Refinement by MEM
MEM/Rietveld Analysis Application to MOF Chemistry
To see mechanism of gas adsorption
in nano space
O2 in CPL-1

Science 298 (2002) 2358

C2H2 in CPL-1

Nature 436 (2005) 238

17

Gas Adsorption in MOF
O NaN

S.Kitagawa

O

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

H2O

N

pyz

O
N

L
L

L
L

L
L

L
L

L= N

O Na

N

PyrazineNa2pzdc[Cu(pzdc)] n[Cu 2(pzdc)2(pyz)]nCu2(pzdc)2
sheet

CH4

O2

H

C

N

O

Cu

Ｇas import system for Powder Diffractometry
Gonio head

Imaging Plate
Slit

CapillaryX-ray = 0.8 Å

Gas import tube

Ion chamber
Shutter
Collimator
SampleN2 gas flow nozzle

Pressure gauge

V
Cold
head

V

He leak
detectorVO2 gas18

Powder diffraction patterns of [Cu2(pzdc)2(pyz)]nwith Oxygen gas adsorption
Pressure of O2 gas is 600Torr

Structure Model Refinement
by MEM/Rietveld Analysis

19MEM/Rietveld Analysis

1st Rietveld fitting based on the initial model[Cu2(pzdc)2(pyz)]n (90K)
RWP = 18.5 %RI = 54.2 %20

Model Refinement in MEM/Rietveld Analysis
O2 ?

Model without O2 molecule
Rwp =18.5%, RI =54.2%,

O2
Model with O2 molecule
Rwp =2.1%, RI =4.0%,

Rietveld fitting based on the final model[Cu2(pzdc)2(pyz)]n‐O2 (90K)P21/c, Monoclinic:
a=4.68759(4)Å, b=20.4373(2) Å, c=10.9484(1) Å, =96.9479(6)º21

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
After
Adsorption

1.0e/Å 3

1-D Array of O2, N2, CO2, CH4 ,C2H2 and …
O2

N2

CO2

CH4

C ２H ２Channel

4.8Å

11.８o

19.6o

29.8o

61.2o

22

MEM Charge Density of C2H2 in CPL‐1
First Visualization of One Dimensional Arrayof Chemisorbed O2 Molecules in MOF
Nature vol.436 (2005) 238

High‐Density Compression!!
200 times higher compressionof the lower explosive limit:2 atom. @ R.T.

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

23

Refinement by Omit‐Difference MEM Mapping Method∙ The reliability factor
R1 = 17.9 % ⇒ 9.6 %∙ The number of water molecules
151 ⇒ 203 + 0.6 Cl ion∙ The number of disordered residues
0⇒9∙ H atoms
93.9 % of total H atoms were found.

Eiji Nishibori, et al., Acta Cryst. D.(2008)

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

24MEM Nuclear Density of TiO2

Ti：negative scattering length

Complementary Information of Bonding Nature
in Nuclear Density
Be Nuclear Density

Be Charge Density

Squarish deformation
due to4th order anharmonicity

Triangular deformation
due to
3rd order anharmonicity

25

5. Application

Orbital Order Model

Direct Observation of
Orbital & Charge Order in Manganite
by MEM/Rietveld Method

M.Takata et al. JPSJ Lett. 68(1999)2190

Finding Variety of Orbital/Charge Orderin Manganite
CE-type (Nd0.5Sr0.5MnO3)
A-type (NdSr2Mn2O7)

C-type (Nd0.35Sr0.65MnO3)

Prof. Y.Moritomo

Dr. K.Kato

Charge Order visualized in MEM Electrostatic Potential
18 K

Electrostatic potential on 0.8 e/Å3 iso‐surface

300 K

K. Kato et al., Phys. Rev. B 77, 081101(R) (2008).

26

Nuclear Density of KDP (KH2PO4 )investigation of hydrogen bond
10K

Paraelectric Phase

Feroelectric Phase2n (n＝2,3,4…） ×10-13 cm/Å3

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)27