Generalisation of the LEED approach to full potential + Mole

cular Dynamics approach

Keisuke Hatada

Dipartimento di Fisica, Università Camerino

3d images of MT & NMTFe(CN)6

Ge K-edge of GeCl4

1 Ge + 4 Cl + 38 EC (Empty cells)

For full potential case with homogeneous local harmonic oscillation

Tk,k ' =Tk,k ' ei(k−k ')⋅u =e−k2σ 2 (1−cos(k̂⋅̂k '))Tk,k '

Fujikawa ‘93

LEED type averaging

TLL '(u)LEED

=4πe−k2σ 2

i l1 jl1 (−ik2σ 2 ) G(ΛL1 |L)G(Λ 'L1 |L ')TΛΛ 'ΛΛ '∑

L1

∑

T (u) ≡ J (−u)TJ (u)

Exact spherical wave case

TLL '(u) =4πi l−l 'e−k2σ 2

i l1 jl1 (−ik2σ 2 ) iλ−λ 'G(ΛL1 |L)G(Λ 'L1 |L ')TΛΛ 'ΛΛ '∑

L1

∑

Hatada unpublished

QuickTime™ and a decompressor

are needed to see this picture.

For simple anisotropic case

σ x2 = σ y

2 = σ ⊥2

d =14

1σ z

2 −1

σ⊥2

⎛

⎝⎜⎞

⎠⎟

Just

s =14

1σ z

2 +1

σ⊥2

⎛

⎝⎜⎞

⎠⎟

TLL '(u) = M l1 ,l2l3 (−1)l1 G(L1L2 |L3)ΩLL '

L1L2

L1L2 L3

∑

ΩLL 'L1L2 = (4π )2 i l−l '+l1 +l2 iλ −λ 'G(LL1 | Λ)G(Λ'L2 | L ')TΛΛ'

ΛΛ'∑

M l1 ,l2

l3 =2π

1σ⊥

2σ z

al '(l )K l1 ,l2

l ' (s,d)l '=0

∞

∑

The lowest contribution for anisotropy is l’=2

Pair oscillation case

τpair

= T −1 + J(uij ) G⎡⎣ ⎤⎦−1

oscillation should be treated as EXAFS likefor all pairs of sites

τloc

= J(ui ) T−1

+G⎡⎣

⎤⎦

−1

J(−u j )

no correlation

Correlation between displacements

R

R+δ

Δ

Δ Pair: R+δ

Uncorrelated : 2Δ

Probably small moleculeneglecting the correlation is bad,but for solid might be less problematic

Goal is for geometrical and electronic structure fitting of Nano cluster with ligands.

=> anisotropy of potential is big, many atoms, continuum state, thermal vibration, charge fluctuation, etc.

average of thousands of MD models

Ni2+ in water – Ni Kedge

Calculations for some particular snapshots

including the second shellaverage

first shell average

first + second shells average

one shell fit

two shells fit

sizeable effects in the energy range 0 - 30 eV

P. D’Angelo et al. JACS 128 (2006)

The water solvation of I-

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

-20 0 20 40 60 80 100

Energy (eV)

L1 - L3 XAS data

as in the previous case we analyze those data by MD snapshots generate by

QM/MM and DFT methods

in collaboration with Chergui’s groupHayakawa, Benfatto, unpublished

We have used more than 1000 frames - three of them at L3 edge

Very disordered system!

The calculation includes atoms (H and O) up to 7 Å

L3 Fits

DFT

QM/MM

0

0,5

1

1,5

2

-20 0 20 40 60 80 100 120

L3-QM/MM simulation

fit-l3-mm-4Angexp-l3-mm-4Angfit-l3-mm-5.5Angexp-l3-mm-5.5Ang

energy (eV)

0

0,5

1

1,5

2

-20 0 20 40 60 80 100 120

L3-DFT simulation

fit-l3-dft-4Angexp-l3-dft-4Angfit-l3-dft-5.5Angexp-l3-dft-5.5Ang

fit-l3-dft-4Ang

energy (eV)

L1 Fits

0

0,5

1

1,5

2

-20 0 20 40 60 80 100 120

fit-l1-mm-4Angexp-l1-mm-4Angfit-l1-mm-5.5Angexp-l1-mm-5.5Ang

energy (eV)

L1-QM/MM simulation

0

0,5

1

1,5

2

-20 0 20 40 60 80 100 120

fit-l1-dft-4Angexp-l1-dft-4Angfit-l1-dft-5.5Angexp-l1-dft-5.5Ang

fit-l1-dft-4Ang

ene

L1-DFT-simulation

QM/MM

DFT

The QM/MM calculations reproduces better than DFT the experimental data for both L1 and L3 edges - Increasing the cluster size DFT becomes worse than QM/MM

It seems that DFT introduces a partial order that is not verify in the reality

Optical theorem

dk̂∫ BL 'i (k) BL '

j (k)*=−e−2k2σ 2 1

πℑ τ LL '

ij

Totally homogeneous vibration case,