Electronic Structure Electrons in Momentum Space

7/29/2014
FKF
FKF
Electron Density Matrices
Density matrices Γ, an alternative to the wavefunction Ψ,
for the description of a quantum system
Electronic Structure
The N-particle density matrix
Electrons in Momentum Space
can be reduced to the 2-particle-
Ulrich Wedig
or to the 1-particle-level
Max Planck Institute for Solid State Research - Stuttgart
Dept. Quantum Materials / Takagi
*
→ 1-particle properties
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ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
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FKF
FKF
Electron Density Matrices
Electron Density in Position Space
Electron position density ρ(r) and form factor F(χ)
Interrelationships between various representations
diagonal of the
1-particle density matrix
←
←
momentum
space
→
position
space
momentum
space
position
space
→
X-ray diffraction
FD 6D Fourier-Dirac transformation
FD 3D Fourier-Dirac transformation
P Projection into a subspace
S Selection of a subspace
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ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
Wolf Weyrich
Lecture Notes in Chem. 67
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FKF
Electron Density in Momentum Space
Electron momentum density ϖ(r) and reciprocal form factor B(s)
Compton scattering
momentum
space
X-ray diffraction
diagonal of the
1-particle density matrix
position
space
projections parallel
to the diagonal r
into the subspace {s}
orthogonal to r
FD 3D Fourier-Dirac transformation
P Projection into a subspace
S Selection of a subspace
ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
Electron Density in Momentum Space
Electron momentum density ϖ(r) and reciprocal form factor B(s)
Compton scattering
momentum
space
X-ray diffraction
X-ray diffraction and Compton scattering
are complementary methods
to get information about the
1-particle density matrix
diagonal of the
1-particle density matrix
position
space
projections parallel
to the diagonal r
into the subspace {s}
orthogonal to r
FD 3D Fourier-Dirac transformation
P Projection into a subspace
S Selection of a subspace
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Compton Scattering - Principles
k1
Compton Profiles
Projection of
the electron’s initial momentum
onto the scattering vector
inelastic scattering of photons (X-ray, γ)
p1
q
k = k2 – k 1
p2 – p1
k2
p2
momentum transfer from photon to electron
(energy and momentum conservation)
Probing several directions
yields information on the
k1 wave vector of photon before collision
k2 wave vector of photon after collision
k = k2 – k1 scattering vector
p1 electron momentum before collision
p2 electron momentum after collision
q
projection of p onto k
electrons at rest → Compton line
Electron Momentum Density
02
 1  h
mec
 1  cos  
moving electrons → Compton profile
(Doppler broadening)
The formula is valid within the impulse approximation
- Energy transfer is much larger than the binding energy of the electron
- Corrections for multiple scattering processes
P. Eisenberger, P. M. Platzman, Phys. Rev. A 2, 415 (1970)
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Compton Profiles
Projection of
the electron’s initial momentum
onto the scattering vector
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Computed Compton Profiles
Directional Compton Profiles can be computed in two ways:
1) 2D-integration of the Electron Momentum Density π(p)
Probing several directions
yields information on the
2) Via the reciprocal form factor B(r)
Electron Momentum Density
In the AO basis set
The formula is valid within the impulse approximation → synchrotron-, γ-radiation
Beamline ID15B at ESRF, Grenoble
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ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
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FKF
Computed Compton Profiles
Directional Compton Profiles can be computed in two ways:
FKF
Computed Compton Profiles
Directional Compton Profiles can be computed in two ways:
BRG, interfaced to CRYSTAL98:
A. Saenz, T. Asthalter, W. Weyrich, Int. J. Quant. Chem. 65, 213 (1997)
1) 2D-integration of the Electron Momentum Density π(p)
1) 2D-integration of the Electron Momentum Density π(p)
Since CRYSTAL09: Keyword BIDIERD
A. Erba, C. Pisani, S. Casassa. L. Maschio, M. Schütz, D. Usvyat, Phys. Rev. B 81, 165108 (2010)
(use of corrected density matrices (MP2))
2) Via the reciprocal form factor B(r)
CRYSTAL: Keyword PROF
2) Via the reciprocal form factor B(r)
In the AO basis set
- analytical integration
valence contribution only for molecules and non conducting polymers
In the AO basis set
- numerical integration
dense k-mesh required
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Computed Compton Profiles
Hartree-Fock vs. Correlated wavefunction vs. Kohn-Sham
Hartree-Fock:
FKF
Computed Compton Profiles
Hartree-Fock vs. Correlated wavefunction vs. Kohn-Sham
Hartree-Fock:
Electron correlation is missing.
→ 1-particle density matrix is idempotent.
eigenvalues (occupation numbers) of the spin-orbitals either 1 or 0
Correlated wavefunction:
Electron correlation is missing.
→ 1-particle density matrix is idempotent.
eigenvalues (occupation numbers) of the spin-orbitals either 1 or 0
Correlated wavefunction:
→ 1-particle density matrix is non-idempotent.
eigenvectors (natural orbitals) and non-integer eigenvalues 0 < n < 1
→ 1-particle density matrix is non-idempotent.
eigenvectors (natural orbitals) and non-integer eigenvalues 0 < n < 1
DFT – electron position density minimizing the energy for a given functional
Kohn-Sham: Total density is the sum of orbital densities (noninteracting electrons).
The Kohn-Sham approach is incorrect for momentum related properties
“correlated density” from “uncorrelated density matrix”
DFT – electron position density minimizing the energy for a given functional
Kohn-Sham: Total density is the sum of orbital densities (noninteracting electrons).
→ 1-particle density matrix constructed from KS-orbitals is idempotent.
→ 1-particle density matrix constructed from KS-orbitals is idempotent.
Error in the kinetic energy term, compensated in the exchange-correlation functional Exc
Corrections: L. Lam, P. M. Platzman, Phys. Rev. B 9, 5122 (1974)
Error in the kinetic energy term, compensated in the exchange-correlation functional Exc
Corrections: L. Lam, P. M. Platzman, Phys. Rev. B 9, 5122 (1974)
Kohn-Sham:
Kohn-Sham:
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Theory vs. Experiment
We have to consider:
Theory vs. Experiment
We have to consider:
- Spectrometer specific corrections
- Spectrometer specific corrections
- Corrections for multiple scattering
expt. data
- Corrections for multiple scattering
expt. data
- Deviations from the impulse approximation (photon energy)
- Deviations from the impulse approximation (photon energy)
- Limited resolution
- Limited resolution
if differences of directional CP (anisotropies in the EMD)
Most effects are less relevant
are examined.
theor. data
- 0K vs. finite temperature → statistically averaged density matrix
(Alessandro Erba)
ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
theor. data
- 0K vs. finite temperature → statistically averaged density matrix
(Alessandro Erba)
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An Example – CP of Zn and Mg
FKF
An Example – CP of Zn and Mg
FKF
Results from a project within the DFG priority program 1178:
Results from a project within the DFG priority program 1178:
Deviations from Ideal Structures in Metallic Elements and Simple Intermetallics:
Combined Experimental and Theoretical Studies of Electron Distributions
Deviations from Ideal Structures in Metallic Elements and Simple Intermetallics:
Combined Experimental and Theoretical Studies of Electron Distributions
J. Nuss, H. Nuss, D. Fischer,
H. Schlenz, K. Friese, W. Morgenroth,
Ch. Busch, Ch. Hauf, W. Scherer
J. Nuss, B. Boldrini,
Th. Buslaps
M. Jansen, U. Wedig
Stuttgart
Electron density
Momentum space
W. Weyrich
Konstanz
Electron density
Position space
Experiments
and
Theory
Momentum space
Position space
Experiments
and
Theory
A. Kirfel
B. Paulus
N. Gaston, D. Andrae, E. Voloshina
P. Sony, K. Rosciszewski
Berlin
U. Wedig, H. Nuss, J. Nuss, M. Jansen, D. Andrae, B. Paulus, A. Kirfel, W. Weyrich, Z. Anorg. Allg. Chem, 639, 2036 (2013)
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22nd
Juli 2014 - Ulrich Wedig
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7/29/2014
An Example – CP of Zn and Mg
FKF
FKF
Anisotropies in the directional CP in the range of 0.1
Hexagonal close packed elements – unusual structures
1.9
Computed CP of Zn and Mg
Cd
Zn
1.8
+ 9.3 %
c/a
─ BIDIERD: TOLINTEG 42 10 10 10 15 vs. 16 24 16 24 32
─ BRG:
TOLINTEG 42 10 10 10 15 vs. 24 32 24 32 48
─ BIDIERD vs. BRG: TOLINTEG 42 10 10 10 15
1.7
Crucial is the first parameter (ITOL1) in keyword TOLINTEG
Mg
overlap threshold for coulomb integrals
1.6
but also
governs the truncation of the lattice sums with the construction
of the density matrix
1.5
10
20
30
40
volume /Å3
50
60
70
anisotropic in-plane and out-of-plane bonding
→
tiny effects in the electron density
Truncation errors may have the same magnitude
as the physical effects
Thanks to Alessandro Erba for valuable hints
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ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
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Zn 7 x 3 x 0.7 mm
(Z3M [001])
FKF
Directional CP of Zn and Mg
Anisotropy of the Momentum Density
Mg 7 x 3 x 1 mm
(M1M [423])
Samples:
Single crystals, one for each direction
sheets perpendicular to the scattering vector
[423] direction
1 a.u. = 1 DuMond = ħ / Bohr
Dispersion-compensating scanning spectrometer
Beamline ID15B, ESRF, Grenoble
Directions considered:
• [100] ‘intraplanar bond’
• [423] ‘interplanar bond’
• [001] ‘nonbonding direction’ as reference
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Juli 2014 - Ulrich Wedig
Experimental resolution: Zinc 0,07 – 0,09 a.u.; Magnesium 0.09 – 0.11 a.u.
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Anisotropy of the Momentum Density
Comparing the ‘bonding directions’ [100] – [423] (B3PW)
Anisotropy of the Momentum Density
Comparing the ‘bonding directions’ [100] – [423] (B3PW)
[423]
[100]
Zinc in momentum space:
4s2-valence shell: Reduction of the anisotropy (change of the Fermi surface)
3d10-shell:
Enhancement of the anisotropy (dynamical correlation)
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Anisotropy of the Momentum Density
FKF
The Method of Increments (MOI) for Metals
Comparing the ‘bonding directions’ [100] – [423] (B3PW)
The method of increments for metals
Many-body expansion of the
correlation energy per unit cell
with
E  E HF  E corr
E corr    i    ij 
i
i j
 ij   ij   i   j 
 
i  j k
ijk
...
etc.
the indices i,j,k denote groups of orthogonal localized orbitals (e.g. atoms)
in finite embedded clusters (i: unit cell; j,k: whole system)
a proper embedding scheme allows for:
Does the dynamical behaviour of the
electrons govern the unusual structure
of zinc in position space?
-improved localizability
-prevention of surface charging
-a disappearing band gap
Correlation treatment: CCSD(T)
Zinc in momentum space:
4s2-valence shell: Reduction of the anisotropy (change of the Fermi surface)
3d10-shell:
Enhancement of the anisotropy (dynamical correlation)
ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
H. Stoll, Phys. Rev. B 46 (1992) 6700
B. Paulus, Phys. Rep. 428 (2006) 1
E. Voloshina, B. Paulus, SPR Chemical Modelling: Application and Theory 6, M (2009)
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Zinc, Cadmium – Cohesive Energies
FKF
Zn
Zinc – Energy Landscape beyond DFT
FKF
Cd
expt.
-1.35
-1.16
MOI s2d10-correlation
-1.35
-1.19
MOI s2-only-correlation
-0.90
-0.67
DFT (LDA)
-1.65
-1.46
DFT (PBE)
-0.97
-0.69
DFT (B3PW)
-0.82
-0.59
DFT (B3LYP)
-0.28
-0.17
Correlation energy computed with the Method of Increments
Phys. Rev. Lett. 100, 226404 (2008)
4s23d10-correlation
only
4s2-correlation
The bonding in zinc is exclusively due to
dynamical correlation of the electron
3-body terms lead to different bonding interactions
within und between the hexagonal planes.
The closed 3d10-shell contributes significantly to
the bonding.
The cohesive energy is obtained quantitatively by
the Method of Increments.
(MOI: -1.35 eV; expt.: -1.35 eV)
MOI: Method of Increments, values given for the expt-like minimum
Phys. Rev. B 75, 205123 (2007)
Phys. Rev. Lett. 100, 226404 (2008)
Phys. Chem. Chem. Phys. 12, 681 (2010)
ISAMS2014 Regensburg - 22nd Juli 2014 - Ulrich Wedig
high pressure data:
K. Takemura Phys. Rev. B56 (1997) 5170
A modification of zinc with an ideal c/a ratio may
exist.
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