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Correlated Electrons
Erik Koch
Computational Materials Science German Research School for Simulation Sciences
outline
many-body physics for atoms
• self-consistent field methods• density matrices• exchange-correlation hole• density-functional theory• exercises: DFT code for atoms
• multiplets in open shells• second quantization• Hamiltonian matrix elements• angular momenta• diagonalizing the interaction• Hund’s rules
www.cond-mat.de/teaching/correl
exercises
Correlated Electrons SS 2020, E. Koch
Exercise Sheet 1 due 16 April
prepare your solutions so you are ready to present them in classif you cannot solve a problem, explain where you got stuck.
1. atomic units
Express the speed of light and the Bohr magneton in atomic units.What unit of temperature do we have to choose to also make the numerical valueof the Boltzmann constant to equal 1?
2. magnetic moment
From classical magnetostatics we know that the magnetic moment due to anelectrical current density ~je is given by
~m =1
2
Z~r ⇥~je d
3r .
i. Given the quantum-mechanical probability current density
~j =h
2ime
⇣ (~r ) ~r (~r )� (~r ) ~r (~r )
⌘,
calculate the corresponding magnetic moment. Compare to the expectationvalue of the angular momentum operator ~L.
ii. What is the z-component of the magnetic moment for the following or-bitals of the hydrogen atom |n, l ,mi: |1, 0, 0i, |2, 0, 0i, |2, 1,�1i, |2, 1, 0i,|2, 1, 1i, and |5, 3, 2i. Express your results using the Bohr-magneton
µB =eh
2me.
3. charge states
What formal charge do you expect for the transition metal in KCrF3? Which forthe manganese ions in SrMnO3?
4. atomic radii
Read the article W.L. Bragg, Phil. Mag. 40, 169 (1920) and try to understandhow atomic radii are derived from crystal structure data.
Historical notes: First crystal structures were determined 1912 by Max von Laue (Nobel
Prize in Physics 1914 for his discovery of the di↵raction of X-rays by crystals) and by
the Braggs (father and son, Nobel Prize 1915 for their services in the analysis of crystal
structure by means of X-rays). The Schrodinger equation was published 1926 (Nobel Prize
1933). For a readable account of the discoveries, you can browse the Nobel lectures at
http://www.nobelprize.org/nobel˙prizes/physics/laureates/1914 etc..
Lecture Notes
E. Pavarini, E. Koch, F. Anders, M. JarrellCorrelated Electrons: From Models to Materials
• Exchange Mechanisms• Multiplets in Transition Metal Ions• Estimates of Model Parameter• Crystal Field Theory, Tight-Binding
Autumn School on Correlated Electronswww.cond-mat.de/events/correl.html
Übungsaufgabe
Gegeben: Atome der Ordnungszahl Zα an den Positionen Rα.
Lösen Sie
The underlying laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that exact applications of these laws lead to equations which are too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. P.M.A. Dirac, Proceedings of the Royal Society A123, 714 (1929)
Theory of (almost) Everything
H = �~22m
NeX
j=1
⇥2j �1
4⇥�0
NeX
j=1
NiX
�=1
Z�e2
|rj � R�|+1
4⇥�0
NeX
j<k
e2
|rj � rk |+1
4⇥�0
NiX
�<⇥
Z�Z⇥e2
|R� � R⇥ |
typical units
http://physics.nist.gov/cuu/Constants/index.html h = 6.626068 10-34 Js mel = 9.109382 10-31 kg e = 1.602176 10-19 C E =
~2 k22mel
why use Å and eV?
1 Å = 10-10 m 1 eV = 1.602176 10-19 J
E [in J] = 6.10 10-39 (k [in m-1])2
E [in eV] = 3.81 (k [in Å-1])2
atomic units
http://physics.nist.gov/cuu
solve
~ = 1 a20me/t0me = 1mee = 1 e4�⇥0 = 1 t20e
2/a30me
to obtain
1 a.u. length = a0 =4�⇥0~2mee2
� 5.2918·10�11 m1a.u. mass = me = � 9.1095·10�31 kg1 a.u. time = t0 =
(4�⇥0)2~3mee4
� 2.4189·10�17 s1 a.u. charge = e = � 1.6022·10�19 C
~ = 1.0546·10�34 Js [ML2T�1]me = 9.1094·10�31 kg [M]e = 1.6022·10�19 C [Q]4�⇥0 = 1.1127·10�10 F/m [M�1L�3T 2Q2]
H. Shull and G.G. Hall, Nature 184, 1559 (1959)
Übungsaufgabe
Gegeben: Atome der Ordnungszahl Zα an den Positionen Rα.
Lösen Sie
The underlying laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that exact applications of these laws lead to equations which are too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. P.M.A. Dirac, Proceedings of the Royal Society A123, 714 (1929)
Theory of (almost) Everything
H = �~22m
NeX
j=1
⇥2j �1
4⇥�0
NeX
j=1
NiX
�=1
Z�e2
|rj � R�|+1
4⇥�0
NeX
j<k
e2
|rj � rk |+1
4⇥�0
NiX
�<⇥
Z�Z⇥e2
|R� � R⇥ |H = �
1
2
NeX
j=1
r2j �NeX
j=1
NiX
↵=1
Z↵
|rj � R↵|+NeX
j<k
1
|rj � rk |+NiX
↵<�
Z↵Z�
|R↵ � R� |
More is Different
… the reductionist hypothesis does not by any means imply a ``constructionist'' one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. Sometimes, as in the case of superconductivity, the new symmetry — now called broken symmetry because the original symmetry is no longer evident — may be of an entirely unexpected kind and extremely difficult to visualize. In the case of superconductivity, 30 years elapsed between the time when physicists were in possession of every fundamental law necessary for explaining it and the time when it was actually done. Thus with increasing complication at each stage, we go up the hierarchy of the sciences. We expect to encounter fascinating and, I believe, very fundamental questions at each stage in fitting together less complicated pieces into the more complicated system and understanding the the basically new types of behavior which can result. P.W. Anderson: More is Different, Science 177, 393 (1972)
periodic table
H
Li
Na
K
Rb
Cs
Fr
Be
Mg
Ca
Sr
Ba
Ra
Sc
Y
Lu
Lr
Ti
Zr
Hf
Rf
V
Nb
Ta
Db
Cr
Mo
W
Sg
Mn
Tc
Re
Bh
Fe
Ru
Os
Hs
Co
Rh
Ir
Mt
Ni
Pd
Pt
Cu
Ag
Au
B
Al
Ga
In
Tl
C
Si
Ge
Sn
Pb
N
P
As
Sb
Bi
O
S
Se
Te
Po
F
At
Ne
Xe
Rn
Ce
Th
Pr
Pa
Nd
U
Pm
Np
Sm
Pu
Eu
Am
Gd
Cm
Tb
Bk
Dy
Cf
Ho
Es
Er
Fm
Tm
Md
Yb
No
La
Ac
He
Zn
Cd
Hg
I
Kr
ArCl
Br
atomic radii
E. Clementi, D.L.Raimondi, and W.P. Reinhardt J. Chem. Phys. 47, 1300 (1967)
periodic table
1s 1s
2s
3s
4s
5s
6s
7s
2p
3p
4p
5p
6p
4d
5d
3d
6d
4f
5f
real harmonics
s Y0,0 =q14�
pz Y1,0 =q34� cos �
px
q12 (Y1,�1 � Y1,1) =
q34� sin � cos⇥
py
q12 i (Y1,�1 + Y1,1) =
q34� sin � sin⇥
d3z2�1 Y2,0 =q
516�
�3 cos2 � � 1
�
dzx
q12 (Y2,�1 � Y2,1) =
q1516� sin 2� cos⇥
dyz
q12 i (Y2,�1 + Y2,1) =
q1516� sin 2� sin⇥
dx2�y2q12 (Y2,�2 + Y2,2) =
q1516� sin
2 � cos 2⇥
dxy
q12 i (Y2,�2 � Y2,2) =
q1516� sin
2 � sin 2⇥
real harmonics
fz(5z2�3) Y3,0 =q
716�
�5 cos2 � � 3
�cos �
fx(5z2�1)
q12 (Y3,�1 � Y3,1) =
q2132�
�5 cos2 � � 1
�sin � cos⇥
fy(5z2�1)
q12 i (Y3,�1 + Y3,1) =
q2132�
�5 cos2 � � 1
�sin � sin⇥
fz(x2�y2)
q12 (Y3,�2 + Y3,2) =
q10516� cos � sin
2 � cos 2⇥
fxyz
q12 i (Y3,�2 � Y3,2) =
q10516� cos � sin
2 � sin 2⇥
fx(x2�3y2)
q12 (Y3,�3 � Y3,3) =
q3532� sin
3 � cos 3⇥
fy(3x2�y2)
q12 i (Y3,�3 + Y3,3) =
q3532� sin
3 � sin 3⇥
atom in spherical mean-field approximation
-100
-80
-60
-40
-20
0He Ne Ar Kr Xe Rn
� in
eV
atom in spherical mean-field approximation
1s2
−8000
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
0 0.2 0.4 0.6 0.8 1
� n,l i
n eV
r in Å
Fe : [Ar] 3d6 4s2 4p0
atom in spherical mean-field approximation
−1000
−800
−600
−400
−200
0
0 0.5 1 1.5 2
� n,l i
n eV
r in Å
Fe : [Ar] 3d6 4s2 4p0
2s2
2p6
why no 2s-2p degeneracy?
atom in spherical mean-field approximation
0.1
1
10
100
0 1 2 3 4 5 6 7 8
Z eff(
r)
r in Å
Fe : [Ar] 3d6 4s2 4p0
atom in spherical mean-field approximation
−100
−80
−60
−40
−20
0
0 0.5 1 1.5 2 2.5 3 3.5 4
� n,l i
n eV
r in Å
Fe : [Ar] 3d6 4s2 4p0
3s2
3p6
atom in spherical mean-field approximation
−10
−8
−6
−4
−2
0
2
0 1 2 3 4 5 6 7 8
� n,l i
n eV
r in Å
Fe : [Ar] 3d6 4s2 4p0
3d6
4s2
why not 3d8 4s0 ??
atom in spherical mean-field approximation
3d7
4s1
−10
−8
−6
−4
−2
0
2
0 1 2 3 4 5 6 7 8
� n,l i
n eV
r in Å
Fe : [Ar] 3d7 4s1 4p0
total energy
Fe
Fe+
Fe2+
Fe3+
tools: pse#!/bin/bashif [ "$1" == "" ]thencat << EOF1 2 3a La/Ac 4a5a6a7a8a8a8a1a2a3 4 5 6 7 8H HeLiBe B C N O F NeNaMg AlSiP S ClArK CaSc TiV CrMnFeCoNiCuZnGaGeAsSeBrKrRbSrY ZrNbMoTcRuThPdAgCdInSnSbTeI XeCsBaLaCePrNdPmSmEuGdTbDyHoErTmYbLuHfTaW ReOsIrPtAuHgTlPbBiPoAtRnFrRaAcThPaU NpPuAmCmBkCfEsFmMdNoLrusage: pse [nuclear charge | element name | element symbol ]EOFexitfi
for pattern in $@dogrep -iw “ $pattern “ << EOF 1 Hydrogen Wasserstoff H 1s1 2 Helium He 1s2 3 Lithium Li [He] 2s1 4 Beryllium Be [He] 2s2 5 Bor B [He] 2s2 2p1 6 Carbon Kohlenstoff C [He] 2s2 2p2 7 Nitrogen Stickstoff N [He] 2s2 2p3
crystals
KCuF3: orbital-ordering
d-bands; eg orbitals
-5
0
5
10
15
NPX!Z
"k in
eV
orbital ordering
eg
s2-z2
3l2-1
Cu d9
k-space vs real space
H = �~22m
NeX
j=1
⇥2j �1
4⇥�0
NeX
j=1
NiX
�=1
Z�e2
|rj � R�|+1
4⇥�0
NeX
j<k
e2
|rj � rk |+1
4⇥�0
NiX
�<⇥
Z�Z⇥e2
|R� � R⇥ |
single-electron terms diagonal in k-space (band structure)
interaction terms diagonal in real (configuration) space
intinerant vs. localized
magnetite Fe3O4
spinel structure AB2O4: O on fcc sites Fe in tetrahedral (1) and octahedral (2) sites
8 f.u. per unit cell space group Fd-3m
MS Senn et al. Nature 481, 173 (2012) doi:10.1038/nature10704
Charge, orbital and trimeron order in the low-temperature magnetite structure.
oxydation states
−10
−8
−6
−4
−2
0
2
0 1 2 3 4 5 6 7 8
� n,l i
n eV
r in Å
O : [He] 2s2 2p4
−10
−8
−6
−4
−2
0
2
0 1 2 3 4 5 6 7 8
� n,l i
n eV
r in Å
Fe : [Ar] 3d6 4s2 4p0
O2- : [He] 2s2 2p6 = [Ar] Fe2+ : [Ar] 3d6
Fe3+ : [Ar] 3d5
typical charge states
H
Li
Na
K
Rb
Cs
Fr
Be
Mg
Ca
Sr
Ba
Ra
Sc
Y
Lu
Lr
Ti
Zr
Hf
Rf
V
Nb
Ta
Db
Cr
Mo
W
Sg
Mn
Tc
Re
Bh
Fe
Ru
Os
Hs
Co
Rh
Ir
Mt
Ni
Pd
Pt
Cu
Ag
Au
B
Al
Ga
In
Tl
C
Si
Ge
Sn
Pb
N
P
As
Sb
Bi
O
S
Se
Te
Po
F
At
Ne
Xe
Rn
Ce
Th
Pr
Pa
Nd
U
Pm
Np
Sm
Pu
Eu
Am
Gd
Cm
Tb
Bk
Dy
Cf
Ho
Es
Er
Fm
Tm
Md
Yb
No
La
Ac
He
Zn
Cd
Hg
I
Kr
ArCl
Br
+1 +2 -1-2al
kali
met
als
alka
line-
earth
s
pnic
toge
ns
chal
coge
ns
halo
gens
nobl
e ga
sses
transition metals
ionic radii
