Quantum Cluster MethodsErik Koch

German Research School for Simulation Sciences, Jülich

Übungsaufgabe

gegeben Ne Elektronen, Ni Atomkerne der Masse Mα und Kernladungszahl Zα,

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)

H = ��22m

Ne�

j=1

⇤2j �Ni�

�=1

�22M�

⇤2� �1

4⇥�0

Ne�

j=1

Ni�

�=1

Z�e2

|rj � R�|+1

4⇥�0

Ne�

j<k

e2

|rj � rk |+1

4⇥�0

Ni�

�<⇥

Z�Z⇥e2

|R� � R⇥ |

The Theory of EverythingR. B. Laughlin* and David Pines†‡§

*Department of Physics, Stanford University, Stanford, CA 94305; †Institute for Complex Adaptive Matter, University of California Office of the President,Oakland, CA 94607; ‡Science and Technology Center for Superconductivity, University of Illinois, Urbana, IL 61801; and §Los Alamos Neutron Science CenterDivision, Los Alamos National Laboratory, Los Alamos, NM 87545

Contributed by David Pines, November 18, 1999

We discuss recent developments in our understanding of matter,broadly construed, and their implications for contemporary re-search in fundamental physics.

The Theory of Everything is a term for the ultimate theory ofthe universe—a set of equations capable of describing all

phenomena that have been observed, or that will ever beobserved (1). It is the modern incarnation of the reductionistideal of the ancient Greeks, an approach to the natural world thathas been fabulously successful in bettering the lot of mankindand continues in many people’s minds to be the central paradigmof physics. A special case of this idea, and also a beautifulinstance of it, is the equation of conventional nonrelativisticquantum mechanics, which describes the everyday world ofhuman beings—air, water, rocks, fire, people, and so forth. Thedetails of this equation are less important than the fact that it canbe written down simply and is completely specified by a handfulof known quantities: the charge and mass of the electron, thecharges and masses of the atomic nuclei, and Planck’s constant.For experts we write

i!!

!t !" " # !!" " [1]

where

! # $ "j

Ne !2

2m #j2 $ "

%

Ni !2

2M%#%

2

$ "j

Ne "%

Ni Z%e2

!r!j $ R! %!& "

j$$k

Ne e2

!r!j $ r!k! & "%$$'

Nj Z%Z'e2

!R! % $ r!'!. [2]

The symbols Z% and M% are the atomic number and mass of the%th nucleus, R% is the location of this nucleus, e and m are theelectron charge and mass, rj is the location of the jth electron, and! is Planck’s constant.

Less immediate things in the universe, such as the planetJupiter, nuclear fission, the sun, or isotopic abundances ofelements in space are not described by this equation, becauseimportant elements such as gravity and nuclear interactions aremissing. But except for light, which is easily included, andpossibly gravity, these missing parts are irrelevant to people-scale phenomena. Eqs. 1 and 2 are, for all practical purposes, theTheory of Everything for our everyday world.

However, it is obvious glancing through this list that theTheory of Everything is not even remotely a theory of everything (2). We know this equation is correct because it has beensolved accurately for small numbers of particles (isolated atomsand small molecules) and found to agree in minute detail withexperiment (3–5). However, it cannot be solved accurately whenthe number of particles exceeds about 10. No computer existing,or that will ever exist, can break this barrier because it is acatastrophe of dimension. If the amount of computer memoryrequired to represent the quantum wavefunction of one particleis N then the amount required to represent the wavefunction ofk particles is Nk. It is possible to perform approximate calcula-tions for larger systems, and it is through such calculations that

we have learned why atoms have the size they do, why chemicalbonds have the length and strength they do, why solid matter hasthe elastic properties it does, why some things are transparentwhile others reflect or absorb light (6). With a little moreexperimental input for guidance it is even possible to predictatomic conformations of small molecules, simple chemical re-action rates, structural phase transitions, ferromagnetism, andsometimes even superconducting transition temperatures (7).But the schemes for approximating are not first-principlesdeductions but are rather art keyed to experiment, and thus tendto be the least reliable precisely when reliability is most needed,i.e., when experimental information is scarce, the physical be-havior has no precedent, and the key questions have not yet beenidentified. There are many notorious failures of alleged ab initiocomputation methods, including the phase diagram of liquid 3Heand the entire phenomenonology of high-temperature super-conductors (8–10). Predicting protein functionality or the be-havior of the human brain from these equations is patentlyabsurd. So the triumph of the reductionism of the Greeks is apyrrhic victory: We have succeeded in reducing all of ordinaryphysical behavior to a simple, correct Theory of Everything onlyto discover that it has revealed exactly nothing about many thingsof great importance.

In light of this fact it strikes a thinking person as odd that theparameters e, !, and m appearing in these equations may bemeasured accurately in laboratory experiments involving largenumbers of particles. The electron charge, for example, may beaccurately measured by passing current through an electrochem-ical cell, plating out metal atoms, and measuring the massdeposited, the separation of the atoms in the crystal being knownfrom x-ray diffraction (11). Simple electrical measurementsperformed on superconducting rings determine to high accuracythe quantity the quantum of magnetic f lux hc#2e (11). A versionof this phenomenon also is seen in superfluid helium, wherecoupling to electromagnetism is irrelevant (12). Four-pointconductance measurements on semiconductors in the quantumHall regime accurately determine the quantity e2#h (13). Themagnetic field generated by a superconductor that is mechani-cally rotated measures e#mc (14, 15). These things are clearlytrue, yet they cannot be deduced by direct calculation from theTheory of Everything, for exact results cannot be predicted byapproximate calculations. This point is still not understood bymany professional physicists, who find it easier to believe that adeductive link exists and has only to be discovered than to facethe truth that there is no link. But it is true nonetheless.Experiments of this kind work because there are higher orga-nizing principles in nature that make them work. The Josephsonquantum is exact because of the principle of continuous sym-metry breaking (16). The quantum Hall effect is exact becauseof localization (17). Neither of these things can be deduced frommicroscopics, and both are transcendent, in that they wouldcontinue to be true and to lead to exact results even if the Theoryof Everything were changed. Thus the existence of these effectsis profoundly important, for it shows us that for at least some

The publication costs of this article were defrayed in part by page charge payment. Thisarticle must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.§1734 solely to indicate this fact.

28–31 ! PNAS ! January 4, 2000 ! vol. 97 ! no. 1

PNAS 97, 28 (2000)

most important question

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

what atoms to pick?what materials are interesting?

simpler question

electronic Hamiltonian in Born-Oppenheimer approximation

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� |

given a material { Zα, Rα }solve

HΨ(x1, ..., xN) = EΨ(x1, ..., xN)

antisymmetrize wavefunction

antisymmetric wave-functions

(anti)symmetrization of N-body wave-function: N! operations

S± (x1, . . . , xN) :=1pN!

X

P

(±1)P �xp(1), . . . , xp(N)

�

antisymmetrization of products of single-particle states

S� '↵1(x1) · · ·'↵N (xN) =1pN!

���������

'↵1(x1) '↵2(x1) · · · '↵N (x1)'↵1(x2) '↵2(x2) · · · '↵N (x2)...

.... . .

...'↵1(xN) '↵2(xN) · · · '↵N (xN)

���������

much more efficient: scales only polynomially in N

Slater determinant: �↵1···↵N (x1, . . . , xN)

Slater determinants

simple examplesN=1:

N=2:

�↵1(x1) = '↵1(x1)

�↵1···↵N (x) =1pN!

���������

'↵1(x1) '↵2(x1) · · · '↵N (x1)'↵1(x2) '↵2(x2) · · · '↵N (x2)...

.... . .

...'↵1(xN) '↵2(xN) · · · '↵N (xN)

���������

expectation values need only one antisymmetrized wave-function:

remember: M(x1, ..., xN) symmetric in arguments

corollary: overlap of Slater determinants:Zdx1 · · · dxN �↵1···↵N (x1, . . . , xN)��1···�N (x1, . . . , xN) = det

⇣h'↵n |'�mi

⌘

Zdx (S± a(x))M(x) (S± b(x)) =

Zdx

⇣pN! a(x)

⌘M(x) (S± b(x))

�↵1↵2(x1, x2) =1p2

⇣'↵1(x1)'↵2(x2)� '↵2(x1)'↵1(x2)

⌘

basis of Slater determinantsZdx1 · · · dxN �↵1···↵N (x1, . . . , xN)��1···�N (x1, . . . , xN) = det

⇣h'↵n |'�mi

⌘

Slater determinants of ortho-normal orbitals φa(x) are normalized

a Slater determinant with two identical orbital indices vanishes (Pauli principle)

Slater determinants that only differ in the order of the orbital indices are (up to a sign) identical

define convention for ordering indices, e.g. α1 < α2 < ... < αN

given K (ortho-normal orbitals) { φa(x) | α ∈ {1, ..., K} }the K! / N! (K−N)! Slater determinants

are an (ortho-normal) basis of the N-electron Hilbert space

n

�↵1···↵N (x1, . . . , xN)�

�

�

↵1 < ↵2 < · · · < ↵N 2 {1, . . . , K}o

second quantization: motivation

keeping track of all these signs...

�↵�(x1, x2) =1p2('↵(x1)'�(x2)� '�(x1)'↵(x2))Slater determinant

|↵,�i =1p2(|↵i|�i � |�i|↵i)corresponding Dirac state

use operators |↵,�i = c†�c†↵|0i

position of operators encodes signs

product of operators changes sign when commuted: anti-commutation

c†�c†↵|0i = |↵,�i = �|�,↵i = �c†↵c

†� |0i

anti-commutator {A,B} := AB + BA

second quantization: motivation

specify N-electron states using operators

N=0: |0i (vacuum state)

normalization: h0|0i = 1

N=1: |↵i = c†↵|0i (creation operator adds one electron)

overlap:

normalization: h↵|↵i = h0|c↵c†↵|0i

h↵|�i = h0|c↵c†� |0i

adjoint of creation operator removes one electron: annihilation operator

c↵|0i = 0 and c↵c†� = ±c†�c↵ + h↵|�i

N=2: |↵,�i = c†�c†↵|0i

antisymmetry: c†↵c†� = �c

†�c†↵

second quantization: formalism

vacuum state |0⟩and

set of operators cα related to single-electron states φα(x)defined by:

c↵|0i = 0�c↵, c�

= 0 =

�c†↵, c

†�

h0|0i = 1�c↵, c

†�

= h↵|�i

see alsowww.cond-mat.de/events/correl13/manuscripts/koch.pdf

second quantization: field operators

creation/annihilation operators in real-space basis

creates electron of spin σ at position r

†(x) with x = (r,�)

then c

†↵ =

Zdx '↵(x)

†(x)

complete orthonormal set{'↵n(x)}X

j

'↵j (x)'↵j (x0) = �(x � x 0)

(x) =X

n

'↵n(x) c↵n

put electron at x with amplitude φa(x)

� (x), (x 0)

= 0 =

�

†(x), †(x 0)

� (x), †(x 0)

= �(x � x 0)

they fulfill the standard anti-commutation relations

second quantization: Slater determinants

�↵1↵2...↵N (x1, x2, . . . , xN) =1pN!

⌦0�� (x1) (x2) . . . (xN) c

†↵N . . . c

†↵2c

†↵1

�� 0↵

proof by induction

N=0: �() = h0|0i = 1⌦0�� (x1) c

†↵1

�� 0↵=

⌦0��'↵1(x1)� c†↵1 (x1)

�� 0↵= '↵1(x1)N=1:

N=2:

using { (x), c†↵} =Rdx

0'↵(x 0)

� (x), †(x 0)

= '↵(x)

⌦0�� (x1) (x2) c

†↵2c

†↵1

�� 0↵

=⌦0�� (x1)

�'↵2(x2)� c†↵2 (x2)

�c

†↵1

�� 0↵

=⌦0�� (x1)c

†↵1

�� 0↵'↵2(x2)�

⌦0�� (x1)c

†↵2 (x2)c

†↵1

�� 0↵

= '↵1(x1)'↵2(x2)� '↵2(x1)'↵1(x2)

second quantization: Slater determinants

general N: commute Ψ(xN) to the right

Laplace expansion in terms of N-1 dim determinants wrt last row of

=

���������

'↵1(x1) '↵2(x1) · · · '↵N (x1)'↵1(x2) '↵2(x2) · · · '↵N (x2)...

.... . .

...'↵1(xN) '↵2(xN) · · · '↵N (xN)

���������

D0��� (x1) . . . (xN�1) (xN) c†↵Nc

†↵N�1 . . . c

†↵1

��� 0E=

+D0��� (x1) . . . (xN�1) c†↵N�1 . . . c

†↵1

��� 0E

'↵N (xN)

�D0��� (x1) . . . (xN�1)

Qn 6=N�1 c

†↵n

��� 0E

'↵N�1(xN)

...

(�1)ND0��� (x1) . . . (xN�1) c†↵N . . . c

†↵2

��� 0E

'↵1 (xN)

second quantization: Dirac notation

product state c

†↵N · · · c

†↵2c

†↵1 |0i

corresponds to

Slater determinant �↵1↵2...↵N (x1, x2, . . . , xN)

as

Dirac state |↵icorresponds to

wave-function '↵(x)

second quantization: expectation values

Zdx1 · · · dxN ��1···�N (x1, · · · , xN)M(x1, · · · , xN)�↵1···↵N (x1, · · · , xN)

=D0��� c�1 · · · c�N M c

†↵N · · · c

†↵1

��� 0E

collecting field-operators to obtain M in second quantization:

expectation value of N-body operator wrt N-electron Slater determinants

M =1

N!

Zdx1 · · · xN †(xN) · · · †(x1)M(x1, · · · , xN) (x1) · · · (xN)

apparently dependent on number N of electrons!

Zdx1 · · · dxN

1pN!

D0��� c�1 · · · c�N

†(xN) · · · †(x1)��� 0

EM(x1, · · · , xN)

1pN!

⌦0�� (x1) · · · (xN) c†↵N · · · c

†↵1

�� 0↵

=

⌧0

���� c�1 · · · c�N1

N!

Zdx1 · · · dxN †(xN) · · · †(x1)M(x1, · · · , xN) (x1) · · · (xN) c†↵N · · · c

†↵1

���� 0�

|0ih0| = 1 on 0-electron space

result independent of N

second quantization: zero-body operator

M0 =1

N!

Zdx1dx2 · · · xN †(xN) · · · †(x2) †(x1) (x1) (x2) · · · (xN)

=1

N!

Zdx2 · · · xN †(xN) · · · †(x2) N (x2) · · · (xN)

=1

N!

Zdx2 · · · xN †(xN) · · · †(x2) 1 (x2) · · · (xN)

...

=1

N!1 · 2 · · · N = 1 using

Zdx

†(x) (x) = N

only(!) when operating on N-electron state

zero-body operator M0(x1,...xN)=1 independent of particle coordinates

second quantized form for operating on N-electron states:

second quantization: one-body operators

result independent of N

M1 =1

N!

Zdx1 · · · dxN †(xN) · · · †(x1)

X

j

M1(xj) (x1) · · · (xN)

=1

N!

X

j

Zdxj

†(xj)M1(xj) (N � 1)! (xj)

=1

N

X

j

Zdxj

†(xj)M1(xj) (xj)

=

Zdx

†(x) M1(x) (x)

expand in complete orthonormal set of orbitals

M1 =X

n,m

Zdx '↵n(x)M(x)'↵m(x) c

†↵nc↵m =

X

n,m

h↵n|M1|↵mi c†↵nc↵m

one-body operator M(x1, . . . , xN) =Pj M1(xj)

second quantization: two-body operators

two-body operator M(x1, . . . , xN) =Pi<j M2(xi , xj)

result independent of N

expand in complete orthonormal set of orbitals

M2 =1

2

X

n,n0,m,m0

Zdxdx

0'↵n0 (x

0)'↵n(x)M2(x, x0)'↵m(x)'↵m0 (x

0) c†↵n0 c†↵nc↵mc↵m0

=1

2

X

n,n0,m,m0

h↵n↵n0 |M2|↵m↵m0i c

†↵n0c

†↵nc↵mc↵m0

M2 =1

N!

Zdx1 · · · dxN †(xN) · · · †(x1)

X

i<j

M2(xi , xj) (x1) · · · (xN)

=1

N!

X

i<j

Zdxidxj

†(xj) †(xi)M2(xi , xj) (N � 2)! (xi) (xj)

=1

N(N � 1)X

i<j

Zdxidxj

†(xj) †(xi)M2(xi , xj) (xi) (xj)

=1

2

Zdx dx

0

†(x 0) †(x) M2(x, x0) (x) (x 0)

BO Hamiltonian

electronic Hamiltonian in Born-Oppenheimer approximation

solve HΨ(x1, ..., xN) = EΨ(x1, ..., xN) and antisymmetrize

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� |

solve H |Ne⟩ = E |Ne⟩, where

|Nei =Xa↵1,...,↵Ne

Y

i

c†↵i |0iH = �X

↵�

t↵� c†↵c� +

1

2

X

↵���

U↵���c†↵c

†�c�c�

t↵� =X

↵0�0

(S�1)↵↵0

Zdx '↵0(x)

✓1

2~r2 � V

ext

(~r )

◆'�0(x) (S

�1)�0�

U↵���=

X

↵0�0�0�0

S

�1↵↵0 S

�1��0

Zdx

Zdx

0'↵0(x)'�0(x 0)

1

|~r � ~r 0| '�0(x 0)'�0(x) S

�1� 0� S

�1�0�

approximation: restrict basis set

formulations in 1st and 2nd quantization equivalent,as long as basis set complete

restrict basis set to K functions φαN-electron Hilbert space restricted to

binom(K,N)-dimensional variational space

all-electron approach:increase K until convergence

pseudized approach:variational space only for interesting electrons

how?perturbation theory

renormalization

100105101010151020102510301035

102 103

dim(H)

K

N=10N=16

perturbation theory: atomic multiplets

H =X

j

✓�1

2r2j �

Z

rj

◆+

X

k<j

1

|rj � rk |

=X

j

✓�1

2r2j �

Z

rj+ Uscf(rj)

◆

| {z }=:H0

+X

j

✓X

k<j

1

|rj � rk |� Uscf(rj)

◆

| {z }=:H1

H1 should be small perturbationUscf must describe electron-electron repulsion well

accurate density gives accurate Hartree termDFT orbitals!

open shell - degenerate perturbation theory

atomic multiplets

Q. Zhang:Calculations of Atomic Multiplets across the Periodic Table MSc thesis, RWTH Aachen 2014www.cond-mat.de/sims/multiplet

Hund’s rule: d5 ground state 6S

|0, 0, 5/2, 5/2i = c†�2"c†�1"c

†0"c†1"c†2" |0i

|0, 0, 5/2, 3/2i =1p5

⇣c†2#c

†�2"c

†�1"c

†0"c†1" � c

†1#c†�2"c

†�1"c

†0"c†2" + c

†0#c†�2"c

†�1"c

†1"c†2" � c

†�1#c

†�2"c

†0"c†1"c†2" + c

†�2#c

†�1"c

†0"c†1"c†2"

⌘|0i

|0, 0, 5/2, 1/2i =1p10

⇣c†1#c

†2#c†�2"c

†�1"c

†0" � c

†0#c†2#c†�2"c

†�1"c

†1" + c

†�1#c

†2#c†�2"c

†0"c†1" � c

†�2#c

†2#c†�1"c

†0"c†1" + c

†0#c†1#c†�2"c

†�1"c

†2"

�c†�1#c†1#c†�2"c

†0"c†2" + c

†�2#c

†1#c†�1"c

†0"c†2" + c

†�1#c

†0#c†�2"c

†1"c†2" � c

†�2#c

†0#c†�1"c

†1"c†2" + c

†�2#c

†�1#c

†0"c†1"c†2"

⌘|0i

|0, 0, 5/2,�1/2i =1p10

⇣c†0#c

†1#c†2#c†�2"c

†�1" � c

†�1#c

†1#c†2#c†�2"c

†0" + c

†�2#c

†1#c†2#c†�1"c

†0" + c

†�1#c

†0#c†2#c†�2"c

†1" � c

†�2#c

†0#c†2#c†�1"c

†1"

+c†�2#c†�1#c

†2#c†0"c†1" � c

†�1#c

†0#c†1#c†�2"c

†2" + c

†�2#c

†0#c†1#c†�1"c

†2" � c

†�2#c

†�1#c

†1#c†0"c†2" + c

†�2#c

†�1#c

†0#c†1"c†2"

⌘|0i

|0, 0, 5/2,�3/2i =1p5

⇣c†�1#c

†0#c†1#c†2#c†�2" � c

†�2#c

†0#c†1#c†2#c†�1" + c

†�2#c

†�1#c

†1#c†2#c†0" � c

†�2#c

†�1#c

†0#c†2#c†1" + c

†�2#c

†�1#c

†0#c†1#c†2"

⌘|0i

|0, 0, 5/2,�5/2i = c†�2#c†�1#c

†0#c†1#c†2# |0i

Slater integrals

Spin-Orbit coupling

E6S = 10F(0) �

5

7

⇣F (2) + F (4)

⌘

HSO =1

2c2

X

i

1

ri

dV

dri`i · si

F (k)nl =

Z 1

0dr |unl(r)|2

✓1

r k+1

Z r

0dr r k |unl(r)|2 + r k

Z 1

rdr

1

r k+1|unl(r)|2

◆

LS coupling: Co 4+

LS coupling: Co 4+

6S character

non-LS coupling: Ir 4+

non-LS coupling: Ir 4+

6S character

non-LS coupling: Ir atom

modeling correlated electrons: renormalization

H = �1

2

Ne�

j=1

⇤2j �Ne�

j=1

Ni�

�=1

Z�|rj � R�|

+Ne�

j<k

1

|rj � rk |

complete orbital basis φn,i

distinguish two types of electrons/orbitals: correlated and uncorrelated

assume no hybridization between them: product ansatz

H = ��

n,i ;m,j ;�

tn,i ;m,j a†m,j� an,i� +

�

i

Vn,i�;m,j��nn,i�nm,j��

⇥({nn,i⇥}) =�

n

�

µ

�i ,µ⇥n({nc,i⇥}) �µ({nc,i⇥}; {na,j⇥�}) � ⇥0�({nc,i⇥})

instant-screening approx.: slow correlated, fast other electronsH|⇥⇤ = E|⇥⇤ � ⇥�|H|�⇤|�⇤ = E|�⇤

simple example: 3-band Hubbard cluster

U11 = 1.5U22 = 0.6U33 = .2

U12 = 0.7U23 = 0.05

t22 = 0.1t13 = 0.5

Δ = 0.5

spectral function: projected vs. 1-band

0

2

4

6

8

10

12

14

0.4 0.6 0.8 1 1.2 1.4

Spectr

al F

unction

!

G221-band Ubare, tbare1-band Ueff, tbare1-band Ueff, tinst

screening U

instantaneous screening approximation

spectral function: projected vs. 1-band

0

2

4

6

8

10

12

14

0.4 0.6 0.8 1 1.2 1.4

Spectr

al F

unction

!

G221-band Ubare, tbare1-band Ueff, tbare1-band Ueff, tinst

hopping reduction

overlap between different screening states reduces hopping: teff

spectral function: projected vs. 1-band

0

2

4

6

8

10

12

14

0.4 0.6 0.8 1 1.2 1.4

Spectr

al F

unction

!

G221-band Ubare, tbare1-band Ueff, tbare1-band Ueff, tinst

screening reduces U and t

could justify that constrained calculations tend to give “too small U”

t/U not as sensitive to screening as one might think

C. Adolphs:Renormalization of the Coulomb Interaction in the Hubbard Model

Diploma Thesis, RWTH Aachen 2010

surface effects

10×10×10 cluster 83=29=512 atoms inside

almost 50% of atomson surface...

how to simulate bulk?

periodic boundary conditions

finite size scaling

-1.28

-1.275

-1.27

-1.265

-1.26

0 0.01 0.02 0.03 0.04 0.05

E tot

/L

1/L

half-filled one-dimensional L-site chainEpbc

= �2tX

�

�bN�/2c+N�X

n=�bN�/2c

cos(2⇡n/L)

Eobc

= �2tX

�

N�X

n=1

cos(2⇡n/(L+ 1))

periodic boundary conditions

additional interaction with periodic images

Hpbc

= �1

2

NX

i=1

~r2i +X

~n2Z3

X

i

V Cext

(~ri�~R~n) +1

2

X

~n2Z3

X

i ,j

0 1

|~ri � ~rj�~R~n|

finite-size correction " =�Ecorr

(L)� EMF

(L)�/L+ "

MF

Hartree term

exchange-correlation hole

electron density: Γ(x; x) = n(x)

conditional electron density: 2Γ(x, x’; x, x’) = n(x, x’) electron density at x’ given that an electron is at x

Coulomb repulsion hUi =Zdx dx

0 �(2)(x, x 0; x, x 0)

|r � r 0| =1

2

Zdx dx

0 n(x, x0)

|r � r 0|

rewrite in terms of Hartree energy(how ⟨U⟩ differs from mean-field)

n(x, x 0) = n(x)n(x 0) g(x, x 0) = n(x)n(x 0) + n(x)n(x 0) (g(x, x 0)� 1)

pair correlation function exchange-correlation hole

sum ruleRdx

0n(x 0)

�g(x, x 0)� 1

�= �1

Rdx

0n(x, x 0) = n(x) (N � 1)

exchange-correlation holes from QMC

0

0.4

0.8

1.2 r = 0.8s

↑↓

↑↑

r = 1s

↑↓

↑↑

0

0.4

0.8

1.2 r = 2s

↑↓

↑↑

r = 3s

↑↓

↑↑

0

0.4

0.8

1.2 r = 4s

↑↓

↑↑

r = 5s

↑↓

↑↑

0

0.4

0.8

1.2

0 0.5 1 1.5 2 2.5

r = 8s

↑↓

↑↑

0 0.5 1 1.5 2 2.5

r = 10s

↑↓

↑↑

** *

**

**

0.95 0.85 0.75 0.65 0.55

0.5

0.75

1

(b)

** *

**

**

-0.025 -0.05

-0.075 -0.1

-0.1

0

(a)

homogeneous electron gas g��0

xc

(r/rs

)

G. Ortiz, M. Harris, P. Ballone, Phys. Rev. Lett. 82, 5317 (1999)P. Gori-Giorgi, F. Sacchetti, G.B. Bachelet, Phys. Rev. B 61, 7353 (2000)

R.Q. Hood, M.Y. Chou, A.J. Williamson, G. Rajagopal, R.J. Needs, W.M.C. Foulkes, Phys. Rev. B 57, 8972 (1998)

(110) plane of Si, electron at bond center

gVMCc (~r )gVMCx

(~r )

lattices

A = (a1, a2, . . . , ad)LA =

n

X

i

ni ai

�

�

�

ni 2 Zo

primitive lattice vectors

not unique

Vc =��det(A)

��

canonical choice:vectors to nearest neighbors

(LLL algorithm)

allowed transformations of A:•exchange vectors•change sign of vector•add int multiple of other vector

LA =�An

�� n 2 Zd

d → ∞ applications to cryptography

SVPfind shortest vector

in lattice

lattice periodic functions

Fourier transform

V (r) =

Zdk V (k) e ik·r

lattice periodicity

only modes that contribute

V (r + An) =

Zdk V (k) e ik·r e ik·An| {z }

=1

= V (r)

k 2�Gm

�� m 2 Zd = RL with G = (2⇡A�1)T

Gm · An = mT GTA|{z }2⇡

n = 2⇡ ⇥ integer

Bloch theorem

Hsingle =X

k

k2

2c†k,� ck,� +

X

k

X

m2ZdVGm c

†k+Gm,� ck,�

lattice-periodic potential

eigenstates: Bloch waves

'n,k(r) =X

m2Zdcn,m e

i(k+Gm)·r 'n,k(r + An) = eik·An 'n,k(r)

reduce eigenvalue problem to single unit cell with pbc✓1

2

�� irr + k

�2+ V (r)

◆un,k(r) = "n,k un,k(r)

'n,k(r) = eik·r un,k(r)vector potential

many-electron Bloch theorem

H =X

k,�

0

@k2

2c†k,� ck,� +

X

m

VGm c†k+Gm,�ck,� +

1

2

X

k0,�0;q

c†k+q,�c†k0�q,�0

1

|q |2 ck0,�0ck,�

1

A

couples all states with given total crystal momentum(invariance under translation of all electrons by lattice vector)

to move all electrons into a simulation cell C, need to postulateBloch-like theorem

Cn,k(r1, r2, . . .) = e

i k·Pi ri UC

n,k(r1, r2, . . .)

k enters eigenvalue equation for UC as a vector potential

susceptibility distinguish metals from (Mott) insulators

Kohn, Phys. Rev. 133, A171 (1964)

d2E(k)

dk2

����k=0

supercells

C = AL V =��det(C)

��

Hermite normal form

⇤ =

0

BBB@

�11 0 0 · · ·�21 �22 0�31 �32 �33...

. . .

1

CCCA

gcd(a, b) =

8<

:

|a| if b = 0 (change sign of column)

gcd(b, a) if |a| < |b| (exchange columns)gcd(a � ba/bcb, b) otherwise (add integer multiple of col)

0 �i j < �i i

Euclidean algorithm

Hermite normal form

L :

✓2 04 8

◆

HNF

✓4 20 4

◆ ✓0 4�4 �2

◆ ✓4 8�2 0

◆ ✓4 02 4

◆

HNF

supercell: k-point sampling

KS = (2⇡C�1)T = G(L�1)T

supercell Brillouin zone

supercell: k-point sampling

KS = (2⇡C�1)T = G(L�1)T

supercell Brillouin zone

Monkhorst-Pack grid

particularly suited for Brillouin-zone integrals

L =

0

@n1 0 00 n2 00 0 n3

1

A k =X

i

(ni � 1)ki2ni

PHYSICAL REVIE%' B VOLUME 13, NUMBER 12 15 JUNE 1976

Special points for Brillonin-zone integrations*

Hendrik J. Monkhorst and James D. PackDepartment of Physics, University of Utah, Salt Lake City, Utah 84112

(Received 21 January 1976)A method is given for generating sets of special points in the Brillouin zone which provides an efficient meansof integrating periodic functions of the wave vector. The integration can be over the entire Brillouin zone orover specified portions thereof. This method also has applications in spectral and density-of-state calculations.The relationships to the Chadi-Cohen and Gilat-Raubenheimer methods are indicated.

I. INTRODUCTION

Many calculations in crystals involve integratingperiodic functions of a Bloch wave vector overeither the entire Brillouin zone (BZ) or overspecified portions. The latter case arises, forexample, in averages over states within the Fermisurface, and the calculation of the dielectric con-stant and generalized susceptibilities. To optimizethe calculations it is helpful only to compute thesefunctions at a carefully selected set of points inthe BZ. This becomes more urgent in sophisti-cated calculations where the computational effortfor each BZ point is substantial.Methods for finding such sets of "special" points

have been discussed by Chadi and Cohen' (here-after referred to as CC). This paper presents analternate approach which yields sets of pointsidentical to those given by CC and additional setswith the same properties. It will be shown thatthe use of such points simply generates an expan-sion of the periodic function in reciprocal-spacefunctions with the proper symmetries. This sug-gests an obvious, and rather accurate, interpola-tion of the function between the special pointswhich is intrinsically more satisfactory than linearor quadratic methods. The relation to Gilat-Raubenheimer methods will be pointed out.

k~„, =u~b, +u„5, +u, b, . (4)

This gives q' distinct points uniformly spaced inthe BZ. Let A (k} be given by

A (k) Iq-)h Q e(k R

I Rl = c(5)

where the sum is taken over all R vectors relatedby the operations of the lattice point group. Thisset of vectors is usually called a star. The Care in ascending order, starting with C, =0. Nis the number of members in the mth star of R[or the number of terms in the sum in Eq. (5}].Note that A (k) is totally symmetric under allpoint-group operations.Let us now consider the quantity S „(q) given by

v is the unit cell volume, and 5,-5, span the re-ciprocal lattice with BZ volume 8v'/v.Let us define the sequence of numbers

u, =(2r —q —1)/2q ( r1, 2, 3, . . . , q),where q is an integer that determines the numberof special points in the set. With the above u„'swe now define

II. DERIVATION S „(q) =—Q A (k „,)A„(k „,).p.r.s=i (6)

In this section we will prove the existence of aset of periodic functions which are orthonormalon a uniformly spaced set of special points inthe BZ.Consider a lattice defined by the primitive trans-

lation vectors ti, t„ts. A general lattice point isgiven by

R =R,t, +R272+R3t, ,where R,-R, are integers.The associated primitive reciprocal-lattice

vectors are given by

Substituting Eq. (5}for A „and A„we can reduceS „(q) to

(7)

where

1~+('(q) — ~ e(&(/c)(2r-a-()( - s))s)q ~ (8}

In Eq. (7) the a and b sums are over the membersin the stars m and n, respectively. It can noweasily be seen that the 8'&bean assume the follow-

13 5188

top cited Phys. Rev. papers

S. Redner: Citation Statistics from 110 years of Physical ReviewPhysics Today June 2005, p. 49

Table 1. Physical Review Articles with more than 1000 Citations Through June 2003Publication # cites Av. age Title Author(s)PR 140, A1133 (1965) 3227 26.7 Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn, L. J. Sham

PR 136, B864 (1964) 2460 28.7 Inhomogeneous Electron Gas P. Hohenberg, W. Kohn

PRB 23, 5048 (1981) 2079 14.4 Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems J. P. Perdew, A. Zunger

PRL 45, 566 (1980) 1781 15.4 Ground State of the Electron Gas by a Stochastic Method D. M. Ceperley, B. J. Alder

PR 108, 1175 (1957) 1364 20.2 Theory of Superconductivity J. Bardeen, L. N. Cooper, J. R. Schrieffer

PRL 19, 1264 (1967) 1306 15.5 A Model of Leptons S. Weinberg

PRB 12, 3060 (1975) 1259 18.4 Linear Methods in Band Theory O. K. Anderson

PR 124, 1866 (1961) 1178 28.0 Effects of Configuration Interaction of Intensities and Phase Shifts U. Fano

RMP 57, 287 (1985) 1055 9.2 Disordered Electronic Systems P. A. Lee, T. V. Ramakrishnan

RMP 54, 437 (1982) 1045 10.8 Electronic Properties of Two-Dimensional Systems T. Ando, A. B. Fowler, F. SternPRB 13, 5188 (1976) 1023 20.8 Special Points for Brillouin-Zone Integrations H. J. Monkhorst, J. D. Pack

PR, Physical Review; PRB, Physical Review B; PRL, Physical Review Letters; RMP, Reviews of Modern Physics.

top cited Phys. Rev. papers

S. Redner: Citation Statistics from 110 years of Physical ReviewPhysics Today June 2005, p. 49

Table 1. Physical Review Articles with more than 1000 Citations Through June 2003Publication # cites Av. age Title Author(s)PR 140, A1133 (1965) 3227 26.7 Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn, L. J. Sham

PR 136, B864 (1964) 2460 28.7 Inhomogeneous Electron Gas P. Hohenberg, W. Kohn

PRB 23, 5048 (1981) 2079 14.4 Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems J. P. Perdew, A. Zunger

PRL 45, 566 (1980) 1781 15.4 Ground State of the Electron Gas by a Stochastic Method D. M. Ceperley, B. J. Alder

PR 108, 1175 (1957) 1364 20.2 Theory of Superconductivity J. Bardeen, L. N. Cooper, J. R. Schrieffer

PRL 19, 1264 (1967) 1306 15.5 A Model of Leptons S. Weinberg

PRB 12, 3060 (1975) 1259 18.4 Linear Methods in Band Theory O. K. Anderson

PR 124, 1866 (1961) 1178 28.0 Effects of Configuration Interaction of Intensities and Phase Shifts U. Fano

RMP 57, 287 (1985) 1055 9.2 Disordered Electronic Systems P. A. Lee, T. V. Ramakrishnan

RMP 54, 437 (1982) 1045 10.8 Electronic Properties of Two-Dimensional Systems T. Ando, A. B. Fowler, F. SternPRB 13, 5188 (1976) 1023 20.8 Special Points for Brillouin-Zone Integrations H. J. Monkhorst, J. D. Pack

PR, Physical Review; PRB, Physical Review B; PRL, Physical Review Letters; RMP, Reviews of Modern Physics.

top 10 cited Phys Rev papers (July 2007)

10 PR 108 1175 (1957) 1650 Theory of Superconductivity Bardeen, Cooper, Schrieffer

cites !age" impactarticle title author(s)

3 PRB 23 5048 (1981) 3007 Self-Interaction Correction to.. J. P. Perdew & A. Zunger4 PRL 45 566 (1980) 2514 Ground State of the Electron.. D. M. Ceperley & B. J. Alder

1 PR 140 A1133 (1965) 4930 Self Consistent Equations.. W. Kohn & L. J. Sham 2 PR 136 B864 (1965) 3564 Inhomogeneous Electron Gas.. P. Hohenberg & W. Kohn

6 PRB 13 5188 (1976) 2277 Special Points for Brillouin.. H. J. Monkhorst & J. D. Pack5 PRL 77 3865 (1996) 2478 Generalized Gradient Approx... Perdew, Burke, Ernzerhof

7 PRB 54 11169 (1996) 1933 Efficient Iterative Schemes.... G. Kresse & J. Furthmuller8 PRB 43 1993 (1991) 1776 Efficient Pseudopotentials for .... N. Troullier & J.L. Martins9 PRB 41 7892 (1990) 1749 Soft Self-Consistent Pseudopotentials... D. Vanderbilt

S. Redner: A Physicist’s Perspective on Citation AnalysisTrends in Research Measurement Metrics, Washington DC, Oct 2010

http://physics.bu.edu/~redner/talks/citations-10.pdf

Conclusions

c↵|0i = 0�c↵, c�

= 0 =

�c†↵, c

†�

h0|0i = 1�c↵, c

†�

= h↵|�i

second quantization

�↵1↵2...↵N (x1, x2, . . . , xN)

c†↵N · · · c†↵2c

†↵1 |0i

degenerate perturbation theory

renormalized modelsmany-body Bloch supercell &