Physics 250-06 “Advanced Electronic Structure”

Frozen Phonon and Linear ResponseCalcuations of Lattice Dynamics

Contents:

1. Total Energies and Forces

2. Density Functional Linear Response.

3. Applications.

Frozen Phonon Method

Given vector q generates displacements field

*iqR iqRR de d e

If a supercell can be found such that the vector q becomesreciprocal lattice vector in the new structure, one restoresperiodicity and can study lattice dynamics with the frozenphonon method

22

0 02

1[ ( ), ] [ ( ),0] | |

2DFT DFT

DFT DFT d d

E EE r d E r d d

d d

Evaluating the forces

0( ) |DFTd

EF d

d

Trying to evaluate first derivative of the energy analytically

would mean that

2

0 02

( )| |DFTd d

E F d

d d

i.e numerically more stable

Hellmann-Feynman Theorem

1[ ] [ ]

2 | ' |DFT i i eff ext xci

E f V V Er r

DFT expression for the total energy

would assume

[ ]( ) [ ]

[ ] ( )

effDFT i i exti i

i

exteff ext H xc HF

VE f VF d f

d d d d d

VV V V V F d

d d

Non-self-consistency forcewhich is difficult to evaluate!

Pulay forces due to incomplete basis sets

Change in the eigenvalues

2

2 2

2 2

| | | |

| | | |

| | | |

effii i eff i i i

i i i

i ieff i i eff

i i

eff i ieff i i i eff i

i i

eff

Vf V

d d d

V Vd d

VV V

d d d

V

d

only when 2( ) | 0eff i iV

(Pulay, 1967)

However, if2

( ) ( )

| | 0

ii

ieff i

r A r

V A

2

2

2

| | .

( )( ) | | .

( )| | .

ieff i i

ii

eff i i

ieff i i

V c cd

r AA r V c c

d d

rA V c c

d

We obtain

If the basis does not depend on atomic positions Pulayforce disappears! If it is centered on atoms

( ) ( )ikRR

R

r e r R d ( ) ( )

( )ikR ikRRR

R R

r r R de e r R d

d d

Outcome: Pulay forces need to be kept when workingwith atom centered basis sets (TB, LAPWs, LMTOs, LCAOs, etc)

They disappear for plane wave basis sets, not becausePlane waves are complete, but because they don’t dependon atomic positions (even obviously incomplete basiswith a single plane wave will not have Pulay force!)

Pulay forces are very large! If neglected, the Hellman-FeynmanForce alone can be 1-2 order of magnitude larger than the real force. On the other hand, since numericalderivatives of the total energy produce correct numericalforce, the HF+Pulay force obtained analytically is very accurate!

Linear Response Lattice Dynamics

Main advantage: no need for supercells, works for any q!

Consider external perturbation2 2 2

( ) . .| | | | | |

iqRext R

R R R

Ze Ze ZeV r d e c c

r R R r R r R

made of two traveling waves2

( )| |

( ) ( )

iqRext R

R

iqRext ext

ZeV r e

r R

V r R e V r

External perturbation causes change in the electronic density

where change in the electronic (KS) wave function is

*( ) ( ) ( ) . .kj kj kjkj

r f r r c c

' ' '' ' '' ' '

' ' | | ' | |( ) ( ) ( )kj k j k qj

k j jkj k j kj k qj

k j V kj k qj V kjr r r

One then obtains (Sham, 1969):

0

'0 * *' '

' '

( ) ( , ') ( ') '

( , ') ( ) ( ) ( ') ( ')

q

kj k qjq kj k qj k qj kj

kjj kj k qj

r r r V r dr

f fr r r r r r

One needs to assume self-consistency because

which changes

( )( ) ( ) ( )

| ' |eff ext xc

rV r V r V r

r r

'' '

' | |( ) ( ) eff

kj k qjj kj k qj

k qj V kjr r

and obtains new

0( ) ( , ') ( ') 'q effr r r V r dr In other words:

0

1

1

0 1

( ) [ ( ) ]

( )

eff ext

eff ext C xc C xc ext

eff ext

C xc

V V

V V v I I v I V

V V

I v I

Problem with standard perturbation theory

Main problem is the perturbation theory using original basis set:

'' '

' | |( ) ( )kj k qj

j kj k qj

k qj V kjr r

Consider rigid shift of the lattice: V V

' '' ''

' | |( ) ( ) ( ) ' | | ( )kj kj kj kj

j jkj kj

kj V kjr r r kj kj r

Only if basis is complete

How to repair perturbation theory?

If force is given by

( ) ( )extF r V r dr The dynamical matrix is the next variation (Sham, 1969)

2

2

( ) ( ) ( ) ( ) ( )

( ) ( , ') ( ') ' ( ) ( )

ext ext

ext q ext ext

q r V r dr r V r dr

V r r r V r drdr r V r dr

We obtained a compact linear response expression for the dynamical matrix valid for any wave vector q.

Unfortunately, it is only valid for plane wave basis sets.when using atom centered basis sets, this Hellmann-Feynmanbased formula has huge errors (~1000%)

Perturbation Theory with adjustable basis

If

'' '

' | |( ) ( )kj k qj

j kj k qj

k qj V kjr r

( ) ( )kj kkj r A r

we expect

( ) ( ) ( )kj k kj kkj r A r A r

Where the important term is given by the change in the basis!

This is absent in standard perturbation theory

(SS, PRL 1992)

We obtain

'' '

'' '

'' '

( ) ( )

( ) | |( )

' | | ( )( )

' | |( )

kj kkj

kj k qkj

k qjj kj k qj

kj kkj

k qjj kj k qj

k qjj kj k qj

r A r

A r H kjr

k qj H A rr

k qj V kjr

which goes back to standard perturbation theoryif basis is complete: second term is equal zero, while firstand third terms cancel out!

Consider acoustic sum rule

'' '

'

' '

| | | | | |( ) ( ) ( )

( )( ) [ ( )( ) ( )]

kj kj effkj kj kj

j kj kj

kjkj kj kj kj

j kj kj

kj H kj kj H kj kj V kjr r r

rr r H r dr

Equal zero automatically!

We never used completeness property to show thatacoustic sum rule is satisfied.

Expression for dynamical matrix is more complicated. BasicallyOne needs to start from the energy and gets the force which accounts for the Pulay part

Dynamical Matrix

1[ ] [ ]

2 | ' |DFT i i eff ext xci

E f V V Er r

[ ] [ ] [ | | . .]DFT ext kj kj kj kjkj

F E V f H c c Now the dynamical matrix is the second order variation

2 2

2

[ ] [ ]

[ | | . .] 2 | |

DFT ext ext

kj kj kj kj kj kj kj kj effkj kj

E V V

f H c c f H V

Dynamical matrix can be thought as a functional of first orderChange in charge density. It is stationary with respect tovariations of this quantity as directly follows from DFT! If thatis the case, any change in the dynamical matrix with Respect to external parameter can be computed

2n+1 Theorem (Gonze, et.al PRL 1992)

[ ] [ ] [ ] [ ]

[ ]

d d d

d d d

Equal to zero!

Thus, here is the 2n+1 theorem: knowledge of ground state Density allows to evaluate energy and force (n=0), Knowledge of first order change in density (n=1) allowsto evaluate dynamical matrix and third-order anharmonicityconstants (2n+1=3 here), knowledge of second-order densityvariation will allow to evaluate anharmonicity coefficents ofForth and fifth order (2n+1=5), etc.

(after SS, PRB1996)

(after SS, PRB1996)

(after SS, PRB1996)

(after SS, PRB1996)

(after SS, PRB1996)

(after SS, PRB1996)

Superconductivity in MgB2 was recently studied using density functional linear response (after Y. Kong et.al. PRB 64, 020501 (R) 2002)

Phosphorus under pressure and its implicationsfor spintronics. In collaboration with Ostanin, Trubitsin, Staunton, PRL 91, 087002 (2003)

Phonons and Linewidths in MgCNiPhonons and Linewidths in MgCNi33

<- Carbon modes

<- Mg based modes

<- Ni based modes

Harmonic ~ 0.9 from all modes except for soft mode.

Soft mode cannot be neglected.

(after Ignatov, SS, Tyson, PRB 2004)

Nearly Instable Mode at q=(0.5,0.5,0.0)Nearly Instable Mode at q=(0.5,0.5,0.0)Soft Mode view

Octahedral interstitial view Soft mode as antibreating

Antiperovskite Lattice

EXAFS in MgCNiEXAFS in MgCNi33

Recent EXAFS data (Ignatov et.al, PRB 2003) can be used:

• EXAFS does not show static distortions in MgCNiMgCNi33• EXAFS shows a double well with depth =20 K, 0 =70K, and 1

=150K

M point Frozen PhononM point Frozen Phonon

(after Ignatov, SS, Tyson, PRB 2004)

Phonons in Phonons in -Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

(after Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Phonons in doped CaCuO2Phonons in doped CaCuO2

(after SS, Andersen, PRL 1996)

(after SS, Andersen, PRL 1996)