Relativistic QC: BDF

Wenjian Liu(Sept. 11, Beijing)

Outline

Introduction to RQC

Salient features of BDF

http://www.chem.pku.edu.cn/itcc

Why do we need RQC ?

Fundamental theories

1905 : STR Einstein : “E = mc2”

1926 : QM Schrödinger

1928 : RQM Dirac

1949 : QED Tomonaga, Schwinger &

Feynman

Special Theory of Relativity

The laws of physics take the same form in all inertial frames (Galileo)

In any inertial frame, the speed of light is the same whether the light is emitted by a body at rest or by a body in uniform motion (Einstein)

Michelson-Morley(1887)

c=137.0359895 a.u. or 8 13 10 m s

Time-Space

Galilean transformation (time, 3D)

Lorentzian transformation( FitzGerald 1889, Larmor 1898, Lorentz 1899 )

,( , ),ict x y z

' ' '

'

, ,x x vt y y z z

t t

2

2

' ' '

'

2

2 2

,1

1

,x vt

x y y z

vc

vz

t

c

c

t

v

x

Rules of velocity addition

Galilean

Lorenzian

" "u v u v

2

" "1

u vu v

uvc

" "u c c

" "c c c

Consequences of Relativity

Time delayed

Length contracted

Mass increased

'

2

21

dtd

vc

t

'2

21

cL L

v

0

2

21c

MM

v

Equation of motion

Mass-Energy

Correspondence principle Klein-Gordon:

Dirac:

2E mc2 2 4 2 2

0E m c c p

( , )E i p it

22 2 2 2 2 4

02( , ) ( ) ( , )r t c m c r t

t

( 0, 0)s m

20( , ) ( ) ( , )i r t i c m c

tr t

Dirac Matrices

0

0

1 0

0 1z

0 1

1 0x

0

0y

i

i

0

0

I

I

2 2[ , ] 2 ,[ , ] 0, 1i j ij i i

v c [ , ] ( )

[ , ] , ] 02 2

[

D

D D

l H i c p

j H l sH

Dirac spinors (nodeless !)

L

L L

S S

S

Spectrum of Dirac operator

Electronlike continuum solutions

Positronlike continuum solutions

Electronlike bound solutions

Dirac sea

1933 positron

QED

Relativistic Hamiltonian

One-electron:Two-electron:

Coulomb: spin-same-orbitGaunt : spin-other-orbit, orbit-orbit, spin-spinRetardation: spin-independent

Dh

2

12

1(1,2) (1,2)

| |g dc

r

1 2

12| |Gd r

1 12 2 12

312

( )(1

2

)

2 | |B Gd dr r

r

Fundamental problems

BO approximation (preferred reference)T=0kH is not fully Lorenzian invariantExplicit correlation ? (Kutzelnigg’s conjec

ture)

STR (locality) and QM (nonlocality) not fully compatible

Relativistic H: Dirac Equation

22

L L

S S

V c p

c p V mc

22

1

2

L S L

S L L

mc V

V c p

cp p

mc

:L L

S ScDPT

2 2

1 0

2 0

V p

p m c cV

Kinetic balance

Expansion

Lévy-Leblond ( ) (spin not relativistic)

ZORA ( )

(0) (0),

(0)1,

0 0 0 02

0 0 0

0 0

1( )

22

V pVp

m

0 00

0 0

1 0

2 0 0

V p

p m

0 0 0 0

00 0

| |

|

V

2

2

2

1 0

2 0 0

( )2

ZORA ZORA

ZORAZORA ZORA

ZORA ZORAZORA

V p

p m Vc

cp p V

c V

DPT

Why 4c is preferred ?

Complicated operators in transformed H

ZORA: (DFT, gauge dependent)

2

22

mcH p p V

m Vc

22 2 2 4

2, ,2

iii ii i i

ii

cpE mcE p c m c A p

E mcE

1 1 1 1( , ) ( )i j i i j j i j i j i j

i i j i j i j

V i j A A p p p p p p p p A Ar r r r

(basis of T, assuming p2-dependent terms diagonal)

DKH :

Why 4c is preferred ?

Computational effort: 2c only slightly cheaper than 4c Analytic energy gradient: 4c: straightforward (stationarity condition) 1c,2c: difficult (DKH) Properties: picture change ! 4c: one term (magnetic) 1c,2c: four terms Core states: 2c not applicable

(LL|SS), (SS|SS)

Relativistic Effects

Dirac (1929): ”…of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions’’

Direct

Indirect

Penetration of electron to the core

1 0.58sv z c 6 0.02eff

s eff

Zv c

n Au:

Relativistic effects

Alkali metals

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

Relativistic effects

Group 13

0.0

0.1

0.2

0.3

0.4

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

relativistic

Relativistic effects

Group 12

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

relativistic

Relativistic effects

Alkali metals

1

2

3

4

5

6

7

8

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

Relativistic effects

Group 13

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 50 100 150Nuclear charge

nonrelativisticrelativisticrelativistic

Relativistic effects

Group 12

0.8

1.0

1.2

1.4

1.6

1.8

2.0

20 70 120Nuclear Charge

nonrelativisticrelativisticrelativistic

Relativistic effects on

Electronic and molecular structuresReaction energies and mechanismsProperties: NMR, (hyper-)polarizability,

electric field gradient,

hyperfine interaction

Cf. P. Pyykkö, Chem. Rev. 88, 563 (1988)

W. Kutzelnigg, Chem. Phys. 225, 203 (1997)

What is BDF ?

ADFAmsterdamE. J. Baerends1973-NR, BP, ZORA

BDFBeijingW. Liu1993-NR, DPT, ZORA, 4c

Methodologies in BDF (JTCC 2, 257, 2003)

Benchmarking

basis

min

DCB

HF FCI

BDF

NR

Comment on

“Four-component relativistic density functional calculations of heavy diatomic molecules” [Fricke et al. ,

J. Chem. Phys. 112, 3499 (2000)]

Wenjian Liu and Christoph van Wüllen,

J. Chem. Phys. 113, 2506 (2000)

Features of BDF

NR, DPT,1c-, 2c-, and 4c-DFTNumerical spinors+STF Full symmetry (Double&Single)Multipolar decomposition of densityDirect evaluation of De (GTS) Analytic energy gradient Open-shell (NCOL, COL, KU) A large shop for (meta-)GGA

Future plans

New functional for f- and d-elementsNR/R hybrid H for large systems Excited states: RDFT, GWMechanism of “relativistic reactions”Relativistic ab initio Langevin MD for gro

und and excited statesRelativistic band energyMagnetic and electric properties

Crying for …

BDF teamworkNSFC: a possible special grant ? Industry: money and problems ?

Our goal is, in the foreseeable future, to kick out some overseas products !

BDF Best DF

Acknowledgments

Prof. Lemin Li, Dr. Fan Wang

BDF-group Drs. Bo Song, Wei Li, Wenli Zou

Mrs. Yong Zhang, Daoling Peng,

Yunlong Xiao, Jun Gao

Alternative expansion

Exact relationship

Wang-Li expansion (TCA 108, 53 (2002))

2

1

2S L

c Vc p

0 02 2 2 1

0

1 1 ( )

2 2 (2 )

k

kk

V

c V c V cV V

10 0 0 0, {1 exp[ ( )]}A A A A

N AA

V V V V r r

00

( )A A AN A

rV V erf

r

SEAX

Starting from total Energy

Making stationary

10[ ] [ ; ]Exact L SEAX LkE E V

SEAXE

20

2 20

2

2 2

2

0

2

0

( )[ ]

(2 )2

[ ](2 )

1

Li

Li i

c V Vp p

c V

cp p V

c

c

V

cp p

V

SEAX down to ZORA and IORA

ZORA(SLF) ( ) (Wang and Li, Acta Chim Sinica 12, 1499 (2002) )

IORA ( ; no stationarity condition)

2 20(2 ) 0c V

2

20

[ ]2

L Li i i

cp p V

c V

0V V

2 2

2 2 2[ ] [1 ]

2 (2 )L Li i i

c cp p V p p

c cV V

Properties of SEAX

Variationally stableStationarity condition4c method ! Property evaluated like 4c-method !More accurate but more expensive than

ZORAComputationally cheaper than IORAMuch less gauge dependent

0202

s L Lc pX

c V

Relativistic SDFT (KU, COL, NONCOL)

Wenjian Liu(REHE2003, July 27-30, Berlin)

Contents

Sketch of RDFT “Spin-density” Selected examples Conclusions

Sketch of RDFT

3[ ]A ext gs

eE j AF j d r E

c

0 0( ) | ( )extA r j r

,[ [] [ ]]FF Jj

No-virtual-pairElectrostatic limit DFT

QED CDFT

Dirac-Kohn-Sham

[ ] [ ] [ ] [ , [ ]] [ , [ ]]s ext H xc NNE T E E J E J V

[ ] [ ] [ ] [ [ ]]s ext H NNNRxcEE T E E V

3( , , , , )NRc cx x dE r

Dirac-Coulomb-Breit

Dirac-Coulomb

“spin-density”

?

“Spin-density”

Gordan decomposition of

( )J r

1 1( 2 )

2 2J S ML

[ , [ ]] [ , ]xc xcE J E M

†1

2 k kk

S

2M S LocalNonlocal

“Spin-density” (noncollinear)

†( ) ( )k kk

M r tr

* *( )k kk

k k

1( )

2s 2 2| | ( ) 4 | |s M

3[ , ] ( , , , , )NCOLxc xcE s d r

Local spin quantization

“Spin-density” (collinear)

( ) (0,0, )zM r M Global spin quantization

†k z k

k

s

3[ , ] ( , , , , )COLxc xcE s d r

“Moment-density” (KU)

{ , } { , }k k ku kd { , }k k

† ,k kk

,u d

u d M u d

3( , , , , )KUxc xc u d uu dd udE d r

Disjoint !

No ( )M r

Comparison of KU, COL, NCOL

2. Fixed quantization axis: COL&KU Local quantization axis: NCOL

| | 0 0

5. COL&NCOL break symmetry [ , ] 0Dh 6. KU treats orbital and spin magnetisms equally

3. NCOL invariant w.r.t. spin and spacial rotations

7. CPU: KU<COL<<NCOL

1. All converge to the same NRL & closed-shells

4. COL&NCOL: cannot be fully polarized if 0small

XC potential

[ , ] [ , ]NCOL xc xcxc

E s E s MV

s s

[ , ] [ , ]COL xc xc z zxc

E s E s MV

s s

2 2N N

[ , ] [ , ]KU xc xcxc

E s E sV

s

2( )N N

KU:COL:NCOL=1:2:4

x,y,z

Examples

2 2 21(2 ,2 , )

3 x z y z yz xM p p p p p p p 2 2 21

( )3 x y zp p p

1( )

3z

x y

pr

p ip

11/ 2,1/ 2 :np

COL: 2 2 21( )

3z z x ys M p p p 3 1

3sd r

NCOL: (KU)

2 2 2x y zs M M M 3 1sd d

Examples

1 13/ 2,1/ 2 3/ 2,3/ 2 :np np 1

21

6

z

x y

p

p ip

2

1

02

x yp ip

2 2 21( 2 , 2 )

32,x z y z x yz pp pp p p pM

2 2 22

( )3 x y zp p p

COL: 3 3 4

3zsd r M d r

NCOL: 3 3| | 1.5097sd r M d r

KU:3 3 2sd r d r Fully polarized !

Numerical Comparison of NCOL, COL and KU

ΔE(eV) Ns

KU COL NCOL COL NCOL

Bi6p13/21/2 0.17 0.12 0.18 0.57 1.10

Bi6p13/23/2 0.18 0.31 0.31 1.98 2.02

Bi2p13/21/2 9.89 5.70 9.53 0.34 0.98

(ZORA) 10.15 6.19 10.15 0.34 1.00

Bi2p13/23/2 10.12 10.14 10.14 1.01 1.01

(ZORA) 10.39 10.55 10.55 1.02 1.02

2 1 1 21/ 2 3/ 2 1/ 2 3/ 2(76%6 6 19%6 6 )p p p p

Summary for the whole p-block(J. Chin. Chem. Soc. (Taiwan))

KU NCOL KU NCOL

Sn 0.78 0.85 2.00 1.86Sb 1.39 1.48 3.00 2.89Bi 2.08 2.21 1.00 2.02Te 1.15 1.03 2.00 1.87Po 2.19 2.01 2.00 1.71

E sN

First IP (eV): BPSCC

KU NCOL Expt.

Bi 6.84 6.96 7.29

Se 9.69 9.60 9.75

Te 9.12 8.79 9.01

Po 8.34 8.12 8.42

Spectroscopic constants for Pt2

Ns (e) Re(Å) De (eV)

KU 0 2.334 3.26 231

NCOL 0 2.334 3.33 231

KU 1 2.344 3.28 223

NCOL 1 2.352 3.44 222

KU 2 2.373 3.36 221

NCOL 2 2.370 3.42 222

Expt. 0 ? 2.333 3.15 222

1( )e cm

Conclusions

KU is better than COLKU is very close to NCOL if SOC is either very small or very large;

otherwise: p-orbitals ca. 0.15 eV d-orbitals ca. 0.05 eV

KU is much cheaper than NCOLKU greatly facilitates RTDDFTNCOL may meet severe convergence diffic

ulty for complex open-shells

Acknowledgments

Dr. Fan Wang

Prof. Lemin Li

NSFC for RMB