l. bu činský - unice.frcassam/workshop11/presentations/presentati… · relativistic electron and...

Relativistic electron and spin densities

L. Bučinský

Faculty of Chemical Technology and Food Processing, Institute of Physical Chemistry and Chemical Physics, Radlinského 9, SK-812 37, Bratislava,

Slovak University of Technology, Slovakia

[email protected]

(What is Old in Relativistic Quantum Chemistry)

Outline of the talk

• Dirac Hamiltonian

• Diagonalization of the Dirac Hamiltonian– DKH2 and IOTC approaches– Picture Change Error

• Electron densities– Rn atom (electron and orbital densities)– Compound containing copper and ruthenium

• Spin density (unpaired electron density)– 1-component: Cu, Ag, Au and Cu2+

– 2-component: Ar/Cl, HCl·, C6H5O·, Os complex• Kramers pairs and spin contamination

Schrödinger equation

ˆ ˆ ˆH T V= +H EΨ = Ψ

2ˆˆ2

pT

m= p i j k

i x y z

∂ ∂ ∂= + + ∂ ∂ ∂

ℏ

2 2 4 2 2ˆ ˆT m c c p= +

Theory of relativity:

Classical or Newtonian physics:

2 4 2 2ˆ ˆ ????T m c c p= ± + =

Relativistic quantum chemistry

ˆ ˆˆ ˆ ˆ.( ) .( . )x y zT c mc p p p c mc= + + + = +x y zβ α α α β α p

Dirac Hamiltonian:

; ; ;

0 1 0 1 0 1 0; ; ;

1 0 0 0 1 0 1

i

i

= = = =

− = = = = −

yx z 2x y z

yx z 2

x y z 2

0 σ0 σ 0 σ 1 0α α α β

σ 0σ 0 σ 0 0 -1

σ σ σ 1

2 2 2 4 2 2ˆ ˆ ˆ ˆ ˆ ˆ.( ) .( . )x y zT c mc p p p c mc p m c c p= + + + = + = +2x y zβ α α α β α

2i j j i ijδ+ =α α α α 0i i+ =α β βα 2 2 1i = =α β

Free particle Dirac equation:

1 2 3 4; ; ;+ ∗ ∗ ∗ ∗ Ψ = Ψ Ψ Ψ Ψ

( ) PEpcmcE ±=+±= 222

ˆ.( . ) ( ) ( )c mc E+ Ψ = Ψβ α p r r1

2

3

4

L

L

S

S

α

β

α

β

ΨΨ ΨΨ Ψ = = Ψ Ψ ΨΨ

ˆˆ.( . ) ( ) ( )c mc V E + + Ψ = Ψ β α p r r

Dirac equation of hydrogen like atom

( , , , )zn j jκΨ = Ψ

Dirac Coulomb Hamiltonian

DHF DHF DHFF c Sc ε=

2 2 12

1 1 1

ˆ ˆ. ( )n n n

i iA iA iji A i j

H c mc mc V r g= = =

= + − + +

∑ ∑ ∑∑iα p β 1

2

0

1

4A

iAij

Z eV

rπε= −

2

0

1

4ijij

eg

rπε=

Many electron Hamiltonian:

Dirac package Jensen-Saue-Visscher triangleBDF W. Liu & co.Respect Malkin clan & M. Kaupp Utd.

Coulomb (NR) potential:

SCF:

1 1

1

2

n n n

D

j j k

E j H j jk g jk jk g kj= =

= + − ∑ ∑∑ENERGY:

DHF equations (rel. SCF)

( )L iL i Li

j c g rα α=∑

β

α

β

α

S

S

L

L

S

L

j

j

j

j

j

jj ==

•The kinetic balance conditions:

( ) ( )( )2 211 2 ll r l rx e lx x ex

α αα +− − −∂ = −∂

1ˆ.

2S Lpmc

Ψ = Ψσ

1-component basis sets: lDHF DHF DHFF c Sc ε=

p s 2 dlx

α∂ = −∂

Hg atom ( ) LL S L L S S

S

+ + + + + ΨΨ Ψ = Ψ Ψ = Ψ Ψ + Ψ Ψ Ψ

Bučinský, S. Biskupič, M. Ilčin, V. Lukeš, V. Laurinc J. Comp. Chem. 30 (2009) 65-74.(using the Dirac04 package, see http://dirac.chem.sdu.dk)

A bit of (quasi)relativistic theory

Dirac Hamiltonian Dirac equation

−= 22ˆ.

ˆ.ˆmcVpc

pcVH D

σ

σ

=

− S

L

S

L

mcVpc

pcV

ψψ

εψψ

22ˆ.

ˆ.

σ

σ

“… the first chicken did indeed come from a chicken egg, even though that egg didn't come from chickens.“ David Papineau at King's College London.http://www.guardian.co.uk/science/2006/may/26/uknews

h+

h-

A bit of (quasi)relativistic theory

Dirac Hamiltonian Dirac equation

−= 22ˆ.

ˆ.ˆmcVpc

pcVH D

σ

σ

=

− S

L

S

L

mcVpc

pcV

ψψ

εψψ

22ˆ.

ˆ.

σ

σ

The unitary transformation (block diagonalization) of the Dirac Hamiltonian:

0ˆ ˆ0

decoupledD

hH UH U

h++

−

= =

The picture change:

ˆ ˆ ˆ ˆ decoupledD D DH U UH U U U UH U U H+ + +Ψ Ψ = Ψ Ψ = Ψ Ψ = Ψ Ψɶ ɶ

i.e.

0U + Ψ

Ψ = Ψ =

ɶɶ

W. Liu, Mol. Phys. 108 (2010) 1679-1706

1-electron potential (V)2-ele potential: J. Seino and M. Hada, Chem. Phys. Lett. 461 (2008) 327

BLOCK DIAGONALIZATION

3 2 1 0U U U U U=…

, 2 221 0 0 1

2 , 2

DKHDKHD

DKH

E OH U U H U U

O E++ +

−

= =

Douglas Kroll Hess

2, 2 0 1 2

DKHDKHH E E E E+= = + +

U0 free particle Foldy-Wouthuysen transformationU1 first order Douglas-Kroll transformation 0

ˆ1 .

ˆ. 1p

PU A

P

=

−

σ

σ

…+++= 212

111 1 WWU

B. A. Hess, Phys. Rev. A32 (1985) 756.A. Wolf, M. Reiher, B. A. Hess, J. Chem. Phys. 117 (2002) 9215.

10 0 0 1 1DH U H U E E O+= = + +

Infinite Order Two Component

1 0 0 1 1 1 1

0ˆ0IOTC D

hH U U H U U U H U

h++ + +

−

= = =

ΩΩΩ−Ω

=

ΩΩΩΩ

=−++

−+++

−++

−−+

R

R

R

RU1

++− −= RR ( ) 2/1

1−

++++ +=Ω RR

1 1 1

(1,1) (1,2)ˆ(2,1) (2,2)

IOTC IOTCIOTC

IOTC IOTC

H HH U H U

H H+

= =

++++++++++

+++++ =ΩΩ+ΩΩ+ΩΩ+ΩΩ= hRHRRHHRHH IOTC 22122111)1,1(

0)1,2( 22122111 =ΩΩ+ΩΩ−ΩΩ+ΩΩ−= ++−+++−+−++− RHRHRHHRH IOTC

M. Barysz, A. J. Sadlej, J. Chem. Phys. 116 (2002) 2696.M. Barysz, A. J. Sadlej, J. Mol. Struct. (THEOCHEM) 573 (2001) 181.

None iterative way: Iliaš, Saue, J. Chem. Phys. 126 (2007) 064102.Kedziera, Barysz, Chem. Phys. Lett. 446 (2007) 176.W. Liu, Mol. Phys. 108 (2010) 1679-1706.

Implementation of DKH/IOTC

1. Primitive basis set space φ, 1=YSY 2. T→YTY→(YU)+T(YU) where (YU)+T(YU) = p2/2 3. i = φ YU 4. (YU)+V(YU), (YU)+pVp(YU) 5. Evaluation of momentum space prefactors 6. Get the R matrix in the IOTC approach 7. Build the 1 electron Fock matrix Fi 8. F1φ=(YSU).Fi.(YSU)+ 9. F1e=C+. F1φ.C

How to treat momentum space?

Hess and coworkers:B. A. Hess, Phys. Rev. A32 (1985) 756-763. R. J. Buenker, P. Chandra, B. A. Hess, Chem. Phys. 84 (1984) 1.

1R p Y−= σ.p

Picture Change Error

ˆ ˆ ˆ ˆX X U UXU U U UXU U UXU+ + + += Ψ Ψ = Ψ Ψ = Ψ Ψ = Ψ Ψɶ ɶ

ˆ ˆ ˆ ˆDecoupledD D Dh U Uh U U U Uh U U h+ + +Ψ Ψ = Ψ Ψ = Ψ Ψ = Ψ Ψɶ ɶ

UΨ = Ψɶ

0+ Ψ

Ψ =

ɶɶ

Rigorous:

PCE contaminated: ˆ( , )X n NR X= Ψ Ψɶ ɶ

( ) ( )0 0

ˆ( , ) ... ...n nm m

LLX n m U U XU U+ +

+ + = Ψ Ψ

ɶ ɶ

A. Wolf, M. Reiher, J. Chem. Phys. 124 (2006) 064102.A. Wolf, M. Reiher, J. Chem. Phys. 124 (2006) 064103.

X(H,PCE corr)

• PCE– picture change of the wavefunction

– neglected transformation of 2-electron terms

– 1-component vs. 2-component picture

UΨ = Ψɶ

J. Seino and M. Hada, Chem. Phys. Lett. 461 (2008) 327

spin-orbit effects

R. Mastalerz, R. Lindh, M. Reiher, Chem. Phys. Lett. 465 (2008) 157.R. Mastalerz, P. O. Widmark, B. O. Roos, R. Lindh, M. Reiher, J. Chem. Phys. 133 (2010) 144111.S. Knecht, S. Fux, R. van Meer, L. Visscher, M. Reiher, T. Saue, Theor. Chem. Acc. 129 (2011) 631.

• Requirement:– Primitive integral classes of X, pXp

• Considering only even operatorsˆ 0ˆ

ˆ0

E

E

XX

X

=

Implementation of PCE correction

1. Primitive basis set space φ, 1=YSY 2. T→XTX→(YU)+T(YU) where (YU)+T(YU) = p2/2 3. i = φYU 4. (YU)+X(YU), (YU)+pXp(YU)

DKH: + (YU)+V(YU), (YU)+pVp(YU) 5. prefactors 6. IOTC: R matrix → Xi

DKH: Xi 7. X1φ=(YSU).Xi.(YSU)+ 8. X1e=C+. X1φ.C 9. evaluate the expectation value X

1R p Y−= σ.p

1 ,( ) ( )eP Xµν µνµν

ρ =∑r r

PCE correction in electron density

PCE in electron density ( )1

ˆ ˆ ( )en

ii

X ρ δ=

→ = −∑r r r :

NR level: ˆ( ) ( ) ( ) ( )P g gµν µ νµν

ρ ρ ∗= Ψ Ψ = Ψ Ψ =∑r r r r

PCE corr: 1 ,ˆ( ) ( ) ( )eU U P Xµν µνµν

ρ ρ += Ψ Ψ =∑r r rɶ ɶ

( ) ( )a bg gr r → ( ) ( )E

+=X YU ρ r YU

( ). ( )a bg g∇ ∇r r → ( ) ( )E

+=pX p YU pρ r pYU

ˆ ˆ ˆ ˆX X U UXU U U UXU U UXU+ + + += Ψ Ψ = Ψ Ψ = Ψ Ψ = Ψ Ψɶ ɶ

LL

SS

Results:Put the mask down, Mr. Worf!

In memoriam of Stoka!

Remaining program:•Radon atom

Radial distributions of electron densityRadial distributions of s-, p-orbital density

•[CuL2] complex Electron (1D / 2D)

•[Ru Cl3 NO (Hind)2] complex2D: REL. effects

PCE SO coupling

•Contact spin densities Cu, Ag, Au•Kramers pairs (2-componentish)

•UHF vs. GCHF•2-component spin densities

(H2O+, phenyl radical, Os-cmpx)

IOTC/DKH2 + PCE

Rn atom

Figure of radial distributions of electron density of radon atom (UTZ+10s) in the vicinity

of the nucleus at different levels of theory

Point charge nucleus model Gaussian charge nucleus model

X(2,1) = U1U0δU0+U1

+ = XE,0 + XE,1 (missing XE,2 from U2)

Bučinský, Biskupič, Jayatilaka, Theor Chem Acc 129 (2011) 181–197.


X(2,0)=XE,0X(2,1)=XE,0 + XE,1X(2,2)=XE,0 + XE,1 + XE,2

X(2,0)=U0δU0+

X(2,1)=U1U0δU0+U1

+

X(2,2)=U2U1U0δU0+U1

+U2+

Radial distribution of s, p orbital density of Rn atom

d

d5/2

d3/2

p

p3/2

p1/2

αα αβ

βα ββ

cLα

cLβ

RHF 1-component; scalar; spinfree

UHF+SO 2-component = GCHFJayatilaka, J. Chem. Phys. 108 (1998) 7587

h+

2s orbital densities of Rn (UTZ+10s)

point charge Gaussian charge distribution nodal behavior

2p1/2 orbitals 2p1/2 orbitals 2p orbitals


The Bis[bis(methoxycarbimido)aminato]copper(II) complex [CuL2]

(a) Relativistic effects in electron density (b) Relativistic effects in spin density

Hudák et al. (2010). Acta Cryst. A 66, 78–92

UHFDoubletPoint charge nucleus1-componentish

Electron density along the Cu-N bond of [CuL2]

Bučinský L., Biskupič S., Jayatilaka D. (2011).Chem. Phys., 10.1016/j.chemphys.2011.04.026

2D difference electron densities at the Cu atom of [CuL2]

[ size 1 x 1 bohr2, 41x 41 grid points, cutoff: -0.05 (blue) – 0.05 (red) ] relativistic effects relativistic effects + PCE PCE

Radial distributions of spin density Cu atom Cu2+ atom


Spin Density

[Ar]3d104s1[Ar]3d9

Spin density along the Cu-N bond of [CuL2]


2D difference spin densities at the Cu atom of [CuL2] [ size 1 x 1 bohr2, 41x 41 grid points, cutoff: -0.005 (blue) – 0.005 (red) ] relativistic effects relativistic effects + PCE PCE

V. B. Arion, G. E. Büchel, R. Ponec, D. Jayatilaka, P. Rapta., M. Breza, L. Bučinský, M. Fronc, J. Kožíšek, S. Biskupič, M. Gall, in preparation.

Electron density at Ru atom, direction Ru → Cl1

Electron density at Cl1 atom, direction Ru → Cl1

UDZPCHRHF/GCHF

The mer,trans-[RuCl3(Hind)2(NO)] complex


(IOTC,IOTC) vs. NR (4 x 4 bohr2)

(IOTC,NR) vs. NR (4 x 4 bohr2)

PCE (RHF) (4 x 4 bohr2)

PCE

[CuL2]

2D densities

ρ(gchf-IOTC) – ρ(rhf-IOTC) ρ(rhf-IOTC) – ρ(rhf-DKH2)

The quasirelativistic electron densities are not PCE corrected, units: e.bohr-3


GCHF vs. RHF (IOTC,IOTC) (1 x 1 bohr2)

GCHF vs. RHF (IOTC,NR)

(1 x 1 bohr2)

Significance of PCE In SO effects

PCE in mer,trans-[RuCl3(Hind)2(NO)] complex

Spin densities

1-component level sz is a good quantum number

Contact spin densities of Cu, Ag and Au•Relativistic effects (IOTC vs. NR)•Picture change error•Effects of the size of nucleus

•Comment: •the Fermi contact term expressed as contact spin density is a n.r.l. limit!

Pyykkö, Pajanne, Phys. Rev. A 35 (1971) 53Kutzelnigg Theor. Chim. Acta 73 (1988) 173van Lenthe et al. J Chem. Phys. 108 (1998) 4783I. Malkin et al., Chem. Phys. Lett. 396 (2004) 268S. Komorovsky et al., J. Chem. Phys. 124 (2006) 084108

Malček, Bučinský, Biskupič, Jayatilaka, in preparation

αs-AO> αFN

0ˆ ˆ

yK i Kσ= − ˆi iKΨ = Ψ 0i iΨ Ψ =

Energy expression using Kramers pairs:

( ) ( )1 1

1ˆ2

n n n

Dj j k

E j h j jj kk jk kj= =

= + − ∑ ∑∑General energy expression without using Kramers pairs (GCHF):

( ) ( ) ( )/ 2 / 2 / 2

1 1 1 / 2

2 2n n n n

Dj j k k n

E j h j jj kk jk kj jk k j= = = =

= + − − ∑ ∑∑ ∑

2-component spin densitiesαα αβ

βα ββ

cLα

cLβ

RHF 1-component; scalar; spinfree

UHF+SO 2-component = GCHF h+

i

ii

α

β

Ψ Ψ = Ψ

Jayatilaka, J. Chem. Phys. 108 (1998) 7587.

*

*i

i

i

β

α

−ΨΨ = Ψ

Kramers pairs & spin contamination

( )

2

2

| in the case of a closed shell system

| - in the case of an open shell system

e e

e e

N N

ei j

N N

e ui j

i j N

i j N N

=

≤

∑∑

∑∑

: 0

: 0

UHF i j

GCHF i j

≠ ⇔

≠ ⇔

ℝ

ℂ

2

1

2

1

| 1 in the case of a closed shell system

| 1 in the case of an open shell system

e

e

N

j

N

j

i j

i j

=

=

=

≤

∑

∑2

2

|

|

NN

i j

NN

i j

i j N

i j N

βα

βα

α β β

α β β

=

≤

∑∑

∑∑

UHF

22 2NN

ijUHF EXACTi j

S S N Sβα

αββ= + −∑∑

Cassam-Chenai, Chandler, IJQC 46 (1993) 593

Overlap of Kramers pairs

2

1

|eN

j

i j=∑

2

1

|N

j

i jβ

α β=∑

Chlorine atoms

2-component spin densities

Collinear approach

Noncollinear approach

SC approach UHF:

GCHF:

H. Eschrig, V. D. P. Servegio, J. Comp. Chem. 20 (1999) 23.C. van Wüllen, J. Comp. Chem. 23 (2002) 779.F. Wang, W. Liu, J. Chin. Chem. Soc. 50 (2003) 597.F. Wang, T. Ziegler, J. Chem. Phys. 121 (2004) 12191.R. Bast, H. J. A. Jensen, T. Saue, Int. J. Quantum Chem. 109 (2009) 2091.

Cassam-Chenai, Chandler, IJQC 46 (1993) 593

1c/2c spin densities of H2O+


collinear noncollinear

1c/2c spin densities of phenoxyl radical




1c / 2c spin densities of Os complex


Conclusions• IOTC - DKH2 electron densities are very sensitive to PCE at nucleus

• FN models avoid any problems with the singularity of electron/spin density at the nucleus and/or basis set artifact

• The extent of PCE is much less significant in the case of 2D densities of [CuL2] than the Ruthenium compound

• The (scalar) relativistic contact spin density is not a Fermi contact term

• 2-component level of theory– Spin contamination analog can be introduced using Kramers pairs– The proposed scheme for spin density seems to have the following shortcomings

• Although ħ=1au it contains the prefactor ħ2

• The expression contains a double summation (scales like N2)• Not reliable if spin contamination is huge

• I am the worst programmer in the world! Dylan, sorry that I am destroying your code and thank you for your help!

Acknowledgements

• APVV – APVV-0093-07– APVV-0202-10

• VEGA (contract No. 1/0127/09) • FWF (project I 374-N19)• ARC and CNRS• TONTO package

Jayatilaka, Grimwood Computational Science - ICCS 2003, 4, 142-151wiki: Welcome to Tonto

Dylan JayatilakaTonto bossimplementation of pVp integralsmatrix representations in primitive basis sets spherical basis sets

Kenneth G. DyallGrasp code including Gaussian model of nucleus

Michal Malček (Cu, Ag, Au calculations) Stanislav Biskupič (boss) Martin Hudák

THANK YOU, FOR YOUR ATTENTION

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