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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
(What is Old in Relativistic Quantum Chemistry)
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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
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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= ± + =
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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
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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
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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:
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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∂ = −∂
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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)
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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-
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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
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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+= = + +
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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.
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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
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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)
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• 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.
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• 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
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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
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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
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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.
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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+
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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+
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2s orbital densities of Rn (UTZ+10s)
point charge Gaussian charge distribution nodal behavior
2p1/2 orbitals 2p1/2 orbitals 2p orbitals
Bučinský, Biskupič, Jayatilaka, Theor Chem Acc 129 (2011) 181–197.
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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
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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
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Radial distributions of spin density Cu atom Cu2+ atom
Bučinský L., Biskupič S., Jayatilaka D. (2011).Chem. Phys., 10.1016/j.chemphys.2011.04.026
Spin Density
[Ar]3d104s1[Ar]3d9
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Spin 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 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
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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
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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.
(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
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ρ(gchf-IOTC) – ρ(rhf-IOTC) ρ(rhf-IOTC) – ρ(rhf-DKH2)
The quasirelativistic electron densities are not PCE corrected, units: e.bohr-3
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.
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
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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
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Malček, Bučinský, Biskupič, Jayatilaka, in preparation
αs-AO> αFN
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Malček, Bučinský, Biskupič, Jayatilaka, in preparation
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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
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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
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Overlap of Kramers pairs
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2
1
|eN
j
i j=∑
2
1
|N
j
i jβ
α β=∑
Chlorine atoms
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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
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1c/2c spin densities of H2O+
Malček, Bučinský, Biskupič, Jayatilaka, in preparation
collinear noncollinear
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1c/2c spin densities of phenoxyl radical
Malček, Bučinský, Biskupič, Jayatilaka, in preparation
collinear noncollinear
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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.
1c / 2c spin densities of Os complex
collinear noncollinear
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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!
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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
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THANK YOU, FOR YOUR ATTENTION