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The Molecular Orbitals in DMFTTheoretical Chemistry in Rio
Meeting to honor the 65th birthday of Prof. Marco Antonio Chaer Nascimento
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga,G. Merino, J. M. Mercero, E. Matito, J. M. Ugalde
Euskal Herriko Unibertsitatea, Kimika Fakultatea, P.K. 1072, 20080 Donostia.
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
June 12, 2012
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Outline
1 Introduction to the DMFT (NOFT)
2 PNOF5 Results
- examples of systems, whereDFT yields pathological failures
- potentiality of the NOF theory.
3 PNOF5 Molecular Orbitals
ISSN 1463-9076
Physical Chemistry Chemical Physics
www.rsc.org/pccp Volume 13 | Number 45 | 7 December 2011 | Pages 20023–20482
COVER ARTICLE
Matxain et al.Homolytic molecular dissociation in natural orbital functional theory
HOT ARTICLE
De Kepper et al.Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems
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View Online / Journal Homepage / Table of Contents for this issue
−0.3552 (1.838)−0.3324 (1.811)
−0.2486 (1.397)
−0.2010 (1.875)
−0.1326 (0.603)
−0.0608 (0.189)
−0.0196 (0.125)
−0.0551 (0.162)
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
The electronic energy E for N-electron systems
E =∑ik
HikΓki +∑ijkl
< ij |kl > Dkl ,ij
Γki : 1-RDM
Dkl ,ij : 2-RDM
Hik : core-Hamiltonian
< ij |kl >: Coulomb integrals
E [N, Γ,D] is an explicitly known functional of the 1- and 2-RDMs!
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
1-RDM Functional
Last term in the Energy: U [N,D] =∑ijkl
< ij |kl > Dkl ,ij can be
replaced by an unknown functional of the 1-RDM:
Vee [N, Γ] = minD∈D(Γ)
U [N,D]
D (Γ): family of N-representable 2-RDMs which contract to the Γ
E [N, Γ,D]⇒ E [N, Γ] =∑ik
HikΓki + Vee [N, Γ]
T. L. Gilbert, Phys. Rev. B 12, 2111 (1975); M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979)
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
Natural Orbital Functional
The 1-RDM can be diagonalized by a unitary transformation of thespin-orbitals φi (x):
Γki = niδki , Γ(x′1|x1
)=∑i
niφi(x′1
)φ∗i (x1)
φi (x) is the natural spin-orbital with the corresponding
occupation number ni
E [N, Γ]⇒ E [N, ni , φi] =∑i
niHii + Vee [N, ni , φi]
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
Ansatz for singlet states |S = 0〉 Int. J. Quantum Chem. 106, 1093 (2006)
Spin-blocks of the 2-RDM:
Dσσ,σσpq,rt = 1
2 (npnq −∆pq) (δprδqt − δptδqr ) (σ = α, β)
Dαβ,αβpq,rt = 1
2 (npnq −∆pq) δprδqt +Πpr
2 δpqδrt
∆ np , Π np : real symmetric matrices
Sum Rule:Pq
′∆pq = np (1− np)
Conserving rule for S2 ⇒ diagonal elements
∆pp = n2p , Πpp = np
J. Chem. Phys. 131, 021102 (2009)
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
The N-representability and o-diagonal elements
N-representability
RDMs must be derivable from an N-particle wave function Ψ
N-representability of Γ: 0 ≤ ni ≤ 1 (∑
ini = N)
One may approximate the unknown ∆ [n] and Π [n], in terms of theoccupation numbers, considering the analytic constraints imposedby necessary N-representability conditions of the 2-RDM.
D ≥ 0,Q ≥ 0 ⇒ ∆qp ≤ nqnp, ∆qp ≤ (1− nq) (1− np)
G ≥ 0 ⇒ Π2qp ≤ nqhqnphp + ∆qp (nqhp + hqnp) + ∆2
qp
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix FunctionalNatural Orbital FunctionalSinglet states: 2-RDM, Spin, N-representability
PNOF5: ∆- and Π-matrices for singlet states |S = 0〉
∆pq = n2pδpq + npnp δpq
Πpq = npδpq −√npnpδpq
np + np = 1
E = 2N∑
p=1npHpp +
N∑p,q=1
′′ nqnp (2Jpq − Kpq)
+N∑
p=1
[npJpp −
√npnpKpp
](p = N − p + 1 ;
P ′′ : q 6= p, p)
J. Chem. Phys. 134, 164162, 2011
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
Planar trimethylenemethane
HOMO LUMO
Relative energy to its cyclic isomer (kcal/mol)
TMM IMA OXA
CAS(12,12) 34.4 34.0 26.2
PNOF5 40.8 37.2 26.5
CASPT2(12,12) 43.3 39.7 32.6
Occupation numbers of (quasi)degenerate orbitals
TMM IMA OXA
PNOF4 1.07/0.97 1.36/0.71 1.57/0.50
PNOF5 1.00/1.00 1.26/0.74 1.46/0.54
CAS(12,12) 1.01/0.99 1.25/0.75 1.45/0.55
ChemPhysChem 12, 1673, 2011
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
Ethylene Torsion J. Chem. Phys 134, 164102, 2011
90120150180γ
0.0
0.2
0.5
0.8
1.0
1.2
1.5
1.8
2.0
Occ
upat
ion
E (Hartrees) ∆E(kcal/mol)
Min(D2h)† TS (D2d )
†
CASPT2(12,12) -78.342567 -78.238122 65.5
PNOF5 -78.136524 -78.032063 65.6
B3LYP‡ -78.591976 -78.490308 63.8
PBE0‡ -78.485589 -78.388529 60.9
M06-2X‡ -78.543689 -78.437072 66.9
† cc-pVDZ Basis Set, ‡ Broken symmetry energies for TS.DS2
E= 1.01
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
cc-pVTZ dissociation curves for diatomic molecules
0.5 1 1.5 2 2.5 3 3.5 4 4.5R (A)
-250
-200
-150
-100
-50
0
Energies (kcal/mol)
H2LiHBHFHN2CO
BONDS
covalent with dierentpolarity H2, FH, BH
multiple bond CO, N2
electrostatic LiH
In all cases, dissociation limit implies an homolytic cleavage of thebond, high degree of near-degeneracy at the dissociation asymptote
J. Chem. Phys. 134, 164102, 2011
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
Homolytic Dissociations: 14-electron isoelectronic series
N2 CN−
Re De BO µe qN Re De BO µe qN
PNOF5 1.099 229.9 2.87 0.000 7 1.180 247.6 2.89 0.900 7
CAS(10,8) 1.117 205.0 2.85 0.000 7 1.200 220.0 2.86 2.241 7
CAS(14,14) 1.115 210.4 2.85 0.000 7 1.196 235.4 2.86 2.360 7
Exptl. 1.098 225.1 - 0.000 7 1.177 - - 0.630 7
NO+ CO
Re De BO µe qN Re De BO µe qC
PNOF5 1.059 228.2 2.87 0.337 6/7 1.130 221.0 2.92 0.209 6
CAS(10,8) 1.077 229.0 2.84 2.368 7 1.143 249.9 2.88 -0.259 6
CAS(14,14) 1.076 261.7 2.83 2.260 6 1.145 247.0 2.86 -0.059 6
Exptl. 1.066 - - - 7 1.128 256.2 - 0.112 6
Re in Å, De in kcal/mol and µe in Debyes Phys. Chem. Chem. Phys. 13, 20129, 2011
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
Dissociation of transition metal dimers (ECP, 6s5p3d)
Cr2 W2
2 3 4 5 6 7 8 9R in Å
-1
0
1
2
Rel
ativ
e E
nerg
y (e
V)
CASSCFCASPT2PNOF5
2 3 4 5 6 7 8 9R in Å
-4
-3
-2
-1
0
1
2
Rel
ativ
e E
nerg
y (e
V)
CASSCFCASPT2PNOF5
Mo2 PNOF5: Cr2, CrMo, Mo2, W2
2 3 4 5 6 7 8 9R in Å
-4
-3
-2
-1
0
1
2
Rel
ativ
e E
nerg
y (e
V)
CASSCFCASPT2PNOF5
2 3 4 5 6 7R (Å)
-4
-3
-2
-1
0
1
2
De (
eV)
Cr2
CrMoMo
2
W2
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Diradicals and Diradicaloids. Ethylene TorsionHomolytic Dissociations. N2 and 14-e− seriesDissociation of transition metal dimers
Dissociation of transition metal dimers
XY (1Σg )→ X(7S3) + Y(7S3)
Binding Energies (eV)
PNOF5 CASPT2 Exp.
Cr2 0.95 1.50a 1.56
CrMo 1.80 2.62b 2.09
Mo2 3.26 4.41c 4.28
W2 3.48 5.37d 5.00
a J. Chem. Theory Comput. 7, 1640 (2011)
b Inorg. Chem. 50, 9219 (2011)
c Chem. Phys. 343, 210 (2008)
d Chem. Phys. Lett. 490, 24 (2010)
CASSCF/CASPT2: ANO-RCC-QZ, PNOF5:ECP
Cr2 at Re(EXP) = 1.679Å
−0.3552 (1.838)−0.3324 (1.811)
−0.2486 (1.397)
−0.2010 (1.875)
−0.1326 (0.603)
−0.0608 (0.189)
−0.0196 (0.125)
−0.0551 (0.162)
Eective Bond Order:
PNOF5=4.16, CASPT2=4.45
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Molecular orbital representations
Natural Orbitals ϕp (r)
Orthonormality: 〈ϕp|ϕq〉 = δpq
Ω = E − 2Ppq
εqp [< ϕp|ϕq > −δpq]
Euler Eqs.:
npVp |ϕp〉 =Pq
εqp |ϕq〉
εqp = np 〈ϕq| Vp |ϕp〉
Λ = εqp, Γ = npδpq
[Λ, Γ] 6= 0⇒ solution cannot be reduced
to diagonalization of Λ
Self-consistent iterative diagonalizationprocedure: JCC 30, 2078 (2009)
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Molecular orbital representations
Natural Orbitals ϕp (r)
Orthonormality: 〈ϕp|ϕq〉 = δpq
Ω = E − 2Ppq
εqp [< ϕp|ϕq > −δpq]
Euler Eqs.:
npVp |ϕp〉 =Pq
εqp |ϕq〉
εqp = np 〈ϕq| Vp |ϕp〉
Λ = εqp, Γ = npδpq
[Λ, Γ] 6= 0⇒ solution cannot be reduced
to diagonalization of Λ
Self-consistent iterative diagonalizationprocedure: JCC 30, 2078 (2009)
Canonical Orbitals χp (r)
E =Pp
[npHpp + εpp]
εpp = np 〈ϕp| Vp |ϕp〉 6= IPs
EKT: νqp = − εqp√nqnp
E = Tr (HΓ + Λ) , Λ = εqp
U : X ′ = U†XU, U† = U
−1
Tr (HΓ + Λ) = Tr (H′Γ′ + Λ′)
A Λ′ = U†ΛU, Γ′ = U
†ΓU
Λ′ =˘ε′pδqp
¯, Γ′ =
˘n′qp¯
⇒ Λ′ |χp〉 = ε′p |χp〉
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
PNOF5 valence orbitals of methane (CH4)
−0.0126 (0.0160)
−0.6517 (1.9840)
E
Natural Orbital Representation
−0.9458 (1.9840)
−0.5521 (1.9840)
−0.0141 (0.0160)
−0.0120 (0.0160)
Canonical Orbital Representation
ChemPhysChem 2012, J. Chem. Theor. Comp. 2012
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Valence vertical ionization energies, in eV, for methane
cc-pVDZ
T2 A1
B3LYP 10.57 18.79
BLYP 9.13 16.66
BP86 9.33 16.93
M06-2X 12.22 21.20
M06L 9.56 17.76
M06 10.74 18.98
MPWPW91 9.27 16.87
O3LYP 9.98 18.06
Experiment 14.40 23.00
cc-pVDZ
T2 A1
OLYP 9.17 16.88
PBEPBE 9.22 16.83
PBEHPBE 9.23 16.82
PW91PW91 9.29 16.88
HF 14.76 25.62
−εCanOrbpp 15.02 25.74
EKT-PNOF5 15.14 25.89
OVGF 14.21 23.47
Experiment 14.40 23.00
Chem. Phys. Lett. 531, 272, 2012.
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
PNOF5 valence orbitals of water (H2O)
−0.8538 (1.9818)
−0.7238 (1.9931)
−0.0179 (0.0182)−0.0084 (0.0069)
E
Natural Orbital Representation Canonical Orbital Representation
−1.3458 (1.9868)
−0.7167 (1.9816)
−0.5821 (1.9869)−0.5057 (1.9915)
−0.0186 (0.0184)−0.017 (0.184)−0.0085 (0.0085)−0.0075 (0.0079)
ChemPhysChem 2012, J. Chem. Theor. Comp. 2012
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Valence orbitals of carbon dioxide (CO2). Banana Orbitals.
E
−0.9679 (1.9880)
−0.0137 (0.0120)−0.0086 (0.0071)
−0.8090 (1.9929)
Natural Orbital Representation Canonical Orbital Representation
−0.0168 (0.0120)−0.0155 (0.0111)−0.0109 (0.0119)
−0.5472 (1.9895)
−0.7169 (1.9898)−0.7469 (1.9907)−0.8035 (1.9911)
−1.4806 (1.9895)
−1.5333 (1.9892)
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Aromaticity: PNOF5 valence orbitals of benzene (C6H6)
Natural Orbital Representation
−0.3938 (1.9587)
−0.0155 (0.0413)
−0.3403 (1.9573)
E
−0.5012 (1.9587)
−0.0169 (0.0427)−0.0122 (0.0413)
Canonical Orbital Representation
ChemPhysChem 2012, J. Chem. Theor. Comp. 2012
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Valence orbitals of carbon dimer (C2)
E
−1.0218 (1.9975)
−0.4563 (1.8420)
−0.0594 (0.1580)−0.0033 (0.0025)
−1.0136 (1.9956)
−0.5164 (1.8420)
−0.4261 (1.8420)
−0.0730 (0.1600)−0.0521 (0.1581)−0.0034 (0.0024)
Natural Orbital Representation Canonical Orbital Representation
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Valence orbitals of silicon diatomic (Si2)
E
−0.4270 (1.9320)
−0.6121 (1.9942)
−0.2895 (1.9100)
−0.0043 (0.0058)−0.0216 (0.0900)−0.0234 (0.0680)
−0.2895 (1.9100)−0.3004 (1.9467)
−0.4567 (1.9319)
−0.7067 (1.9794)
−0.0223 (0.0681) −0.0241 (0.0680)−0.0226 (0.0900)−0.0043 (0.0059)
Natural Orbital Representation Canonical Orbita Representationl
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Closing Remarks
PNOF5 can describe in a balanced way chemical bondingsituations that evolve gradually from non-degenerate todegenerate states.
Integer number of electrons have been found on thedissociated atoms.
Two equivalent orbital representations are possible.PNOF5 could be a practical tool for the interpretation of thechemical bonding.
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT
Introduction to DMFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in DMFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Acknowledgement
Financial support comes from the Basque Government and theSpanish Oce for Scientic Research.
The SGI/IZO-SGIker UPV/EHU is greatfully acknowledged forgenerous allocation of computational resources.
Thank you for your attention !!!
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in DMFT