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The Molecular Orbitals in NOFTWorkshop on ab initio valence bond theory
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga,G. Merino, J. M. Mercero, E. Matito, J. M. Ugalde
Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU),
Donostia International Physics Center (DIPC), 20080 Donostia.
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
July 18, 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Outline
1 Introduction to the 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
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
N-representabilty ⇒ inequalities for o-diagonal elements
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
Minimization of the functional E [N, np, ϕp]
Constraints:
1 Löwdin's normalization: 2∑
p np = N (np + np = 1)
2 N representability of the 1-RDM: 0 ≤ np ≤ 1
=⇒ np = cos2 γp, np = sin2γp : Conjugate Gradient Method
3 Orthonormality of natural orbitals: 〈ϕp|ϕq〉 = δpq
=⇒ Method of Lagrangian multipliers
Ω = E − 2∑pq
εqp [< ϕp|ϕq > −δpq]
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Density Matrix and Natural Orbital FunctionalsSinglet states: 2-RDM, Spin, N-representability, PNOF5Minimization of the energy functional and Euler equations
Euler equations for the natural orbitals ϕp (r)
npVp |ϕp〉 =∑q
εqp |ϕq〉, εqp = np 〈ϕq| Vp |ϕp〉
Vp (1) = H (1) + Jp (1)−√
npnpKp (1) +
N∑q=1
′′ nq
[2Jq (1)− Kq (1)
]
[Λ, Γ] 6= 0⇒ solution cannot be reduced to diagonalization of Λ
Λ = εqp, Γ = npδpq
Self-consistent iterative diagonalization procedure
J. Comp. Chem. 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 NOFT
Introduction to NOFTPNOF5 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 NOFT
Introduction to NOFTPNOF5 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 NOFT
Introduction to NOFTPNOF5 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 NOFT
Introduction to NOFTPNOF5 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 NOFT
Introduction to NOFTPNOF5 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
1 2 3 4 5 6R in Å
-1
-0.8
-0.6
-0.4
-0.2
0
Rel
ativ
e E
nerg
y (e
V)
CASSCF (ecp)CASPT2 (ecp)PNOF5 (ecp)PNOF5 (all-electron)CASPT2 (all-electron)
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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 13, 2297, 2012, Phys. Chem. Chem. Phys. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 13, 2297, 2012, Phys. Chem. Chem. Phys. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 13, 2297, 2012, Phys. Chem. Chem. Phys. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Valence orbitals of silicon dimer (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 NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. Methane. WaterDelocalization eectsCarbon dimers vs Silicon dimers
Closing Remarks
Two orbital pictures are possible: NOs and COs.
The COs can be obtained only after solving the problem in theNO representation.
PNOF5 NOs are localized orbitals that nicely agree with thechemical intuition of chemical bonding, VB and VSEPRbonding pictures.
PNOF5 COs are symmetry-adapted delocalized orbitals similarto those obtained by molecular orbital theories.
NO and CO representations are unique one-particle pictures ofthe same solution ergo complement each other in thedescription of the electronic structure.
M. Piris, J. M. Matxain, X. Lopez, F. Ruipérez, J. Uranga, G. Merino, J. M. Mercero, E. Matito, J. M. UgaldeThe Molecular Orbitals in NOFT
Introduction to NOFTPNOF5 Results
PNOF5 Molecular Orbitals
Two orbital pictures in NOFT. 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 NOFT