Can NOFT bridge the gap between DFT and WFT?

Transcription

1 Kathmandu Workshop on Theoretical Chemistry Euskal Herriko Unibertsitatea, Kimika Fakultatea, P.K. 1072, Donostia. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain. May 3, 2012

2 The exact energy functional of RDMs Outline The electronic energy E for N-electron systems E = ik H ik Γ ki + ijkl < ij kl > D kl,ij Γ ki : 1-RDM D kl,ij : 2-RDM H ik : core-hamiltonian < ij kl >: Coulomb integrals E[N, Γ, D] is an explicitly known functional of the 1- and 2-RDMs! Variational Methods: DFT (reconstruction) ρ (r) = RDMs Γ, D (contraction) = Γ N Ψ CI, MCSCF, CCSD,...

3 Volume 13 Number 45 7 December 2011 Pages COVER ARTICLE Matxain et al. Homolytic molecular dissociation in natural orbital functional theory HOT ARTICLE De Kepper et al. View Online / Journal Homepage / Table of Contents for this issue Sustained self-organizing ph patterns in hydrogen peroxide driven aqueous redox systems Outline Introduction The exact energy functional of RDMs Outline 1 Introduction to the DMFT and NOFT 2 - computational eciency of the method 3 - examples of systems, where DFT yields pathological failures - potentiality of the NOF theory. Downloaded by UNIVERSIDAD DEL PAIS VASCO on 15 November 2011 Published on 09 September 2011 on doi: /c1cp21696a Physical Chemistry Chemical Physics 4 Closing Remarks ISSN

4 1-RDM Functional Introduction 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices Last term in the Energy: U [N, D] = ijkl < ij kl > D kl,ij can be replaced by an unknown functional of the 1-RDM: V ee [N, Γ] = min U [N, D] D D(Γ) D (Γ): family of N-representable 2-RDMs which contract to the Γ E[N, Γ, D] E[N, Γ] = ik H ik Γ ki + V ee [N, Γ] T. L. Gilbert, Phys. Rev. B 12, 2111 (1975); M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979)

5 Natural Orbital Functional 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices The 1-RDM can be diagonalized by a unitary transformation of the spin-orbitals {φ i (x)}: Γ ki = n i δ ki, Γ ( x 1 x 1 ) = i n i φ i (x ) 1 φ i (x 1 ) φ i (x) is the natural spin-orbital with the corresponding occupation number n i E[N, Γ] E [N, {n i, φ i }] = i n i H ii + V ee [N, {n i, φ i }]

6 Cumulant expansion of the 2-RDM D σσ,σσ pq,rt D αβ,αβ pq,rt 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices = n σ p nσ q (δ 2 pr δ qt δ pt δ qr ) + λ σσ,σσ pq,rt (σ = α, β) = n α p nβ q δ 2 pr δ qt + λ αβ,αβ pq,rt λ σσ,σσ pq,rt λ αβ,αβ pq,rt = σσ pq 2 (δ pr δ qt δ pt δ qr ) = αβ pq 2 δ pr δ qt + Πpr 2 δ pqδ rt : real symmetric matrix ( σ 1σ 2 pq = σ 2σ 1 P Sum Rules: σσ pq = n σ p `1 σ P np, q q Π : spin-independent Hermitian matrix (Π αα pr = Π αβ pr = Π βα pr = Π ββ pr qp ) αβ pq = Π pr, Π pr = Π rp) = Π pp Int. J. Quantum Chem. 106, 1093 (2006)

7 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices Conserving rule for Ŝ 2 and diagonal elements Assume N α N β and high-spin multiplet state M S = S Ŝ 2 = S (S + 1) + N β n α p nβ p 2 λ αβ,αβ pq,qp p pq conservation of the total spin 2 pq λ αβ,αβ pq,qp = N β p n α p nβ p ( λ αβ,αβ pq,qp = 1 2 Π pp αβ pp ) δ pq J. Chem. Phys. 131, (2009). for our reconstruction n α p = n p + m p, n β p = n p spin conserving rule: αβ pp = n α p nβ p = n 2 p + n p m p, Π pp = n p

8 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices 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 n i 1 ( i n i = N) lack of sucient conditions for N-representability of D One may approximate the unknown [n] and Π [n], in terms of the occupation numbers, considering the analytic constraints imposed by necessary N-representability conditions of the 2-RDM. D 0, Q 0 σ 1σ 2 qp n σ 1 q nσ 2 p, σ 1σ 2 qp h σ 1 q hσ 2 p G 0 Π 2 qp n q h q n p h p + qp (n q h p + h q n p ) + 2 qp

9 Implemented Approximations 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices PNOF1: Int. J. Quantum Chem. 106, 1093, PNOF2: J. Chem. Phys. 126, , PNOF3: J. Chem. Phys. 132, , PNOF4: J. Chem. Phys. 133, , PNOF5: J. Chem. Phys. 134, , 2011.

10 1-RDM Functional Natural Orbital Functional Two-particle cumulant - and Π-matrices PNOF5: - and Π-matrices for singlet states S = 0 n 2 p, q = p n p, q = p qp = 0, q p, Π qp = 0, q p n p n p, q = p n p n p, q = p n p + n p = 1 E PNOF 5 = N [ np (2H pp + J pp ) ] n p n p K p p p=1 + N p,q=1 n q n p (2J pq K pq ) ( p = N p + 1 ; P : q p, p) J. Chem. Phys. 134, , 2011

11 Minimization of the energy functional and Euler equations The Hermitian matrix F General overview of the program architecture Parallelization of the bottleneck Minimization of the functional E PNOF 5 under constraints 1 Löwdin's normalization: 2 p n p = N (n p + n p = 1) 2 N representability of the 1-RDM: 0 n p 1 = n p = cos 2 γ p, n p = sin 2 γ p : Conjugate Gradient Method 3 Orthonormality of natural orbitals: ϕ p ϕ q = δ pq = Method of Lagrangian multipliers Ω = E 2 ε qp [< ϕ p ϕ q > δ pq ] pq

12 Minimization of the energy functional and Euler equations The Hermitian matrix F General overview of the program architecture Parallelization of the bottleneck Euler equations for the natural orbitals {ϕ p (r)} n p ˆVp ϕ p = q ε qp ϕ q, ε qp = n p ϕ q ˆVp ϕ p ˆV p (1) = Ĥ (1) + Ĵ p (1) n p n p ˆK p (1) + N q=1 ] n q [2Ĵ q (1) ˆKq (1) [Λ, Γ] 0 solution cannot be reduced to diagonalization of Λ Λ = {ε qp }, Γ = {n p δ pq } Self-consistent iterative diagonalization procedure J. Comp. Chem. 30, 2078 (2009)

13 Minimization of the energy functional and Euler equations The Hermitian matrix F General overview of the program architecture Parallelization of the bottleneck The Hermeticity of Λ and the Aufbau Principle The Lagrangian is Hermitian at the extremum: ε pq = ε qp - Dene a new Hermitian matrix F: (o-diagonal elements) F pq = θ (q p) [ε pq ε qp] + θ (p q) [ε qp ε pq ] - {F pp } cannot be determined from the Hermiticity of Λ First order perturbative theory (Hillier 1970, Saunders 1973) F 0 pq 2 E = E p<q pqf 0 = E pq p<q Fpp 0 F qq 0 { F 0 > F 0 qq pp} E is bound to drop upon diagonalization of F 0 Aufbau Principle for diagonal elements J. Comp. Chem. 30, 2078 (2009)

14 Minimization of the energy functional and Euler equations The Hermitian matrix F General overview of the program architecture Parallelization of the bottleneck

15 Parallel Eciency (H 2 O, cc-pv6z, 290 GBF) Minimization of the energy functional and Euler equations The Hermitian matrix F General overview of the program architecture Parallelization of the bottleneck cores (n) nodes Time (h) Total # Iter. Eciency (%) Maximum speedup that can be achieved is 1/(1-P) 40 P is the proportion of the program made parallel ( 97.5%) E n = T 1 nt n IT n IT1 100%

16 Planar trimethylenemethane TMM IMA OXA HOMO LUMO TMM: trimethylenemethane IMA: iminoallyl diradicaloid OXA: oxyallyl diradicaloid diradical character TMM > IMA > OXA

17 Relative Energies and Occupation Numbers Relative energy with respect to its cyclic isomer, in kcal/mol TMM IMA OXA CAS(12,12) PNOF CASPT2(12,12) Occupation numbers of the (pseudo)degenerate orbitals TMM IMA OXA PNOF4 1.07/ / /0.50 PNOF5 1.00/ / /0.54 CAS(12,12) 1.01/ / /0.55 ChemPhysChem 12, 1673, 2011

18 Ethylene Torsion Introduction Occupation γ 90 J. Chem. Phys. 134, , 2011

19 Ethylene Torsion. Energetics E (Hartrees) E (kcal/mol) Min(D2h) TS (D2d ) CASPT2(12,12) PNOF B3LYP PBE M06-2X cc-pvdz Basis Set. Optimized at the CASSCF(4,4)/cc-pVDZ level of theory. Broken symmetry energies for TS. S 2 = 1.01 J. Chem. Phys. 134, , 2011

20 cc-pvtz dissociation curves for diatomic molecules 0 Energies (kcal/mol) H2 LiH BH FH N2 CO BONDS covalent with dierent polarity H 2, FH, BH multiple bond CO, N 2 electrostatic LiH R (A) In all cases, dissociation limit implies an homolytic cleavage of the bond, high degree of near-degeneracy at the dissociation asymptote J. Chem. Phys. 134, , 2011

21 Dissociation for multiply bonded molecule: N 2 E(a.u.) N 2 ( 1 Σ) N( 4 S) + N( 4 S) PNOF5 CASSCF(10,8) CASSCF(14,14) Occupation PNOF5 - CASSCF(14,14) σ π π σ R (Å) R (Å) Phys. Chem. Chem. Phys. 13, 20129, 2011

22 cc-pvtz Ionization Potentials of N 2, in ev EKT: diagonalization of the matrix ν whose elements are ν qp = εqp nqn p Molecule MO KT PNOF5-EKT EXP N 2 σ g ( 1.63) (1.09) π u ( 0.00) (0.82) σ u ( 2.40) (1.67) J. Chem. Phys., 2012 (DOI: / )

23 14-electron isoelectronic series N 2 CN R e D e BO µ e q N R e D e BO µ e q N PNOF CAS(10,8) CAS(14,14) Exptl NO + CO R e D e BO µ e q N R e D e BO µ e q C PNOF / CAS(10,8) CAS(14,14) Exptl R e in Å, D e in kcal/mol and µ e in Debyes Phys. Chem. Chem. Phys. 13, 20129, 2011

24 Dissociation curves for O 2+ 2 E( a.u.) PNOF5 CAS (6,6) CAS (10,8) CAS (14,14) R (Å) R e (Å) BO R (Å) E ( kcal mol ) De ( kcal mol ) PNOF (6,6) (10,8) (14,14) MRCI R. H. Nobes, et. al. Chem. Phys. Lett. 182, 216 (1991) q O Phys. Chem. Chem. Phys. 13, 20129, 2011

25 : Cr 2, CrMo, Mo 2 Cr 2 at R e = Å (0.125) (0.162) (0.189) (0.603) (1.875) (1.397) XY ( 1 Σ g ) X( 7 S3) + Y( 7 S3) CASSCF PNOF5 CASPT2 Exp. Cr a 1.56 CrMo b 2.09 Mo c (1.811) (1.838) a J. Chem. Theory Comput. 7, 1640 (2011) b Inorg. Chem. 50, 9219 (2011) c Chem. Phys. 343, 210 (2008) Eective Bond Order: CASSCF/CASPT2: ANO-RCC-QZ, PNOF5: 6-31G PNOF5=4.16, CASPT2=4.45

26 Canonical Orbitals Introduction E = N [n p H pp + ε pp ], ε pp = n p ϕ p ˆVp ϕ p p=1 the trace of square matrix is the sum of its diagonal elements, E = Tr (HΓ + Λ), Λ = {ε qp } the trace of a matrix is invariant under U (X = U X U) Tr (HΓ + Λ) = Tr ( H Γ + Λ ), U : Λ = U ΛU ε qp = ε p δ qp, Γ = U ΓU n qp n p δ qp {χ p (r)}: canonical orbitals

27 Valence orbitals of methane (CH 4 ) Natural Orbital Representation Canonical Orbital Representation (0.0160) E (0.0160) (0.0160) (1.9840) (1.9840) (1.9840) ChemPhysChem 2012

28 Valence vertical ionization energies, in ev, for methane cc-pvdz cc-pvdz T 2 A1 T 2 A1 B3LYP BLYP BP M06-2X M06L M MPWPW O3LYP Experiment OLYP PBEPBE PBEHPBE PW91PW HF ε CanOrb pp EKT-PNOF OVGF Experiment Chem. Phys. Lett. 531, 272, 2012.

29 Valence orbitals of Benzene (C 6 H 6 ) Natural Orbital Representation Canonical Orbital Representation (0.0413) E (0.0413) (0.0427) (1.9573) (1.9587) (1.9587) ChemPhysChem 2012

30 Closing Remarks Introduction it is now feasible to perform expensive NOFT calculations. The parallelization of the bottlenecks of our code allows us to achieve an execution 37 times faster than the sequential one, in 48 processors, with an eciency of 86%. the functional N-representabilty plays a crucial role towards achieving chemical accuracy. The PNOF5 can describe in a balanced way chemical bonding situations that evolve gradually from non-degenerate to degenerate states. Integer number of electrons have been found on the dissociated atoms. two equivalent orbital representations are possible. PNOF5 could be a practical tool for the interpretation of the chemical bonding.

31 Acknowledgement Financial support comes from the Basque Government and the Spanish Oce for Scientic Research. The SGI/IZO-SGIker UPV/EHU is greatfully acknowledged for generous allocation of computational resources. Thank you for your attention!!!