Introduction and Overview of the Reduced Density Matrix Functional Theory

Transcription

1 Introduction and Overview of the Reduced Density Matrix Functional Theory N. N. Lathiotakis Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens April 13, 2016

2 Outline 1 2 Density matrices and N-representability 3 Reduced density matrix functional theory () 4 Functionals minimization/performance 5 Comparison with DFT 6 Application to prototype systems: Correlation energies, Molecular dissociation, Homogeneous electron gas, Spectra, IPs, Electronic Gap 7 Conclusion

3 Hartree Fock Wave function is one Slater Determinant: Φ(x 1,x 2, x N ) = ϕ 1 (x 1 ) ϕ 1 (x 2 ) ϕ 1 (x N ) ϕ 2 (x 1 ) ϕ 2 (x 2 ) ϕ 2 (x N ) ϕ N (x 1 ) ϕ N (x 2 ) ϕ N (x N ) We need to minimize: E tot = Φ Ĥ Φ Φ Φ Minimization chooses N orbitals out an infinite dimension space (or of dimension M > N for practical applications).

4 Energy in Hartree-Fock Spin-orbitals ϕ(x) = ϕ(r)α(ω). For spin compensated systems: N/2 N/2 E tot = 2 h (1) jj +2 h (1) jj = J jk = K jk = j=1 j,k=1 J jk N/2 j,k=1 K jk [ d 3 rϕ j(r) 1 ] 2 2 r +V(r) ϕ j (r) d 3 r d 3 r d 3 r ϕ j(r) 2 ϕ k (r ) 2 r r d 3 r ϕ j(r)ϕ j (r )ϕ k (r )ϕ k (r) r r

5 Energy in Hartree-Fock Spin-orbitals ϕ(x) = ϕ(r)α(ω). For spin compensated systems: E tot = 2 j=1 h (1) jj = J jk = K jk = n j h (1) jj +2 j,k=1 n j n k J jk j,k=1 [ d 3 rϕ j(r) 1 ] 2 2 r +V(r) ϕ j (r) d 3 r d 3 r Where n j and n k occupation numbers d 3 r ϕ j(r) 2 ϕ k (r ) 2 r r n j n k K jk d 3 r ϕ j(r)ϕ j (r )ϕ k (r )ϕ k (r) r r

6 Hartree Fock Functional in E tot = 2 j=1 n j h (1) jj +2 j,k=1 n j n k J jk j,k=1 n j n k K jk Assume that this functional is minimized w.r.t. n j, ϕ j. It is not bound! n j should satisfy extra conditions. Ensemble N-representability conditions of Coleman: 0 n j 1, and 2 n j = N j=1 The first reflects the Pauli principle and the second fixes the number of particles. No extrema between 0 and 1: collapses to HF Theory

7 Density matrices Density Matrices N-representability Foundations N-body density matrix (NRDM) Γ (N) (r 1,r 2..r N ;r 1,r 2..r N ) = Ψ (r 1,r 2..r N )Ψ(r 1,r 2..r N ) Reduce the order of the density matrix (prdm) Γ (p) (r 1,..r p ;r 1,..r p ( ) ) = N d 3 r p p+1..d 3 r N Ψ (r 1,..r p,r p+1..r N )Ψ(r 1,..r N )

8 Density matrices Density Matrices N-representability Foundations N-body density matrix (NRDM) Γ (N) (r 1,r 2..r N ;r 1,r 2..r N ) = Ψ (r 1,r 2..r N )Ψ(r 1,r 2..r N ) Reduce the order of the density matrix (prdm) Γ (p) (r 1,..r p ;r 1,..r p ( ) ) = N d 3 r p p+1..d 3 r N Ψ (r 1,..r p,r p+1..r N )Ψ(r 1,..r N ) Recurrence relation Γ (p 1) (r 1,..r p 1 ;r 1,..r p 1) = p d 3 r p Γ (p) (r 1,..r p ;r 1 N p+1,..r p 1,r p)

9 Density matrices Density Matrices N-representability Foundations One-body reduced density matrix (1RDM) Γ (1) (r,r ) = 2 d 3 r 2 Γ (2) (r,r 2 ;r,r 2 ) =: γ(r;r ) N 1 Expectation value of p-body operator: } {Γ Ô = Tr (p) Ô Total energy: expectation value of the Hamiltonian (2-body) The e-e interaction energy simple functional of 2RDM: E ee = d 3 r 1 d 3 ρ (2) (r 1,r 2 ) r 2 r 1 r 2 ) ρ (2) (r 1,r 2 ) = Γ (2) (r 1,r 2 ;r 1,r 2 ) (second reduced density) Why don t we minimize the total energy with respect to Γ (2)?

10 N-representability of the 2RDM Density Matrices N-representability Foundations Remember Γ (2) (r 1,r 2 ;r 1,r 2) = N(N 1) 2 d 3 r 3..d 3 r N Ψ (r 1,r 2,r 3..r N )Ψ(r 1..r N ) with Ψ: antisymmetric, normalized wave function For Γ (2) several necessary N-representability conditions are known 1. These conditions are not sufficient. If they were sufficient they would provide the solution of the many electron problem. 1 work and talk of D. A. Mazziotti.

11 N-representability of the 1RDM Density Matrices N-representability Foundations For γ ensemble N-representability, necessary and sufficient conditions were proven by Coleman 2 : 0 n j 1, n j = N occupation numbers n j and the natural orbitals ϕ j : γ(r,r ) = Plus orthonormality of ϕ j j n j ϕ j(r )ϕ j (r) j=1 2 Rev. Mod. Phys. 35, 668 (1963)

12 N-representability of the 1RDM Density Matrices N-representability Foundations For γ ensemble N-representability, necessary and sufficient conditions were proven by Coleman 2 : 0 n j 1, n j = N occupation numbers n j and the natural orbitals ϕ j : γ(r,r ) = j n j ϕ j(r )ϕ j (r) j=1 Plus orthonormality of ϕ j 2RDM vs 1RDM functional theory: (simple functional but complicated N-rep) vs (totally unknown functional but simple N-rep.) 2 Rev. Mod. Phys. 35, 668 (1963)

13 N-representability of the 1RDM Density Matrices N-representability Foundations Coleman s conditions are also sufficient for pure state N-rep for even number of electrons and spin compensated systems. Given the exact functional of the 1RDM the ensemble N-rep conditions are enough. Are there pure state N-rep conditions? Generalized Pauli constrains 3 for (N,M); N: number of electrons, M: size of the Hilbert space. Recently: a method to generate them for all pairs (M,N). Unfortunately, their number explodes as N, M increase. Their incorporation in calculations for 3-electron systems improves the results for approximate 1RDM functionals. 3 Talks on Thursday

14 Foundations Density Matrices N-representability Foundations Gilbert s Theorem (T. Gilbert Phys. Rev. B 12, 2111 (1975)): γ gs (r;r ) 1 1 Ψ gs (r 1,r 2...r N ) Every ground-state observable is a functional of the ground-state 1RDM. The exact total energy functional E tot = E kin +E ext +E ee E ee [γ] = min Ψ γ Ψ V ee Ψ (Domain of γ: Pure state N-representable (Lieb 1979)) E ee [γ] = min Ψ V ee Ψ Γ (N) γ (Domain of γ: Ensemble state N-representable (Valone 1980))

15 Foundations Total energy Density Matrices N-representability Foundations E tot = E kin +E ext +E ee E kin = E ext = d 3 rd 3 r δ(r r ) d 3 rv ext (r)γ(r;r) ( 2 2 ) γ(r;r ) E ee = E H +E xc E H = d 3 r d 3 r γ(r;r) γ(r ;r ) r r Exchange-correlation energy does not contain any kinetic energy contributions

16 Müller type functionals Functionals Minimization Comparison with DFT E xc = 1 2 j,k=1 f(n j,n k ) d 3 rd 3 r ϕ j(r)ϕ j (r )ϕ k (r )ϕ k (r) r r Hartree-Fock: f(n j,n k ) = n j n k Müller functional 4 : f(n j,n k ) = n j n k Goedecker-Umrigar 5 : f(n j,n k ) = n j n k (1 δ jk )+n 2 j δ jk Power functional 6 f(n j,n k ) = (n j n k ) α, α 0.6. ML: Pade approximation for f, fit for a set of molecules 7 4 A. Müller, Phys. Lett. 105A, 446 (1984); M. A. Buijse, E. J. Baerends, Mol. Phys. 100, 401 (2002) 5 S. Goedecker, C. J. Umrigar, Phys. Rev. Lett. 81, 866 (1998). 6 Sharma et al, PRB 78, R (2008) 7 Marques, et al, PRA 77, (2008).

17 BBC functionals Functionals Minimization Comparison with DFT A hierarchy of 3 corrections 8 BBC3: n j n k j kbothweakly occupied j k bothstronglyoccupied n f(n j,n k ) = j n k j(k) anti bonding, k(j) not bonding n 2 j j = k nj n k otherwise. AC3: Similar to BBC3 with C2,C3 corrections analytic 9 8 O. Gritsenko, et al, JCP 122, (2005) 9 Rohr, et al, JCP 129, (2008).

18 PNOF PNOFn, n=1,6: Functionals Minimization Comparison with DFT Γ (2) pqrs = n p n q (δ pr δ qs δ ps δ qr )+λ pqrs [γ] Approximations for the cumulant part λ pqrs [γ]. PNOF5: Satisfies sum rule, positivity, particle-hole symmetry 10 Pairs (p, p): n p +n p = 1, N E PNOF5 = [n p (2H pp +J pp ) n p n p K p p ] + p=1 N p,q=1 q p,q p n p n q (2J pq K pq ) J pq,k pq : Coulomb and Exchange integrals; H pp single particle term 10 Piris et al, JCP 134, (2011).

19 Phase dependent functionals Functionals Minimization Comparison with DFT Löwdin-Shull 11 (exact for 2 electrons ): E ee = 1 f j f k nj n k K jk 2 j,k where f j = ±1. Usually f 1 f j = 1. Antisymmetrized product of strongly orthogonal geminals (APSG) 12. Phase dependent functionals are useful in TD-: occupation numbers from BBGKY TD equations are time dependent for phase dependent functionals. 11 LS, JCP 25, 1035 (1956). 12 Surjan, Topics in Current Chem , (1999)

20 Minimization j=1 Functionals Minimization Comparison with DFT We need to minimize the quantity ( F = E tot µ n j N ǫ jk j,k=1 d 3 rϕ j (r)ϕ k(r) δ jk ) Minimize with respect to n j and ϕ j Minimization with respect to n j can have border minima (pinned states) E 0 1 n j At the solution, µ = de/dn j, fractional n j

21 Minimization j=1 Functionals Minimization Comparison with DFT We need to minimize the quantity ( F = E tot µ n j N ǫ jk j,k=1 d 3 rϕ j(r)ϕ k (r) δ jk ) Minimize with respect to n j and ϕ j Minimization with respect to n j can have border minima (pinned states) At the solution, µ = de/dn j, fractional n j Minimization with respect to ϕ j is complicated; not a diagonalization problem. The Lagrangian Matrix ǫ jk is Hermitian at the extremum. Orbital optimization remains the bottleneck of.

22 Scaling of Functionals Minimization Comparison with DFT Scaling depends only on the Hilbert space size M (all orbitals are occupied) and not the number of electrons. Nominal scaling M 5. For comparison, DFT, HF: M 4 ; CI, CCSD(T), MP4: M 7 Orbital minimization can be extremely expensive. Several iterative schemes have been developed (effective Hamiltonian schemes), however the problem still remains.

23 Comparison with DFT Functionals Minimization Comparison with DFT Similarly to DFT, is not a variational method. The kinetic energy is a known functional of the 1RDM. On the contrary, it is NOT a functional of the density. Without the exact kinetic energy functional,dft resorts on the fictitious Konh-Sham (KS) system to restore the quantum mechanics. In no KS systems is required and does not exist. the quasiparticle picture (like KS) is not built in the theory. In KS-DFT, kinetic energy parts pollute the xc energy term. In, no such terms exist. Easier to create functionals. The KS system with a single Slater determinant is difficult to describe non-dynamic correlations. Due to the non-idempotency of the 1RDM, can describe more naturally non-dynamic correlation.

24 The price to pay! Functionals Minimization Comparison with DFT is much less efficient than DFT due to: Nominal scaling M 5 vs M 4. In principle, all orbitals are occupied. Extra occupation number optimization. Orbital optimization does not reduce to eigenvalue iterative problem. For routine calculations that DFT works well, there is no need to use. Where can be useful? In systems with strong non-dynamic correlations where single Slater description is poor: Molecular dissociation, diradicals, band gaps of semicunductors, insulators and highly correlated periodic systems.

25 H 2 dissociation Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap -0.7 E tot [a.u.] RHF CI Müller GU BBC d [Å] Energy of the H 2 molecule

26 Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Molecular dissociation with PNOF5 N 2 Molecule

27 Molecular dissociation with APSG Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap K. Pernal, J. Chem. Theory and Comput. 10, 4332 (2014).

28 Homogeneous electron gas Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Correlation Energy [Ha] Monte Carlo (Ortiz-Ballone) Müller Functional (Csányi-Arias) Müller Functional (Cioslowski-Pernal) Csányi-Arias CHF Csányi-Goedecker-Arias (CGA) BBC1 BBC r s [a.u.] Correlation energy of the HEG as a function of r s N.N.L. et al, Phys. Rev. B 75, (2007)

29 Power-functional for HEG Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap 0 Correlation Energy [a.u.] Exact Müller (α=0.50) α=0.55 α=0.60 α= r s Correlation energy for different α

30 Benchmark for finite systems Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Benchmark for 150 molecules and radicals (G2/97 test set) G* basis set, Comparison with CCSD(T) Method max δ δmax δe Müller (C 2 Cl 4 ) 135.7% 438.3% (Na 2 ) GU (C 2 Cl 4 ) 51.63% 114.2% (Si 2 ) BBC (C 2 Cl 4 ) 69.91% 159.1% (Na 2 ) BBC (C 2 Cl 4 ) 45.02% 125.0% (Na 2 ) BBC (SiCl 4 ) 18.37% 50.8% (SiH 2 ) PNOF (SiCl 4 ) 20.84% 59.1% (SiCl 4 ) PNOF (SiCl 4 ) 17.11% 46.0% (Cl 2 ) ML(cl. shell) (pyridine) 11.02% 35.7% (Na 2 ) MP (C 2 Cl 4 ) 11.86% 35.7% (Li 2 ) B3LYP (SiCl 4 ) 305.0% % (Li 2 ) N.L. et al, J. Chem. Phys. 128, (2008)

31 E c for finite systems Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap (a) -1.0 E c (a.u) (ref) E c (a.u.) Muller GU BBC1 BBC3 Correlation energy for finite systems

32 E c for finite systems Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap (b) E c (a.u) Muller GU PNOF PNOF (ref) E c (a.u.) Correlation energy for finite systems

33 E c for finite systems Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap 0.0 (c) -1.0 E c (a.u) MP2 B3LYP BBC3 PNOF (ref) E c (a.u.) Correlation energy for finite systems

34 Atomization energies Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap set 1 set 2 δ (%) δ max (%) δ (%) δmax (%) R(O)HF (F 2 ) (F 2 ) Mueller (Na 2 ) (Na 2 ) GU (ClF 3 ) (F 2 ) BBC (ClF 3 ) (O 2 ) BBC (ClO) (F 2 ) BBC (Li 2 ) (Li 2 ) PNOF (ClF 3 ) (F 2 ) PNOF (Li 2 ) (Cl 2 ) MP (Na 2 ) (Na 2 ) B3LYP (BeH) (F 2 ) set 1: G2/97 test set, 6-31G*-basis set 2: subset of 50 molecules, cc-pvdz-basis

35 One-electron spectrum Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap In there is no Kohn-Sham scheme which offers one-electron (quasiparticle) spectrum. Exact KS-DFT ǫ HOMO = IP; Koopmans theorem in HF. Two proposals and an approximate framework: Extended Koopmans theorem (EKT). IP is an eigenvalue of λ ij = ǫ jk nj n k Energy derivative at half occupancy ǫ i = E n i {nj}=optimal,n i=1/2 Local : Minimize functionals under the additional constraint that the orbitals satisfy a KS-like equation with a local potential.

36 Local Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Minimize functionals under the additional constraint that the orbitals are eigenfunctions of a single particle Hamiltonian with a local potential. ] [ 2 2 +v ext(r)+v rep (r) ϕ j (r) = ǫ j ϕ j (r). Local potential in is an approximation: true natural orbitals do not come from a local potential Total energy will be higher than full. Optimization of a local potential: smaller scale problem than orbital optimization. Are the obtained single-electron properties reasonable?

37 Ionization potential Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap All methods give errors of a few % in the calculation of IP s for a set of atoms/molecules (He, H2, LiH, H2O, HF, CH4, CO2, NH3, Ne, C2H4, C2H2). Local for two aromatic molecules 15 : 15 NNL et al, JCP 141, (2014)

38 Fundamental gap Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Extend the whole theory to fractional particle number Lagrange multiplier µ is the chemical potential µ(n) is a step function with step at integer N The fundamental gap is given as the discontinuity of µ at integer particle number 16 = I A = lim η 0 (µ(n +η) µ(n η)) 16 Helbig et al, Europhys. Lett. 77, (2007)

39 Results for LiH Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap 0 µ (Ha) GU Müller BBC1 BBC2 BBC3 BBC3 without SI PNOF M The discontinuity of µ at N = 4 electrons for LiH Helbig et al, Phys. Rev. A 79, (2009)

40 Fundamental gap Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap System Other Experiment µ(m) step I A theoretical Li Na F LiH 0.269,

41 Band gaps for solids Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap δ ( % ) DFT-LDA GW γ α 2 (α = 0.7) γ α 2 (α = 0.65) Ge Si GaAs BN C LiF LiH MnO NiO FeO CoO Sharma et al, PRB 78, R (2008)

42 Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap Equilibrium lattice parameter for solids Solid Expt. DFT-LDA Müller α = 0.7 Diamond Si BN : Absolute percentage deviation : Sharma et al, PRB 78, R (2008).

43 Conclusion Prorotype systems Correlation energies Quasiparticle spectrum Fundamental gap : a framework for electronic correlations where the 1RDM is the fundamental variable. Kinetic energy is an explicit functional of the 1RDM. No kinetic energy contributions in the xc energy term. No need for a KS auxiliary system. is not computationally efficient (M 5 scaling, Orbital optimization bottleneck). is a promising alternative to DFT Target: not to replace DFT but to give answers for problems the DFT results are not satisfactory is promising in systems with static correlations (molecular dissociation) Present results show that potentially it can be successful in strongly correlated periodic systems