To Freeze or Not to Freeze: playing with electron density

Transcription

1 To Freeze or Not to Freeze: playing with electron density T.A. Wesolowski, University of Geneva, Switzerland One-electron equations for embedded orbitals basic concepts, challenges for theory, possible applications Representative applications solids, biomolecules, liquids Orbital-free embedding potential vs. pseudpotential The origin of a possible confusion Rising the temperature interaction energies from fully variational calculations Summary

2 Motivation and objectives A sound formal framework to unify the quantum mechanical (orbital) and classical (orbital-free) levels of description of a complex system Its practical implementation of well-defined range of applicability Multi-level numerical simulations of complex materials

3 Key elements Basic descriptors of a multi-electron system: ρ B ρ B and {φ i emb(a), i=1,n A } Universal functionals of ρ B and ρ A = Σ i NA φ i emb(a) 2 E xc [ρ] and T s [ρ] Obtaining ρ A = Σ i NA φ emb(a) i 2 one-electron equations (reference system of non-interacting electrons as in the Kohn-Sham formulation of DFT) Second Hohenberg-Kohn theorem Wesolowski and Warshel, J. Phys. Chem. 1993, 97, 8050

4 The orbital-free embedding formalism formal framework One-electron equations * for embedded orbitals φ i emb(a) ρ A =Σ i=1 n i φ A i 2 The total energy: * Eqs in [Wesolowski and Warshel, J. Phys. Chem. 1993, 97, 8050]

5 The orbital-free embedding formalism formal framework One-electron equations for embedded orbitals φ i emb(a) A new challenge! Approximation to T s nad [ρ A,ρ B ] The total energy:

6 The orbital-free embedding formalism formal framework One-electron equations for embedded orbitals φ i emb(a) A new challenge! Approximation to T s nad [ρ A,ρ B ] The total energy:

7 The orbital-free embedding formalism formal framework One-electron equations for embedded orbitals φ i emb(a) The total energy: The starting point LDA: T s nad(lda) [ρ A,ρ B ] = C TF ((ρ A +ρ B ) 5/3 - ρ A 5/3 -ρ B 5/3 )dr

8 The orbital-free embedding formalism formal framework One-electron equations for embedded orbitals φ i emb(a) The total energy: T s nad(gga97) [ρ A,ρ B ]= T s LC94 [ρ A +ρ B ] - T s LC94 [ρ A ] - T s LC94 [ρ B ] Wesolowski et al., J. Chem. Phys. 105 (1996) 9182 Wesolowski, J. Chem. Phys. 106 (1997) 8516

9 Can one-electron equations for embedded orbitals lead to the exact ground-state electron density? ρ Ao =Σ i=1 n i φ A i 2 To satisfy the relation ρ exact = ρ B + ρ A0 the sufficient conditions are * : i) The functional derivatives of exact functionals (E xc [ρ] and T s nad [ρ A,ρ B ]) are used. ii) The chosen ρ B is such that ρ exact - ρ B is pure-state non-interacting v-representable. * T.A. Wesolowski One-electron equations for embedded electrons In: Computational Chemistry Review of Current Trends, vol. 10, J. Leszczynski Ed., World Scientific, pp. 1-82

10 To Freeze or Not to Freeze: playing with electron density T.A. Wesolowski, University of Geneva, Switzerland One-electron equations for embedded orbitals basic concepts, challenges for theory, possible applications Representative applications solids, biomolecules, liquids Orbital-free embedding potential vs. pseudpotential The origin of a possible confusion Rising the temperature interaction energies from fully variational calculations Summary

11 Complexation induced CD (ICD) USC Los Angeles, August 13, 2007 Courtesy of Johannes Neugebauer (ETHZ) Neugebauer&Baerends, J. Phys. Chem. A., 110 (2006) 8786

12 Complexation induced CD (ICD) USC Los Angeles, August 13, 2007 Courtesy of Johannes Neugebauer (ETHZ) Neugebauer&Baerends, J. Phys. Chem. A., 110 (2006) 8786

13 Complexation induced CD (ICD) USC Los Angeles, August 13, 2007 Courtesy of Johannes Neugebauer (ETHZ) Neugebauer&Baerends, J. Phys. Chem. A., 110 (2006) 8786

14 ligand-field theory parameters from first principles Lanthanide centres in chloroelpasolites M. Zbiri, M. Atanasov, C. Daul, J.-M. Garcia Lastra, T.A. Wesolowski, Chem. Phys. Lett. 397 (2004) 441 M. Zbiri, C. Daul, T.A. Wesolowski, J. Chem. Theor. & Comput., 2 (2006) 1106

15 Embedding case: LR-TDDFT using embedded orbitals in one subsystems dynamic response (and/or static) of one subsystem neglected (δρ B (ω)=0): USC Los Angeles, August 13, 2007 One-electron equations for embedded orbitals: Generalized formalism: LR-TDDFT using embedded orbitals in all subsystems [Casida and Wesolowski, Intl.J.Quant.Chem. 2004, 96, 577] working equations in the case when dynamic responses in all subsystems are treated on equal footing δρ A (ω) 0 and δρ B (ω) 0 [Wesolowski, J.Am. Chem. Soc. 2004,126,11444] extension to excited states

16 Multi-level simulations of solvates aminocoumarin c151 High level: For each instantaneous ρ B, the electronic excitation energies obtained from Casida equation and orbitals derived from oneelectron equations for embedded orbitals assuming that δρ B (ω)=0. Low level: ρ B is generated for each instantaneous geometry of the environment in a statistical ensemble or along the trajectory (Car- Parrinello or classical). Further simplifications to ρ B such as superposition of molecular electron densities. shift n-hexane water shift n-hexane-water ev (exp) ev (calc) J. Neugebauer, C.R. Jacob, T.A. Wesolowski, E.J. Baerends, J.Phys.Chem. A., 109 (2005) 7805 J. Neugebauer, M.J. Louwerse, E.J. Baerends, T.A. Wesolowski, J. Chem. Phys. 122 (2005)

17 Other applications of the orbital-free embedding potential [v eff emb [ρ A,ρ B ](r) as defined in Wesolowski&Warshel, 1993] Observable System Ref. Hyperfine tensor Mg + in rare gas matrices Wesolowski, Chem.Phys. Lett Ligand-field parameters Hyperfine tensor Mn impurity in fluoroperovskites Garcia-Lastra et al., J. Phys.: Condens. Matter 2006 Dipole moments Rg CO 2 complexes (Rg=He,..Xe) Jacob et al., J. Chem. Phys Hyperfine tensor H 2 NO-liquid water Neugebauer et al., J. Chem. Phys Singlet-triplet gap Mo-containing enzyme Leopoldini et al, Chem. Eur, J See also recent works by Warshel (EVB/PES), Visscher (NMR), Neugebauer (UV/Vis), and Carter, Wang, Kluner (related approach using also v eff emb [ρ A,ρ B ](r)].

18 To Freeze or Not to Freeze: playing with electron density T.A. Wesolowski, University of Geneva, Switzerland One-electron equations for embedded orbitals basic concepts, challenges for theory, possible applications Representative applications solids, biomolecules, liquids Orbital-free embedding potential vs. pseudpotential The origin of a possible confusion Rising the temperature interaction energies from fully variational calculations Summary

19 4 1 6 r2 + V eff [ρ A;~r] + V emb 2 Nuc X ~r] = [ρ; eff V emb Nuc B X ρ B ;~r] = A; [ρ eff V nadd s [ρ A ; ρ B ] ffit ffiρ A USC Los Angeles, August 13, 2007 ρ ~r 0 j~r 0 ~rj d~r0 + V xc [ρ (~r)] j~r 0 ~rj d~r ffi Ai = ffl Ai ffi Ai eff [ρ A; ρ B ;~r] i Z Z ~ R i j + j~r Orbital-free effective embedding potential Wesolowski and Warshel, J. Phys. Chem. 1993, 97, 8050 i=1 ~r 0 i Z Z ~ R i j + j~r ρ B i=1 + Vxc [ρ A (~r) + ρ B (~r))] Vxc [ρ A (~r)] +

20 emb Nuc B X ρ B ;~r] = A; [ρ eff V ffiρ A j~r 0 ~rj d~r0 How the orbital-free embedding potential looks like? USC Los Angeles, August 13, 2007 Example I ~r 0 i Z Z ~ R i j + j~r ρ B i=1 + Vxc [ρ A (~r) + ρ B (~r))] Vxc [ρ A (~r)] nadd s [ρ A ; ρ B ] ffit ρ A =F -, ρ B =H 2 O + Example II Example III ρ A =F -, ρ B =H 2 O ρ A =Ln 3+, ρ B =Cl 6 6-

21 emb Nuc B X ρ B ;~r] = A; [ρ eff V nadd s [ρ A ; ρ B ] ffit ffiρ A j~r 0 ~rj d~r0 Contributions to the orbital-free embedding potential USC Los Angeles, August 13, 2007 ρ A =F -, ρ B =H 2 O ~r 0 i=1 i Z Z ~ R i j + j~r ρ B electrostatic part + Vxc [ρ A (~r) + ρ B (~r))] Vxc [ρ A (~r)] + entire v eff emb [ρ A,ρ B ]

22 Case from the literature: Is the short range attraction reported by Truong and Stefanovitch* related to the wrong distribution of charge density (charge-leak problem)? *Truong and Stefanovitch, J. Chem. Phys., 104 (1996) 2946

23 T s nad [ρ A,ρ B ] in a model system 1) We consider a closed-shell system comprising 4 electrons. 2) Let φ 1 and φ 2 denote the canonical Kohn-Sham orbitals for this system. ρ o =2 φ φ 2 and <φ 1-2 φ 1 > + <φ 2-2 φ 2 > is minimal and equals to T s [ρ o ] 3) Subsystem densities: ρ α = 2α φ (1 α) φ 2 2 ρ 1 α = 2(1 α) φ α φ 2 2

24 T s nad [ρ A,ρ B ] in a model system 4) Embedded orbitals: φ α = 2-1/2 ρ α 1/2 φ 1 α = 2-1/2 ρ 1 α 1/2 Wesolowski, Mol Phys. 103 (2005) 1165

25 T s nad [ρ A,ρ B ] in a model system 4) Embedded orbitals: φ α = 2-1/2 ρ α 1/2 φ 1 α = 2-1/2 ρ 1 α 1/2 α=1 (or α=0): φ α and φ 1 α are just two orthogonal Kohn-Sham orbitals for this system ρ α and ρ 1 α are orbital densities ρ α =2 φ 1 2 and ρ 1-α =2 φ 2 2 T nad s [ρ 1,ρ 2 ]=T nad s [ρ α,ρ 1-α ]=0 Wesolowski, Mol Phys. 103 (2005) 1165

26 T s nad [ρ A,ρ B ] in a model system 4) Embedded orbitals: φ α = 2-1/2 ρ α 1/2 φ 1 α = 2-1/2 ρ 1 α 1/2 0<α<1: φ α and φ 1 α do not have to be orthogonal (they are identical for α=0.5) What is T s nad [ρ α,ρ 1-α ] in this case? Wesolowski, Mol Phys. 103 (2005) 1165

27 T s nad [ρ A,ρ B ] in a model system The exact formula for T s nad [ρ α,ρ 1-α ] reads: Wesolowski, Mol Phys. 103 (2005) 1165

28 T s nad [ρ α,ρ 1-α ] for non-orthogonal orbitals φ α and φ 1 α T s [ρ o ] = <φ 1-2 φ 1 > + <φ 2-2 φ 2 > = <φ α - 2 φ α > + <φ 1-α - 2 φ 1-α > + T s nad [ρ α,ρ 1-α ] Li + -H 2 T s nad [ρα,ρ 1-α ] [hartree] φ 1 ks φ 2 ks Bernard, Dulak, Kaminskim &TAW, to appear d(li-h)= R e

29 T s nad [ρ α,ρ 1-α ] for non-orthogonal orbitals φ α and φ 1 α T s [ρ o ] = <φ 1-2 φ 1 > + <φ 2-2 φ 2 > = <φ α - 2 φ α > + <φ 1-α - 2 φ 1-α > + T s nad [ρ α,ρ 1-α ] Li + -H 2 T s nad [ρα,ρ 1-α ] [hartree] Here φ α = φ 1 α!!! φ 1 ks φ 2 ks d(li-h)= Re

30 F - H 2 O Kohn-Sham reference Truong and Stefanovitch, J. Chem. Phys., 104 (1996) 2946 Dulak and Wesolowski, J. Chem. Phys. 124 (2006)

31 F - H 2 O Kohn-Sham reference The strange behaviour of the interaction energy at short intermolecular distances reported by Truong and Stefanovich is probably the results of an inadequate numerical implementation of the orbital-free embedding potential (LDA level)!!! Truong and Stefanovitch, J. Chem. Phys., 104 (1996) 2946 Dulak and Wesolowski, J. Chem. Phys. 124 (2006)

32 To Freeze or Not to Freeze: playing with electron density T.A. Wesolowski, University of Geneva, Switzerland One-electron equations for embedded orbitals basic concepts, challenges for theory, possible applications Representative applications solids, biomolecules, liquids Orbital-free embedding potential vs. pseudpotential The origin of a possible confusion Rising the temperature interaction energies from fully variational calculations Summary

33 Energy minimization: USC Los Angeles, August 13, 2007 What we want to optimize? No embedding: everything is optimized

34 Energy minimization: USC Los Angeles, August 13, 2007 What we want to optimize? Embedding: structurally rigid environment

35 Energy minimization: USC Los Angeles, August 13, 2007 What we want to optimize? Born-Oppenheimer approximation (Cortona formulation of DFT)

36 Energy minimization: USC Los Angeles, August 13, 2007 What we want to optimize? Embedding: frozen density of the environment (Wesolowski&Warshel)

37 Energy minimization: USC Los Angeles, August 13, 2007 What we want to optimize? Gordon-Kim model

38 Fully variational calculations at rigid geometry freeze-and-thaw cycle to get ground-state energy in Born-Oppenheimer approximation [- ½ 2 +v eff emb [ρ A,ρ B ](r)]φ ia =ε ia φ ia ρ B =Σ i=1 n B i φb i 2 [- ½ 2 +v eff emb [ρ B,ρ A ](r)]φ ib =ε ib φ ib ρ A =Σ i=1 n A i φa i 2 Subsystems = atoms in solids: [Cortona, Phys. Rev. B, 44, (1991) 8454] Subsystems = interacting molecules: [Wesolowski & Weber, Chem. Phys. Lett., 248, (1996) 71]

39 Pathways for reaching the minimum * on the Born- Oppenheimer potential energy surface Geometry Born-Openheimer ** Partial optimisation *** Partial optimisation *** and linearization {R AN } v eff emb [ρ in,ρ jn ] ρ AN, ρ B N v eff emb [ρ AN,ρ B N-1 ] ρ A N v eff emb [ρ BN,ρ AN ] ρ B N v eff emb [ρ A N-1,ρ B N-1 ] ρ A N v eff emb [ρ B N-1,ρ A N-1 ] ρ B N {R A N+1 } v eff emb [ρ i N+1,ρ j N+1 ] ρ A N+1, ρ B N+1 v eff emb [ρ A N+1,ρ BN ] ρ A N+1 v eff emb [ρ BN,ρ A N+1 ] ρ B N+1 v eff emb [ρ AN,ρ BN ] ρ A N+1 v eff emb [ρ BN,ρ AN ] ρ B N+1 * Example for two consecutive steps in minimization of only {R A } at fixed {R B } ** denotes the freeze-and-thaw cycle of self-consistent calculations to obtain ρ i and ρ j *** denotes one self-consistent calculation to obtain ρ i (i=a or B)

40 Three pathways to optimize the total geometry (both subsystems): 1) Always on the Born-Oppenheimer potential energy surface (freeze-and-thaw at each geometry) 2) Above the Born-Oppenheimer potential energy surface (for each geometry of one subsystem the electron density of the other together with its geometry are frozen) 3) Above the Born-Oppenheimer potential energy surface (for each geometry of one subsystem the electron density of the other together with its geometry are frozen and linearized effective embedding potential is used ) Practically identical equilibrium geometries* at the end of the optimization of {R A }, {R B }, ρ A, and ρ B. *For two cases studied in detail (water dimer and HCl-CH 3 SH), atomic positions agree within 0.01Å. Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

41 yz-projection Equilibrium geometries (CED-LDA) HCONH 2 dimer xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int = (-14.94) kcal/mol d NO =2.89 (2.88) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

42 yz-projection Equilibrium geometries (CED-LDA) HCOOH-HCOOH xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int = (-16.15) kcal/mol d OO =2.81 (2.70) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

43 Equilibrium geometries (CED-LDA) H 2 O-NH 3 yz-projection xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int =-6.72 (-6.41) kcal/mol d NO =2.97 (2.97) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

44 Equilibrium geometries (CED-LDA) H 2 O dimer yz-projection xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int =-4.97 (-4.97) kcal/mol d OO =2.96 (2.94) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

45 Equilibrium geometries (CED-LDA) HF-HF yz-projection xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int =-4.12 (-4.57) kcal/mol d FF =2.87 (2.78) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

46 Equilibrium geometries (CED-LDA) NH 3 -NH 3 yz-projection xz-projection wave-function benchmark * subsystem DFT (LDA) benchmark ** E int =-3.99 (-3.15) kcal/mol d NN =3.14 (3.27) Å * Zhao & Truhlar database: J. Chem. Theor.& Comput., (2005) ** Dulak et al, J. Chem. Theor.& Comput., 3 (2007) 735

47 Summary USC Los Angeles, August 13, 2007 One-electron equations for embedded orbitals: Only universal density functionals Approximating the non-additive kinetic energy functional: A new challenge for theory Simple approximations to the orbital-free effective embedding potential: (and the parent functional): LDA and GGA97 have large range of applicability (small overlap between ρ A and ρ B such as in the case of hydrogen-bonded complexes) Multi-level modelling: The interface between a subsystem described at the Kohn-Sham resolution and the subsystem described without orbitals (density only). Orbital-free effective embedding potential V eff emb [ρ A,ρ B ](r): V eff emb [ρ A,ρ B ](r) is different for different guests in the same ρ B environment. V eff emb [ρ A,ρ B ](r) changes during SCF. During SCF, small changes in V eff emb [ρ A,ρ B ](r). Linearized V eff emb [ρ A,ρ B ](r), frequently applicable.

48 Orbital-free effective embedding potential vs. conventional embedding techniques q 1 ε Conventional embedding calculations: i) To obtain Ψ emb, only electrostatic terms * in V emb. ii) The non-electroststatic contributions to energy are added a posteriori. Ψ emb q 2 q 3 (H 0 +V emb )Ψ emb = E emb Ψ emb iii) Extensive use of empirical information about the environment: atomic charges, group dipole moments, polaizability for V emb and London-type formula for atom-atom interactions for a posteriori energy corrections. * Some methods use a generalization of the idea of pseudopotental to account for othogonality in V emb.

49 4 1 6 r2 + V eff [ρ A;~r] + V emb eff 2 Nuc X ~r] = [ρ; eff V emb Nuc B X ρ B ;~r] = A; [ρ eff V ρ ~r 0 nadd s [ρ A ; ρ B ] ffit j~r 0 ~rj d~r0 USC Los Angeles, August 13, 2007 Orbital-free effective embedding potential vs. conventional embedding techniques 2 3 q 1 ε 7 A; ρ B ;~r] 5 ffi Ai = ffl Ai ffi Ai [ρ i Z Z ~ R i j + j~r j~r 0 ~rj d~r0 + V xc [ρ (~r)] i=1 Ψ emb ~r 0 i Z Z ~ R i j + j~r ρ B i=1 q 2 q 3 + Vxc [ρ A (~r) + ρ B (~r))] Vxc [ρ A (~r)] (H 0 +V emb )Ψ emb = E emb Ψ emb + Information about the environment built in using empirical descriptors (q, μ, α, pseudopotential ) ffiρ A - environment characterized by ρ B only - universal density functionals - virtually exact

50 Acknowledgements M. Casida (Grenoble): general formalism LR-TDDFT for embedded systems E.J. Baerends, J. Neugebauer, and the ADF team (Amsterdam): ADF implementation, testing properties, multi-level simulations Y. Bernard, R. Kevorkiants, M. Dulak, J. Kaminski (Geneva): demon2k implementation, model systems, tests $ FNRS