Subsystem Quantum Chemistry

Transcription

1 Subsystem Quantum Chemistry Johannes Neugebauer Graduate Course on Theoretical Chemistry and Spectroscopy Han-sur-Lesse, December 2009

2 I. Non-Separability of Quantum Systems!?

3 Description of Quantum Systems To describe a quantum system, we have to construct a Hamiltonian: add kinetic-energy operators for all elementary particles (in chemistry: electrons and nuclei) add interaction operators for all interactions between elementary particles (in chemistry: Coulomb interactions)

4 Description of Quantum Systems To describe a quantum system, we have to construct a Hamiltonian: add kinetic-energy operators for all elementary particles (in chemistry: electrons and nuclei) add interaction operators for all interactions between elementary particles (in chemistry: Coulomb interactions) find wavefunction Ψ that fulfills Schrödinger Equation with this Hamiltonian

5 Description of Quantum Systems To describe a quantum system, we have to construct a Hamiltonian: add kinetic-energy operators for all elementary particles (in chemistry: electrons and nuclei) add interaction operators for all interactions between elementary particles (in chemistry: Coulomb interactions) find wavefunction Ψ that fulfills Schrödinger Equation with this Hamiltonian note: Ψ = Ψ(x 1, x 2,..., x n, X 1, X 2,..., X N, t) x = electronic coordinates (incl. spin), X = nuclear coordinates, t = time

6 Non-Separability of Quantum Systems all charged particles interact via Coulomb forces in a strict sense, there are no isolated systems (and we have to find the wavefunction of the universe) but: experimental evidence shows that systems can be treated as if they were isolated Can we separate a quantum system into parts (electrons, atoms, functional groups, molecules,...)? a posteriori approach Can we calculate the electronic structure of a big quantum chemical system based on its constituents? a priori approach

7 Simplification of the Wavefunction for stationary states, we can separate t from x, X Born Oppenheimer approximation: allows to separate x from X harmonic oscillator approximation: allows to obtain 3N separate equations for nuclear coordinates common problem in quantum chemistry: find Ψ(x 1, x 2,..., x n ) for fixed coordinates X 1, X 2,..., X N find eigenfunctions of Ĥ e = ˆT e + ˆV ne + ˆV ee = n 2 i 2 i I n i Z I n R I r i + 1 r i r j i<j

8 Hartree Atomic Units... are used throughtout e 1 m el 1 1 4πε 0 1

9 Hartree Atomic Units... are used throughtout e 1 a.u. of charge m el 1 a.u. of mass 1 a.u. of action 4πε 0 1 a.u. of permittivity

10 Simplification of the Electronic Wavefunction Hartree product: Π H (x 1, x 2,..., x n ) = φ 1 (x 1 ) φ 2 (x 2 )... φ n (x n ) electrons have individuality (can be labeled) in line with intuitive chemical concepts violates Pauli principle Hartree Fock wavefunction: antisymmetrized Hartree product Φ HF (x 1, x 2,..., x n ) = ÑÂφ 1 (x 1 ) φ 2 (x 2 )... φ n (x n ) electrons have no individuality (indistinguishable) in line with Pauli principle introduces difficulties for intuitive concepts

11 Hartree Fock Equations canonical Hartree Fock equations: ĥ + j (Ĵ j ˆK j ) φ i = ɛ i φ i with n ĥ = 2 2 Z I r R I Ĵ j = φ j 1/r 12 φ j ˆK j φ i = φ j 1/r 12 φ i φ j I solution: basis set expansion φ i = µ c µi χ µ

12 Solution of the Hartree Fock Equations insert ansatz into Hartree Fock equations, ˆf c µi χ µ = ɛ i c µi χ µ µ µ where ˆf = ĥ + j (Ĵ j ˆK j ) multiply from the left with χ ν χ ν ˆf χ µ c µi = ɛ i c µi χ ν χ µ }{{}}{{} µ µ f νµ S νµ or fc i = ɛ i Sc i

13 Population Analysis: Are There Atoms in Molecules? Integral over sum of orbital densities: n i φ i 2 dr = = n i m c αi c βi αβ χ α χ β dr ( m n ) c αi c βi S αβ = αβ i m P αβ S αβ = n atoms are no elementary units in quantum chemistry but: we can use several partitioning schemes to assign effective charges to atomic centers Mulliken electron population of atom A: n A = P αβ S αβ α A gross charges: Q A = Z A n A β αβ

14 Population Analysis Density-based schemes avoid basis set problems: Bader charges: Q A = Z A ρ(r)dr atomic basin (defined through ρ(r) n(r) = 0) Hirshfeld charges: Q A = Z A p A (r)ρ(r)dr with p A (r) = ρ promolecule (r) = A ρfree atom A (r) ρ promolecule (r) ρ free atom A (r) similar: Voronoi charges (hard boundaries)

15 Orbitals in Hartree Fock Theory canonical HF orbitals are often delocalized energy and properties remain unchanged under unitary transformation among occupied orbitals φ i (x) = n U ik φ k or φ = U φ k=1 localized orbitals: often more useful for chemical concepts Boys Foster localization: find U that minimizes distance between electrons in a spatial orbital φ i φ i (r 12 ) 2 φ i φ i. i Pipek Mezey localization: maximize sum of Mulliken charges

16 Canonical vs. Localized Orbitals canonical orbitals localized orbitals (Boys Foster)

17 Quasi-Electrons Edminston Ruedenberg (ER) localization: find U such that exchange is minimized, φ i (x 1 ) φ j (x 1 ) 1 φ r j (x 2 ) φ i (x 2 )dx 1 dx 2 = min 12 i<j (alternative formulation: maximize self-energy) ER orbitals: often similar to Hartree orbitals (exchange completely missing) Hartree orbitals describe the quasi-electrons of chemists H. Primas, U. Müller Herold, Elementare Quantenchemie

18 II. Subsystem Approaches in Quantum Chemistry

19 Group Functions subsystems A and B will be distinct if each is described by its own wavefunction, Ψ(x 1, x 2,..., x na +n B ) = Ψ A (x 1,..., x na )Ψ B (x na +1,..., x na +n B ) Pauli principle requires: Ψ(x 1, x 2,..., x na +n B ) = ÑÂ [Ψ A (x 1,..., x na )Ψ B (x na +1,..., x na +n B )], (unless separation A B becomes very large; exact for non-interacting subsystems) can be starting point for accurate calculations (if Ψ A, Ψ B are highly accurate) note: Slater determinant is included as a special case

20 Group Functions E E b Ψ 1b Ψ 2b E a Ψ 1a Ψ 2a monomer 1 monomer 2 example for group functions: discussion of energy-transfer phenomena: DA, D A, DA basis for exciton models

21 Hybrid Methods General idea: calculate energy of active subsystem (1) with accurate method (M 1 ) add correction for interaction with environment (2) based on less accurate method (M 2 ) E tot = E M E M E M (additive scheme) E tot = E M E M 2 (1+2) EM 2 1 (subtractive scheme) since E M 2 (1+2) = EM E M E M can be extended to several layers examples: QM/MM, WF/DFT, ONIOM, QM/PCM

22 Hybrid Methods simple hybrid methods only partition the energy (mechanical embedding; wavefunction/density (1) unchanged) electronic/hamiltonian embedding: wavefunction (1) polarized by environment [ 1 ] v HF/KS (r) + v emb (r) φ i (r) = ɛ i φ i (r) e.g. v emb (r) = A MM q A r R A more advanced schemes consider back-polarization of the environment by the active system

23 Increment Methods General idea: largest part of total energy is due to energies of (isolated) subsystems interaction energy: mainly due to pair interactions general energy expression: E tot = I E I + I<J E I J = E IJ E I E J E I J + I<J<K E I J K +... E I J K = E IJK E I E J (E IJ E I E J ) (E IK E I E K ) (E JK E J E K ) nth-order increment method: exact for n subsystems

24 Other Subsystem Methods methods that partition the correlation energy ( increment methods for E corr ) correlation methods based on localized orbitals conventional methods (e.g., CI-type): Ψ = c HF Φ HF + ia c a i Φ a i + ijab c ab ij Φ ab ij +..., local methods: create a domain [i] of virtual orbitals spatially close to φ i (r) include all excitations from φ i to [i] include all double excitations from φ i,j to pair domain [i, j] group pair domains into strong, weak, and distant pairs methods that partition the density matrix What about density-based schemes?

25 III. Subsystem Density-Functional Theory

26 Some Basics of DFT General Idea: calculate energy based on observable real-space quantity ρ(r) Formal basis: 1st HK theorem: The ground-state electron density determines the external potential (and thus Ĥ, Ψ i, E i, and all properties). 2nd HK theorem: We can obtain the electronic ground-state energy E 0 from a variational procedure: E 0 = min ρ {ρ} v E[ρ]

27 Minimization of the Energy Functional minimize energy subject to constraint: ρ(r)dr = n construct Lagrange functional: [ L[ρ] = E[ρ] µ ] ρ(r)dr n and minimize unconstrained minimization condition: δl[ρ] = or δl[ρ] δρ(r) δρ(r)dr = 0 δl[ρ] δρ(r) = 0 δe[ρ] δρ(r) µ = 0 µ = δe[ρ] δρ(r) (Euler Lagrange equation)

28 Some Basics of DFT Partitioning of the energy: E[ρ] = V ext [ρ] + V ee [ρ] + T[ρ] where V ext [ρ] = ρ(r)v ext (r)dr Z A v ext (r) = (most often) r R A A V ee [ρ] = Ψ 1 r i<j ij Ψ = 1 ρ(r1 )ρ(r 2 ) dr 1 dr 2 2 r }{{ 12 } J[ρ] T[ρ] = Ψ i 2 i 2 Ψ =??? ddd } {{ } T TF [ρ] + V ee,nc [ρ] }{{}??? (very crude)

29 Some Basics of DFT Partitioning of the energy: E[ρ] = V ext [ρ] + V ee [ρ] + T[ρ] where V ext [ρ] = ρ(r)v ext (r)dr Z A v ext (r) = (most often) r R A A V ee [ρ] = Ψ 1 r i<j ij Ψ = 1 ρ(r1 )ρ(r 2 ) dr 1 dr 2 + V ee,nc [ρ] 2 r }{{ 12 }{{}}??? T[ρ] = Ψ i 2 i 2 J[ρ] Ψ C F ρ 5/3 (r)dr }{{} T TF [ρ] (very crude)

30 Functional Derivatives functional derivative: s(x) = δf[f ] δf (x) is defined through: F[f (x) + δf (x)] = F[f (x)] + s(x)δf (x)dx + O(δf 2 ) simple example: V ext [ρ(r) + δρ(r)] = = v ext (r)[ρ(r) + δρ(r)]dr v ext (r)ρ(r)dr + v ext (r) δρ(r)dr }{{}}{{} δvext[ρ] V ext[ρ] δρ(r)

31 Functional Derivatives Coulomb energy J[ρ + δρ] = 1 (ρ(r1 ) + δρ(r 1 ))(ρ(r 2 ) + δρ(r 2 )) dr 1 dr 2 2 r 1 r 2 = 1 ρ(r1 )ρ(r 2 ) 2 r 1 r 2 dr 1dr ρ(r1 )δρ(r 2 ) dr 1 dr 2 2 r 1 r δρ(r1 )ρ(r 2 ) dr 1 dr δρ(r1 )δρ(r 2 ) dr 1 dr 2 2 r 1 r 2 2 r 1 r 2 ( ) ρ(r 2 ) = J[ρ] + r 1 r 2 dr 2 δρ(r 1 )dr 1 + O[(δρ(r)) 2 ] }{{} v Coul (r 1 ) δj[ρ] δρ(r) = v Coul(r)

32 Thomas Fermi Theory T TF [ρ]: exact kinetic energy for homogeneous electron gas energy functional and Lagrangian: E TF [ρ] = V ext [ρ] + J[ρ] + T TF [ρ] [ ] L TF [ρ] = E TF µ ρ(r)dr n minimization condition: µ = δe TF[ρ] δρ(r) = 5 3 C Fρ 2/3 (r) + v ext (r) + v Coul (r) very simple DFT model extension: Dirac exchange Ex Dirac [ρ] = C x ρ 4/3 (r)dr

33 Kinetic-Energy Functionals: Another Example assume one-electron system, ρ(r) = φ(r) 2 T = T[φ] = 1 φ(r) 2 φ(r)dr 2 = 1 ( φ(r))( φ(r))dr 2 = 1 ( ) ( ρ(r) ) ρ(r) dr 2 = 1 ( ) ( ) ρ(r) ρ(r) 1 2 ρ(r) ρ(r) dr = 1 ρ(r) 2 dr 8 ρ(r) von Weizsäcker functional T vw [ρ]

34 Kohn Sham DFT Non-interacting System consider system of non-interacting electrons : Ĥ s = ˆT s + ˆV s = n i=1 [ 1 ] 2 2 i + v s (r i ) exact wavefunction for this system: antisymmetrized product (Slater determinant) of orbitals φ i, [ 1 ] v s (r) φ i (r) = ɛ i φ i (r) exact kinetic energy of this system: T s [{φ i }] = φ 2 i 2 φ i i

35 Kohn Sham DFT Non-interacting System ground-state density of this system: occ ρ(r) = φ i (r) 2 this density fulfills (2nd HK theorem): (Euler Lagrange eq.) µ = δe s[ρ] δρ(r) = δt s[{φ i }] δρ(r) i + v s (r)

36 Kohn Sham DFT Central assertion in Kohn Sham theory: For any interacting system of electrons, there is a potential v s (r), so that the ground-state density of the interacting system equals that of a non-interacting system with external potential v s (r). ρ 0 is v s -representable use T s [{φ i }] as approximation for T[ρ] E[ρ] = T s [{φ i }] + V ext [ρ] + J[ρ] + (V ee [ρ] J[ρ] + T[ρ] T s [{φ i }]) = T s [{φ i }] + V ext [ρ] + J[ρ] + E xc [ρ]

37 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r)

38 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r) = δt[{φ i}] δρ(r) + δj[ρ] δρ(r) + δv ext[ρ] δρ(r) + δe xc[ρ] δρ(r)

39 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) + δj[ρ] δρ(r) + δv ext[ρ] δρ(r) + δe xc[ρ] δρ(r) + v Coul (r) + v ext (r) + v xc (r)

40 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) + δj[ρ] δρ(r) + δv ext[ρ] δρ(r) + δe xc[ρ] δρ(r) + v Coul (r) + v ext (r) + v xc (r) + v eff (r)

41 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) + δj[ρ] δρ(r) + δv ext[ρ] δρ(r) + δe xc[ρ] δρ(r) + v Coul (r) + v ext (r) + v xc (r) + v eff (r) is fulfilled by the density of the non-interacting system choose v s (r) = v eff (r) and solve [ 1 ] v eff (r) φ i (r) = ɛ i φ i (r) (Kohn Sham equations)

42 Kohn Sham DFT Euler Lagrange equation for the interacting system: µ = δe[ρ] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) = δt[{φ i}] δρ(r) + δj[ρ] δρ(r) + δv ext[ρ] δρ(r) + δe xc[ρ] δρ(r) + v Coul (r) + v ext (r) + v xc (r) + v eff (r) is fulfilled by the density of the non-interacting system choose v s (r) = v eff (r) and solve [ 1 ] v eff [ρ](r) φ i (r) = ɛ i φ i (r) (Kohn Sham equations)

43 Kohn Sham DFT Remaining problem: E xc [ρ] often partitioned into E xc [ρ] = E x [ρ] + E c [ρ] several strategies for the development of approximations exactly solvable models (local density approximation, LDA) generalized gradient approximation (GGA; may involve exact constraints) Meta-GGAs empirical fits Hybrid functionals generalized random phase approximation (RPA) G.E. Scuseria, V.N. Staroverov, in: C.E. Dykstra, G.Frenking, K.S. Kim, G.E. Scuseria (Eds.) Theory and Applications of Computational Chemistry: The First Forty Years, Elsevier, Amsterdam 2005, pp

44 Density-Partitioning Schemes General idea: HK theorems: E 0 = E[ρ 0 ] = min ρ E[ρ] electron density is a real-space quantity additive for different subsystems, ρ tot (r) = I p I (r)ρ tot (r) = I ρ I (r) problems: how can we obtain the (optimum) ρ I (r)? how can we approximate the energy functional?

45 Density Partitioning: Early Attempts Zeroth-order approximation: start from unperturbed densities (Gombas, 1949) ρ tot (r) = I ρ 0 I (r) Gordon Kim model (1972): Hartree Fock monomer densities, ρ(r) = ρ tot (r) = ρ HF A (r) + ρ HF B (r) Thomas Fermi kinetic energy functional, E GK [ρ] = T TF [ρ] + E Coul [ρ] + Ex Dirac [ρ] + Ec LDA [ρ] + ρ(r)v ext (r)dr

46 Gordon Kim Model Equilibrium distances of rare gas dimers: (in units of Å) calc. exp. Ne Ne Ar Ar Kr Kr interaction energies: quite good, better than KS-DFT R.G. Gordon, Y.S. Kim, J. Chem. Phys. 56 (1972), 3122.

47 Gordon Kim Model Equilibrium distances of rare gas dimers: (in units of Å) calc. exp. Ne Ne Ar Ar Kr Kr interaction energies: quite good, better than KS-DFT R.G. Gordon, Y.S. Kim, J. Chem. Phys. 56 (1972), Why did this work?

48 The Gordon Kim Model was mainly applied to rare gas dimers very weak density perturbations was not used self-consistently avoids problems of kinetic-energy potentials to reproduce shell-structure in atomic densities was applied to study interaction energies only, E int GK = V nn + E GK [ρ] (E GK [ρ a ] + E GK [ρ b ]) atomic densities taken from Hartree Fock calculations both kinetic-energy and exchange correlation energy have wrong distance dependence cancellation of several error sources

49 Energy Functional without T[ρ] KS orbital energies: occ ɛ i = occ i i occ = φ i 2 /2 φ i + dr ] φ i (r) 2 (v Coul (r) + v xc (r) + v ext (r)) [ occ φ i 2 /2 φ i + dr [ρ(r)(v Coul (r) + v xc (r) + v ext (r))] i i

50 Energy Functional without T[ρ] KS orbital energies: occ ɛ i = occ i i occ = i φ i 2 /2 φ i + dr ] φ i (r) 2 (v Coul (r) + v xc (r) + v ext (r)) [ occ φ i 2 /2 φ i + dr [ρ(r)(v Coul (r) + v xc (r) + v ext (r))] i total electronic energy: E[ρ] = = occ i occ φ i 2 /2 φ i ɛ i 1 2 i ρ(r)v Coul (r)dr ρ(r)v Coul (r)dr + E xc [ρ] + ρ(r)v xc (r)dr + E xc [ρ] ρ(r)v ext (r)dr

51 Energy Functional without T[ρ] KS orbital energies: occ ɛ i = occ i i occ = i φ i 2 /2 φ i + dr ] φ i (r) 2 (v Coul (r) + v xc (r) + v ext (r)) [ occ φ i 2 /2 φ i + dr [ρ(r)(v Coul (r) + v xc (r) + v ext (r))] i total electronic energy: E[ρ] = = occ i occ φ i 2 /2 φ i ɛ i 1 2 i ρ(r)v Coul (r)dr ρ(r)v Coul (r)dr + E xc [ρ] + ρ(r)v xc (r)dr + E xc [ρ] ρ(r)v ext (r)dr kinetic energy is implicitly contained in orbital energies

52 The Harris Functional assume density change is small: ρ(r) = ρ 0 (r) + δρ(r) define change in effective potential δv eff (r) = v eff [ρ](r) v eff [ρ 0 ](r) = [ v Coul [ρ](r) v Coul [ρ 0 ](r) ] + [ v xc [ρ](r) v xc [ρ 0 ](r) ] determine orbitals for fixed potential (no SCF) ( 1 ) v eff [ρ 0 ](r) φ i (r) = ɛ i φi (r) 1st order approximation for true orbital energies: occ i occ ɛ j i ɛ i + ρ(r)δv eff (r)dr

53 The Harris Functional E[ρ] approximation for total energy: = occ j occ j ɛ j + E xc [ρ] + ɛ j + E xc [ρ] + [ ρ(r) δv eff (r) 1 ] 2 v Coul[ρ](r) v xc [ρ](r) dr [ ] 1 ρ(r) 2 v Coul[ρ](r) v Coul [ρ 0 ](r) v xc [ρ 0 ](r) dr. XC term: E xc [ρ] ρ(r)v xc [ρ 0 ](r)dr E xc [ρ 0 δexc [ρ 0 ] ] + δρ(r) δρ(r)dr ρ 0 (r)v xc [ρ 0 ](r)dr = E xc [ρ 0 ] ρ 0 (r)v xc [ρ 0 ](r)dr δρ(r)v xc [ρ 0 ](r)dr

54 The Harris Functional Coulomb term: [ ] 1 ρ(r) 2 v Coul[ρ](r) v Coul [ρ 0 ](r) dr [ 1 = (ρ 0 (r) + δρ(r)) 2 (ρ0 (r ) + δρ(r )) r r ρ0 (r ] ) r r dr dr 1 ρ 0 (r)v Coul [ρ 0 ](r)dr 2 energy (to first order in δρ): E[ρ] occ i ɛ i only density ρ 0 appears! J. Harris, Phys. Rev. B 31 (1985), [ ] 1 ρ 0 (r) 2 v Coul[ρ 0 ](r) + v xc [ρ 0 ](r) dr + E xc [ρ 0 ].

55 Harris Functional: Results E b (ev) R e (bohr) ω e (mev) Harris KS exp. Harris KS exp. Harris KS exp. Be N F Cu J. Harris, Phys. Rev. B 31 (1985), 1770.

56 Subsystem DFT total density: ρ(r) = i φ super i (r) 2 partition into subsystem contributions: ρ(r) = I ρ I (r) write each ρ I in terms of subsystem orbitals ρ I (r) = i φ ii (r) 2 assume all φ ii are known (but not φ super i )

57 Subsystem DFT problem for calculations of KS energy: T s [{φ super i }] write formally exactly as [ or T s [{φ super i }] = I T s [{φ ii }] + T s [{φ super i }] I T s [{φ ii }] ] T s [{φ super i }, {{φ ij }}] = I T s [{φ ii }] + Ts nadd [{φ super i }, {{φ ij }}] introduce density-dependent approximation, T nadd s [{φ super i }, {{φ ij }}] Ts nadd [{ρ J }] = T s [ρ] I T s [ρ I ]

58 One-Particle Equations in Subsystem DFT energy functional: E[{ρ J }] = E ext [ρ] + E Coul [ρ] + E xc [ρ] + I T s [{φ ii }] + T nadd s [{ρ J }], choose no. of electrons per subsystem (N J ) construct Lagrangian L[{ρ J }] = E[{ρ J }] + ( ) µ I d 3 rρ I (r) N I I and minimize w.r.t. all ρ K Euler Lagrange equations: 0 = v ext (r) + v Coul [ρ](r) + v xc [ρ](r) + δt s[{φ ik }] δρ K (r) + δtnadd s [{ρ K }] + µ K δρ K (r)

59 Frozen-Density Embedding assume all subsystem densities are v s -representable subsystem orbitals can be obtained from ( 1 ) v sub eff [ρ, ρ I ](r) φ ii = ɛ ii φ ii, if we choose v sub eff [ρ, ρ I ](r) = v eff [ρ](r) + δt s[ρ] δρ(r) δt s[ρ I ] δρ I (r), the systems of non-interacting particles fulfill v sub eff [ρ, ρ I ](r) + δt s[{φ ii }] + µ I = 0 δρ I (r) these are the sought-for densities

60 Some References on Subsystem (TD)DFT Subsystem DFT: P. Cortona, Phys. Rev. B 44 (1991), 8454 Frozen density embedding (FDE): T.A. Wesolowski, A. Warshel, J. Phys. Chem. 97 (1993), FDE-TDDFT: M.E. Casida, T.A. Wesolowski, Int. J. Quant. Chem. 96 (2004), 577; T.A. Wesolowski, J. Am. Chem. Soc. 126 (2004), Subsystem TDDFT: JN, J. Chem. Phys. 126 (2007), FDE: C.R. Jacob, L. Visscher, J. Chem. Phys. 128 (2008), ADF implementation: JN, C.R. Jacob, T.A. Wesolowski, E.J. Baerends, J. Phys. Chem. A 109 (2005), 7805; C.R. Jacob, JN, L. Visscher, J. Comput. Chem. 29 (2008), overview over subsystem methods for spectroscopy: JN, ChemPhysChem (2009), DOI: /cphc

61 Embedding Potential define complementary density to ρ I (r) ρ compl. I (r) = ρ J (r) = ρ(r) ρ I (r) J,J I one-particle equations become ( 1 ) v eff [ρ I ](r) + v emb [ρ I, ρ compl. I ] φ ii = ɛ ii φ ii (Kohn Sham equations with constrained electron density, KSCED) embedding potential v emb [ρ I, ρ compl. I ](r) = J,J I v J ext(r) + v Coul [ρ J ](r) J,J I + {v xc [ρ](r) v xc [ρ I ](r)} + δt s[ρ] δρ(r) δt s[ρ I ] δρ I (r)

62 Solution of the KSCED equations 1 define subsystems (R A, Z A, A I, and N I ) 2 provide density guess for each subsystem; most common: solve KS equations for all isolated subsystems 3 loop over all subsystems: calculate embedding potential due to all other subsystems solve KSCED equations for currently active subsystem I update density ρ I if density change in system I is negligible: stop otherwise: next cycle in loop Note: step 2 is crucial for the definition of the subsystems

63 Frozen-Density Embedding common type of partitioning: active subsystem and environment environment has small (though non-negligible) influence on properties of active part construct density guess of the environment (ρ 2 (r)) determine embedding potential due to ρ 2 (r) calculate density of active part (ρ 1 ) orbital-free embedding, frozen DFT, FDE

64 Monomer vs. Supermolecular Expansion of subsystem densities main computational advantage with monomer basis sets: φ i1 = ν 1 c i1 ν 1 χ ν1, φ i2 = ν 2 c i1 ν 2 χ ν2 occ 1 ρ = c i1 ν 1 χ ν1 i 1 ν 1 2 occ 2 + c i2 ν 2 χ ν2 properties often converge faster with supermolecular basis: i 2 ν 2 2 φ i1/2 = ν 1 c i1/2 ν 1 χ ν1 + ν 2 c i1/2 ν 2 χ ν2 occ 1 2 occ 2 ρ = c i1 ν 1 χ ν1 + c i1 ν 2 χ ν2 + c i2 ν 1 χ ν1 + c i2 ν 2 χ ν2 ν 1 ν 1 i 1 (computationally [much] more demanding) ν 2 i 2 ν 2 2

65 Electron Densities from FDE: F H F F1 H F2 BCP1 BCP2 strong, symmetric hydrogen bonds ρ 1 : H F2, ρ 2 : F1 asymmetric fragments

66 Electron Densities from FDE: F H F F1 H F2 BCP1 BCP2 strong, symmetric hydrogen bonds ρ 1 : H F2, ρ 2 : F1 asymmetric fragments Density from FDE: K. Kiewisch, G. Eickerling, M. Reiher, JN, J. Chem. Phys. 128 (2008),

67 F H F : negative Laplacian L L frag L emb L super K. Kiewisch, G. Eickerling, M. Reiher, JN, J. Chem. Phys. 128 (2008),

68 v s -Representability Conditions 1 1 a) b) c) a) initial F fragment, b) fully relaxed, c) fully relaxed, ghost basis all ρ J must be v s -representable test case: ρ emb,exact water := ρ tot ρ F for H 2 O F negative areas in target density reduced upon relaxation K. Kiewisch, G. Eickerling, M. Reiher, JN, J. Chem. Phys. 128 (2008),

69 IV. (Subsystem) TDDFT

70 Time-Dependent DFT Formal Basis: Runge Gross theorem (1984): one-to-one correspondence between v ext (r, t) and ρ(r, t) Runge and Gross also developed an effective one-electron equation ( 1 ) v eff (r, t) ψ i (r, t) = i t ψ i(r, t) TDKS equations, where v eff (r, t) = v ext (r, t) + v Coul (r, t) + v xc (r, t) solutions (unperturbed system): ψ (0) i (r, t) = e iɛit φ i (r),

71 Linear Response TDDFT apply small perturbation δv eff (r, t) = δv pert [ e iωt + e iωt] = 2δv pert cos(ωt), write perturbed wavefunction as ψ i (r, t) = ψ (0) i (r, t) + δψ i (r, t) insert into TDKS equation, subtract unperturbed TDKS equations [ 1 ] v eff (r) + δv eff (r, t) δψ i (r, t) + δv eff (r, t)ψ (0) i (r, t) = i t δψ i(r, t) expand δψ i (r, t) into unperturbed functions δψ i (r, t) = r c ir (t)ψ (0) r (r, t)

72 Linear Response TDDFT solve for c is (to first order) ( ) c is (t) = 1 e [i(ω si+ω)]t 2 (ω si + ω) + e[i(ωsi ω)]t φ s δv pert φ i (ω si ω) with ω si = ɛ s ɛ i first-order change in the density δρ(r, t) = ρ(r, t) ρ (0) (r, t) = [ ] n r ψ r (0) (r, t)δψ r (r, t) + ψ r (0) (r, t)δψr (r, t) r (n r = occupation number of orbital r)

73 Frequency-Dependent Response insert expression for c rs into ansatz for δψ r (r, t) insert ansatz for δψ r (r, t) into δρ(r, t) identify Fourier components of δρ δρ(r, ω) = rs n r [ φr δv pert φ s φ s (r)φ r (r) ω ω sr φ r (r) can always be chosen real for real perturbations (electric fields): φ s δv pert φ r = δv pert sr φ s δv pert φ r φ ] r (r)φ s (r) ω + ω sr = δv pert rs occ occ and virt virt pairs do not contribute to δρ(r, ω)

74 The Perturbed Density Matrix rewrite δρ as δρ(r, ω) = ia = ia = ia [ ] 1 1 ω ω ai ω + ω ai [ ω + ωai ω 2 ωai 2 ω ω ] ai ω 2 ωai 2 [ ] ω ai 2 ω 2 ωai 2 δv pert ia φ i(r)φ a (r) δv pert ia φ i(r)φ a (r) δv pert ia φ i(r)φ a (r) = ia 2P ia (ω)φ i (r)φ a (r) where P ia (ω) = χ s ia(ω)δv pert ia χ s ω ia ia(ω) = ωia 2 = ω ai ω2 ω 2 ωai 2

75 Linear Response in the Potential external perturbation δv ext induces δρ δρ induces change in potential, δv eff δv ind δv pert ia = δv ext ia + δv ind ia, δv ind ia = φ i δv ind φ a = d 3 r 1 {φ i (r 1 )δv ind φ a (r 1 )} [ ] } = d 3 r 1 {φ i (r 1 ) d 3 δv eff (r 1 ) r 2 δρ(r 2 ) δρ(r 2) φ a (r 1 ) [ ( ) ] } = d 3 r 1 {φ i (r 1 ) d 3 1 r 2 r 1 r 2 + f xc(r 1, r 2 ) δρ(r 2 ) φ a (r 1 )

76 Exchange Correlation Kernel f xc (r 1, r 2 ) = Fourier transform of f xc (r, r, t, t ) = δv xc(r, t) δρ(r, t )... gives rise to causality problems typical approximation exchange part f xc (r, r ) = δv xc(r) δρ(r ) f x (r, r ) 4 9 C xρ 2/3 (r )δ(r r )

77 Linear Response in the Potential matrix elements of the induced potential: δv ind ia = jb 2K ia,jb P jb (ω) coupling matrix [ K ia,jb = d 3 r 1 {φ i (r 1 ) ( ) d 3 1 r 2 r 1 r 2 + f xc ] } φ j (r 2 )φ b (r 2 ) φ a (r 1 ) combine results for δv ind ia P ia (ω) = and δvext ia ω ia ω 2 ia ω2 δv ext ia 2K ia,jb P jb (ω) + jb }{{} δv pert ia

78 Response Equations in Matrix Form solve for δv ext ia jb ω jb δ ij δ ab 2K ia,jb }{{} M ia,jb ω 2 δ ij δ ab P jb (ω) = δv ext ia ω jb introduce matrix S S ia,jb = 1 (ɛ b ɛ j ) δ ijδ ab, re-write equation in matrix vector form: [ M + ω 2 S ] P(ω) = δv ext, S 1/2 [ M + ω 2 S ] } S 1/2 {{ S 1/2 } P(ω) = S 1/2 δv ext 1

79 Response Equations in Matrix Form formal solution for P: where P(ω) = S 1/2 S } 1/2 MS {{ 1/2 } +ω 2 1 Ω P(ω) = S 1/2 [ ω 2 1 Ω ] 1 S 1/2 δv ext 1 S 1/2 δv ext, Ω ia,jb = (ɛ a ɛ i ) 2 δ ij δ ab + 2 (ɛ a ɛ i )K ia,jb (ɛ b ɛ j )

80 Casida s Equation for Excitation Energies re-write matrix equation as [ Ω ω 2 1 ] S 1/2 P(ω) }{{} F consider δv ext 0: [ Ω ω 2 k 1 ] F k = 0 = S 1/2 δv ext identical to constrained variational treatment (Ziegler et al.)

81 Subsystem TDDFT partition density response δρ(r) = I δρ I (r) expand response density in products of subsystem orbitals δρ I (r) = (ia) I 2δP (ia)i φ ii (r)φ ai (r), perturbed density matrix P (ia)i (ω) = χ s (ia) I (ω)δv pert (ia) I δv pert (ia) I = δv ext (ia) I + δv ind (ia) I

82 Local Response Approximation total induced potential: ( δv ind I (r 1 ) = dr 2 fcxck tot δ2 T s [ρ] δρ 2 ρi ) δρ I (r 2 ) + f tot δρ J (r 2 ) Cxck J,J I

83 Local Response Approximation total induced potential: ( δv ind I (r 1 ) = dr 2 fcxck tot δ2 T s [ρ] δρ 2 ρi ) δρ I (r 2 ) + f tot δρ J (r 2 ) Cxck J,J I approximation for local excitations

84 Effective Kernel in Subsystem TDDFT matrix elements of the induced potential δv ind (jb) I = 2 (ia) J K eff (jb) I,(ia) J δp (ia)j where K eff (jb) I,(ia) J = d 3 r 1 { φ ji (r 1 )φ bi (r 1 ) ( ) } d 3 r 2 fcxck(r tot 1, r 2 ) δ2 T s [ρ I ] δρ(r 2 )δρ(r 1 ) δ IJ φ ij (r 2 )φ aj (r 2 ) total effective response kernel: f tot Cxck(r 1, r 2 ) = 1 r 1 r 2 + δ2 E xc [ρ] δρ(r 2 )δρ(r 1 ) + δ2 T s [ρ] δρ(r 2 )δρ(r 1 )

85 Subsystem TDDFT in Matrix Form perturbed density matrix δp (jb)i = χ s (jb) I δv ext (jb) I + 2 (ia) J K eff (jb) I,(ia) J δp (ia)j matrix equation for excitation energies [ Ω s ωk 2 ] F s k = 0

86 Approximate Solutions full problem: Ω AA Ω AB Ω AZ Ω BA Ω BB Ω BZ Ω ZA Ω ZB Ω ZZ ω2 k F A k F B k.. F Z k = 0 A 0 B. 0 Z 1 diagonalize subsystem problems yields U A, U B,... (solution factor matrices) within local response approximation 2 construct supermatrix U containing U A, U B blocks on the diagonal 3 transform equation by U but: that still requires all couplings JN, J. Chem. Phys. 126 (2007),

87 Approximate Solutions full problem: Ω AA Ω BB... ΩZZ ω2 k F A k F B k.. F Z k = 0 A 0 B. 0 Z 1 diagonalize subsystem problems yields U A, U B,... (solution factor matrices) within local response approximation 2 construct supermatrix U containing U A, U B blocks on the diagonal 3 transform equation by U but: that still requires all couplings JN, J. Chem. Phys. 126 (2007),

88 Approximate Solutions full problem: Ω AA Ω AB Ω AZ Ω BA Ω BB Ω BZ Ω ZA Ω ZB Ω ZZ ω2 k F A k F B k.. F Z k = 0 A 0 B. 0 Z 1 diagonalize subsystem problems yields U A, U B,... (solution factor matrices) within local response approximation 2 construct supermatrix U containing U A, U B blocks on the diagonal 3 transform equation by U but: that still requires all couplings JN, J. Chem. Phys. 126 (2007),

89 Approximate Solutions full problem: ω 2 A,0 Ω AB Ω AZ Ω BA ω 2 B,0 Ω BZ Ω ZA Ω ZB ω 2 Z,0 ω2 k F A k F B k.. F Z k = 0 A 0 B. 0 Z 1 diagonalize subsystem problems yields U A, U B,... (solution factor matrices) within local response approximation 2 construct supermatrix U containing U A, U B blocks on the diagonal 3 transform equation by U but: that still requires all couplings JN, J. Chem. Phys. 126 (2007),

90 Approximate Solutions full problem: ω 2 A,0 Ω AB Ω AZ Ω BA ω 2 B,0 Ω BZ Ω ZA Ω ZB ω 2 Z,0 ω2 k F A k F B k.. F Z k = 0 A 0 B. 0 Z 1 diagonalize subsystem problems yields U A, U B,... (solution factor matrices) within local response approximation 2 construct supermatrix U containing U A, U B blocks on the diagonal 3 transform equation by U but: that still requires all couplings JN, J. Chem. Phys. 126 (2007),

91 Selective Inclusion of Couplings FDE often works fine without intersystem couplings

92 Selective Inclusion of Couplings FDE often works fine without intersystem couplings critical cases: identical chromophores, degenerate excitations

93 Selective Inclusion of Couplings FDE often works fine without intersystem couplings critical cases: identical chromophores, degenerate excitations new protocol: 1 calculate most important eigenvectors for subsystems to be coupled

94 Selective Inclusion of Couplings FDE often works fine without intersystem couplings critical cases: identical chromophores, degenerate excitations new protocol: 1 calculate most important eigenvectors for subsystems to be coupled 2 calculate intersystem couplings only for these states (or a subset of them)

95 Selective Inclusion of Couplings FDE often works fine without intersystem couplings critical cases: identical chromophores, degenerate excitations new protocol: 1 calculate most important eigenvectors for subsystems to be coupled 2 calculate intersystem couplings only for these states (or a subset of them) 3 diagonalize the corresponding subblock of Ω sub

96 Selective Inclusion of Couplings FDE often works fine without intersystem couplings critical cases: identical chromophores, degenerate excitations new protocol: 1 calculate most important eigenvectors for subsystems to be coupled 2 calculate intersystem couplings only for these states (or a subset of them) 3 diagonalize the corresponding subblock of Ω sub requires matrix elements between subsystem excitations µ A, ν B : Ω µa ν B = U (ia)a µ A Ω (ia)a (jb) B U (jb)b ν B (ia)a (jb) B

97 A Closer Look at the Coupling Matrix Elements explicit expression for Ω µa ν B : Ω µaν B = dr 1 2 U (ia)aµ ω(ia)a A φ ia φ aa (ia) A = dr 1 dr 2 2ρ t µ A (r 1 )f tot Cxckρ t ν B (r 2 ) dr 2 fcxck tot U (jb)bν ω(jb)b B φ jb φ bb (jb) B

98 A Closer Look at the Coupling Matrix Elements explicit expression for Ω µa ν B : Ω µaν B = dr 1 2 U (ia)aµ ω(ia)a A φ ia φ aa (ia) A = dr 1 dr 2 2ρ t µ A (r 1 )f tot Cxckρ t ν B (r 2 ) dr 2 fcxck tot U (jb)bν ω(jb)b B φ jb φ bb (jb) B interaction between transition densities of local excitations

99 A Closer Look at the Coupling Matrix Elements explicit expression for Ω µa ν B : Ω µaν B = dr 1 2 U (ia)aµ ω(ia)a A φ ia φ aa (ia) A = dr 1 dr 2 2ρ t µ A (r 1 )f tot Cxckρ t ν B (r 2 ) dr 2 fcxck tot U (jb)bν ω(jb)b B φ jb φ bb (jb) B interaction between transition densities of local excitations 2 k EET Ω µa ν B

100 Coupled FDE: Interacting Dimers 4.95 excited state energy / ev isolated monomer distance / Angstroem π π in benzaldehyde (BP86/TZP, 20 monomer states):

101 Coupled FDE: Interacting Dimers 4.95 excited state energy / ev isolated monomer FDE uncoupled distance / Angstroem π π in benzaldehyde (BP86/TZP, 20 monomer states): uncoupled FDE: slight shift in monomer excitation

102 Coupled FDE: Interacting Dimers 4.95 excited state energy / ev isolated monomer FDE uncoupled supermolecule distance / Angstroem π π in benzaldehyde (BP86/TZP, 20 monomer states): uncoupled FDE: slight shift in monomer excitation supermolecular TDDFT: splitting between monomer excitations

103 Coupled FDE: Interacting Dimers 4.95 excited state energy / ev isolated monomer FDE uncoupled supermolecule FDE coupled distance / Angstroem π π in benzaldehyde (BP86/TZP, 20 monomer states): uncoupled FDE: slight shift in monomer excitation supermolecular TDDFT: splitting between monomer excitations coupled FDE: splitting is reproduced JN, J. Chem. Phys. 126 (2007),

104 Natural Light-Harvesting Complexes Structure of LH2 of Rhodopseudomonas acidophila main pigments: Bchl a G. McDermott et. al, Nature 374 (1995), 517.

105 Natural Light-Harvesting Complexes Structure of LH2 of Rhodopseudomonas acidophila main pigments: Bchl a 9 pairs of {α, β}-bchl a in B850 unit, 9 Bchl a in B800 unit G. McDermott et. al, Nature 374 (1995), 517.

106 Natural Light-Harvesting Complexes Structure of LH2 of Rhodopseudomonas acidophila main pigments: Bchl a 9 pairs of {α, β}-bchl a in B850 unit, 9 Bchl a in B800 unit 9 carotenoid pigments (rhodopin glucoside) G. McDermott et. al, Nature 374 (1995), 517.

107 Natural Light-Harvesting Complexes Structure of LH2 of Rhodopseudomonas acidophila main pigments: Bchl a 9 pairs of {α, β}-bchl a in B850 unit, 9 Bchl a in B800 unit 9 carotenoid pigments (rhodopin glucoside) 9 α- and β-apoproteins G. McDermott et. al, Nature 374 (1995), 517.

108 Protein Pigment Interactions in LH2 Rhodopseudomonas acidophila excitation energies (ev; SAOP/TZP/FDE) for B800 Bchl a: environment Q y Q x exp α-met β-arg in vitro, R.E. Blankenship, Molecular Mechanisms of Photosynthesis Blackwell Science, Oxford, shifts from FDE agree with conventional TDDFT results Z. He, V. Sundström, T. Pullerits, J. Phys. Chem. B 106 (2002), protein environment can be described by FDE

109 Photosynthetic Light-Harvesting Complexes B850 ε / m 2 mol Q x B850 + B800 Q y wavelength / nm (SAOP/TZP/FDEc; ca atoms) strong excitonic coupling in B850 B850 and B800 spectra do not interact strongly JN, J. Phys. Chem. B, 112 (2008), 2207.