Quantum Chemical Embedding Methods Lecture 3

Transcription

1 Quantum Chemical Embedding Methods Lecture 3 Johannes Neugebauer Workshop on Theoretical Chemistry, Mariapfarr,

2 Structure of This Lecture Lecture : Subsystems in Quantum Chemistry subsystems in wave-function- and DFT- based Quantum Chemistry basics of subsystem density-functional theory Lecture 2: Exact Density-Based Embedding potential reconstruction projection-based embedding external orthogonality and the Huzinaga equation Lecture 3: Embedded Wavefunctions, Density Matrices, etc. wave function-in-dft embedding: ground states wave function-in-dft embedding: excited states density-matrix embedding theory and bootstrap embedding

3 I. Wavefunction-in-DFT Embedding

4 Background: sdft vs. DFT/DFT sdft energy bifunctional for given v ext : Ev sdft ext [ρ A, ρ B] = T s[ρ A] + T s[ρ B] + J[ρ A + ρ B] + E xc[ρ A + ρ B] + V ext[ρ A + ρ B] + Ts nad [ρ A, ρ B] note: KS energy functional for external potential v K ext is hence, where E sdft E KS v K ext [ρk] = Ts[ρK] + J[ρK] + Exc[ρK] + VK ext[ρ K], v ext [ρ A, ρ B] = E KS E OFDFT v A ext = E KS v A ext [ρa] + EKS v B ext [ρb] + Jint[ρA, ρb] + VA ext[ρ B] + V B ext[ρ A] +E nad xc [ρ A, ρ B] + T nad s [ρ A, ρ B] [ρa] + EKS v B ext [ρb] + EOFDFT A B [ρ A, ρ B], A B [ρ A, ρ B ] = E OFDFT [ρ A + ρ B ] E OFDFT [ρ A ] E OFDFT [ρ B ] v ext v A ext v B ext

5 Background: sdft vs. DFT/DFT this energy expression has the typical additive hybrid-method form, E hybrid A+B = EA method + EB method2 + EA B method3 for the special case method = method2 it can also be brought into the form E sdft v ext [ρ A, ρ B ] = Ev OFDFT ext [ρ A + ρ B ] + (E KS v [ρ A A ] E OFDFT ext v [ρ A A ]) + ext (E KS v [ρ B] E OFDFT [ρ B ext B ]) typical for subtractive hybrid-method approaches in the limit of exact functionals, all approaches give identical results but: sdft can in fact be used in hybrid-method fashion v B ext

6 Subsystem DFT as a DFT/DFT Hybrid Method Typical examples: Note: orbital-dependent XC functionals for intra-subsystem contributions, e.g., E hybrid A+B = EA B3LYP + EB B3LYP + EA B BLYP meta-gga for active system (A), GGA description for environment (B), and LDA for interaction, e.g. E hybrid A+B = EA TPSS + EB PBE + EA B LDA For orbital-dependent functionals like (double) hybrids we have to use different approximations for intra- and inter-subsystem contributions, since E A B needs to be evaluated with OFDFT

7 Subsystem DFT as a DFT/DFT Hybrid Method Free variables to be optimized? for an energy functional E sdft v ext [ρ A, ρ B ] = Ev OFDFT ext [ρ A + ρ B ] + (E KS v [ρ A A ] E OFDFT ext v [ρ A A ]) + ext (E KS v [ρ B] E OFDFT [ρ B ext B ]) it seems natural to optimize ρ A (r) and ρ B (r) more often, subtractive hybrid methods have a form like this: v B ext E KSDFT/OFDFT A+B = EA+B OFDFT + (EA KS EA OFDFT ) can be considered a functional of ρ(r) = ρ A (r) + ρ B (r) and ρ A E KSDFT/OFDFT v ext [ρ A, ρ] = Ev OFDFT ext [ρ] + (E KS v [ρ A A ] E OFDFT ext v [ρ A A ]) ext in the latter case, ρ B (r) is obtained as the difference ρ(r) ρ A (r) can be negative in certain regions, which can cause severe problems!

8 WF/DFT as a Hybrid Method Hybrid WF/DFT energy expression: or E WF/DFT (A+B) E WF/DFT (A+B) = E WF A + E DFT B + E DFT (A B) = E(A+B) DFT + (EWF A EA DFT ) (embedding potential can be identified as functional derivative δe DFT (A B) /δρ A(r) [for ρ B fixed]) N. Govind, Y.A. Wang, A.J.R. da Silva, E.A. Carter, Chem. Phys. Lett. 295 (998), 29. Questions: separate calculations or self-consistent energy minimization? if self-consistent: what are the free variables? which approximations enter in practice? (e.g. DFT part: KS or OF?)

9 WF/DFT: Energy Functional WF/DFT energy functional: E WF/DFT [Ψ A, ρ B ] = Ψ A Ĥ A Ψ A +EB DFT [ρ B ]+E(A B) DFT [ρ A, ρ B ] (ρ A : density obtained from Ψ A ) minimization w.r.t. Ψ A yields (ρ B fixed): ( ) n A ˆ H A Ψ A = Ĥ A + v A emb[ρ A, ρ B ](r i ) Ψ A = Ẽ A Ψ A where i= v A emb[ρ A, ρ B ](r) = v B ext(r) + v Coul [ρ B ](r) + v xc [ρ](r) v xc [ρ A ](r) + v nad t [ρ A, ρ B ](r) same form as in sdft!

10 WF/DFT: Energy Functional WF/DFT energy functional: E WF/DFT [Ψ A, ρ B ] = Ψ A Ĥ A Ψ A +EB DFT [ρ B ]+E(A B) DFT [ρ A, ρ B ] minimization w.r.t. ρ B yields (Ψ A fixed): ( ) v eff [ρ B ](r) + v B emb[ρ B, ρ A ] φ ib = ɛ ib φ ib, (density ρ B is obtained as n B i= φ ib (r) 2 ) full optimization: iterative freeze-and-thaw between WF and DFT part expensive!

11 WF/DFT in Practice Simple Standard Recipie: perform simple FDE or full sdft optimization of ρ A (r) [and ρ B (r)] store final embedding potential v A,final emb (r) point-wise on integration grid covering system A evaluate χ µ v A,final emb (r) χ ν numerically add this to one-electron part of f µν in HF calculation on system A ignore ρ A -dependence of v A emb in SCF make sure to remove this contribution from one-electron part in energy evaluation get embedded orbitals and corresponding MO integrals use those in correlated WF treatment on system A

12 Exact WF/DFT Embedding? Background: if DFT is good for densities, but not accurate enough for energies: derive highly accurate density-based embedding potential and transfer to WF part typical target application: transition-metal complexes difficult electronic situation at metal center still: DFT accurate enough for ligand system Embedding potential by reconstruction: create exact embedding potential in sdft calculation store final embedding potential v A,reconstr emb (r) point-wise on integration grid covering system A proceed as before problem: convergence of reconstruction step bad for larger systems

13 Exact WF/DFT Embedding? Embedding through projection: determine projection operator ˆP = i B φ i B φ ib from DFT calculation (as discussed before) again, this can be added to one-electron part in WF step F.R. Manby, M. Stella, J.D. Goodpaster, T.F. Miller III, J. Chem. Theory Comput. 8 (202), problem: accurate projector requires supermolecular basis fewer occupied orbitals in WF step, but all virtual orbitals of (A + B) problem for scaling behavior of correlated WF methods [e.g., CCSD(T): n 3 occ n4 virt ] another problem: DFT overdelocalization error may affect projector can be solved by using HF for orbitals, DFT for energies of system B R.C.R. Pennifold, S.J. Bennie, T.F. Miller III, F.R. Manby, J. Chem. Phys. 46, 0843.

14 Basis Set Truncation in Projection-WF/DFT Try to reduce no. of virtual orbitals: define region of border atoms divide system-b orbitals into {φ distant } and {φ border (based on population on border atoms) only include {φ border i B } in projector i B i B } truncate basis for WF step to include only AOs in system A and at border atoms orthogonality with system B not fully enforced T.A. Barnes, J.D. Goodpaster, F.R. Manby, T.F. Miller III, J. Chem. Phys. 39 (203), Alternative without border atoms : truncate basis set based on population in system A specifically: include only basis functions with Mulliken net populations q net α := D A ααs αα > 0 4 (D A αα = density matrix of system A) S.J. Bennie, M. Stella, T.F. Miller III, F.R. Manby, J. Chem. Phys. 43 (205),

15 Basis Set Truncation in Projection-WF/DFT Correction for neglected orbitals in projector: define density ρ distant B (r) = i B,distant φdistant i B (r) 2 add density-based correction: v nad t [ρ A, ρ distant B ](r) = δt s[ρ] δρ(r) ρ=ρa+ρ distant B δt s[ρ] δρ(r) ρ=ρa R.C.R. Pennifold, S.J. Bennie, T.F. Miller III, F.R. Manby, J. Chem. Phys. 46, 0843.

16 WF/DFT as a Special Kind of sdft Consider Multi-Configurational DFT: el. el.interaction is separated into long- and short-range parts using r 2 = erf(r 2) r 2 + erfc(r 2) r 2 short-range part is treated, e.g., by LDA long-range part is treated through WF for interacting particles

17 WF/DFT as a Special Kind of sdft Relation to WF/DFT embedding: can be considered as DFT method with representation of ρ A through a multi-determinant wave function T.A. Wesolowski, Phys. Rev. A 77 (2008), energy functional: Ev WF/DFT ext [Ψ MD A, ρ B ] = Ψ MD A ˆT + ˆV ee Ψ MD A + Vext[ρ A A ] + T s [ρ B ] + J[ρ B ] +E xc [ρ B ] + Vext[ρ B B ] + Vext[ρ A B ] + Vext[ρ B A ] +J int [ρ A, ρ B ] + Ts nad [ρ A, ρ B ] + Exc nad [ρ A, ρ B ], full minimization w.r.t. Ψ MD A (in the limit of exact functionals) but: only if search space for Ψ MD A otherwise, Ψ MD A min Ψ MD A ρa and ρ B will lead to true ground-state energy includes exact wave function ˆT + ˆV ee Ψ MD A T[ρ A ] + V ee [ρ A ] which needs to be corrected by an additional density functional

18 Perturbation-Theory Perspective on WF/DFT if only changes in system A properties due to environment are of interest, consider perturbation, perturbation theory tells us E () A N A Ĥ A Ĥ A + ṽ A emb(r i ), Ψ (0) A i= N A ṽ A emb(r i ) i= Ψ(0) A connection to hybrid-method viewpoint (assume ρ 2 fixed): E WF/DFT v ext [Ψ A, ρ B ] = Ψ A Ĥ A + n A i=, v elstat (r i ) Ψ A + E xc [ρ A + ρ B ] E xc [ρ A ] +T s [ρ A + ρ B ] T s [ρ A ] + const. with const. = E KS v B [ρ B ] + Vext A [ρ B] E xc[ρ B ] T s[ρ B ] ext

19 Perturbation-Theory Perspective on WF/DFT if we want to write this as E WF/DFT v ext [Ψ A, ρ B ] = Ψ A Ĥ A + we also have to replace n A i= v A emb(r i ) Ψ A + const. E xc [ρ A + ρ B ] E xc [ρ A ] + T s [ρ A + ρ B ] T s [ρ A ] by v nad xc [ρ A, ρ B ](r)ρ A (r)dr + v nad t [ρ A ](r), ρ B ]ρ A (r)dr, which is an additional approximation!

20 Excitation Energies in WF/DFT General expression for excitation energies: E = E WF/DFT (+2)e E WF/DFT (+2)g Case A: no differential polarization, ρ 2 unchanged ( ) E = E WF [Ψ e ] + EKS 2 [ρ 2 ] + E( 2) OFDFT [ρe, ρ 2] ( ) E WF [Ψ g ] + EKS 2 [ρ 2 ] + E( 2) OFDFT [ρg, ρ 2] = E WF [Ψ e ] + EOFDFT ( 2) [ρe, ρ 2] E WF [Ψ g ] EOFDFT ( 2) [ρg, ρ 2]

21 Excitation Energies in WF/DFT Expression for pure electrostatic embedding: with E elstat = (E WF [Ψ e ] EWF [Ψ g ]) +V nuc,2 [ρ e ] V nuc,2[ρ g ] + J[ρe, ρ 2] J[ρ g, ρ 2], = (E WF [Ψ e ] EWF [Ψ g ]) + Ψ e v elstat. emb,(r i ) Ψe i Ψ g v elstat. emb,(r i ) Ψg, i v elstat. emb,(r) = v nuc,2 (r) + ρ2 (r ) r r dr

22 Excitation Energies in WF/DFT Approximate expression based on embedding operators : E (A) approx = E WF [Ψ e ] + Ψe i v emb, (r i ) Ψ e E WF [Ψ g ] Ψg i v emb, (r i ) Ψ g, = E WF [Ψ e ] EWF [Ψ g ] +V nuc,2 [ρ e ] V nuc,2[ρ g ] + J[ρe, ρ 2] J[ρ g, ρ 2] + v nad xc [ρ g, ρ 2]( r) [ ρ e ( r) ρg ( r)] d r + v nad t [ρ g, ρ 2]( r) [ ρ e ( r) ρg ( r)] d r

23 Excitation Energies in WF/DFT True excitation energy expression for case A E (A) = E WF [Ψ e ] EWF [Ψ g ] +V nuc,2 [ρ e ] V nuc,2[ρ g ] + J[ρe, ρ 2] J[ρ g, ρ 2] +Exc nad [ρ e, ρ 2] Exc nad [ρ g, ρ 2] +T nad s [ρ e, ρ 2] Ts nad [ρ g, ρ 2] approximation made with embedding operators: {v nad xc [ρ g, ρ 2]( r) + v nad t [ρ g, ρ 2]( r) } [ ρ e ( r) ρg ( r)] d r Exc nad [ρ e, ρ 2] Exc nad [ρ g, ρ 2] + Ts nad [ρ e, ρ 2] Ts nad [ρ g, ρ 2] similar to linearization approximation M. Dulak, T.A. Wesolowski, J. Chem. Theory Comput. 2 (2006), 538.

24 Differential Polarization in WF/DFT Case B: ρ 2 polarized in state-specific manner E (B) = E WF [Ψ pol,e ] E WF [Ψ g ] + E2 KS [ρ pol 2 ] EKS 2 [ρ 2 ] + V nuc, [ρ pol 2 ] V nuc,[ρ 2 ] + V nuc,2 [ρ pol,e ] V nuc,2 [ρ g ] + J[ρpol,e, ρ pol 2 ] J[ρg, ρ 2] + Exc nad [ρ pol,e, ρ pol 2 ] Enad xc [ρ g, ρ 2] + Ts nad [ρ pol,e, ρ pol 2 ] Tnad s [ρ g, ρ 2]

25 Differential Polarization in WF/DFT Use of embedding operators in state-specific embedding E (B) approx = E WF [Ψ pol,e ] + Ψ pol,e i v pol,e emb, (r i) Ψ pol,e E WF [Ψ g ] Ψg i v emb, (r i ) Ψ g = E WF [Ψ pol,e ] E WF [Ψ g ] +V nuc,2 [ρ pol,e ] V nuc,2 [ρ g ] + J[ρpol,e, ρ pol [ + v nad xc [ρ pol,e, ρ pol 2 ]( r) + vnad t [ρ pol,e, ρ pol [v nad xc [ρ g, ρ 2]( r) + v nad t [ρ g, ρ 2]( r) ] ρ g ( r)d r 2 ] J[ρg, ρ 2] ] 2 ]( r) ρ pol,e ( r)d r C. Daday, C. König, O. Valsson, J. Neugebauer, C. Filippi, J. Chem. Theory Comput. 9 (203), 2355.

26 Differential Polarization in WF/DFT... implies the following error: (already disregarding differential kinetic-energy and XC terms in E WF ) E KS 2 [ρ pol 2 ] EKS 2 [ρ 2 ] + V nuc, [ρ pol 2 ] V nuc,[ρ 2 ] 0 Error in Coulomb terms: {v Coul [ρ 2 + ρ 2 /2]( r) + v nuc, ( r) + v nuc,2 ( r)} ρ 2 ( r)d r 0 with ρ 2 ( r) = ρ pol 2 ( r) ρ 2( r) Use correction: δe (B) simple = E KS 2 [ρ pol 2 ] EKS 2 [ρ 2 ] + v nuc, ( r) ρ 2 ( r)d r

27 Differential Polarization in WF/DFT Improved excitations from state-specific embedding: E (B) E WF [Ψ pol,e ] + Ψ pol,e i v pol,e emb, ( r i) Ψ pol,e E WF [Ψ g ] Ψg i v emb, ( r i ) Ψ g + δe(b) simple Remaining error: (similar to case A) δe (B) nadd = Exc nad [ρ pol,e, ρ pol 2 ] Enad xc [ρ g, ρ 2] +Ts nad [ρ pol,e, ρ pol 2 ] Tnad s [ρ g, ρ 2] [ v nad xc [ρ pol,e, ρ pol 2 ]( r) + vnad t [v nad xc [ρ g, ρ 2]( r) + v nad t [ρ pol,e ], ρ pol 2 ]( r) [ρ g, ρ 2]( r) ] ρ g ( r)d r ρ pol,e ( r)d r

28 II. Density-Matrix Embedding Theory

29 Decomposing Total-System States Consider active system A and bath B: states of combined system can be expanded as, n A states n B states n A states n B states Ψ A B = c ij A i B j = A i c ij B j i j i j = n A states n B states A i B i with B i = c ij B j i j B i are not necessarily orthogonal; use B orth i A nstates = i c ii B i such that B orth i B orth j = δ i j

30 Decomposing Total-System States for simplicity, we write b i := B orth for these bath states, j Ψ A B = n A states n A states i i A i b i even if n B n A, exact embedding of A is possible considering only n A bath states but: constructing these states requires knowledge of full Ψ A B idea of DMET: construct { b i } from simple approximation Φ to Ψ A B use these bath states to embedd high-quality approx. for { A i } try to match one-particle density matrices between low- and high-level approximations G. Knizia, G.K.-L. Chan, J. Chem. Theory Comput. 9 (203), 428.

31 DMET in Practice () perform HF calculation on full system (A + B) yields N occ occupied orbitals {φ i } (2) perform orbital rotation among {φ i }; specifically: calculate S ij = p A φ i p p φ j (overlap matrix of orbitals projected onto the L A sites p of fragment A) eigenvectors of S define a rotation matrix; there are N occ L A eigenvectors with zero eigenvalue environment orbitals {φ env i the remaining L A entangled orbitals {φ ent i } without overlap with A } do have overlap with A (3) project {φ ent q } onto environment sites yields L A bath orbitals b { } { b } = r r φ ent i r B G. Knizia, G.K.-L. Chan, J. Chem. Theory Comput. 9 (203), 428.

32 DMET in Practice the many-body wavefunction is expressed as a CAS-CI wavefunction with (half-filled) active space of { p } (L A sites of fragment A) and { b } (L A bath orbitals) (4) project total Hamiltonian Ĥ into active space, Ĥ A emb = ˆPĤˆP, and find high-accuracy approx. to Ĥ A emb Ψ A = E A Ψ A (e.g., FCI) (5) adjust one-particle density matrix D to match that of Ψ A as close as possible on fragment A. Do that for all fragments and iterate. Note: This adjustment minimizes = Φ â r âs Φ Ψ A â r âs Ψ A 2 A rs A by changing the Fock operator ˆf ˆf + A ˆµ A with the correlation potentials ˆµ A = rs A µa rsâ r âs in the calculation of the mean-field wavefunction Φ (i.e., is minimized w.r.t. all µ A rs) G. Knizia, G.K.-L. Chan, J. Chem. Theory Comput. 9 (203), 428.

33 DMET in Practice total DMET energy = sum of fragment energies, E = E A = D ps h ps + d pstu g pstu 2 A A p A,s p A,stu (one- and two-particle density matrices D and d here include contributions from the pure environment orbitals) non-integer number of electrons per fragment; total number of electrons is recovered important conceptual question: Can the density matrix of the high-level wave function be represented through the chosen low-level wave function? for HF-in-HF-embedding, the results are exact G. Knizia, G.K.-L. Chan, J. Chem. Theory Comput. 9 (203), 428.

34 Bootstrap Embedding concept similar to DMET, but different matching condition use overlapping fragments match one-particle density matrix of the edge of one fragment to the center of a partially overlapping fragment M. Welborn, T. Tsuchimochi, T. Van Voorhis, J. Chem. Phys. 45 (206),

35 General References Recent Embedding reviews: M. Banafsheh, T.A. Wesolowski, Int. J. Quantum Chem. 8 (208), e2540 T. Dresselhaus, JN, Theor. Chem. Acc. 34 (205), 97. T.A. Wesolowski, S. Shedge, X. Zhou, Chem. Rev. 5 (205), 589 C.R. Jacob, JN, WIREs Comput. Mol. Sci. 4 (204), 325. F. Liebisch, C. Huang, E.A. Carter Acc. Chem. Res. 47 (204), A.P.S. Gomes, C.R. Jacob, Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 08 (202), 222. T.A. Wesolowski, Y.A. Wang (eds.), Recent Progress in Orbital-Free Density Functional Theory, World Scientific, Singapore (203).