Unexpected and expected features of the Lembarki-Chermette ap. functional for the kinetic energy T s [ρ]

Transcription

1 Unexpected and expected features of the Lembarki-Chermette approximation to the density functional for the kinetic energy T s [ρ] Tomasz A. Wesolowski, Université de Genève, Switzerland DFT-2017 : Development and Application of Density Functional Theory. A meeting in honor of Prof. emeritus Henry Chermette Nov 10, 2017, Lyon-Villeurbanne (France)

2 1 Frozen-Density Embedding Theory based simulations Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications 2 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture 3 Linearization of the FDET energy functional Linearized FDET with ADC(n) for Hˆ A 4

3 Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Arbitrary partitioning of the external potential and the total number of electrons {v ext (r) = v A (r) + v B (r) and N AB = N A + N B } ĤA = ˆT NA + ˆV ee N A + ˆV A Total energy as a functional of independent variables: Ψ A and ρ B ( r) E EWF AB [Ψ A, ρ B ] = Ψ A ĤA Ψ A + + ρ A ( r)v B ( r)d r + ρ A ( r)ρ B ( r ) r r d r d r + ExcT nad [ρ A, ρ B ] + ρ B ( r)v A ( r)d r + Ev HK [ρ B B ] Euler-Lagrange equation for stationary Ψ I A (and ρi A ( r) = Ψ A ˆρ Ψ A ) δe AB EWF [ΨI A,ρ B ] δψ I λ I Ψ I A A EWF = 0 by construction EAB [ΨI A, ρ B ] = E HK [ρ I A + ρ B ] E o for all ρ B The embedded SE form of the necessary condition for Ψ I A with a local embedding potential ( ˆT NA + ˆV ee N A + ˆV A + ˆv emb ) Ψ I A = ɛi Ψ I A with v emb [ρ A, ρ B, v B ]( r) = v B ( r) + ρ B ( r ) r r d r + δenad xct [ρ A,ρ B ] δρ A ( r) TAW & Warshel. J. Phys. Chem. 97 (1993) 8050; TAW, Phys. Rev.A. 77 (2008) ; Pernal & TAW, IJQC, 109 (2009) 2520

4 Density functionals in FDET Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Constrained search definition of E xct [ρ A, ρ B ] E nad xct [ρ A, ρ B ] = T nad s [ρ A, ρ B ] + E nad xc [ρ A, ρ B ] + F MD [ρ A ] T nad s [ρ A, ρ B ] = min Ψ s ˆT Ψ s min Ψ s ˆT Ψ s min Ψ s ˆT Ψ s Ψs ρ A +ρ B Ψs ρ A Ψs ρ B E nad xc [ρ A, ρ B ] = E xc [ρ A + ρ B ] E xc [ρ A ] E xc [ρ B ] with E xc [ρ B ] as in Kohn Sham DFT F MD [ρ A ] = min Ψ A ˆT NA + ˆV ee Ψ A ρ N Ψ A A A min Ψ WF A ρ A Ψ WF A ˆT NA + ˆV ee N Ψ WF A A Approximating E xct [ρ A, ρ B ] using explicit density functionals Ts nad [ρ A, ρ B ] T s [ρ A + ρ B ] T s [ρ A ] T s [ρ B ] Lembarki Chermette ( T s [ρ], for instance) Exc nad [ρ A, ρ B ] Ẽ xc [ρ A + ρ B ] Ẽ xc [ρ A ] Ẽ xc [ρ B ] (one of the common functionals Ẽ xc [ρ]) F MD [ρ A ] (usually neglected Aquilante & Wesolowski, J. Chem. Phys. 135 (2011) ])

5 Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Retaining the wavefunction resolution for a small number of electrons (N A ) and represent the remaining ones (even if their number is macroscopic) by means of an embedding operator - ˆV emb. The Nobel Prize in Chemistry 2013 was awarded jointly to Martin Karplus, Michael Levitt and Arieh Warshel for the development of multiscale models for complex chemical systems.

6 FDET among embedding methods Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Probably any method developed in Quantum Chemistry has been combined with some embedding potential: atomistic (QM/MM, QM/MD, QM/QM ) or continuum models (PCM, COSMO) for the environment. In these methods: ˆV emb is a local operator (potential) or a non-local operator (projection operator) Parameters such as charges, van der Waals parameters, polarizabilities are deduced from experimental data or generated by Quantum Mechanical methods (non-empirical force-fields). The exchange-repulsion and van der Waals interactions - in QM/MM methods are added a posteriori, i.e., after solving (ĤA + ˆV emb ) Ψ A = E Ψ A. In Frozen-Density Embedding Theory, we use the key ideas of Density Functional Theory to obtain universal-system independent and self-consistent expressions for the total energy and ˆV emb as functionals of charge densities. constrained search, multiplicative potentials as functional derivatives, Euler-Lagrange equation

7 Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications FDET is: Variational principle : E o E min [ρ B ] = min Ψ A E EWF AB [Ψ A, ρ B ] Euler Lagrange equation : (ĤA + ˆv emb ) Ψ I A = ɛi Ψ I A ρb ( r ) Embedding potential : v emb [ρ A, ρ B, v B ]( r) = v B ( r) + r r d r + δe nad xct [ρ A, ρ B ] δρ A ( r) Approximations made in FDET based simulations Choice of a method to solve many-body quantum problem ĤA ˆ HA Not really the problem of embedding Exc nad [ρ A, ρ B ] Ẽ xc nad [ρ A, ρ B ] and F MD [ρ A ] Ts nad nad [ρ A, ρ B ] T s [ρ A, ρ B ] Are sufficiently accurate approximations available? How good is vt nad [ρ A, ρ B ]( r) obtained form the Lembarki-Chermette approximation to T s [ρ]? Choice of ρ B ( r) Investigation of the sensitivity of the FDFT results on ρ B needed. Monomer vs. supermolecular basis set expansions Localization of embedded wavefunction imposed or not?

8 Frozen-Density Embedding Theory based simulations nad [ρa, ρb ](~ r ) challenge Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Zhou et al. J. Am. Chem. Soc., 136 (2014) 2723 Tuning the visible light absorption of retinal by protein environment!!"#$%&%'("%)*+*,)*,-+.%/,)*0,12*,$3%$4%*("%)(,4*%$4%"+-(%"5-,*+*,$3%"3"067%89:9;<%$4%4.2$0"$3"%,3%="$.,*"% ''''JK*K'?"2160%3'"L061%168#'"#"27H'F?MMI' Zhou et al. Phys. Chem. Chem. Phys., 15 (2013) 159 '("0"%99%D,-*20")%+)%4$..$?,36& 9<%*("% *"/%+1)$0D*,$3%)D"-*02#%$4%4.2$0"3$3"%,3%?"*%E"$.,*"%>F% C:99<%'("%)(,4*%$4%"5-,*+*,$3%"3"06,")%$4%.$?")*%9;%*0+3),*,$3)F% the lowest excitation Molecular structures of fluorenone and zeolite L E. Fois, G. Tabacchi, G. Calzaferri, J. Phys. Chem. C, 2010, 114, Left: ball-and-bonds representation of the fluorenone dye (gray, C; red, O; white, H). Right: representation of the fluorenone-zeolite L composite evidencing the one-dimensional channel system typical of the LTL framework. Color code: brown, Si; red, O; green, Al; gray, C; white, H; yellow, K; blue, potassium cation interacting with OFl. Case II: Structurally rigid environment (explicit model) Absorption spectrum of fluoronenone in zeolite L Tuning Mechanisms:!"#$%&!$'$%!()*+!!!"#$% "!%,!!!"#$% " &'()(*+,$ -./0# :8;;< 568=9:8=;< &!) -./0# >?9:757< 565@9:7?A< 1B!!$ -./0# >A9:7>=< 56>59:7>8<.23! -.#123/ 56=@9:85?< Electrostatic interactions with the protein in Rhodopsin, Red and Green pigments. Derpotonation of the retinal chromophore Blue pigment. 56=59:857< X. Zhou, D. Sundholm, T.A. Wesolowski, V.R.I. Kaila, J. Am. Chem. Soc., 136, (2014) 2723 &!)C D69E'(3F9G69-3$)'(2%F9H6I69J!+(2(K+L,9M$)9"69NM,2MF9OIP-9P(%%3$,QMR,($9:,$9*B!++<6 C65D"#'E;85'FG#6$"2=61H'89'!"#"$%I./"012%3'45#6#7'89':;8<8/=6#'>'?6=5%3'@8#"'A67B"#1=!"#"$%&'()*+,)*,*-''''''*( Lyaktonov et al. PCCP, 18 (2016) ! Tomasz A. Wesolowski, Universite de Gene ve, Switzerland

9 Key elements of Frozen-Density Embedding Theory (FDET) FDET as a basis for multi-level simulation methods Some applications Methods related to FDET Subsystem DFT (Cortona, Wesolowski) Subsystem LR-TDDFT (Casida-Wesolowski, Neugebauer) ONIOM schemes (Carter) Beyond density embedding (Miller-Manby, Carter, Chan, Neugebauer, Della Sala, Pavanello, Jacob) Fractional occupancies in subsystems (Cohen-Wasserman-Burke,..)

10 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations - the first sign of a trouble Ts nad [ρ A, ρ B ] T nad s [ρ A, ρ B ] = T s [ρ A + ρ B ] T s [ρ A ] T s [ρ B ]

11 Ts nad [ρ A, ρ B ] vs. approximating δt nad s [ρ A,ρ B ] δρ A ( r) Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture in analytically solvable system A series of parent functionals T s [ρ tot ] (W, LDA=TF, GEA2=TF+1/8W, TFW, CW) used to get T s [ρ tot ], T s nad [ρ A, ρ B ] and δ T s nad [ρ A,ρ B ]. Reference values for the analytically solvable model system (Wesolowski-Savin). δρ A ( r) Approximation Ts [ρ o ] T nad s [ρ o ρ B, ρ B ] ϕ exact ϕ none W TF GEA TFW exact No correlation between accuracies of the three quantities! [Wesolowski & Savin, A. Recent Progress in Computational Chemistry. In Recent Progress in Orbital-Free Density Functional Theory; Wesolowski, T., Wang, Y., Eds.; World Scientific: Singapore, 2013; Vol. 6, pp ]

12 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Exact properties of T nad s [ρ A, ρ B ] Exact properties of the bi-functional T s nad [ρ A,ρ B ] 1) T s nad [ρ A,ρ B ] = 0 for ρ A ρ B dr=0 [1] 2) T s nad [ρ A,ρ B ] = C TF ((ρ A +ρ B ) 5/3 - ρ A 5/3 -ρ B 5/3 )dr for uniform electron gas 3) T s nad [ρ A,ρ B ] 0 for a certain class of pairs pairs ρ A,ρ B [1,2] 4) for ρ A 0 and ρ B 5/3 dr=2 for ρ A 0 and ρ B dr=2 [3] 5) Analytically available for some systems (Bohrium) [4] [1] Wesolowski In: Computational Chemistry Review of Current Trends, vol. 10, J. Leszczynski Ed., World Scientific, 2006, pp [2] Wesolowski, Journal of Physics A: Mathematical and General, 36 (2003) [3] Garcia Lastra, Kaminski & TAW, J. Chem. Phys. 129 (2008) [4] Savin & Wesolowski Prog. Theor. Chem.& Phys, 19 (2009) 327,

13 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] Approximating T s nad [ρ A,ρ B ] - decomposable strategy (1)! The dominant component in T s [ρ] (Thomas-Fermi or von Weizsäcker?) 1) Thomas-Fermi functional (exact for uniform-electron gas expression) T s appr [ρ] = C TF ρ 5/3 dr + something small 2) von Weizsäcker functional (exact for one or two electrons) T s appr [ρ] = T W [ρ] + Pauli potential (also small) 1 ρ 2 where T W [ρ] = dr 8 ρ

14 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] Approximating T s nad [ρ A,ρ B ] - decomposable strategy (2)! First strategy (uniform electron gas):! Second strategy (spin compensated system of one or two electrons):! W 9

15 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Exact properties of T nad s [ρ A, ρ B ] δt s nad [ρ A,ρ B ]/δρ A near nuclei in the environment! 1) If the aproximant to v t nad [ρ A,ρ B ] is obtained from local density Approximation (Thomas-Fermi) the term: v t nad [ρ A,ρ B ] is not represented at all!!! 2) GEA2 and GGA functionals take this condition only partially into account. But this term leads to the +1/r behaviour near the nucleus! Garcia Lastra, Kaminski & TAW, J. Chem. Phys. 129 (2008)

16 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Exact properties of T nad s [ρ A, ρ B ] NDSD approximation for δt s nad [ρ A,ρ B ]/δρ A and T s nad [ρ A,ρ B ] where, f =f[ρ B ] is density-based switching function non-zero only near nuclei [Garcia Lastra, Kaminski, & TAW, J. Chem. Phys. 129 (2008) ]

17 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Exact properties of T nad s [ρ A, ρ B ] NDSD approximant to δt s nad [ρ A,ρ B ]/δρ A! Interaction induced dipole moments! Eq. 11 denotes the kinetic-energy dependent! part of the embedding potential derived! from Thomas-Fermi functional.! Eq. 22 is Eq. 11 with addition of the! the cusp-condition derived expression.! [Garcia Lastra, Kaminski & TAW, J. Chem. Phys. 129 (2008) ]

18 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Conjointness conjecture [Lee, Lee, & Parr Phys. Rev. A, 44 (1991) 768] Gradient expansion T [ρ] = 2 2/3 C F V σ K[ρ] = 2 1/3 C x V σ ρ 5/3 σ ( r) ( 1 + αx 2 σ +... ) d r ρ 4/3 σ ( r) ( 1 + βx 2 σ +... ) d r The conjecture The kinetic- and exchange energy can be expressed by one-particle density matrix γ( r, r) with x σ = ρ4/3 σ ( r) ρ 4/3 σ ( r) C F = 3 10 (3π2 ) 2/3 C x = 3 4 ( 3 π )1/3 the coefficients α and β are known. where the function G( r, t) is simply related to γ( r, r ).

19 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Complexation induced deformation of density for H 2..NCH T.A. Wesolowski& J. Weber, IJQC, 61 (1997) 303

20 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture

21 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] T.A: Wesolowski, H. Chermette, J. Weber, J. Chem. Phys. vol. 105 (1996) 9182

22 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] T.A. Wesolowski, H. Chermette, J. Weber, J. Chem. Phys. vol. 105 (1996) 9182

23 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] T.A. Wesolowski, H. Chermette, J. Weber, J. Chem. Phys. vol. 105 (1996) 9182

24 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] T.A: Wesolowski, J. Chem. Phys. vol. 106 (1997) 8516

25 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Tran&Wesolowski, IJQC, vol. 86 (2002)441

26 Decomposable approximations: The challenge Lembarki-Chermette approximation to T s [ρ] More on E x [ρ] and T s [ρ] conjointness conjecture Decomposable approximations for T nad s [ρ A, ρ B ] GGA97 approximation to vt nad [ρ A, ρ B ]( r) Using the Lembarki-Chermette functional to approximate vt nad [ρ A, ρ B ]( r) is a pragmatic solution. It replaces the construction of a well-balanced* approximation for T s [ρ]. In volume elements, where such imbalances are likely to occur, the non-additive ikinetic energy density (and potentials) approach smoothly zero or the LDA limit. This is achieved by means of using the PW91 enhancement factor which approaches zero at large reduced density gradients, i.e., where the von Weizsaecker term is expected to dominate. *Similar errors in T s [ρ A + ρ B ], T s [ρ A ] and their functional derivatives.

27 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ The Euler-Lagrange equation in FDET: ( ˆT NA + ˆV ee N A + ˆV ext + ˆv emb ) Ψ I A = ɛi Ψ I A with ρb ( r ) v emb [ρ A, ρ B, v B ]( r) = v B ( r) + r r d r + δe nad xct [ρ A, ρ B ] δρ A ( r) Gene on of I Genera of may have several solutions (I). Thanks to the Levy-Perdew theorem on excited states [Perdew and Levy, Phys. Rev. B 31, 6264 (1985) they correspond to excited states Solv I I Const ion of em Conventional FDET see also [Khait &Hoffman, J. Chem. Phys. 133, (2010); Daday, Koenig, Neugebauer, Filippi, ChemPhysChem 15, 3205 (2014); Wesolowski, J. Chem. Phys. 140, 18A530 (2014); Zech, Aquilante, TAW, J. Chem. Phys. 143, (2015); Dresselhaus & V Energy evaluation of density functionals Conventional: Linearized: or Neugebauer, 2015]

28 Conventional FDET E EWF A, ρ B, v B ] = Ψ A Ĥ A Ψ A + AB + ρ A( r)ρ B ( r ) r r d r d r ρ A( r)v B ( r)d r + E nad [ρ xct A, ρ B ] + ρ B ( r)v A( r)d r + E HK [ρ v B ] B Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Linearized FDET E EWF AB A, ρ B, v B, ρ ref A = Ψ A Ĥ A Ψ A + +E nad xct [ρref + A, ρ B ] + + ρ A( r)ρ B ( r ) r r d r d r ρ A( r)v B ( r)d r (ρa( r) [ ρ ref ( r)) v nad ref ρ, ρ ] B ( r)d r A xct ρ B ( r)v ref HK ( r)d r + E [ρ A v B ] B A ρ ( r ) v emb[ρ A, ρ, v ]( r) = v ( r) + B d r + B B B r r + v nad [ρ A, ρ B ] ( r) xct v emb[ρ A, ρ B, v B, ρ ref ]( r) = ρ B ( r ) v A B ( r) + r d r + r [ + v nad ref ρ, ρ ] B ( r) xct A where v nad xct [ρ A, ρ B ] ( r) = δe nad xct [ρ, ρ B ] δρ( r) ρ( r)=ρ A( r) TAW, J. Chem. Phys., A530 (2014) TAW, Phys. Rev.A. 77 (2008) Zech, Aquilante, TAW, J. Chem. Phys., (2015) In either cases, the Euler-Lagrange equation for stationary embedded wavefunction reads: (ĤA + ˆv emb ) Ψ I A = ɛi Ψ I A

29 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Uracil : the lowest excitations S 1 n 1 (O) π 1 (CO) O S 2 mostly π 2 (N) π ring S 3 n 2 (O) π 2 (CO) S 4 mostly π 1 (N) π ring N H 1 N H O 2 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

30 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Uracil U1 State EAB EWF [Ψ A, ρ B ] EAB LinFDET [Ψ A, ρ B, ρ ref A ] Difference [Hartree] [Hartree] [ev] S E-07 S E-06 S E-05 S E-03 S E-05 ( ρ A ( r) ρ ref A 0.00 ) δẽ nad ( r) xct [ρ A,ρ B ] δρ A ( r) ρa =ρ ref A d r O N H N H O H O H st order correction [ev] S0 S1 S2 S3 S4 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

31 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Uracil U3 State EAB EWF [Ψ A, ρ B ] EAB LinFDET [Ψ A, ρ B, ρ ref A ] Difference [Hartree] [Hartree] [ev] S E-07 S E-03 S E-06 S E-06 S E-05 O H N H O H ( ρ A ( r) ρ ref A ) δẽ nad ( r) xct [ρ A,ρ B ] δρ A ( r) ρa =ρ ref A d r N H O st order correction [ev] S0 S1 S2 S3 S4 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

32 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Uracil U4 State EAB EWF [Ψ A, ρ B ] EAB LinFDET [Ψ A, ρ B, ρ ref A ] Difference [Hartree] [Hartree] [ev] S E-07 S E-03 S E-05 S E-06 S E-05 H O H H O N H ( ρ A ( r) ρ ref A ) δẽ nad ( r) xct [ρ A,ρ B ] δρ A ( r) ρa =ρ ref A d r N H O st order correction [ev] S0 S1 S2 S3 S4 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

33 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Dipeptide : the lowest excitations S 0 Character of excitation S 1 n 1 (O) π1 (CO) S 2 n 2 (O) π2 (CO) 1 2 S 3 π 1 (CO) π 1 (N) π 2 (CO) π 1 (N) S 4 π 2 (N) π 2 (CO) S 5 n 1 (O) π 1 (N) π 2 (CO) π 2 (N) S 6 π 1 (N) π 1 (CO) Zech, Aquilante, TAW, J. Chem. Phys., (2015)

34 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Dipeptide State EAB EWF [Ψ A, ρ B ] EAB LinFDET [Ψ A, ρ B, ρ ref A ] Difference [Hartree] [Hartree] [ev] S E-07 S E-06 S E-04 S E-05 S E-05 S E-05 S E-05 ( ρ A ( r) ρ ref A 0.00 ) δẽ nad ( r) xct [ρ A,ρ B ] δρ A ( r) ρa =ρ ref A d r 1st order correction [ev] S0 S1 S2 S3 S4 S5 S6 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

35 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Bromine : the lowest excitations S 0 Character of excitation S 1 π x σz S 2 π y σz S 3 π x σz S 4 π y σz S 5 π xy σz (Z-axis along the Br-Br bond). Zech, Aquilante, TAW, J. Chem. Phys., (2015)

36 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Bromine State EAB EWF [Ψ A, ρ B ] EAB LinFDET [Ψ A, ρ B, ρ ref A ] Difference [Hartree] [Hartree] [ev] S E-06 S E-04 S E-04 S E-04 S E-04 S E-05 ( ρ A ( r) ρ ref A ) δẽ nad ( r) xct [ρ A,ρ B ] δρ A ( r) ρa =ρ ref A d r 0.4 1st order correction [ev] S0 S1 S2 S3 S4 S5 Zech, Aquilante, TAW, J. Chem. Phys., (2015)

37 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies S. Prager, A. Zech, F. Aquilante, A. Dreuw, T.A. Wesolowski, J. Chem. Phys., 144 (2016) ɛ I (in [ev]) f benzaldehyde-2h 2 O supermolecule-adc(2) reference results errors of FDET-ADC(2) results

38 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies S. Prager, A. Zech, F. Aquilante, A. Dreuw, T.A. Wesolowski, J. Chem. Phys., 144 (2016) ɛ I (in [ev]) f supermolecule-adc(2) reference results uracil-5h 2 O errors of FDET-ADC(2) results

39 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies S. Prager, A. Zech, F. Aquilante, A. Dreuw, T.A. Wesolowski, J. Chem. Phys., 144 (2016) benzene-hf (top) benzene-hf (side) ɛ I (in [ev]) f ɛ I (in [ev]) f

40 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies uracile-(h 2 O) 5 S. Prager, A. Zech, TAW, A. Dreuw, JCTC (2017) vol. 13, 4711

41 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies benzaldehyde-(h 2 O) 2 S. Prager, A. Zech, TAW, A. Dreuw, JCTC (2017) (2017) vol. 13, 4711

42 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Complexation induced shifts of the lowest five excitation energies benzene-hf S. Prager, A. Zech, TAW, A. Dreuw, JCTC (2017) (2017) vol. 13, 4711

43 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ Dependency of the excitation energies on ρ B and on the approximation for the functional v emb [ρ A, ρ B ]( r) S. Prager, A. Zech, TAW, A. Dreuw, JCTC (2017)(2017) vol. 13, 4711

44 Linearization of the FDET energy functional Linearized FDET with ADC(n) for HA ˆ What we can do now? S. Prager, A. Zech, TAW, A. Dreuw, JCTC (2017) (2017) vol. 13, 4711

45 Pros and cons of density embedding Multi-level par excellence (not only multi-qm but multi-physics). Self-consistent expressions for energy and embedded wavefunction (for calculation of other observables than the energy). Upper bound for the total energy. Simple overlap based criterion for applicability. No need for man-power hungry parametrizations of QM/MM methods. Local embedding potential. Bleak prospects for good approximations for Ts nad [ρ A, ρ B ] at strong density overlaps. This limits the applicability of density embedding to well separated (although overlapping densities) subsystems. Applicable for strongly asymetric situations (electronic structure simple for the environment and difficult for the embedded species). Necessity to put experimental information about partitioning of electrons and generating ρ B. Necessity to investigate the sensitivity of the results on ρ B. Strong correlations between subsystems difficult to describe using local potentials.

46 FDET related activities at University of Geneva FDET related research topics at University of Geneva! De Silva & Wesolowski, J. Chem. Phys. 137 (2012) Approximations for T nad s [ρ A, ρ B ] Wesolowski, J. Chem. Phys. 106, 1997, 8516; Garcia Lastra et al., J. Chem. Phys. 129 (2008) ; Bernard et al. J. Phys. A. 41 (2008) ; Savin & Wesolowski, in Recent Progress in OF-DFT, WORLD SCIENTIFIC 2013 Analytically solvable model systems Savin & Wesolowski Prog. Theor. Chem.& Phys, 19 (2009) 327; Multi-level FDET based continnum solvent model Kaminski et al., J. Phys. Chem A, 114 (2010) 6082; Zhou et al., Phys.Chem.Chem.Phys., 13 (2011) ρ A dependency of V FDET emb [ ρ A,ρ B ;r] linearization Dulak et al., Intl. J. Quant. Chem., 109 (2009) 1883 state-dependency for excited states Wesolowski, J. Chem. Phys, 140, (2014) 18A530 FDET based mullti-level models of molecular environments UV/Vis, ESR, NMR clusters: Fradelos et al., J. Phys. Chem. A 113 (2009) 9766 porous solids: Zhou et al., Phys. Chem. Chem. Phys, 15, (2013) 159 proteins: Zhou et al., J. Am. Chem. Soc., 136, (2014) 2723 COST/CODECS Algorithms and code developments eveluation of particular properties, ρ B - generation, demon, demon2k, ADF, MOLCAS Approximations for ΔF MD [ρ A ] Aquilante & Wesolowski, J. Chem. Phys., 135 (2011) Extracting chemical information from electron density (single exponential decay detector, SEDD) de Silva et al, ChemPhysChem, 13,(2012) 3462; J. Chem. Phys., 140, (2014)

47 Happy retirement Henry! Maybe you will be able to find enough time to work on improving the Lembarki-Chermette approximation to T s [ρ]. Decomposable approximation to vt nad [ρ A, ρ B ]( r) better than [ρ A, ρ B ]( r) would be appreciated. v nad(gga97) t