Multi-level Simulation Methods using the Density- Functional-Theory Derived Embedding Potential
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1 DFT2013, Sept. 9-13, Durham, UK Multi-level Simulation Methods using the Density- Functional-Theory Derived Embedding Potential Tomasz A. Wesolowski Département de Chimie Physique
2 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
3 Embedding Strategy (general idea) Frozen density embedding theory (in a nutshell) 1) All information about the environment is contained in density B 2) The embedding operator is a local potential determined uniquely by the charge density of the environment ( B poz, B) and the embedded system ( A ) V emb V emb( FDET) r, A, B N B Z R ( B) r B ( r') dr' r r' E xc T.A. Wesolowski, A. Warshel, J. Phys. Chem. 1993, 97, 8050 T.A. Wesolowski, One-electron equations for embedded orbitals. In: Computational Chemistry: Reviews of current trends, J. Leszczynski Ed. Vol 10 (2006) 1-83 T.A. Wesolowski, Phys. Rev. A, 2008,77, K. Pernal and T.A. Wesolowski, Intl.J. Quant. Chem. 2009, 109, ) Origin in variational principle in Quantum Mechanics (Euler-Lagrange equations) A B E xc A T nad s, B A 1) Many ways to construct V emb 2) Typically, system-dependent empirical information about environment is needed - discrete models: atomic charges, dipoles, polarizabilities, - contuinnum models: dielectric constant, cavity shape, special energy terms 3) Typically, lack of self-consistency between the energy and the wavefunction 4) Self-consistency of energy, embedding operator, embedded density 5) Energy is the upper bound of the ground state energy of the WHOLE system 6) Although based on Quantum Mechanics, holds also for any level of description needed to obtain. 7) Extension to excited states straightforward through Linear-Response TD- DFT formulation T.A. Wesolowski, J. Am. Chem. Soc., 2004, 126, ) Many computational methods possible: what to chose as H o, how to approximate V emb(fdet), what properties are evaluated, how to obtain B
![) v v v A electron density of the QM subsystem By construction, E emb v[ B ] is](/docs-images/89/98098780/images/4-2.jpg "the upper bound of the exact ground-state energy of the whole system of N")
4 B dependency of the total FDET energy B electron density of the environment (can be macroscopic!) total electron density (can be macroscopic!) v v v A electron density of the QM subsystem By construction, E emb v[ B ] is the upper bound of the exact ground-state energy of the whole system of N electrons in potential v(r)!!
5 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
6 Approximating the FDET embedding potential in practice (1) I. Neglecting the last two terms (electrostatic only) Advantages: - easy to calculate (multipole expansion of the potential) - captures the dominant contribution in many cases Disadvantages: - fails to describe quantum confinement (intermolecular Pauli repulsion) [M. Zbiri, M. Atanasov, C. Daul, J.-M. Garcia Lastra & TAW, Chem. Phys. Lett. 397 (2004) 441] - strong dependency on the basis set (wrong variational limit) [Fradelos & TAW, J. Phys. Chem. A 115 (2011) 10018] - lack of self-consentience between: energy of interaction between embedded system and its environment, embedding potential, embedded wavefunction
7 Approximating the FDET embedding potential in practice (2) ~ ~ II. Replacing exact density functionals in the last two terms by approximants Advantages: All results presented in this talk use this approximation to v loc emb (r)! - self-consentience between: energy of interaction between embedded system and its environment, embedding potential, embedded wavefunction. - intermolecular Pauli repulsion is represented - robust solution with respect to basis set choices Disadvantages: - no good approximations for non-electrostatic terms in case of strong density overlap (chemical bonding between subsystems) - interaction energy functional is not homogeneous in A E int A r)v emb (r)dr
8 Approximating the FDET embedding potential in practice (3) ~ fix ~ fix III Replacing exact density functionals by approximants + linearization of the interaction energy bi-functional Replacing the self-consistent A by some arbitrary chosen A fix (electron density of isolated chromophore for instance) Advantages: - intermolecular Pauli repulsion still accounted for - simplicity in numerical implementation [see Dulak, Kaminski &TAW, Intl. J. Quant. Chem., 109 (2009) 1883] Disadvantages: - lack of self-consistency between energy, potential, and wavefunction - may be inadequate for methods based on response theory embedding potential does not depend on A
9 Approximating the FDET embedding potential in practice (4) fix fix = v inv (r) IV Obtaining the potential via direct inversion procedure + linearization of the interaction energy functional and Replacing the self-consistent A by some arbitrary chosen A fix and obtaining the embedding potential directly (without the use of any approximations for density functional) Roncero, Miller, Jacob, Carter. Advantages: - can treat covalently bound environments - all other advantages of linearization of the interaction energy bi-functional Disadvantages: - all disadvantages of linearization - numerical robustness problems (see the recent work by C. Jacob)
10 Choices to be made in computational methods based on FDET within Approximation II made for the embedding potential ~ ~ A) Approximation for the bi-functional T s nad [ A, B ] (and its functional derivative) a) If the overlap between A and B is small (environment not covalently bound,) use GGA97 or NDSD b) If the overlap between A and B is large (covalently bound environment), no good approximations so far. B) What to choose as H 0 : a) If known approximations for in DFT (or LR-TDDFT for excitations) functionals work well, use non-interacting reference system (Kohn-Sham) b) If known approximations DFT functionals fail, go beyond DFT methods use explicitly interacting Hamiltonian
11 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
12 Targets for FDET based simulations <Ô> = < o Ô o > + <Ô> where: <Ô> = < emb Ô emb > - < o Ô o > derived from FDET and < o Ô o > derived from adequately chosen QM method
13 Targets for FDET based simulations: example I Spectral shifts: - * transitions in 7-hydroxyquinoline -905 cm cm cm cm cm cm cm cm cm -1 Exp.: Manca et al. Intl. Rev. Phys. Chem. 24, (2005) cm -1 Fradelos, Kaminski, Leutwyler, TAW J. Phys. Chem. A 113 (2009) 9766
14 Targets for FDET based simulations: example II Absorption spectra of fluoronenone in zeolite L Zhou et al., Phys. Chem. Chem. Phys. 15 (2013) 159
15 Targets for FDET based simulations: example II Absorption spectra of fluoronenone in zeolite L Zhou et al. Phys. Chem. Chem. Phys. 15 (2013) 159 I f R R i B n B n i B j i B j n
16 Targets for FDET based simulations: example II The only orientation of fluorenone consistent with measured spectra Simulations Measurements Zhou et al. Phys. Chem. Chem. Phys. 15 (2013) 159
17 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
18 Analytical construction of the non-additive kinetic potential The for a given pair of electron densities ( A and B ), the non-additive potential (v t nad (r)) (but not the density functional!) can be obtained from the relation (*): v t nad (r) = v s [ A + B ](r) - v s [ A ](r) (**) There is infinite number of such functions B (r) for which which E emb [ B ]=E o [Wesolowski In: Computational Chemistry Review of Current Trends, vol. 10, J. Leszczynski Ed., World Scientific, 2006, pp. 1-82] Entirely analytical in model systems: [Savin & Wesolowski Prog. Theor. Chem.& Phys, 19 (2009) 327] [Savin & Wesolowski, in Recent Progress in OF-DFT, WORLD SCIENTIFIC 2013] Analytical potential from chemically relevant numerical densities: [de Silva & TAW, J. Chem. Phys (2012)] (*) For the relation between v s [ ](r) and the numerically available Kohn-Sham potential, see [P. De Silva & T.A. Wesolowski, PRA 85 (2012) ] (**) Partition DFT (Cohen, Wasserman) is a closely related formal framework leading to unique partitioning
19 v t nad (r) for two different choices for B in He Li + He(-0.95A) Li(0.95A) B 2 1s He 2 A opt = o He..Li+ - B 2 1s Li 2 B 2 1s Li 2 A opt = o He..Li+ - B 2 1s He 2 Adding v t nad (r) to the Kohn-Sham potential for He Li + leads to the change in the ground state density. The density ( opt A ) constructed from the lowest N A occupied orbitals obtained with this new potential added to the frozen density B is equal to the ground state density of the whole system (He Li + ). [P. De Silva & T.A. Wesolowski, J. Chem. Phys. 137 (2012)
20 Partial summary: Exact embedding potentials 1) For practically all interesting pairs A and the potential v t nad [ A, ] is a smooth function 2) In view of lack of uniqueness of the partitioning, such notion as polarization or chargetransfer are not uniquely defined in FDET 3) The observed in numerical experiments uniqueness of the partitioning obtained in fully variational treatment of electron density in subsystems A and B is the result of errors in used approximations for bi-functional for the non-additive kinetic potential 4) If tot in some domain in R 3 for a given system and chosen, how it affects the properties of the optimal opt?
21 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
22 B as a superposition of spherically symmetric atomic densities fluorenone in Si 50 Al 16 K 16 O 163 H 62 In the absence of environment, the energy of the lowest excitation is 2.75 ev.
23 Dependence of solvatochromic shifts of localized excitations on B Benchmarking FDET against EOM-CC [Fradelos, Lutz, Wloch, Piecuch, Wesolowski, J.Chem.Theor. & Comput., 7 (2011) 1647] Cooperative effects: [Fradelos, Kaminski, Leutwyler, TAW J. Phys. Chem. A 113 (2009) 9766] Importance of non-additive kinetic component of V emb (r): [Fradelos&TAW, J. Phys. Chem. A 115 (2011) 10018] B dependence of the shifts : [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear]
24 Dependence of solvatochromic shifts of localized excitations on B Benchmarking FDET against EOM-CC [Fradelos, Lutz, Wloch, Piecuch, Wesolowski, J.Chem.Theor. & Comput., 7 (2011) 1647] Cooperative effects: [Fradelos, Kaminski, Leutwyler, TAW J. Phys. Chem. A 113 (2009) 9766] Importance of non-additive kinetic component of V emb (r): [Fradelos&TAW, J. Phys. Chem. A 115 (2011) 10018] B dependence of the shifts : [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] B
25 Spectral-shifts for chromophores in hydrogen-bonded clusters FDET vs. EOM-CC The EOM-CC and FDET shifts agree within 150 cm -1 - * transitions for 7-hydroxyquinoline in hydrogen-bonded clusters ~ ~ [Fradelos, Lutz, Wloch, Piecuch, Wesolowski, J.Chem.Theor. & Comput., 7 (2011) 1647]
26 B as a superposition of spherically symmetric atomic densities cis-7hydroxyquinoline + water [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
27 B as a superposition of spherically symmetric atomic densities cis-7hydroxyquinoline + ammonia [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
28 B from Kohn-Sham calculations for the isolated environment with different basis sets and exchange-correlation potentials solvatochromic shifts of -> * excitations in cis-7-hydroxyquinoline+nnh 3 =-820 cm -1 (EOM-CC reference) LDA PBE B LDA PBE B SAOP SAOP Basis set in KS calculations of B : STO-SZ, STO-DZ, STO-DZP, STO-TZ2P, STO-aug-TZP [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
29 B from Kohn-Sham calculations for the isolated environment with different basis sets and exchange-correlation potentials solvatochromic shifts of -> * excitations in cis-7-hydroxyquinoline+2nh 3 LDA PBE SAOP B Basis set in KS calculations of B STO-SZ STO-DZ STO-DZP STO-TZ2P STO-aug-TZP [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
30 B from Kohn-Sham calculations for the isolated environment with different basis sets and exchange-correlation potentials solvatochromic shifts of -> * excitations in cis-7-hydroxyquinoline+3h 2 O LDA PBE SAOP B Basis set in KS calculations of B STO-SZ STO-DZ STO-DZP STO-TZ2P STO-aug-TZP [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
31 B from Kohn-Sham calculations for the isolated environment with different basis sets and exchange-correlation potentials solvatochromic shifts of -> * excitations in cis-7-hydroxyquinoline+nh 2 O LDA PBE SAOP LDA PBE SAOP =-562 cm -1 (EOM-CC reference) =-1446 cm -1 (EOM-CC reference) [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear] and Poster 74
32 B from Kohn-Sham calculations for the isolated environment 7-hydroxyl-4 methylcoumarin [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear]
33 B from Kohn-Sham calculations for the isolated environment p-nitro-aniline + 6H 2 O [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear]
34 B from Kohn-Sham calculations for the isolated environment 4-hydroxybenzylidene- 2,3-dimethylimidazolinone anion + 49 water molecules [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear]
35 B from Kohn-Sham calculations for the isolated environment 4-hydroxybenzylidene- 2,3-dimethylimidazolinone anion + 49 water molecules [M. Humbert-Droz, S. Shedge, X. Zhou, and TAW, (2013) TCA to appear]
36 Partial summary: Dependence of solvatochromic shifts of localized excitations on the choice for B 1) Use inexpensive Kohn-Sham calculations for isolated environment to generate B. 2) The shifts are remarkably independent on the choice of the QM method to generate B - variation of the calculated shift within 0.02 ev (less than 10% of the total shift). 3) In FDET as well as in computational methods based on FDET, the notion of polarization of individual subsystem is not well-defined. It shouldn t be used as a guideline to construct the frozen density in the case of hydrogen-bonded environments. Much more important is the mutual polarization of the molecules in the environment. 4) Taking into account polarization of the environment by the embedded subsystem is still needed if the embedded subsystem is charged. (for the worst scenario case of cations surrounded by anions in the environment, see [M. Zbiri, M. Atanasov, C. Daul, J.-M. Garcia Lastra & TAW, Chem. Phys. Lett. 397 (2004) 441]).
37 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
![Zhou, J. Kaminski, Wesolowski, Phys.Chem.Chem.Phys., 13 (2011) 10565] Solvent Dielectric constant Solvatochromic shift (FDET) [ev] Solvatochromic shift (Exp) [ev] Water 78-0.29-0.27 Methanol 33-0.](/docs-images/89/98098780/images/38-2.jpg "24-0.24 Ethanol 25-0.21-0.24 1-propanol 20-0.20-0.23 2-propanol 20-0.18-0.23 Dimethyl ether 4-0.11-0.10 Cyclohexane 2 0.00 0.")
38 Multi-level simulations based on Frozen Density Embedding Theory FDET for excitation energies of embedded chromophores Solvatochromism Implicit treatment of the solvent in FDET (embedding potential evaluated at averaged solvent density < B > [Kaminski, Gusarov, Wesolowski, Kovalenko, J. Phys. Chem A, 114 (2010) 6082] < B elect >+< B nuc > embedding potential -> * transition in coumarin 153 [X. Zhou, J. Kaminski, Wesolowski, Phys.Chem.Chem.Phys., 13 (2011) 10565] Solvent Dielectric constant Solvatochromic shift (FDET) [ev] Solvatochromic shift (Exp) [ev] Water Methanol Ethanol propanol propanol Dimethyl ether Cyclohexane For more about FDET based non-uniform continuum model, see posters : 139 (S. Shedge) and 164 (X. Zhou)
39 Introduction: Outline What is Frozen Density Embedding Theory (FDET) about? Approximations to FDET made in practical multi-level simulations FDET based methods: aiming to describe environment induced shifts of observables Sensitivity of the FDET results on the frozen density Model considerations for exactly solvable cases Numerical illustration (environment effect on local excitations) Miscellaneous FDET based: non-uniform continuum model for solvatochromism Development of approximations for the bi-functional of the non-additive kinetic energy Acknowledgements
40 Single Exponential Decay Detector (SEDD=log ) Fluorenone P. de Silva, et al., ChemPhysChem, 13, 3462(2012) N. Ram et al., CHIMIA, 67 (2013) 253 For more about SEDD, see poster 37, (P. De Silva)
41 Final Summary A) In Frozen-Density Embedding Theory, the embedded wavefunction (interacting or noninteracting) is obtained from the Euler-Lagrange equation. This assures self-consistency between: i) calculated upper bound for the energy of the total system regardless its size ii) embedding potential, iii) embedded wavefunction B) Weak dependence on the choice for B of the the environment induced shifts of energies of local excitations (intuitions based on results of conventional QM/MM simulations might be misleading!). C) Various computational methods based on FDET are possible differing in: i) approximations for the embedding potential, ii) quantum mechanical descriptor for the embedded system, iii) and choice for B D) We presented a robust FDET based computational procedure applying a few simple system-independent approximations allowing to study environment induced shifts in excitation energies. For shifts in the range from to 2000 cm -1 the error in the 100 cm -1 range.
42 Acknowledledgements Frozen-Density Embedding Theory K. Pernal (Technical University of Lodz), A. Savin (Université Paris VI, CNRS), M. Casida (Joseph Fourier University Grenoble) F. Aquilante (Upsala University/University of Lecce/University of Genève) J. Korchoviec (Jagiellonian University) P. De Silva (Jagiellonian University/University of Genève) J.W. Kaminski (Université de Gevève-> UCLA) Hydrogen-bonding effect on electronic excitations in clusters P. Piecuch and J. Lutz (Michigan State University) S. Leutwyler (University of Bern) M. Humbert-Droz (University of Genève) G. Fradelos (Université de Genève) Non-uniform contuum model for solvatochromism S. Gusarov and A. Kovalenko (NRC of Canada, Edmonton) X. Zhou (University of Genève) S. Shedge (University of Genève) J.W. Kaminski (Université de Genève_> UCLA), Chromphore guests in zeolites G.Calcaferri (University of Bern) G. Tabacchi, E. Fois (University of Como) X. Zhou (University of Genève) Funding: Fonds National Suisse de la Recherche Scientifique, Université de Genève, COST
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