Hamiltonians: HF & DFT

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ISAMS14 International School on Ab initio Modelling of Solids with

dreamed that there was a ladder set up on the earth, the top of it

on it Genesis 8, 10-1 Bartolomeo Civalleri Department of Chemistry NIS

it Quantum Chemistry and Molecules/Crystals Structure and composition

H(r)=E(r) Physical and chemical properties The problem is of evaluating

1 ISAMS14 International School on Ab initio Modelling of Solids with CRYSTAL14 Regensburg, 0 5/07/014 1 Hamiltonians: HF & DFT and he dreamed that there was a ladder set up on the earth, the top of it reaching to heaven; and the angels of God were ascending and descending on it Genesis 8, 10-1 Bartolomeo Civalleri Department of Chemistry NIS Centre of Excellence University of Torino bartolomeo.civalleri@unito.it Quantum Chemistry and Molecules/Crystals Structure and composition Quantum mechanics? H(r)=E(r) Physical and chemical properties The problem is of evaluating quantum mechanically the ground state electronic structure and total energy of a system of interacting electrons for a given nuclear configuration 1

2 3 Questions to answer Which model Hamiltonian for solids? Hartree-Fock, DFT,? Which representation for? Plane waves, Gaussians,? (see next lecture) Outline Brief overview of HF and DFT methods: merits and limits DFT methods in CRYSTAL14 Validation of DFT methods for molecules and solids 4 Fundamental approximations ( r, t) H( r, t) i t Time independent Schrödinger equation Born-Oppenheimer approximation Relativistic effects are usually neglected Neglect of higher order effects (e.g. spin-orbit interaction) No excited states Ground state (E 0, 0, 0 )

3 5 Non relativistic Schrödinger equation H ( r) E( r) H = T e + V ee + V ext + V nn For a system of N electrons at a given nuclear configuration (M nuclei) Kinetic energy: V ext T e N N i M R 1 i Electrons-nuclei interaction: Za x i a a i Electron-electron interaction: V ee V nn N N x Nuclear repulsion: 1 x i j i i j M M R ZZ a b R a ba a b 6 The Variational Principle For any legal wavefunction (N-electron, antisymmetric, normalized) the energy is: E Hˆ dr Hˆ the energy is a functional of : E E E E 0 0 Search all to minimize E the ground state (E 0, 0 ) E 0 N mine 3

4 7 The Hartree-Fock (HF) approximation - I Given a complete orthonormal set of {y i (x)} of one-electron spinorbitals (x = r,a ; r,b), the exact ground state wavefunction can be expressed as a linear combination of all N-electron Slater determinants from {y i (x)} : C SD 0 i i,..., i i i N 1 The HF approximation considers just a single-determinant: SD The Hartree-Fock (HF) approximation - II 8 The HF method provides the N spin-orbitals y HF i(x) (i=1,,,n) which define the best single-determinant approximation of 0 : HF SD ˆ SD SD E H min E E SD N 0 The HF equations (SCF) r r f V d r d y y ˆ 1 j i iy i r ext r y i 1 y j 1 iy i 1 r r 1 r r r r r j r1 r Mean-field theory no electron correlation effects (E c =E 0 -E HF ) 4

5 The electron correlation error Electrons moving through the density swerve to avoid one another, like shoppers in a mall J.P. Perdew. Results of the swerving motion are: a reduction of the potential energy of mutual Coulomb repulsion (negative exchange-correlation energy) a small positive kinetic energy contribution to the correlation energy. In Hartree-Fock theory the real electron-electron interaction is replaced by an average interaction. 9 The HF wave function accounts for 99% of the total energy. The 1% lost correlation energy is however crucial for: Bond energy Reaction intermediates Dispersive interactions??? Beyond Hartree-Fock Many methods/approximations applicable, e.g.: MBPT: MP, MP3, MP4 Configuration interaction (CI): CIS, CISD Coupled Cluster (CC): CCSD(T), CC 1 C SD 0 i i,..., i i i N 10 GW QMC Generally expensive but systematically improvable Some of them are now available for periodic systems see the talk on CRYSCOR D. Usvyat s lecture To learn more on Post-HF methods M. Schütz s lecture 5

The Density Functional Theory (DFT) The electron density contains all the ingredients of a given physical system Given an external potential V ext (r), the exact ground state energy E 0 and the

The energy is a functional of (r) E º E r(r) [ ] under the constraint: Eé ër(r) ù û = F é ë r(r) ù û + ò V (r)r(r)dr ext ò r(r)dr = N the system is uniquely described by V ext (r) F[(r)]=T[(r)]+V ee

6 The Density Functional Theory (DFT) The electron density contains all the ingredients of a given physical system Given an external potential V ext (r), the exact ground state energy E 0 and the corresponding exact electron density 0 (r) are obtained by minimizing the functional: where E 0 ( ) N Hohenberg-Kohn (1964) established the basics of the Density Functional Theory (DFT): min E ( r) r The energy is a functional of (r) E º E r(r) [ ] under the constraint: Eé ër(r) ù û = F é ë r(r) ù û + ò V (r)r(r)dr ext ò r(r)dr = N the system is uniquely described by V ext (r) F[(r)]=T[(r)]+V ee [(r)] is the same for all systems (universal) F[(r)] is NOT KNOWN unfortunately 13 DFT: the Kohn-Sham formalism Kohn and Sham (1965) proposed to write the density in terms of a set of orthonormal single-particle functions for non-interacting particles: r ( r ) º r s ( r ) = å y i ( r ) The kinetic energy (non-interacting electrons) and the classical Coulomb energy (Hartree potential) T s = - 1 N å y i ( r ) Ñ i y i ( r ) E H [r] = 1 r(r 1 )r(r ) ò dr 1 dr r 1 -r i N i=1 15 Allow us to recast the energy functional as: r r r r r KS E TS Eext EH Exc E xc [(r)] is the exchange-correlation energy 6

7 Kohn-Sham equations 16 The ground state density 0 (r) can be obtained by solving selfconsistently a coupled set of one-electron pseudo-schrödinger equations, the Kohn-Sham equations: h y r ˆKS i i i ĥ KS = - 1 Ñ + V ext (r) + y ò r r(r ') r - r ' dr ' + V xc (r) No approximations If we knew E xc [(r)] we could solve for the exact ground state energy and density! KS equations are similar to HF equations (non-local potential) Similar cost (N 3 ) but including correlation energy Kohn-Sham & the unknown XC functional 17 E xc [(r)] is the exchange-correlation energy it contains the rest of the total energy: (i.e. Fermi, Coulomb and kinetic correlation, Self-Interaction) E ( T Ts ) ( Vee EH ) it is usually splitted into two components: xc ( ) ( ) ( ) E r E r E r xc x c it is not known exactly approximations (DFA) XC func. acronym: author s name (usually) B-LYP PW91 B3-LYP PBE, PBE0 A.D. Becke R.G Parr J.P. Perdew 7

Alphabeth Soup Peter Elliott The quest of finding more accurate and reliable E xc [(r)] functionals has originated a swarm of approximations: 19 from K.

8 Alphabeth Soup Peter Elliott The quest of finding more accurate and reliable E xc [(r)] functionals has originated a swarm of approximations: 19 from K. Burke JCP 136 (01) Different ingredients Functional forms Empirical (many fitted parameters) Non-empirical (many constraints) Jacob s Ladder classification of DFT Chemical Accuracy Heaven-GGA RPA, DHyb, Hyper-GGA Meta-GGA GGA LDA J.P. Perdew and K. Schmidt in Density Functional Theory and Its Application to Materials, edited by V. Van Doren et al. (AIP, 001) E E E E E mgga xc GGA xc LDA xc ( ),,,, r DHyb DHyb Exact PT xc xc X C ( ),,, r HGGA HGGA Exact xc xc X mgga ( ) xc,,, r GGA ( ), r xc unif () r xc dr dr dr dr 1 Hybrid dr Hartree world The ascent of the ladder consists in embedding increasingly complex (costly) ingredients into E xc [(r)]. 8

9 The Jacob s Ladder in CRYSTAL14 Chemical Accuracy Heaven-GGA RPA, DHyb, In CRYSTAL14, DFA for all of the five rungs of the Jacob s Ladder are now available 5 th rung: Double-Hybrids new in CRYSTAL14 combining CRYSTAL & CRYSCOR Hyper-GGA Meta-GGA GGA 4 th rung: Hybrids: (since CRYSTAL98). Extended to Range-Separated Hybrids (RSH) Extended to mgga hybrids 3 rd rung: m-gga: new in CRYSTAL14 (M. Causà) LDA Hartree world nd rung: GGA 1 st rung: LDA available since CRYSTAL98 4 th rung: Hyper-GGA (Hybrids) 3 Occupied orbitals are included through a HF-like exact exchange term Global hybrids: include a constant amount of HF exchange: E ae 1 a E E GH HF DFA DFA XC X X C E.g.: B3LYP PBE0 B97 Local hybrids: the HF exchange contribution is included through a positiondependent function, a(r), the so-called local mixing function: E.g.: LH HF 1 DFA DFA r EXC a r EX a r EX E a C r W r Range-separated hybrids: The amount of HF exchange included depends on the distance between electrons 9

10 Range-Separated Hybrids (I) Splitting of the Coulomb operator 1/r into different ranges (A. Savin) The amount of HF exchange included depends on the distance between electrons 1 1erf SRr erf SRr erf LRr erf LRr r r r r SR MR LR is the length scale of separation 4 Range-Separated Hybrids: erf(r) 6 Most common formulations of RSH are based on the error function 1 1erf SRr erf SRr erf LRr erf LRr r r r r SR MR LR,,,,,, E E c E E c E E c E E RSH DFA SR HF SR DFA MR HF MR DFA LR HF LR DFA XC XC SR X X MR X X LR X X Accordingly, three families of RSH can be defined: c SR 0, c MR =0, c LR =0 Screened Coulomb RSH (SC-RSH) [HSE06, HSEsol] c SR =0, c MR 0, c LR =0 Middle-range corrected RSH (MC-RSH) [HISS] c SR 0, c MR =0, c LR 0 Long-range corrected RSH (LC-RSH) [LC-PBE, LC-PBEsol, B97, B97-X ] 10

11 5 th rung: from Global Hybrids to Double Hybrids 7 Global hybrids: include a certain amount of HF-type exchange E ae 1 a E E GH HF DFA DFA XC X X C Double hybrids: not only the HF exchange contribution is included but also a MP-like term for correlation (i.e. virtual orbitals) (Truhlar, Grimme 006) 1 1 E DH ae HF a E DFA b E DFA be PT XC X X C C SCF PT: Gorling-Levy perturbation theory (GLMP) Cost as MP, but less basis set dependent Parameters (a and b) are fitted to thermochemical data (Grimme) Post-SCF Double Hybrid Functional Parameters Exchange Correlation a b B-PLYP B88 LYP BGP-PLYP B88 LYP mpw-plyp mpw LYP A rigorous Double Hybrid: Density-Scaled 1DH 8 Rigorously proved through the adiabatic connection that a=l and b=l (l=coupling strength constant). Only one parameter is needed. DS1DH: Density-Scaled one-parameter Double-Hybrid 1 [Sharkas, Toulouse & Savin JCP 134 (011) ] E le l E E l E l E DS1 DH, l HF DFA DFA DFA PT XC X X C C 1 l C SCF Post-SCF 1DH: by neglecting the density-scaling, the one-parameter double-hybrid approximation is obtained thus making a link with standard DHs 1 1 E le l E l E l E 1 DH, l HF DFA DFA PT XC X X C C Double Hybrid Functional Parameters Exchange Correlation l l DS1DH-BLYP B88 LYP DH-BLYP B88 LYP B-PLYP B88 LYP (0.8) 11

12 Summary of XC functionals available in CRYSTAL LDA GGA/mGGA Hybrids GGA/mGGA (GH & RSH) RSH methods based on the PBE functional have been implemented by using the Henderson-Janesko-Scuseria (HJS) X-hole model (as in VASP) In progress: more recent Minnesota family; TPSS, DS1DH functionals Double-Hybrids Slater PBE-xc PBE0 B-PLYP VWN B88 B3LYP BGP-PLYP PZ LYP PBEsol0 mpw-plyp PWLSD PBEsol-xc B97 VBH-x SOGGA-x WC1LYP WC-x PW9-xc WL-c M06-L B97 family M05, M05-X / M06 family HSE06 / HSEsol HISS LC-wPBE / LC-wPBEsol wb97 / wb97-x RSHXLDA HF and DFT: a few remarks HF Catches an important part of physics (mean-field theory) Does not include correlation energy (E c =E 0 -E HF ) Self interaction free No dispersive interaction DFT It is in principle exact In practice, it makes more or less justified approximations Results may be better than those obtained with HF Which E xc functional? Self interaction error Problems with spin-polarized systems and strongly correlated systems No dispersive interaction (empirical and non-empirical corrections available) 1

13 LDA GGA mgga Hybrids DHyb 9/07/014 Benchmarking DFT methods 34 Is the Jacob s Ladder valid? DFA must be validated Validation of DFT methods is now quite common Large databases are available to benchmark different families of XC functionals (D. Truhlar, S. Grimme, J.M.L. Martin, ) Benchmark data are available from experiments and high level calculations (mainly for molecular systems) Computed properties: thermochemistry, geometries, vibrational frequencies, conformation energy, weak interactions, Recent examples: GMTKN30 (Goerigk-Grimme PCCP 13 (011) 6670) CE345, CS0, PE39, PS47 (Peverati-Truhlar Phil. Trans. R. Soc. A (013) ) Assessing DFT methods: Molecular systems 35 General Main group Thermochemistry Kinetics and Noncovalent interactions Goerigk-Grimme PCCP 13 (011) 6670) 30 datasets with 841 data 47 functionals + HF, MP Chemistry Energetic Database Peverati-Truhlar Phil. Trans. R. Soc. A (013) 15 datasets with 345 data 77 functionals + HF, MP (no DH) GMTKN30 CE Accuracy follows the Jacob s ladder Accuracy improves in each decades 13

14 Rel. Dev. 9/07/014 Assessing DFT methods: Solids 36 Examples: Perdew (004-01), Haas (009), Peverati-Truhlar (013) Mostly limited to first three rungs (i.e. LDA, GGA, mgga) In some cases: SC-RSH methods (Kresse 011, Scuseria 005, 01, Peverati-Truhlar 013), rarely global-hybrids or LC-RSH Because of the high expense of long-range HF exchange, global hybrid and long-rangecorrected hybrid functionals are not included in this evaluation for solids (from Peverati-Truhlar 013) Double-hybrids still to be assessed for solids Very few high-level results to compare with (e.g. CCSD(T) incremental scheme, RPA, ) How to do that for solids? Reference data must be available (e.g. band gaps, lattice parameters, ) Large basis sets should be used (Extrapolation?) Convergence to the CBS for post-scf calculations Testing 4 th and 5 th rungs for solids Band gap of solids Lattice constants Bulk Modulus Cohesive energy Band Gaps - 45 systems - SC- and MC-RSH SC-RS / MC-RS Hybrids Solid NH Calculated Band Gap (ev) Experimental Band Gap (ev) HSE06 HSEsol HISS exp. 45 systems (semiconductors/insulators) 16 functionals: LDA,GGA,GH & RSH Simple periodic model systems 14 functionals: GH, RSH & DH D. Presti MSc Thesis & S. Fabre MSc Thesis 14

Band gap of solids 38 Still a matter of debate within the KS approach Band gap is usually computed as the difference between top VB and bottom CB This is correct for the Generalized KS approach

15 Band gap of solids 38 Still a matter of debate within the KS approach Band gap is usually computed as the difference between top VB and bottom CB This is correct for the Generalized KS approach hybrid functionals Fundamental gap: IP-EA (Koopmans theorem) difficult to measure in solids Optical gap: easily accessible from experiment Usually comparison with optical gaps Dataset: 45 systems (SC/40 [Heyd-Scuseria 004] + insulators + Ne, Ar) From very-narrow to very-wide band gap solids: 0. < BG (ev) < methods (HF, LDA, GGA, 4 GH, SC-RSH, 1 MC-RSH, 5 LC-RSH) Basis Sets of at least TZ quality (mostly from Heyd-Scuseria 004) Experimental geometry; Fundamental band gaps at low T Band gap of solids: Calc. vs Exp. Band Gaps - 45 systems - HF and DFT HF / LDA & GGA 45 systems & 16 methods Global Hybrids Band Gaps - 45 systems - HF/DFT global hybrids Calculated Band Gap (ev) HF LDA PBE PBEsol exp. Calculated Band Gap (ev) PBE0 PBEsol0 B97 B3LYP exp Experimental Band Gap (ev) Experimental Band Gap (ev) LC-RS Hybrids Band Gaps - 45 systems - LC-RSH Band Gaps - 45 systems - SC- and MC-RSH SC-RS / MC-RS Hybrids Calculated Band Gap (ev) LC-ωPBE LC-ωPBEsol RSHXLDA ωb97 ωb97-x exp. Calculated Band Gap (ev) HSE06 HSEsol HISS exp Experimental Band Gap (ev) Experimental Band Gap (ev) 15

16 Mean and Mean Abs. Error (in Å) 9/07/014 Band gap of solids: Statistics 45 systems & 16 methods 40 Which is the best performer? HISS, HSE06, B97,... (MAE=0.7 ev)? GHs and SC/MC-RSHs give very similar results Performance degrades for solids with a very-wide band gap Lattice constants of solids: Statistics systems / 45 data & 16 methods Ref: ZPAE corrected experimental data Method ME MAE PBEsol0 and HSEsol are the best performers, followed by HISS and PBEsol As expected: HF overestimates, LDA underestimates LC, GGAs in between but overstimated GHs and RSHs give results that depend on the adopted XC functional (e.g. PBE vs PBEsol) Computed MAEs agree with previous published data 16

17 Rel. Dev. 9/07/014 Cohesive energy: strategy 43 Cohesive energy prediction of simple periodic model systems for which highly accurate reference data are available: System Features Reference LiF Ionic solid Exp./CCSD(T) HF and HCl infinite chains Weakly bounded CCSD/CBS NH 3 crystal Weak H-bonded solid Exp. 14 functionals: GH, RSH and DH Hierarchical basis sets (cc-pvxz(-gh), X=D,T,Q,5 - Dunning s family) [extrapolation: TQ o Q5] Very tight computational parameters DH: SCF calculation with CRYSTAL + post-scf PT with CRYSCOR + Results are also compared with MP and SCS-MP Cohesive energy of LiF 44 Global H SC-/MC-RSH LC-RSH Double Hybrids DE= mha Extr. Q5 Exp. GHs underestimate cohesive energy as well as SC-/MC-RSH All LC-RSH are overestimated, while B97 and B97-X give good results DHs provide slightly underestimated values Other computed results: CCSD(T)=0.00, RPA=-0.06, RPA + =

Rel. Dev. Rel. Dev. Rel. Dev. 9/07/014 Cohesive energy of HF and HCl infinite chains 45 Extr. Q5 CCSD/CBS Relative deviations wrt CCSD/CBS [Buth-Paulus CPL 004] 0.0 0.15 0.10 0.05 0.00-0.05-0.10-0.

18 Rel. Dev. Rel. Dev. Rel. Dev. 9/07/014 Cohesive energy of HF and HCl infinite chains 45 Extr. Q5 CCSD/CBS Relative deviations wrt CCSD/CBS [Buth-Paulus CPL 004] (HF... HF) DE= kj/mol (HCl.. HCl) DE=-8.80 kj/mol H-bond in the HF infinite chain is stronger than for the HCl one Relative deviation for HCl is almost twice than HF Some GHs and SC/MR-RSHs perform well while LC-RSHs show large errors DHs give good results Cohesive energy of NH3 crystal 46 Extr. TQ Exp DE=-36.3 kj/mol Most of the methods underestimate cohesive energy of solid ammonia B97 and B97-X tend to overestimate the cohesive energy of weakly bound systems highly parametrized functionals DHs contain only a part of PT correction long-range correlation effects (vdw) partly included (Dispersion corrections???) 18

DFT and weak interactions: Starway to Heaven In analogy with the Jacob s ladder classification of functionals the stairway to heaven is used to classify and group DFT-based dispersion correction

19 DFT and weak interactions: Starway to Heaven In analogy with the Jacob s ladder classification of functionals the stairway to heaven is used to classify and group DFT-based dispersion correction schemes. J. Klimes & A. Michaelides JCP 137 (01) Step 4: Many body disp., virtual orbitals Step 3: Long-range, nonlocal DFs Step : Structure-dependent C 6 Step 1: Fitted C 6 coefficients Step 0: Devised on fitted B.E. 4) MBD, RPA, Double-Hybrids 3) LRD, vdw-df ) DFT-D3, vdw (TS), BJ (XDM) 1) DFT-D 0) DCACP, Minnesota 48 London-type C n /R n term: DFT-D/D3 49 C E ' s f ( R ) ij n Disp n n dmp ij ij R, g g ij, g e.g. f dmp damping function 1 Rij, g 1 e d R ij, g / R0, ij 1 Total energy is then computed as: E =E +E DFT-D/D3 DFT Disp DFT-D: (S. Grimme, J. Comput. Chem. 7 (006) 1787) Includes only the -C 6 /R 6 term s 6 : scaling factor for each DFT method DFT-D3: (S. Grimme, J. Antony, S. Ehrlich, H. Krieg, J. Chem. Phys. 13 (010) ) Less empiricism (R 0,ij and C n ij parameters are computed from first principles) Environment-dependent dispersion coefficients (C n AB ) through coordination number (CN) Extension to nth-order dispersion coefficients (orders n=6,8) of two-body term E () Three-body term E (3) Broader range of applicability (for 94 elements H-Pu) and higher accuracy DFT-D is implemented in CRYSTAL09 for energy and gradients (atoms and cell) DFT-D3 will be soon available in CRYSTAL14 (thanks to G. Brandenburg) 19

20 Cohesive energy of NH3 crystal: DH + DFT-D DE=-36.3 kj/mol D D* D D* D D* D D* Exp. DH augmented with an empirical London-type correction (DFT-D [S. Grimme, J.Comp. Chem. (006)]) Inclusion of the dispersion correction remarkably improves the cohesive energy Even better results (D*) can be obtained when vdw radii in the DFT-D correction are rescaled as for B3LYP-D* [ et al. CrystEngComm (008)] 51 Final remarks In CRYSTAL14, DFA for all of the five rungs of the Jacob s Ladder will be available, namely: LDA, GGA, mgga, Hybrids, Double-Hybrids Results indicate that DH provide accurate cohesive energies of simple periodic model system even if they still lack some dispersive contributions, in agreement with evidence from molecules. DH are less basis set dependent than MP (not shown) GH and SC-/MC-RSH perform equally well for band gap prediction of solids, although they are still poor for wide gap insulators. SC/MC-RSH give also good lattice constants and bulk moduli DH still miss some long-range correlation DFT-D correction is needed In perspective, both RSH and DH can be improved either through a reparametrization for solids or combining RSH with a long-range post-scf correction. 0

Hamiltonians: HF & DFT

ISAMS15 International School on Ab initio Modelling of Solids with CRYSTAL14 Regensburg, 19 24/07/2015 1 Hamiltonians: HF & DFT and he dreamed that there was a ladder set up on the earth, the top of it