The Incremental Scheme: A Local Correlation Method for Molecules and Solids

Transcription

1 The Incremental Scheme: A Local Correlation Method for Molecules and Solids Michael Dolg Theoretical Chemistry, University of Cologne, Greinstr. 4, Cologne, Germany Summerschool, Beijing Normal University, August 6, 2010 M.Dolg The Incremental Scheme: A Local Correlation Method for Molecules and Solids Theoretical Chemistry, University of Cologne, Greinstr. 4, Cologne, Germany

2 Group (some who work(ed) on the incremental scheme) S/DD ( ): (Martin Mödl, Ming Yu, Klaus Doll), Friedemann Schautz, Yigui Wang, Stephane Pleutin, Gongyi Hong, Martin Albrecht, Ayjamal Abdurahman, Heinz-Jürgen Flad, Yixuan Wang, M.D., Simon Kalvoda, Peter Reinhardt. DD: Gongyi Hong, Wenjian Liu, Heinz-Jürgen Flad, Friedemann Schautz, Wolfgang Küchle, M.D. BN/K ( ): (Katarzyna Walczak), Sombat Ketrat, Martin Böhler, Xiaoyan Cao, M.D., Birgit Börsch-Pulm, Jun Yang, Michael Hanrath, Atashi Basu Mukhopadhyay, Mark Burkatzki, Rebecca Fondermann, Joachim Friedrich. Simon Kalvoda, Alok Shukla, Stephane Pleutin, Yixuan Wang. The Incremental Scheme

3 Outline 1 Introduction 2 Method I: Basic Equations 3 Results I: Crystalline Solids 4 Method II: Extension to Molecules 5 Results II: Molecules 6 Method III: Recent Improvements 7 Results III: Computational Effort, New Developments 8 Conclusions and Outlook 9 Acknowledgments 10 Appendix: Wannier-orbital Hartree-Fock Approach The Incremental Scheme

4 Ring-opening Polymerisation to Polyhydridophosphazene 1 Desireable: one computational approach for monomers, oligomers H and polymer! P H 2 N P N NH P NH 2 2 NH NH 2 P NH H 2 N NH H2 N H 2 N NH 3 H H2 N kcal/mol P NH 2 H 2 N H 2 N P NH 20b kcal/mol H 2 N H 2 N kcal/mol kcal/mol 18b 19b H 2 N H P N H P NH NH 2 NH 2 21b H H H 2 N P N NH kcal/mol H H 2 N P N P NH 2 H P N P NH 2 NH 2 22b H N P NH kcal/mol H H 2 N H 2 N N P P N 24b H 3* kcal/mol = kcal/mol N H 2 N NH P 2 ROP H H P NH 2 N H P N NH 2 25b H N P PH N P n H 2 N N NH 2 H NH 2 NH 2 NH 2 23b MP2 results, 6-31G** basis sets 1 R. Fondermann, M.D., M. Raab, E. Niecke, Chem. Phys. 325 (2006) 291. The Incremental Scheme

5 DFT vs. WFT Solid State Physics Ε[ ρ ] cheap unsystematic success: large systems? expensive systematic Ε[ ψ] Quantum Chemistry Wannier-type orbital picture: ionic crystals covalent crystals Is it possible to carry out a full CI for an infinite system? (Infinite number of nuclei, electrons, orbitals,...) The Incremental Scheme

6 Coordinate System of Quantum Chemical Methods? rel. (4 k.) e.g. DC,DCB rel. (2 c.) e.g. DKH, CPD scalar rel. (1 c.) e.g. CG nrel. Hamiltonian HF CISD MCHF CCSD MRCI CCSD(T) CCSDT FCI... many particle basis exact solution or experiment VDZ VTZ VQZ... Min. DZ TZ QZ... complete rel. ECP (MP,PP) "nrel." one particle basis MP2 "HF" LDA MP3 GGA... ṂP4 DFT? Wanted: suitable compromise between accuracy and computational effort! hybrid... Definitions: correlation effect: X correlation := X exact X Hartree Fock relativistic effect: X relativity := X relativistic X nonrelativistic Note: The effects are usually not additive and depend on the methods applied! The Incremental Scheme

7 On the Need to Include Coulomb Correlation Independent particle (orbital) model Hartree-Fock/Self-Consistent Field (HF/SCF) approach One electron in the average field of all other electrons and the (fixed) nuclei. Fermi correlation: yes (Slater determinant wavefunction) Coulomb correlation: no ( CI, CC, QMC, DFT,...) R e (Angstroem) F 2 Exp. SCF MP2 CCSD CCSD(T) vdz vtz vqz v5z v6z basis set quality D e (ev) F 2 Exp. HF/SCF MP2 CCSD CCSD(T) F 2 is not stable at the HF level! vdz vtz vqz v5z v6z basis set quality omega e (100 cm -1 ) F 2 Exp. SCF MP2 CCSD CCSD(T) vdz vtz vqz v5z v6z basis set quality The Incremental Scheme

8 Scaling of Computational Effort with System Size computational effort (e.g. # operations) formal scaling of methods (no tricks!) CI (n orbitals, m-fold excitations) scales roughly as O(n 2m+2 ). CCSD(T) (O(n 7 )) CCSD (O(n 6 )) MP2 (O(n 5 )) HF (O(n 4 )) DFT (O(n 3 )) MM (O(n 2 )) linear scaling (O(n)) (ultimate goal for molecules) system size (e.g. # orbitals) Note: 1 year seconds. How to treat large or periodic systems? The Incremental Scheme

9 Coulomb Hole: Basis Set Dependence and Locality 378 K.L. Bak et al. / Journal of Molecular Structure 567± ) 375±384 Fig. 2. The Coulomb hole described in nite basis-set calculations. The ground-state helium wave function plotted on a circle of radius 0.5 a 0 about the nucleus with one electron xed at the origin of the plot. The grey thick line represents the exact wave function and the black lines represent FCI wave functions using cc-pvxz basis sets with xˆd, T, Q and 5, respectively. The black dotted line is the numerical HF wave function. The Incremental Scheme

10 Local Correlation Methods only one citation per approach/group listed... Pulay, Saebo, Theor. Chim. Acta 69, 357 (1986) Stoll, Chem. Phys. Lett. 191, 548 (1992) Hampel, Werner, J. Chem. Phys. 104, 6286 (1996) Maslen, Head-Gordon, J. Chem. Phys. 109, 7093 (1998) W. Li, S. Li, J. Chem. Phys. 121, 6649 (2004) Fedorov, Kitaura, J. Chem. Phys. 121, 2483 (2004) Flocke, Bartlett, J. Chem. Phys. 121, (2004) Subotnik, Head-Gordon, J. Chem. Phys. 123, (2005) Auer, Nooijen, J. Chem. Phys. 125, (2006)... and many others (cf. also the local functionals in DFT!) The Incremental Scheme

11 Energy (resp. Orbital Value) Energy (resp. Orbital Value) Noninteracting Particles in a One-dimensional Box n=2 n=1 canonical orbitals, orbital densities, total density (ϕ 1 1 ϕ1 2 ): Length l Energy n=2 n= Length Density Length localized orbitals, orbital densities, total density (ϕ 1 l ϕ1 r ): 2 r Unitary transformation: 1 r>=( 1>- 2>)/sqrt(2), l>=( 1>+ 2>)/sqrt(2) Note: no eigenfunctions of Hamiltonian! Length Energy l r inter-orbital "repulsion" minimized Length Density Note: wavefunction, density and energy unchanged! Length The Incremental Scheme

12 Canonical and Localized Orbitals for Cysteine 24th canonical vs. 24th localized orbital of cysteine Boys localization minimizes spatial extent of orbitals, i.e. n i=1 < ϕ iϕ i ( r 1 r 2 ) 2 ϕ i ϕ i > The Incremental Scheme

13 The Incremental Scheme (A Simple and General Local Correlation Approach) Nesbet, atoms in the framework of CI (1967) Stoll, 3D periodic systems in the framework of CC (1992) Since 1992, Stoll, Fulde, Dolg, Paulus, Birkenheuer,...: crystals 3D semiconductors/ionic crystals, polymers, metals : structure, stability and bandstructure,..., magnetic coupling A lot of hand work required... Automatization a Goals of our current work: molecules Smooth potential energy surfaces b Properties c Open-shell systems d Energy gradients, computational efficiency, point and space group symmetry,..., treatment of periodic systems,... a J. Friedrich, M. Hanrath, M.D., J. Chem. Phys. 126, (2007) b J. Friedrich, M. Hanrath, M.D., Chem. Phys. 338, 33 (2007) c J. Yang, M.D., J. Chem. Phys. 127, (2007) d J. Friedrich, M. Hanrath, M.D., J. Phys. Chem. A 111, 9830 (2008) The Incremental Scheme

14 Incremental Scheme: Procedure and Basic Equations 2 Solve the Hartree-Fock problem Divide the system into localized orbital domains Calculate the correlation energies for single domains, pairs of domains,... up to a given order Expand the total correlation energy as E corr = i ε i + 1 2! ε ij + 1 3! ij ε ijk +... ijk D 2 φ 4 φ 2 D 3 φ 5 D φ 3 D 1 φ 6 D 4 φ 1 ε i = ε i ε ij = ε ij ε i ε j ε ijk = ε ijk ε ij ε ik ε jk ε i ε j ε k... periodic systems: i reference unit cell, j, k,... all unit cells; finite systems: i,j,k,... all domains 2 H. Stoll, Chem. Phys. Lett. 91 (1992) 548. The Incremental Scheme

15 The WANNIER Code I 3 Self-consistent orbital localization by means of projection operators: F HF = T+U+2 J β K β + λ k γ γ(r k ) >< γ(r k ) β β k>1 γ k = 1 refers to the reference cell. λ k γ enforces < γ(r 1) γ(r k ) >= δ 1k. Frozen-environment self-consistent field procedure. γ(r 1 ) > self-consistently localized in C. Total energies per unit cell and bandstructures virtually identical to Bloch-orbital based results. Regions of Space Used in the WANNIER Code 3 Shukla, M.D., H. Stoll, P. Fulde, Chem. Phys. Lett. 262 (1996) 213. The Incremental Scheme for Solids C N R

16 The WANNIER Code II Partitioning of the real space into three subspaces: (central) reference (unit) cell C. Solution of the modified Hartree-Fock equations for orbitals centered in this region. Interactions with the rest of the infinite system are treated depending on their range, i.e., thresholds for the neglect of vanishingly small contributions are introduced. Contributes to T, U, J, K and the projection operators. short-range environment, i.e., neighborhood of the reference cell N. Contributes to U, J, K and the projection operators. long-range environment, i.e., rest of the infinite system R. Contributes only to U and J. The Incremental Scheme for Solids

17 Full-CI for the 3d crystal LiH 4 Li [2s1p], H [3s1p] (R. Dovesi et al., Phys. Rev. B 29, 3591 (1984)) Wannier-function based HF+FCI Increment Wannier Finite O R Function Cluster Approach Approach [Hartree] [Hartree] nn nn nn log(- two-body increment) Van der Waals like decay of two-group increments decay as 1/distance nn 2nn 3nn HF/FCI/exp.: lc 4.106/4.067/4.060Å, ce 129.4/164.4 (93.4%)/176.0mH, bm 33.4/36.5/ , GPa 4 A. Shukla, M.D., P. Fulde, H. Stoll, Phys. Rev. B 60, 5211 (1999). The Incremental Scheme for Solids

18 Application to Simple Crystals: CCSD(T) E = E HF + E CCSD(T) HF: infinite system, Bloch orbitals (CRYSTAL) CCSD(T): finite cluster, incremental scheme, localized orbitals (MOLPRO) method MgO CaO NiO lc (Å) ce (ev) lc (Å) ce (ev) lc (Å) ce (ev) RHF (72.0 %) (73.4 %) (64.6 %) CCSD(T) (96.3 %) (94.1 %) (93.3 %) Exp O: AE, aug-p-cc-vtz [5s4p3d2f]; Mg: PP(2,MDF)+CPP, [4s4p], implicit CV-correlation; Ca: PP(10,MWB), [6s6p5d2f1g], explicit CV-correlation; Ni: PP(18,MDF), [6s5p4d3f], explicit CV-correlation. The Incremental Scheme for Solids

19 Application to Simple Crystals: CCSD(T) 5 sum of increments (Hartree) Sum of CCSD(T) Correlation Increments Ionic Solids: MgO, CaO, NiO 1-body 2-body 3-body 23% 43% 58% 50% 51% 79% -1% -1% -2% MgO CaO NiO 5 K.Doll, M.D., P.Fulde, H.Stoll, Phys. Rev. B 52 (1995) 4842; Phys. Rev. 55 (1997) 10282; Chemical Papers 51 (1997) 357; K.Doll, M.D., H.Stoll, Phys. Rev. 54 (1996) The Incremental Scheme for Solids

20 A Simple Ansatz for Polymers A finite-cluster approach for polymers Consider the difference of incremental expansions of oligomers (without end groups)... E corr unit cell E cor (n + 1) E cor (n) = ɛ 0 + n+1 i>0 ɛ 0i ɛ (n+1) n+1 j>i>0 ɛ 0ij +... error in E corr unit cell should be smaller than ɛ 0n+1, which decays roughly as 1/(n + 1) 6. log(- ε corr (0,n)) Two-body correlation increments between terminal unit cells in (HCN) n+1 H slope log(n) The Incremental Scheme for Solids

21 Polyacetylene 6 t ct tc H C C H H C C H H C C H 8 H C C H H C C H 8 8 cohesive energy: HF 7.31 ev CCSD(T) 9.33 ev bond alternation 0.10 Ang. relative energy: HF 0.10 ev MP ev bond alternation 0.07 Ang. relative energy: HF 0.10 ev MP ev bond alternation 0.08 Ang. Derived structural parameters agree with experimental values within the error bars, e.g. ± 0.01 Å for bond lengths. t-pmi is most stable in agreement with experimental evidence and previous HF and MP2 results (Karpfen, Höller, 1981; Dovesi, 1984; Suhai, 1992;...). t-pa is 0.60 ev per (CH) 2 -unit less stable than benzene. 6 M. Yu, S. Kalvoda, M.D., Chem. Phys. 224, 121 (1997). The Incremental Scheme for Solids

22 Polymethineimine 7 t ct tc N H N H N H C C C 8 N C H N C H 8 8 relative energy: HF 0.13 ev CCSD(T) 0.17 ev bond alternation 0.09 Ang. cohesive energy: HF 8.99 ev CCSD(T) ev polymerization energy: HF 0.29 ev CCSD(T) 0.37 ev bond alternation 0.03 Ang. relative energy: HF (no minimum) CCSD(T) 0.10 ev bond alternation 0.08 Ang. PMI is a semiconductor and has a bond alternating structure in agreement with previous experimental evidence (ct-pmi?; Wöhrle, 1971, 1974) and early HF results (t-pmi; Karpfen, 1979). ct-pmi is more stable than t-pmi in agreement with DFT-results (Hirata, Iwata, 1997). ct-pmi is 0.22 ev per CNH-unit less stable than 1,3,5-triazine, in qualitative agreement with experimental data (Wöhrle, 1971, 1974). 7 A. Abdurahman, A. Shukla, M.D., Chem. Phys. 257, 301 (2000). The Incremental Scheme for Solids

23 Magnetic Coupling in Na 6 Fe 2 S 6 Solid 8 Embedded Na 10 Fe 2 S 6 cluster calculations with surrounding PPs and point charges (fitted to produce the correct electrostatic potential in the cluster region). 8 M. Mödl, M.D., P. Fulde, H. Stoll, J. Chem. Phys. 105, 2353 (1996) The Incremental Scheme for Solids

24 Magnetic Coupling in Na 6 Fe 2 S 6 Solid CASSCF VTZ Fe occupation numbers, local spin and weights of 3d-occupations in Na 6 Fe 2 S 6 q S N 1 N 2 N2 1 N2 2 Ni 2 N i D E S S 2 1 D E S 2 2 D E S1 S 2 d 4 d 5 d The Incremental Scheme for Solids

25 Magnetic Coupling in Na 6 Fe 2 S 6 Solid Model Hamiltonian Heisenberg Hamiltonian: H = 2 J S 1 S 2 Total and local spin: S = S 1 + S 2 here: S 1 = S 2 = 5/2 Eigenvalues: E(S) = J[S(S + 1) S 1 (S 1 + 1) S 2 (S 2 + 1)] Landé interval rule: E(S) E(S 1) = 2 J S possible values for total spin quantum number: S = S 1 S 2,..., S 1 + S 2 The Incremental Scheme for Solids

26 Magnetic Coupling in Na 6 Fe 2 S 6 Solid Incremental Approach to Magnetic Coupling in Na 6 Fe 2 S 6 Partitioning of the total energy: E(S) = E CAS (S) + E corr (S) Incremental expansion of (dynamical) correlation energy: Ecorr(S) i,j + X Ecorr i,j,k (S) +... E corr (S) = X i E i corr(s) + X i<j i<j<k i = 0 : Fe active Fe 3d orbitals i = 1,..., 8 : S1,..., S8 S orbitals in irrep i Note: we are not interested in the absolute value E corr (S), but rather in the differences E corr (S) E corr (S )! Two (useful) approximations: ( Ecorr Si,Sj )/ S = 0 and ( Ecorr i,j,k )/ S = 0 Simplified (difference-dedicated) incremental expansion: E corr (S) E Fe corr(s) + E S1 corr(s) E S8 corr(s) + E Fe,S1 corr (S) Ecorr Fe,S8 (S) + const The Incremental Scheme for Solids

27 Magnetic Coupling in Na 6 Fe 2 S 6 Solid Number of determinants in CASSCF for Na 6 Fe 2 S 6 (Fe 3d active, i.e., 10 electrons in 10 orbitals; (VTZ+S(1d)+Fe(3f1g) basis sets) S state in D 2h C 1 D 2h 5 11 B 1u A g B 1u A g B 1u A g A dynamical correlation treatment (MRCI, MRACPF) was only possible for S = 5, 4. The experimental data refers to S = 0,1, however. The Incremental Scheme for Solids

28 Magnetic Coupling in Na 6 Fe 2 S 6 Solid direct exchange; CASSCF: E(5) E(4) = a.u. J CASSCF (4 5) = cm 1 direct exchange; CASSCF+CI: n a.u. (MRCI) J E(5) E(4) = MRCI (4 5) = cm a.u. (ACPF) J ACPF (4 5) = cm 1 direct + super exchange; CASSCF: E(5) E(4) = a.u. J CASSCF (4 5) = cm 1 direct + super exchange; CASSCF+CI: n a.u. (MRCI) J E(5) E(4) = MRCI (4 5) = cm a.u. (ACPF) J ACPF (4 5) = cm 1 Assumption: ratio J(S S +1)/J(S + 1 S +2) is independent of the electron correlation treatment! J MRCI (0 1) = = J CAS (0 1)/J CAS (4 5) J MRCI (4 5) = = ( 28.75)/( 18.79) ( 52.09) cm 1 = = cm 1 J ACPF (0 1) = = J CAS (0 1)/J CAS (4 5) J ACPF (4 5) = = ( 28.75)/( 18.79) ( 64.98) cm 1 = = cm 1 Experimental value (neutron scattering): J exp (0 1) = 95 cm 1 The Incremental Scheme for Solids

29 Generation of the One-Site Domains I 9 Boys localization of the MOs LMOs {φ a } Construction of the centers of charge according to the diagonal elements of the dipole-integrals in LMO basis: φ a R φ a x φ a a := φ a y φ a φ a z φ a Build the distance matrix D for the centers of charge Determine the connectivity matrix C according to: 0, if D ij > t con 1 C ij := D ij, if D ij t con 1 D ij < , if D ij t con 1 D ij J. Friedrich, M. Hanrath, M.D., J. Chem. Phys. 126, (2007) The Incremental Scheme for Molecules

30 Generation of the One-Site Domains II The connectivity matrix C represents an edge-weighted graph METIS-graph partitioning a is used to divide the set of centers of charge { R a } into disjoint subsets (corresponding to orbital domains) while minimizing the sum of weights of the cut edges graph and connectivity matrix for 8 LMOs: 3 possible domains are (1,2,3,7), (4), (5,6,8) C = a G. Karypis, V. Kumar, SIAM J. Sci. Comput. 20, 359 (1998) The Incremental Scheme for Molecules

31 MP2 and CCSD(T) I MP2 and CCSD(T) energy expressions not invariant with respect to unitary transformations within the occupied or the virtual space! Account for this by the diagonalization of the ɛ matrix in the subspace of the domain ɛ = C L F(C L)C L = core domain virtual core ɛ cc ɛ cd ɛ cv domain ɛ dc ɛ dd ɛ dv virtual ɛ vc ɛ vd ɛ vv C L := coefficient matrix in the local MO basis F(C L ) := Fock matrix in the local MO basis Diagonalize ɛ dd block! The Incremental Scheme for Molecules

32 MP2 and CCSD(T) II Construct the total unitary transformation U core domain virtual core 0 ½ 0 domain 0 Ũ 0 = U ½ virtual 0 0 Ũ is the unitary matrix which diagonalizes ɛ dd Pseudo-canonical MO s are obtained by C L = C L U The procedure ensures that the energy of the canonical MP2 or CCSD(T) is obtained at the highest order of the expansion The Incremental Scheme for Molecules

33 Computational Effort The number of calculations increases steeply with the order N calc = O ( ) N D calc := number of calculations O := order of the expansion i D := # of domains i=1 Example: D = 10 i N calc Computational effort of individual calculation increases roughly with i n (n = 5 (MP2), 6 (CCSD), 7 CCSD(T)) Truncation at low order desireable, e.g. the optimum localization scheme which minimizes inter-domain interactions is seeked! However, only small differences are observed in practise between various schemes. The Incremental Scheme for Molecules

34 Computational Effort Efficient screening procedures are needed to reduce the scaling Distance screening In a local basis we expect the incremental contributions of far distant domains to be negligibly small, i.e. not all increments of a given order have to be evaluated Use the distance between the one-site domains to decide a priori if a n-site increment is important or not Energy screening Estimate the incremental energy contributions with calculations using a cheaper lower-level method The Incremental Scheme for Molecules

35 Localized Orbitals for C 16 H 2 canonical Hartree-Fock π-orbitals: C 2p x coefficients, π-electron density localized Pipek-Mezey π-orbitals: C 2p x coefficients, π-electron density The Incremental Scheme for Molecules

36 E corr (Hartree) Correlation Increments for C 16 H 2 C 16 H 2 π x -shells: 1-, 2- and 3-body increments, n-th order correlation energy canonical Hartree-Fock π-orbitals: E corr (Hartree) E corr (Hartree) E corr (Hartree) % st order % nd order % 3rd order exact 0.00 localized Pipek-Mezey π-orbitals: E corr (Hartree) E corr (Hartree) distance screening: 1,2>1,3>... etc large increments (i,i+1) out of E corr (Hartree) large increments (i,i+1,i+2) out of E corr (Hartree) % st order 98.9% 99.8% 2nd order 3rd order smoother/faster convergence, lower number of relevant increments exact The Incremental Scheme for Molecules

37 Truncation by Energy Screening: Alkyne Simulated energy screening, CCSD, 4th-order expansion energy threshold E corr error n calc % [au] [au] [kcal/mol] total number of calculations The Incremental Scheme for Molecules

38 Decay of the Two-Body Increments for Decane Two-body increments f/r^6 CCSD(T) correlation contributions n-octane cc-pvdz energy [au] e-04 1e-05 1e-06 1e R [Angstrom] % E corr CCSD(T) % 22.7 % 0.3 % order RI-BP86/SVP optimized geometry cc-pvdz CCSD(T) calculation 10 domains, two-body increments ε 1i (i = 2,...,10) The Incremental Scheme for Molecules

39 Truncation by Distance Screening In a local basis increments decrease with increasing distance Increments decrease with increasing order of the expansion Use an order-dependent distance threshold t dist a) t dist = f O i D a φ a1 φ a2 D d φ d1 φ d2 φ b1 D b φ b2 φ b3 A R min B D c D e φ c2 φ c2 φ e1 b) t dist = f O 2 i X f is an adjustable truncation parameter X D R min > t dist ε X 0 The Incremental Scheme

40 Truncation by Distance Screening: DNA Base-Pair O E E corr % E CCSD corr [kcal/mol] [H] exact Incremental vs. full CCSD/6-31G** correlation energies for the guanine-cytosine dimer. (16 domains, core=19) The canonical CCSD calculation is already impossible on a Pentium IV (1.35 GB RAM) PC 421 of 2516 CCSD calculations were necessary for t dist = 16 O i The Incremental Scheme for Molecules

41 Contribution of Low Order Summations p s,t = ( ) ( D t) (s t) with s > t All possibilities to include the 1st order summation in the 5th order summation 5,1 5,2 5,3 5,4 F(s,t) = s 1 p s,i F(i,t) 4,1 4,2 4,3 F(t,t) = 1 i=t 2,1 3,1 3,2 2,1 2,1 3,1 3,2 2,1 order O example: (20 domains) F(4,O) Prefactors for explicit inclusion of tth order summation in sth order summation for D domains: p s,t ; total prefactor for explicit and implicit inclusion: F(s, t). Need to adapt convergence thresholds according to the desired accuracy! The Incremental Scheme

42 Error Propagation - Simulation by Random Numbers Limited accuracy of the correlation calculations ε X = X 10 6 ε X = X O 1e+06 1e+05 3rd order 4th order 5th order 1e+06 1e+05 3rd order 4th order 5th order N/mH N/mH 1e+04 1e+04 1e+03 1e energy [mh] 1e energy [mh] Results based on 2000 sets of uniformly distributed random numbers X The Incremental Scheme

43 Performance of the Incremental Scheme I 10 E = E corr E CCSD corr [kcal/mol]; E corr [H]. Naphthalene (6-31G**) O E E corr % E CCSD corr Alkyne (6-31G**) O E E corr % E CCSD corr J. Friedrich, M. Hanrath, M.D., J. Chem. Phys. 126, (2007) The Incremental Scheme for Molecules

44 Performance of the Incremental Scheme II Nb 2 Cl 10 (cc-pvdz, ECP28MWB) O E E corr % E CCSD corr E = E corr E CCSD corr [kcal/mol]; E corr [H]. TiCp 2 Cl 2 (6-31G*, ECP10MDF) O E E corr % E CCSD corr The Incremental Scheme for Molecules

45 Performance of the Incremental Scheme III 11 E = E corr E CCSD corr [kcal/mol]; E corr [H]. (H 2 O) 11 (6-31G**) O E E corr % E CCSD corr Methane-Indole (6-31G**) O E E corr % E CCSD corr J. Friedrich, M. Hanrath, M.D., J. Phys. Chem. A 111, 9830 (2007) The Incremental Scheme for Molecules

46 Open Shell Systems: 1,3-dimethylimidazol-2-ylidene (S,T) O E corr error % Ecorr CCSD [Hartree] [kcal/mol] S a exact T b E active exact RCCSD/cc-pVDZ, DFT-optimized structures; a 6 domains, b 7 domains. The Incremental Scheme for Molecules

47 Potential Energy Surfaces I: n-octane 12 n-octane DFT (RI-BP86/SVP) relaxed scan along the C4-C5-bond Incremental CCSD/6-31G** energies (8, 9 domains) Smooth potential curve around R e = Å? 12 J. Friedrich, M. Hanrath, M.D., Chem. Phys. 338, 33 (2007) The Incremental Scheme for Molecules

48 Potential Energy Surfaces II: n-octane f=14 f=16 f=inf exact CCSD f=14 f=16 f=inf exact CCSD f=14 f=16 f=inf exact CCSD energy [H] energy [H] energy [H] R R O = 2 O = 3 O = 4 f=14 f=16 f=inf exact CCSD f=14 f=16 f=inf exact CCSD R f=14 f=16 f=inf exact CCSD energy [H] energy [H] energy [H] R R zoom zoom zoom R The Incremental Scheme for Molecules

49 Potential Energy Surfaces III: n-octane f=14 f=16 f=inf exact CCSD f=14 f=16 f=inf exact CCSD f=14 f=16 f=inf exact CCSD energy [H] energy [H] energy [H] energy [me h ] R R O = 2 O = 3 O = 4 R energy [me h ] R error error error energy [me h ] R R The Incremental Scheme for Molecules

50 Core-valence Correlation 13 Additional independent incremental expansions for each core domain and nearby valence domains, i.e. inter-core correlation neglected. Individual truncation/basis sets possible. N core domains (cores to be correlated), n valence domains E c+cv corr = N ε i I=1 N I=1 n ε Ij + 1 3! j=1 N I=1 n n 1 j=k+1 k=1 ε Ijk +... E v corr = n ε i n n 1 ε ij + 1 3! n n 1 n 2 ε ijk + i=1 i=j+1 j=1 i=j+1 j=k+1 k=1 13 J. Friedrich, K. Walczak, M.D., Chem. Phys. 356, 74 (2009) The Incremental Scheme for Molecules

51 Core-valence Correlation: Cysteine Cysteine HS-CH 2 -CHNH 2 -COOH, cc-pcvdz basis set, CCSD(T), 7 core domains, 5 valence domains E c+cv corr : O = 1 core correlation; O > 1 core-valence correlation O E c+cv corr error % E c+cv corr E v corr error % E v corr exact The Incremental Scheme for Molecules

52 Symmetry Analysis 14 Identify symmetry equivalent orbitals by the application of the symmetry operators of the group Ô v = v, e.g. v:= charge center This can be done for domains in an analog way - A domain D λ is a set of centers of charge ÔD λ = D λ = {Ô v a v a D λ } Build the n-site domains and classify the n-site domains by symmetry operations into equivalence classes At the end we calculate the correlation energy for one element per equivalence class Symmetry-adapted one-site domains required! 14 J. Friedrich, M. Hanrath, M.D., Chem. Phys. 346, 266 (2008) The Incremental Scheme for Molecules

53 Usage of Symmetry: A Mercury Cluster I Hg 20 RI-BP86/SVP geometry in T d CCSD/ECP78MWB energies, core=0 Approximately symmetric LMOs Equivalence classes of n-site domains Correlation energies of the 20 one-site domains [Hartree] Class 1 Class 2 Class accuracy 10 9 Hartree The Incremental Scheme for Molecules

54 Usage of Symmetry: A Mercury Cluster II O n calc E corr E corr E CCSD corr [Hartree] [kcal/mol] % E CCSD corr C domains T d domains exact CCSD The Incremental Scheme for Molecules

55 Usage of Symmetry: Periodic Systems Fully automated implementation for periodic systems Symmetry adaption for space groups 15 The summation of the first index is restricted to the reference cell E cell corr = 1 1! n ε i + 1 2! i=1 n i=1 ε ij + 1 3! j=1 n i=1 j=1 ε ijk +... k=1 The infinite sums are restricted to a finite region close to the reference cell The current version uses a HF-reference from a finite cluster The accuracy is limited due to the size of the cluster 15 Parameterization of the operators downloaded from Bilbao crystallographic server The Incremental Scheme for Solids

56 Usage of Symmetry: Polyacetylene Reference wavefunction from the C 16 H 18 The infinite summations were restricted to next neighbor cells Incremental CCSD/6-31G** energy Order O i O i -Correction E corr [au] The error of the third order expansion is about 1.4 kcal/mol This is due to the finite cluster model, since the inclusion of the second nearest neighbor cells worsens the result The Incremental Scheme for Solids

57 Domain Specific Basis Set Approach 16 In order to reduce the computational cost it is necessary to reduce the virtual space of the domains Use a small(er) basis set in the environment of a domain Determine the main region of a domain by a distance threshold t main Obtain the orbitals by a HF calculation in the domain specific basis set with a subsequent Boys localization Identify the domains by the centers of charge using the mapping R a (B 1 ) R a (B 2 ) with R φ a x φ a a = φ a y φ a φ a z φ a 16 J. Friedrich, M.D., J. Chem. Phys. 129, (2008) The Incremental Scheme for Molecules

58 Accuracy for Decane C 10 H 22 MP2 CCSD CCSD(T) O E corr error E corr error E corr error [Hartree] [kcal/mol] [Hartree] [kcal/mol] [Hartree] [kcal/mol] 10 domains domains Basis: cc-pvdz; Environment basis: H: STO-3G; C: 6-31G; t main =3 Error: E corr - E X corr (X=MP2, CCSD, CCSD(T)) The Incremental Scheme for Molecules

59 Timings for Decane C 10 H 22 method # of # of # of wall total RAM domains calc. slaves time time [GB] [%] [%] canonical incremental rd order incremental CCSD(T)/cc-pVDZ calculations Environment basis: H: STO-3G; C: 6-31G; t main =3 The Incremental Scheme for Molecules

60 Timings etc. for Hexadecaoctaene C 16 H 18 method O # of wall total RAM disk error slaves time time [GB] [GB] [kcal/mol] [%] [%] canonical incremental incremental Conventional vs. incremental CCSD(T)/6-31G** calculations Environment basis: H: STO-3G; C: 6-31G; t main =3 The Incremental Scheme for Molecules

61 Water Cluster (H 2 O) 6 (point group S 6 ) DFT optimized structure in S 6 symmetry (RI-BP86/SVP) Incremental CCSD(T), CCSD and MP2 energies Different basis sets Timings The Incremental Scheme for Molecules

62 Accuracy for (H 2 O) 6 (point group S 6 ) MP2 CCSD CCSD(T) O E corr error E corr error E corr error [Hartree] [kcal/mol] [Hartree] [kcal/mol] [Hartree] [kcal/mol] aug-cc-pvdz cc-pvtz aug-cc-pvtz ? ? environment: O: 6-31G, H: STO-3G; t main =3 The Incremental Scheme for Molecules

63 Timings etc. for (H 2 O) 6 (point group S 6 ) method O # of wall time total time RAM disk slaves [%] [%] [GB] [GB] aug-cc-pvdz canonical STO-3G fit G/STO-3G fit G/STO-3G fit cc-pvtz canonical STO-3G fit G/STO-3G fit Timings, RAM and disk space requirements of incremental and canonical CCSD(T) calculations for (H 2 O) 6, using the MOLPRO quantum chemistry package. The Incremental Scheme for Molecules

64 Low-energy Structures of (H 2 O) 6 Bag Boat Book Cage Cyclic Prism Geometries: Day et al. J. Chem. Phys. 112, 2063 (2000) The Incremental Scheme for Molecules

65 Accuracy of Relative CCSD(T) Energies (H 2 O) 6 O E rel error E rel error E rel error Bag Boat Book canonical Cage Cyclic Prism canonical Basis=aug-cc-pVDZ; environment basis: H: STO-3G; O: 6-31G; t main =3; E rel [kcal/mol] The Incremental Scheme for Molecules

66 Timings etc. for (H 2 O) 18 structure taken from Day et al. J. Chem. Phys. 112, 2063 (2000) The Incremental Scheme for Molecules

67 Timings etc. for (H 2 O) 18 MP2 CCSD CCSD(T) O i ε(i) Ecorr (i) ε(i) Ecorr (i) ε(i) Ecorr (i) aug-cc-pvdz a aug-cc-pvtz b core=18, dsp=4, t con =3, t main =3, fit basis: H=STO-3G; O=6-31G correlated orbitals=72, f 2 =25) a 738 basis functions b 1656 basis functions The Incremental Scheme for Molecules

68 Timings etc. for (H 2 O) conventional MP2 aug-cc-pvdz effort: ca. 48 hours CPU time, 61 GB RAM, 305 GB hard disk error 3rd order incremental MP2 aug-cc-pvdz: 0.4 kcal/mol conventional CCSD, CCSD(T) not feasible on current hardware (64 GB RAM, 1 TB disk) 3rd order incremental CCSD(T) aug-cc-pvdz effort: ca. 12 hours on 51 nodes, using at most 2.4 GB RAM and 13.6 GB disk space 3rd order incremental CCSD(T) aug-cc-pvtz effort: at most 7.6 GB RAM and 160 GB disk space 17 J. Friedrich, M. Hanrath, M.D., Z. Phys. Chem. 224, 513 (2010) The Incremental Scheme for Molecules

69 Timings etc. for Na + (H 2 O) 4 18 basis set error relative time C 1 S 4 ( kcal mol ) (%) (%) aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz relative time wrt canonical CCSD(T) calculation in C 2 symmetry structure optimized at TZVPP RI-BP86 DFT level CCSD(T), aug-cc-pvnz basis sets 18 K. Walczak, J. Friedrich, M.D., Chem. Phys., accepted The Incremental Scheme for Molecules

70 Timings etc. for Na + (H 2 O) 4 dsp D x Orb. i error N(i) time at i rel. time ( kcal mol ) (%) (%) 4 4x (C 1 ) 1 16x (C 1 ) 1 16x (S 4 ) aug-cc-pvdz basis set The Incremental Scheme for Molecules

71 Timings etc. for Na + (H 2 O) 4, cntd. dsp D x Orb. i error N(i) time at i rel. time ( kcal mol ) (%) (%) 4 4x (C 1 ) 4 4x (S 4 ) 3 2x x (C 1 ) 5 2x x (C 1 ) aug-cc-pvdz basis set The Incremental Scheme for Molecules

72 Incremental F12-CCSD and F12-MP2 calculations 19 CCSD-F12 = e T HF with T = T 1 + T 2 + T 2 T 1 = taa i aa i ai and T 2 = 1 4 T 2 = 1 4 ijαβ abij t ij ab a aa b a ja i w ij αβ a αa β a ja i with w ij αβ = S ij αβ ˆQ 12 f (r 12 ) ij f (r 12 ) = ( 1/γ)exp( γr 12 ) is the correlation factor. Q 12 ensures that the F12 basis functions are strongly orthogonal to the HF reference state and that they are orthogonal to the space of the conventional doubles. α,β practically complete orbital set. S ij generates spin-adapted functions. 19 J. Friedrich, D. Tew, W. Klopper, M.D., J. Phys. Chem. 132, (2010) The Incremental Scheme for Molecules

73 E corr error E corr Reaction full NH 3 +4H 2 O 2 HNO 3 +5H 2 O H 2 O 2 +H 2 2H 2 O H 2 CO+H 2 CH 3 OH C 2 H 6 +H 2 2CH C 2 H 4 +H 2 C 2 H SO 2 +H 2 O 2 SO 3 +H 2 O P 4 +6H 2 4PH CH 3 CHO+H 2 O 2 CH 3 COOH+H 2 O HCOOH+CH 3 OH HCOOCH 3 +H 2 O CH 3 COOH+NH 3 CH 3 CONH 2 +H 2 O CH 3 NH 2 +3H 2 O 2 CH 3 NO 2 +4H 2 O C 2 H 4 +H 2 O 2 HOC 2 H 4 OH C 2 H 4 +H 2 O C 2 H 5 OH CH 3 CHO+H 2 C 2 H 5 OH HCOOH+NH 3 HCON 2 +H 2 O CCSD(F12)/cc-pVTZ-F12 correlation contribution to the reaction energies and errors for 2nd- to 4th-order one-orbital domain-based expansions in kj/mol. The Incremental Scheme for Molecules

74 E corr error E corr Reaction exact NH 3 +4H 2 O 2 HNO 3 +5H 2 O H 2 O 2 +H 2 2H 2 O H 2 CO+H 2 CH 3 OH C 2 H 6 +H 2 2CH C 2 H 4 +H 2 C 2 H SO 2 +H 2 O 2 SO 3 +H 2 O P 4 +6H 2 4PH CH 3 CHO+H 2 O 2 CH 3 COOH+H 2 O HCOOH+CH 3 OH HCOOCH 3 +H 2 O CH 3 COOH+NH 3 CH 3 CONH 2 +H 2 O CH 3 NH 2 +3H 2 O 2 CH 3 NO 2 +4H 2 O C 2 H 4 +H 2 O 2 HOC 2 H 4 OH C 2 H 4 +H 2 O C 2 H 5 OH CH 3 CHO+H 2 C 2 H 5 OH HCOOH+NH 3 HCON 2 +H 2 O MP2/cc-pVTZ-F12 correlation contribution to the reaction energies and errors for 2nd- to 4th-order one-orbital domain-based expansions in kj/mol. The Incremental Scheme for Molecules

75 E corr error E corr Reaction exact NH 3 +4H 2 O 2 HNO 3 +5H 2 O H 2 O 2 +H 2 2H 2 O H 2 CO+H 2 CH 3 OH C 2 H 6 +H 2 2CH C 2 H 4 +H 2 C 2 H SO 2 +H 2 O 2 SO 3 +H 2 O P 4 +6H 2 4PH CH 3 CHO+H 2 O 2 CH 3 COOH+H 2 O HCOOH+CH 3 OH HCOOCH 3 +H 2 O CH 3 COOH+NH 3 CH 3 CONH 2 +H 2 O CH 3 NH 2 +3H 2 O 2 CH 3 NO 2 +4H 2 O C 2 H 4 +H 2 O 2 HOC 2 H 4 OH C 2 H 4 +H 2 O C 2 H 5 OH CH 3 CHO+H 2 C 2 H 5 OH HCOOH+NH 3 HCON 2 +H 2 O MP2-F12/cc-pVTZ-F12 correlation contribution to the reaction energies and errors for 2nd- to 4th-order one-orbital domain-based expansions in kj/mol. The Incremental Scheme for Molecules

76 Conclusions Fully automated implementation of incremental MP2, CCSD, CCSD(T) correlation energy calculations to arbitrary order For a wide variety of systems chemical accuracy (1 kcal/mol) obtained with a truncation at low order (2nd-4th) Efficient treatment of core and core-valence correlation Usage of symmetry within a local correlation treatment possible Costly higher order increments require a lower accuracy than cheaper lower order ones Distance or energy thresholds reduce the number of required calculations Reduction of disk space and RAM requirements Parallel implementation Generally applicable, flexible wrt method/code, systematically improvable, simple The Incremental Scheme

77 Outlook Extension to molecular properties a Extension to r 12 -dependent wavefunctions b Energy-difference dedicated incremental expansions c Implementation of energy gradients Design of a CCSD... code especially adapted to the incremental scheme (truncation of virtual space,...) Design of an incremental Hartree-Fock code Incorporate techniques like density fitting Interface to a periodic Hartree-Fock code (WANNIER or CRYSTAL) a J. Friedrich, S. Coriani, T. Helgaker, M.D., J. Phys. Chem. 131, (2009) b J. Friedrich, D. Tew, W. Klopper, M.D., J. Phys. Chem. 132, (2010) c M. Mödl, M.D., P. Fulde, H. Stoll, J. Chem. Phys. 106, 1836 (1997) The Incremental Scheme

78 Acknowledgments Profs. Hermann Stoll (Stuttgart) and Peter Fulde (Stuttgart/Dresden): collaboration on solids and more. Dr. Joachim Friedrich: most of the work since 2004, i.e. automatized implementation for molecules and tests. Dr. Michael Hanrath: PC cluster management, introduction to C++ and various discussions. Dipl.-Chem. Katharina Walczak: extension to core and core-valence correlation treatment; hybrid schemes. Profs. Wim Klopper (Karlsruhe) and Trygve Helgaker (Oslo): F12-schemes and properties, respectively. Financial support of the German Science Foundation (DFG) through the priority programme (SPP) 1145, the special research programme (SFB) 624 and regular funding (Normalverfahren) is gratefully acknowledged. The Incremental Scheme

79 Thank you for your attention! The Incremental Scheme

80 A Wannier-Function-Based Hartree-Fock Approach Two possible Hartree-Fock (HF) Linear Combination of Atomic or Gaussian Type Orbitals (LCAO or LCGTO) approaches for infinitely extended systems, i.e., polymers and solids: 1 Bloch-Orbitals:delocalized program CRYSTAL (Turino, Daresbury) 2 Wannier-Orbitals:localized program WANNIER (Dresden, Stuttgart) Note: As for finite systems, e.g., molecules, one finds a complete equivalence of the two approaches for systems with an even number of electrons per unit cell. The HF wavefunctions are linked by an unitary transformation of the occupied orbitals.

81 Partitioning of the Infinite System into Finite Subsystems Fact: Every perfect polymer or solid consists of an infinite number of identical unit cells. Often we are interested in properties refering to one unit cell, e.g., total energy per unit cell, lattice constant(s), charge distribution or positions of atoms within a unit cell,... Idea: We can reformulate and solve the Hartree-Fock problem in terms of the orbitals belonging to one special unit cell and generate those of all other cells by exploiting the translational symmetry of the polymer or solid! Question: How can a corresponding embedded-cluster Hartree-Fock approach be set up? Can we go beyond Hartree-Fock, i.e., can we include electron correlations?

82 Hartree-Fock Equations Nonrelativistic Hamiltonian of an infinite system: H = i i i I Z I r i R I + r i r j + i>j I>J 1 Z I Z J R I R J (1) Closed-shell Hartree-Fock energy, i.e., wavefunction ansatz as a single Slater determinant: E HF = 2 i < i T i > +2 i < i U i > + i,j (2 < ij ij > < ij ji >) (2) T and U denote the kinetic and potential energy operators, respectively. Relativistic contributions could be added, e.g., by (scalar-relativistic) ab initio pseudopotentials. Note: The sums in eqs. (1) and (2) run to infinity!

83 Translational symmetry condition: α(r i + R j ) >= T(R i ) α(r j ) > (3) α(r j ) > denotes the α-th orbital in the unit cell located at position R j of the lattice. T(R i ) represents the operator which performs a translation by the vector R i. Rewritten closed-shell Hartree-Fock energy expression for the infinite system (N units cells): { E HF = lim N n c n c 2 < α(0) T α(0) > +2 < α(0) U α(0) > N + n c α,β=1j=1 α=1 α=1 (4) N (2 < α(0)β(r j ) α(0)β(r j ) > < α(0)β(r j ) β(r j )α(0) >) α(0) > denotes an orbital centered in the reference cell. Each unit cell comprises n c doubly occupied orbitals.

84 Stationarity condition for the Hartree-Fock energy with respect to changes of the orbitals in the reference cell under the constraint, that they form an orthonormal set leads to the Hartree-Fock operator: F HF = T + U + 2 β J β β K β (5) Coulomb operator: J β α >= j < β(r j ) 1 r 12 β(r j ) > α > (6) Exchange operator: K β α >= j < β(r j ) 1 r 12 α > β(r j ) > (7) Note: U, J and K contain infinite lattice sums! How too guarantee orthogonality to the orbitals in the other cells and how to avoid delocalization during the self-consistent field process?

85 Self-consistent orbital localization by means of projection operators: F HF = T +U +2 J β K β + λ k γ γ(r k ) >< γ(r k ) (8) β β k>1 γ where k = 1 is assumed to refer to the reference cell for which the orbitals are to be evaluated. λ k γ enforces strict orthogonality of the orbitals in the reference cell to the orbital γ(r k ) >. Note: U, J, K and the projection term contain infinite sums! A modification of the standard quantum chemistry methods to evaluate one- and two-electron integrals is required, i.e., spatial trucation of short-range interactions like K and the projector as well as Ewald summation techniques for long-range interactions like U and J.

86 Partitioning of the real space into three subspaces: (central) reference (unit) cell C. Solution of the modified Hartree-Fock equations for orbitals centered in this region. Interactions with the rest of the infinite system are treated depending on their range, i.e., thresholds for the neglect of vanishingly small contributions are introduced. short-range environment, i.e., neighborhood of the reference cell N. Contributes to U, J, K and the projection operators. long-range environment, i.e., rest of the infinite system R. Contributes only to U and J. Note: The partitioning is system-dependent and has to be adjusted individually. For ionic and well localizable covalent systems typically up to the third-nearest neighbor unit cells are included in the short-range environment. In the present form the approach will not work for metallic, i.e., inherently delocalized systems!

87 Hartree-Fock-Roothaan implementation Linear combination of atomic orbitals (LCAO) or linear combination of Gaussian type functions (LCGTF). α >= C p,α p(r j ) > (9) p R j C+N Here C and N denote the (central) reference cell and its short-range neighborhood, respectively. Hartree-Fock-Roothaan equations: F pq C q,α = ɛ α S pq C q,α (10) q Fock matrix: F pq =< p (T+U+2J K) q > + k N q γ p,q λ k γ < p p >< q q > C p,γc q,γ where unprimed functions p > and q > correspond to orbitals of the reference unit cell C while the primed functions p > and q > to those of its neighborhood N. (11)

88 Total energy per reference (unit) cell: E unitcell = 2 trace{(t +U+0.5(2J K))D}+E nuc nuc +0.5 E nuc nuc T, U, J and K denote the matix representation of the corresponding operators. E nuc nuc is the internuclear interaction energy for both nuclei within the reference cell, whereas E nuc nuc is the corresponding expression for the interaction between the nuclei inside and outside the reference cell. Density matrix: (12) D pq = α C p,α C q,α (13)

89 Algorithm used in WANNIER for the solution of the Hartree-Fock problem 1 Generate localized initial guess occupied orbitals for the reference cell. 2 Formally translate reference cell orbitals to the environment. 3 Construct the Fock matrix for the orbitals in the reference cell. 4 Solve the Hartree-Fock equations. 5 Compute the energy per unit cell. 6 Go to step 2 until convergence has been achieved. For further details: Ab initio embedded cluster approach to electronic structure calculations on perfect solids. A Hartree-Fock study of lithium hydride. A. Shukla, M. Dolg, H. Stoll, P. Fulde, Chem. Phys. Lett. 262 (1996) Obtaining Wannier functions of a crystalline insulator within a Hartree-Fock approach: applications to LiF and LiCl. A. Shukla, M. Dolg, P. Fulde, H. Stoll, Phys. Rev. B 57 (1998)

90 Results for LiF and LiCl Comparison between between total energies of LiX obtained using WANNIER and CRYSTAL for different values of lattice constants. The N region included up to third-nearest neighbor unit cells. Lattice constants are in units of Å, and energies are in atomic units. Lithium fluoride LiF: Lattice Constant Total Energy WANNIER CRYSTAL

91 Lithium chloride LiCl: Lattice Constant Total Energy WANNIER CRYSTAL

92 Hartree-Fock Bandstructures from WANNIER Fock matrix elements in real space: F p0,qj =< p(0) T + U + β (2J β K β ) q(r j ) > (14) Transformation of orbitals from real to k-space: p(k) >= 1 N R j e ikrj p(r j ) > (15) Transformation of operators from real to k-space: Q pq (k) = R j e ikr j p(0) Q q(r j ) (16) Use Q = F for the Fock matrix and Q = 1 for the overlap matrix S! Hartree-Fock-Roothaan equations in k-space: F pq (k)c qα (k) = ɛ α (k) S pq (k)c qα (k) (17) q q Quasiparticle energies ɛ α (k) for a given k-point ( bandstructure).

93 Results for NaCl Equilibrium lattice constants (in Å), bulk moduli (in GPa), equilibrium energies (in atomic units), and lattice energies (in kcal/mol) for NaCl, obtained using WANNIER and CRYSTAL. Experimental data are also given for comparison. Quantity Method Result WANNIER Lattice Constant CRYSTAL 5.80 Exp WANNIER Bulk Modulus CRYSTAL 22.3 Exp WANNIER Equilibrium Energy CRYSTAL WANNIER Lattice Energy CRYSTAL Exp

94 Note: Contributions of electron correlations are still missing! Further details: A Hartree-Fock ab initio band structure calculation employing Wannier-type orbitals. M. Albrecht, A. Shukla, M. Dolg, P. Fulde, H. Stoll, Chem. Phys. Lett. 285 (1998)

95 Hartree-Fock band structure of NaCl Comparison of the quasiparticle energies at high symmetry points for the first three occupied and unoccupied bands. Row a CRYSTAL95, row b WANNIER. The energies at the gamma-point are chosen to be zero. (in a.u.) L a b Γ a b X a b Γ: k = (0, 0, 0) X: k = 2π(1, 0, 0)/c L: k = 2π(1, 1, 1)/( 3c) c denotes the lattice constant of the fcc lattice.

96 Further details: A Hartree-Fock ab initio band structure calculation employing Wannier-type orbitals. M. Albrecht, A. Shukla, M. Dolg, P. Fulde, H. Stoll, Chem. Phys. Lett. 285 (1998)

97 Hartree-Fock band structure of NaCl E/meV L Γ X WANNIER (solid lines), CRYSTAL (dotted lines). The highest two valence bands (Cl 3p; the upper one being doubly degenerate) as well as the lowest three conduction bands (Na 3s, 3p; the upper one being doubly degenerate) are displayed.

98 Usage of Symmetry: Anthracene I RI-BP86/SVP geometry in D 2h Incremental CCSD/6-31G** energies core=14 (C 1s 2 ), dsp=3, t con =3 Domains were constructed in C s and C 1 symmetry