THE CRYSCOR PROJECT : STATUS AND PROSPECTS

Transcription

1 THE CRYSCOR PROJECT : STATUS AND PROSPECTS ( # ) Cesare PISANI, Massimo BUSSO, Gabriella CAPECCHI, Silvia CASASSA, Roberto DOVESI, Lorenzo MASCHIO (*) Claudio ZICOVICH-WILSON ( ) Vic R. SAUNDERS ( & ) Martin SCHÜTZ ( # ) Università di Torino (Italy) (*) Universidad de Cuernavaca (Mexico) ( ) CRLC Laboratories, Daresbury (GB) ( & ) Universität Regensburg (Germany) Cesare Pisani - LCC2004 1

2 THE STRATEGY Combining two well assessed, robust, compatible technologies Geometrical and structural analysis of periodic system Accurate HF solution in terms of AOs Local representation of occupied manifold (WF) CRYSTAL On Local Correlation (LC) theory and techniques at various levels of sophistication Meyer-Pulay-Werner ( MOLPRO ) Cesare Pisani - LCC2004 2

3 Local-correlation approach for crystals MEYER, PULAY, SAEBØ,,WERNER, KNOWLES, HETZER, MANBY, SCHÜTZ, (Arkansas, Stuttgart, Birmingham, Bristol,...) Local, orthonormal representation of occupied HF manifold Local Molecular Orbitals Symmetry Adapted Wannier Functions (SAWF) Local, non-orthonormal, redundant representation of virtual manifold Projected Atomic Orbitals (PAO) (PAO) Reformulation of standard approaches ( MPn, CCSD(T), ) TRUNCATION STRATEGY Dynamical correlation effects are short-ranged: Ignore excitations from very distant pairs of SAWFs Exploit translational (and point) symmetry of SAWFs and PAOs N--SCALING n--scaling (n is the size of the irreducible part of the crystalline cell) Explore the role of computational parameters Cesare Pisani - LCC2004 3

4 THE OBJECTIVE To produce (in a reasonably short time) a public domain ab initio code for estimating electron correlation effects in non-conducting crystals, characterized by: generality simplicity robustness acceptable efficiency, and open to improvements As a first step, an MP2 code has been prepared and is here presented. Cesare Pisani - LCC2004 4

5 Is the objective worth the effort? Is the strategy appropriate? Cesare Pisani - LCC2004 5

6 Is the objective worth the effort? In our opinion it is. A number of ab-initio periodic correlation codes are ready, in preparation or in project (Stoll, Bartlett, Scuseria, Birkenheuer, Saunders-Orlando, Malrieu-Evangelisti, Monkhorst.). However, a conceptually simple, reliable, easily accessible code, could serve the purpose of providing reference data, exploring the role of computational parameters, checking the usefulness of alternative techniques, etc. Cesare Pisani - LCC2004 6

7 Is the strategy appropriate? With respect to other approaches in this area, CRYSCOR is characterized by its being strictly founded on the HF program CRYSTAL (language, basic techniques, etc.), so as to be fully compatible with it in a sense, CRYSCOR is the post-hf option of CRYSTAL. Disadvantages Advantages Cesare Pisani - LCC2004 7

8 Disadvantages CRYSTAL s computational technology is efficient but somewhat rigid and outdated. Inserting new parts of code may require a lot of work and attention. Cesare Pisani - LCC2004 8

9 Advantages Full space group symmetry of system and basis functions are fully and efficiently provided by CRYSTAL. Quasi-HF periodic solutions are obtained, and their quality can be easily assessed. Efficient and well tested techniques for generating localized and symmetry adapted WFs are available. Fourier transformation techniques are available (for instance, for calculating the inverse of quasi-diagonal translationally invariant matrices). Cesare Pisani - LCC2004 9

10 CRYSTAL 2003 : Main Features VR Saunders, R Dovesi, C Roetti, R Orlando, CM Zicovich-Wilson, NM Harrison, K Doll, B Civalleri, IJ Bush, Ph D Arco, M Llunell Distributed starting 9/03. Info : (Rev. Computational Chemistry, in press) The code FORTRAN90, fully parallelized, dynamic memory allocation The periodic model Consistent treatment of periodicity : 3D, 2D, 1D, 0D Ewald techniques for lattice sums (specific for 1D, 2D, 3D) Full exploitation of point symmetry in direct and reciprocal space Basis set Bloch functions as Linear Combination of Atomic Orbitals (contractions of Hermite Gaussian Functions) All-Electron or Valence-only-plus-Pseudopotential basis set Hamiltonians RHF, UHF Kohn-Sham techniques with Local and Gradient-corrected exchange and correlation functionals Hybrid DFT-HF exchange functionals Energy derivatives Automated geometry optimisation based on analytical gradient Wave function analysis and manipulation Band structure, PDOS, Charge, spin, electron momentum density, Structure factors, Compton profiles Elastic, dielectric constant, piezoelectric, hyperfine and nuclear quadrupole coupling tensors, [ Vibrational frequencies, based on analytical gradient ] Localized Wannier Functions [Symmetry adapted] Cesare Pisani - LCC

11 A pre-requisite for Local correlation methods in crystals: Efficient generation of Wannier functions (WF) to span the occupied HF space {Ψ n (k)} occ FT + localization criterion { w sg } occ Edmiston-Ruedenberg (1965) : Maximum intra-lo repulsion (N 5 ) Boys (1966) : Maximum distance between LO centroids (N 3 ) Pipek-Mezey (1989) : Minimum number of atoms per LO (N 3 ) What does efficient mean? Computationally inexpensive Well localized WFs Strictly orthonormal WFs Zicovich, Dovesi & Saunders, J. Chem. Phys. 115, 9708 (2001) Symmetry adapted WFs Zicovich, Casassa (2004) Cesare Pisani - LCC

12 CRYSCOR work to date and scheme of presentation 1. Generalization of the LOCALI part of CRYSTAL03, to produce symmetry adapted Wannier Functions (see Casassa-Zicovich poster) 2. Reformulation of LC-MP2 equations, so as to exploit translational and point symmetry 3. CRYSCOR code preparation (from CRYSTAL output to final results) (see Casassa et al. poster) 4. Refinement work on the integral part of the code to obtain 2-el integrals (i a j b ) either exactly or in a multipolar approximation (see Capecchi-Maschio poster) 5. Test of computational parameters (molecular cases + Diamond, Silicon, SiC, BN, BeS) Cesare Pisani - LCC

13 LC Reformulation of LMP2 and use of Symmetry R ij ab = 0 = K ij ab + Σ cd [ f ac T ij cd S db + S ac Tij cd f dc ] + Σ cd [ S ac Σ k (f ik T kj cd + Tik cd f kj )S cb ] E 2 = Σ (ij) Σ ab (ij) (Kij ab + Rij ab ) (2 Tij ab Tij ba ) ( a sum of pair energies) Ψ (1) = Σ (ij) Σ ( 2 ab (ij) Tij ab Tij ba ) 2 Φ ij ab Φij ba K ij ab = ( i a j b) = Σ µρνσ µρνσ c WF iµ cpao aρ cwf jν cpao bσ ( µ ρ ν σ) Symmetry exploitation Translational : The first WF index (i) is always confined in the zero cell Rotational : Only irreducible WF pairs (i j) need to be considered Irreducible pairs may have a residual symmetry which can be exploited in the K integral evaluation (Capecchi-Maschio poster). Cesare Pisani - LCC

14 Use of Symmetry in the Update step R ij ab = Σ cd [ S ac Σ k (f ik T kj cd + Tik cd f kj )S cb ] = Σ cd [ S ac β ij cd S cb ] T ij ab = Rij ab / ε εij ab To update amplitudes of irreducible WF pairs, we need those of all pairs. For each irreducible pair ij do Update T ij For each symmetry operator of i j (n) do Obtain amplitude of rotated pair T injn For each irreducible pair kl do If (k i n If (l j n and l (close-to) j n ) then β kl = β kl + T inj n f jn l and k (close-to) i n ) then β kl = β kl + f k in T injn enddo enddo enddo Cesare Pisani - LCC

15 BLOCK DIAGRAM OF CRYSCOR CRYSCOR (on disk) CRYSTAL MP2MAIN ORIQAO SYMPAIR DOMAINS MULTIPOLES MP2INT MP2CORE MP2LOOP Crystal type and symmetry; AO, S AOg,F AOg, WF (coefficients & symmetry) Computational parameters from input cards Construct PAOs, Calculate F g PAO, S PAO PAO g PAO,F g WF Recognize irreducible ww pairs and wp-wp bi-pairs Domains (Boughton Boughton-Pulay Pulay), Pair domains and their classification: Calculate multipolar expansion of WF-PAO products Calculate 2-el el-integrals ( exact or multipolar ) (see Capecchi-Maschio poster) Solve LMP2 equations (next dia) Cesare Pisani - LCC

16 SOLUTION OF LMP2 EQUATIONS MP2MAIN MP2CORE For each irreducible WW pair, do LON BSETR FWPRIM IGUESS Construct local orthonormal orbitals (lon lon) Organize quantities for iterative loop (close close-by pairs,, etc.) Calculate local pseudo-canonical virtual orbitals Provide initial guess for amplitudes T and for coupling coupling β quantities Enddo MP2LOOP For each irreducible WW pair, do End YES NO conv? ECALC UPDATE Read K, T, β, Update T,Update, E2 Update β Enddo Cesare Pisani - LCC

17 Main computational parameters in LMP2 for crystals R ij ab = Kij ab + Σ cd [ f ac T ij cd S db + S ac Tij cd f dc ] Σ cd [ S ac β ij cd S cb ] E 2 = Σ (ij) Σ ab (ij) (Kij ab + Rij ab ) (2 Tij ab Tij ba ) K ij ab = ( i a j b) = Σ µρνσ µρνσ c WF iµ cpao aρ cwf jν cpao bσ ( µ ρ ν σ) Basis set {µ} (representation of WFs and PAOs) Truncation of WF and PAO tails ( c WF iµ > tow ; cpao aρ > toq) Exact / multipolar treatment of K integrals (Capecchi-Maschio ) Prescreening of exact K integrals (Schwarz+density screening) Size of WF domains (range of occ. to virt. excitations in Σ ) ab (ij) Maximum WF-WF distance (range of Σ (ij) ) Cesare Pisani - LCC

18 C Shell exponent s coefficient p coefficient d coefficient 6-21G* 1s sp sp d Cesare Pisani - LCC

19 Si 6-21G* Shell exponent s coefficient p coefficient d coefficient s sp sp sp d Cesare Pisani - LCC

20 Main computational parameters in LMP2 for crystals R ij ab = Kij ab + Σ cd [ f ac T ij cd S db + S ac Tij cd f dc ] Σ cd [ S ac β ij cd S cb ] E 2 = Σ (ij) Σ ab (ij) (Kij ab + Rij ab ) (2 Tij ab Tij ba ) K ij ab = ( i a j b) = Σ µρνσ µρνσ c WF iµ cpao aρ cwf jν cpao bσ ( µ ρ ν σ) Basis set {µ} (representation of WFs and PAOs) Truncation of WF and PAO tails ( c WF iµ > tow ; cpao aρ > toq) Exact / multipolar treatment of K integrals (Capecchi-Maschio ) Prescreening of exact K integrals (Schwarz+density screening) Size of WF domains (range of occ. to virt. excitations in Σ ) ab (ij) Maximum WF-WF distance (range of Σ (ij) ) Cesare Pisani - LCC

21 PAOs and WFs (SiC and BeS) Cesare Pisani - LCC

22 Main computational parameters in LMP2 for crystals R ij ab = Kij ab + Σ cd [ f ac T ij cd S db + S ac Tij cd f dc ] Σ cd [ S ac β ij cd S cb ] E 2 = Σ (ij) Σ ab (ij) (Kij ab + Rij ab ) (2 Tij ab Tij ba ) K ij ab = ( i a j b) = Σ µρνσ µρνσ c WF iµ cpao aρ cwf jν cpao bσ ( µ ρ ν σ) Basis set {µ} (representation of WFs and PAOs) Truncation of WF and PAO tails ( c WF iµ > tow ; cpao aρ > toq) Exact / multipolar treatment of K integrals (Capecchi-Maschio ) Prescreening of exact K integrals (Schwarz+density screening) Size of WF domains (range of occ. to virt. excitations in Σ ) ab (ij) Maximum WF-WF distance (range of Σ (ij) ) Cesare Pisani - LCC

23 Influence of Computational parameters on Diamond results Basis set: 6-21G* ; valence-only MP2 calculations Standard parameter settings are in bold letters (outlined in yellow in tables) Parameters WF Size: There is only one type of WF (C-C bond localized orbital). The corresponding domain comprises 2 (C C) or 8 ( C 3 C C C 3 ) atoms Tow : threshold of c WF iµ coefficients ( 0.01, 0.005, 0.001, , 0.001) Toq : threshold of c PAO aρ coefficients ( 0.01, 0.005, 0.001, , 0.001) Pairs: n. of irreducible pairs considered [exact/multipolar] (4,19 / 0,154,301,407) The threshold for disregarding 2-el integrals according to Schwarz criterion has been set to 10-8, except in the cases indicated with an asterisk, where it is set to Results [bb] : contribution to -E 2 from [C C C C] pair (microhartree) [bcb ] : contribution to -E 2 from [C C CC ] (microhartree) -E 2 : absolute value of MP2 energy per cell (microhartree) Time: CPU time/10 3 sec on AMD Athlon, 1.6 GHz Cesare Pisani - LCC

24 Influence of Computational parameters on Diamond results Domain size and number of pairs All energies in microhartrees, times in 10 3 sec on AMD Athlon 1.6 GHz Size Tow Toq Pairs [bb] [bcb ] -E2 Time / / (+2363) 19/ / /0 19/ (+3861) Cesare Pisani - LCC

25 Influence of Computational parameters on Diamond results Truncation of tails and number of pairs Size Tow Toq Pairs [bb] [bcb ] -E 2 Time / / / / / / / / / / * / * / * / * Cesare Pisani - LCC

26 Comparison between 6-21G* results for diamond-like molecules (MOLPRO) and Diamond (CRYSCOR, standard setting) [bb] [bcb ] [b b ] [b b ] System -E2 4 C-C Neopentane 12 C-H C 5 H C-C Adamantane 16 C-H C 10 H C 35 H Diamond m=4 m=24 m=24 m= C All energies in µhartree; the best fit estimate of diamond E2 energy per cell from present molecular results is µhartree Cesare Pisani - LCC

27 Cesare Pisani - LCC

28 Comparison between 6-21G* results for diamond-like molecules (MOLPRO) and Diamond (CRYSCOR, standard setting) [bb] [bcb ] [b b ] [b b ] System -E2 Neopentane C 5 H Adamantane C 10 H C-C 36 C-H. C 35 H Diamond m=4 m=24 m=24 m= C All energies in µhartree; the best fit estimate of diamond E2 energy per cell from present molecular results is µhartree Cesare Pisani - LCC

29 Cesare Pisani - LCC

30 Comparison between 6-21G* results for diamond-like molecules (MOLPRO) and Diamond (CRYSCOR, standard setting) [bb] [bcb ] [b b ] [b b ] System -E2 4 C-C Neopentane 12 C-H C 5 H C-C Adamantane 16 C-H C 10 H C-C 36 C-H. C 35 H C-C Diamond m=4 m=24 m=24 m= per cell C All energies in µhartree; the best fit estimate of diamond E2 energy per cell from present molecular results is µhartree Cesare Pisani - LCC

31 Summary of MP2 valence-only results (6-21 G* basis set) BE HF E MP2 -E at MP2 BE TOT BE INCR BE EXP Diamond Silicon * SiC * BN * BeS * * Still not at convergence Cesare Pisani - LCC

32 STATE OF PROJECT AND PROSPECTS Current work : Refinement and standardization of basic program Higher efficiency, generalization of WF symmetry treatment Tests Basis set quality, different systems (MgO, Polymers, Rare gases ) Open problems and lines of development (Schütz, Usvjat, ): Efficient calculation and transformation of K integrals (Density fitting, etc.) Auxiliary basis set (to complement the HF set) Extension to other local correlation schemes, either within the MOLPRO framework (CCSD, MP4,.. ), or different (IEPA + NO ; MP2-R12 (Kutzelnigg, Klopper, Manby) ;?) Cesare Pisani - LCC