Post Hartree-Fock: MP2 and RPA in CP2K

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Post Hartree-Fock: MP2 and RPA in CP2K A tutorial Jan Wilhelm jan.wilhelm@chem.uzh.ch 4 September 2015

Further reading MP2 and RPA by Mauro Del Ben, Jürg Hutter, Joost VandeVondele Del Ben, M; Hutter, J; VandeVondele, J. JOURNAL OF CHEMICAL THEORY AND COMPUTATION 8, 4177-4188 (2012). Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach. http://dx.doi.org/10.1021/ct300531w Del Ben, M; Hutter, J; VandeVondele, J. JOURNAL OF CHEMICAL THEORY AND COMPUTATION 9, 2654-2671 (2013). Electron Correlation in the Condensed Phase from a Resolution of Identity Approach Based on the Gaussian and Plane Waves Scheme. http://dx.doi.org/10.1021/ct4002202 Del Ben, M; Schütt, O; Wentz, T; Messmer, P; Hutter, J; VandeVondele, J. COMPUTER PHYSICS COMMUNICATIONS 187, 120-129 (2015). Enabling simulation at the fifth rung of DFT: Large scale RPA calculations with excellent time to solution. http://dx.doi.org/10.1016/j.cpc.2014.10.021

Motivation: Jacob s Ladder Model of exchange-correlation term improved over time Pure DFT is not enough Hybrid schemes improve Chemical accuracy (1 kcal/mol) not reached Go beyond DFT and hybrids (e. g. MP2, RPA) Jacob s ladder of density functional approximations Accuracy of the density functional approximations Mundy et al. SCIDAC 2010 L. Goerigk, S. Grimme; Phys. Chem. Chem. Phys., 13:6670, 2011

Density functional versus wavefunction methods Energy of Coulomb repulsion E Coul of electrons: E Coul = E Hartree (H) + E exchange (x) + E correlation (c) DFT: exchange and correlation are functionals of the density n( r): E DFT xc = Exc DFT [n] RPA/MP2: exchange and correlation are functional of all electronic wavefunctions: exact exchange and RPA/MP2 Exc = Ex HF [{Ψ j }] + Ec RPA/MP2 [{Ψ j, ε j }] RPA and MP2 total energy calculations are postprocessing procedures: 1. converge an SCF with Hartree-Fock (for MP2) and functional of your choice for RPA (e. g. GGA or hybrid functional) 2. take molecular orbitals and their energies {Ψ j, ε j } of converged SCF and calculate E EXX/RPA xc or E EXX/MP2 xc.

MP2 and RPA correlation energies Second order Møller-Plesset perturbation theory: E MP2 c occ,vir = ij,ab (ia jb)[2(ia jb) (ib ja)] ε a + ε b ε i ε j Random-phase approximation (RPA) Ec RPA = 1 ( d r 1 d r 2 dω ln 1 χ0 ( r 1, r 2, iω) 2π 0 r 1 r 2 ) χ0 ( r 1, r 2, iω) r 1 r 2 χ 0 ( r 1, r 2, iω) = 2 ia Ψ i ( r 1 )Ψ a( r 1 )Ψ i ( r 2 )Ψ a( r 2 ) (ε a ε i ) 2 + ω 2 In both cases 4-center 2-electron repulsion integrals (ia jb) have to be computed (ia jb) = 1 Ψ i ( r 1 )Ψ a( r 1 ) r 1 r 2 Ψ j ( r 2 )Ψ b ( r 2 )d r 1 d r 2

Resolution of Identity Approach (RI) Gaussian Auxiliary Basis {P} with Coulomb Metric: (ia jb) (ia jb) RI = PQ (ia P)(P Q) 1 (Q jb) = S (ia S)(S jb) = S B S ia BS jb Instead of computing 4-center 2-electron repulsion integrals (ia jb) we compute 3-center 2-electron repulsion integrals (ia S) B S ia = P = µ (ia P)(P S) 1/2 C µi ν C νa (µν S) }{{} GPW Integral

GPW RI Integrals B S µν = (µν S) Calculate S) on grid ρ S (R) FFT Multiply with operator to get potential V S (G) = ρ S (G) O(G) FFT Integrate (µν on grid with V S (R) B S µν = R Φ µν (R) V S (R)

RI-MP2 E MP2 c = o Calculate Integrals using RI ij v ab (ia jb) [2(ia jb) (ib ja)] ε a + ε b ε i ε j (ia jb) = S B S ia BS jb (1) Scaling of computational effort iso(n 5 ), see Eq. (??): occ. orbitals (i, j) N virt. orbitals (a, b) N auxiliary basis (S) N Reduction of prefactor (typically one order of magnitude) compared to non-ri MP2

RI-SOS-MP2 E MP2,OS c = iajb (ia jb) 2 ε a + ε b ε i ε j = iajb (ia jb) 2 ab ij Empirical observation [Y. Jung et al., J. Chem. Phys. 121, 9793 (2004)]: E MP2 c 1.3 E MP2,OS c Transform 1/x = 0 e xt dt (I) and numerical integration (II) E MP2,OS c (I) = 0 dt iajb (ia jb) 2 e t ab ij N q (II) w q (ia jb) 2 e tq ab ij q iajb With RI the overall scaling is reduced to O(N 4 ) BiaB P jbb P ia Q BQ jb e tq ab ij = BiaB P ia Q e tq(εa ε i ) PQiajb PQ ia jb B P jbb Q jb e tq(ε b ε j ) 6-8 quadrature points give µh accuracy

RI-RPA RPA correlation energy in matrix representation: E RPA c = 0 dω Tr (ln (1 + Q(ω)) Q(ω)) (2) 2π where Q(ω) is a matrix of size N aux. bf. N aux. bf with entries Q PQ (ω) = 2 ia B P iag ia (ω)b Q ia and G ia (ω) = ε a ε i (ε a ε i ) 2 + ω 2 RI-RPA overall scaling O(N 4 ) 10-15 quadrature points (minimax quadrature) give µh accuracy

Scaling 64 water molecules, cc-tzvp Basis; 256 occupied orbitals, 3648 basis function, 8704 RI basis functions

Timings for a single calculation of Ec MP2, Ec SOS-MP2 and Ec RPA On 64 cores CRAY XC30: System Atoms Basis (RI basis) MP2 SOS-MP2 RPA 64 water molec. 192 3648 (8704) 140 min 64 min 48 min Timings on 64 cores CRAY CX30 (= 8 nodes on Piz Daint, smallest possible number of nodes due to memory requirements), TZVP basis set, quadrature points: 16 (RPA), 10 (OS-MP2). On 32768 cores CRAY XC30: System Atoms Basis (RI basis) RPA 64 water molecules 192 3648 (8704) 22 s 512 water molecules 1536 29184 (69632) 1.7 h Timings on 32768 cores CRAY CX30 (= 4096 nodes on Piz Daint, 78 % of Piz Daint, worldwide 6th largest supercomputer), TZVP basis set, quadrature points: 16.

Input: Basis set and RI basis set for RI-MP2 and RI-RPA &DFT BASIS_SET_FILE_NAME../BASIS_RI_cc-TZ POTENTIAL_FILE_NAME../POTENTIAL &END DFT &SUBSYS.! RI basis sets are available for TZVP basis. &KIND H BASIS_SET cc-tz RI_AUX_BASIS_SET RI_TZ POTENTIAL GTH-PBE-q1 &END KIND &KIND O BASIS_SET cc-tz RI_AUX_BASIS_SET RI_TZ POTENTIAL GTH-PBE-q6 &END KIND &END SUBSYS! For RI calculations, we additionally need RI basis sets.! For RI calculations, we additionally need RI basis sets.

Input non-periodic* MP2 on top of Hartree-Fock-SCF *for periodic calculations, change section PERIODIC NONE and truncate Hartree-Fock &DFT BASIS_SET_FILE_NAME../BASIS_RI_cc-TZ POTENTIAL_FILE_NAME../POTENTIAL &QS METHOD GPW &END QS &POISSON PERIODIC NONE PSOLVER MT &END POISSON &SCF SCF_GUESS RESTART &END SCF &XC &XC_FUNCTIONAL NONE &END XC_FUNCTIONAL &HF FRACTION 1.00 &SCREENING EPS_SCHWARZ 1.0E-8 &END SCREENING &END HF &END XC &WF_CORRELATION METHOD RI_MP2_GPW! do MP2 calculation NUMBER_PROC 2! number of processes for storing grids. In case you run out of memory when! computing integrals, increase number. Large number slows the calculation down. &RI_MP2 &END RI_MP2 &END WF_CORRELATION &END DFT

Input non-periodic SOS-MP2 on top of Hartree-Fock-SCF &DFT BASIS_SET_FILE_NAME../BASIS_RI_cc-TZ POTENTIAL_FILE_NAME../POTENTIAL &QS METHOD GPW &END QS &POISSON PERIODIC NONE PSOLVER MT &END POISSON &SCF SCF_GUESS RESTART &END SCF &XC &XC_FUNCTIONAL PBE &END XC_FUNCTIONAL &HF FRACTION 1.00 &SCREENING EPS_SCHWARZ 1.0E-8 &END SCREENING &END HF &END XC &WF_CORRELATION METHOD RI_SOS_LAPLACE! do SOS-MP2 calculation NUMBER_PROC 2 SCALE_S 1.3! scale the OS-MP2 energy by 1.3 &RI_LAPLACE QUADRATURE_POINTS 10 SIZE_INTEG_GROUP 64 &END RI_LAPLACE &END WF_CORRELATION &END DFT! number of processes dealing with a single frequency grid point. Increase! this number when running out of memory during the frequency integration.! A larger number slows the calculation down.

Input non-periodic RPA on top of PBE-SCF &DFT BASIS_SET_FILE_NAME../BASIS_RI_cc-TZ POTENTIAL_FILE_NAME../POTENTIAL &QS METHOD GPW &END QS &POISSON PERIODIC NONE PSOLVER MT &END POISSON &SCF SCF_GUESS RESTART &END SCF &XC &XC_FUNCTIONAL PBE &END XC_FUNCTIONAL &END XC &WF_CORRELATION METHOD RI_RPA_GPW! do an RPA calculation NUMBER_PROC 2 &RI_RPA MINIMAX_QUADRATURE TRUE QUADRATURE_POINTS 15 SIZE_FREQ_INTEG_GROUP 16.! choose Minimax quadrature grid points. Then, only few grid points are! necessary in order to converge the frequency integration in Eq. (2).! number of grid points for the frequency integration! number of processes dealing with a single frequency grid point. Increase! this number when running out of memory during the frequency integration.! A larger number slows the calculation down.. &HF! Hartree-Fock for exchange energy FRACTION 1.00 &SCREENING EPS_SCHWARZ 1.0E-8 &END SCREENING &MEMORY MAX_MEMORY 0! we need not to store! any HF integrals because there! are no subsequent SCF steps &END MEMORY &END HF &END RI_RPA &END WF_CORRELATION &END DFT

Optimize RI basis (in case you want other basis than cc-tzvp) &XC &XC_FUNCTIONAL &PBE SCALE_X 0.00 SCALE_C 0.00 &END &END XC_FUNCTIONAL &HF FRACTION 1.00 &SCREENING EPS_SCHWARZ 1.0E-8 SCREEN_ON_INITIAL_P FALSE &END &END &WF_CORRELATION METHOD OPTIMIZE_RI_BASIS &OPT_RI_BASIS DELTA_I_REL 5.0E-6 DELTA_RI 5.0E-5 EPS_DERIV 1.0E-3 MAX_ITER 100! BASIS_SIZE MEDIUM! in case you do not want to specify the number of auxiliary basis functions NUM_FUNC 4 4 3 2 1! specify number of basis functions in RI basis for each shell, e.g.! there are 3 d-functions and 1 g-function. Rule of thumb: maximum! l-quantum number of RI basis exceeds the primary basis by one. &END NUMBER_PROC 1 &END &END XC. &END DFT &SUBSYS. &KIND H BASIS_SET cc-qzvp! basis set for which! RI basis is optimized POTENTIAL GTH-PBE-q1 &END KIND &COORD! optimize RI basis for H2 molecule! and take it for arbitrary environ-! ment of H H 0.0000 0.0000 0.0000 H 0.0000 0.0000 1.42000 &END COORD &END SUBSYS

Isobaric Isothermal Monte Carlo Simulation of Liquid Water Goal: Determine density of liquid water at ambient conditions. Method: Isobaric Isothermal Monte Carlo Pre-sampling using fast methods (classical force fields, DFT with GGA functionals)

Model 64 water molecules, 192 atoms, 256 active electrons cc-tzvp Basis, [3s3p2d1f], [3s2p1d], 3648 basis functions, 8704 RI basis functions

RPA density of liquid water NpT simulations (T = 295 K and p = 1 bar) [M. Del Ben, J. Hutter, and J. VandeVondele: Probing the structural and dynamical properties of liquid water with models including non-local electron correlation, J. Chem. Phys. 143, 054506 (2015)]

Density of Water ρ[g/ml] Error estimate ρ BLYP 0.797 0.008 BLYP-D3 1.066 0.007 PBE-D3 1.055 0.006 M062X-D3 (ADMM) 1.004 0.008 MP2 1.020 0.004 RPA 0.994 0.006 Exp. 0.998 [M. Del Ben, J. Hutter, and J. VandeVondele: Probing the structural and dynamical properties of liquid water with models including non-local electron correlation, J. Chem. Phys. 143, 054506 (2015)]

Very recently: MP2 gradients Del Ben, M; Hutter, J; VandeVondele, J. JOURNAL OF CHEMICAL PHYSICS 143, 102803 (2015). Forces and stress in second order Møller-Plesset perturbation theory for condensed phase systems within the resolution-of-identity Gaussian and plane waves approach. http://dx.doi.org/10.1063/1.4919238

Summary MP2 and RPA: two different approximate approaches for the correlation energy post-scf methods: first, converge SCF with Hartree-Fock (for MP2) or method of your choise (for RPA), then calculate total energy with MP2, RPA accurate total energies, e. g. for MD to determine density of water High computational cost: MP2: O(N 5 ), SOS-MP2, RPA: O(N 4 ) RPA for 64 water molecules on 64 CRAY CX30 cores: 48 min largest RPA calculation so far: 512 water molecules

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