Quantum chemistry: wave-function and density-functional methods

Quantum chemistry: wave-function and density-functional methods Trygve Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo Electronic Structure e-science Meeting, Swedish e-science Research Centre (SeRC), Engsholms Slott, Mörkö, Sweden, April 7 8, 2011 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 1 / 41

Quantum chemistry Wave-function vs. density-functional methods Quantum mechanics has been applied to chemistry since the 1920s early accurate work on He and H 2 semi-empirical applications to larger molecules The idea of ab-initio theory developed in the 1950s early work in the 1950s, following the development of digital computers Hartree Fock (HF) self-consistent field (SCF) theory (1960s) configuration-interaction (CI) theory (1970s) multiconfigurational SCF (MCSCF) theory (early 1980s) many-body perturbation theory (1980s) coupled-cluster theory (late 1980s) Coupled-cluster theory is the most successful wave-function technique introduced from nuclear physics size extensive and hierarchical the exact solution can be approached in systematic manner high cost, near-degeneracy problems Density-functional theory (DFT) emerged during the 1990s Kohn Sham theory introduced from solid-state physics evaluation of dynamical correlation from the density cost broadly similar to HF theory semi-empirical in character cannot be systematically improved Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 2 / 41

Quantum chemistry The calculation of high-resolution NMR spectra by different electronic-structure methods 200 MHz NMR spectra of vinyllithium (C 2 H 3 Li) experiment RHF 0 100 200 0 100 200 MCSCF B3LYP 0 100 200 0 100 200 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 3 / 41

From Hartree Fock to coupled-cluster theory The Hartree Fock approximation The Hartree Fock model the fundamental approximation of wave-function theory each electron moves in the mean field of all other electrons provides an uncorrelated description: average rather than instantaneous interactions gives rise to the concept of molecular orbitals typical errors: 0.5% in the energy; 1% in bond distances, 5% 10% in other properties forms the basis for more accurate treatments The Hartree Fock and exact wave functions in helium: 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.5 0.0 0.0 1.0 0.5 0.0 0.5 1.0 0.5 1.0 0.5 0.0 0.5 1.0 0.5 concentric Hartree Fock contours, reflecting an uncorrelated description in reality, the electrons see each other and the contours becomes distorted Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 4 / 41

From Hartree Fock to coupled-cluster theory Electron correlation and virtual excitations For an improved description, we must describe the effects of electron correlation in real space, the electrons are constantly being scattered by collisions in the orbital picture, these are represented by excitations from occupied to virtual spin orbitals the most important among these are the double excitations or pair excitations Consider the effect of a double excitation in H 2 : 1σ 2 g uu (1 + tuu gg ˆX gg ) 1σ2 g = 1σ2 g 0.11 1σ2 u the one-electron density ρ(z) is hardly affected: -2-1 0 1 2-2 -1 0 1 2 the two-electron density ρ(z 1, z 2) changes dramatically: 2 2 0 0-2 0.04-2 0.04 0 2 0.00 0 2 0.00-2 -2 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 5 / 41

From Hartree Fock to coupled-cluster theory Coupled-cluster theory In coupled-cluster (CC) theory, the starting point is the HF description this description is improved upon by the application of excitation operators ( ) ( ) ( ) CC = 1 + ti a ˆX i a 1 + tij ab ˆX ij ab 1 + t abc abc ijk ˆX ijk HF }{{}}{{}}{{} singles doubles triples with each virtual excitation, there is an associated probability amplitude t abc ijk single excitations represent orbital adjustments rather than interactions double excitations are particularly important, arising from pair interactions higher excitations should become progressively less important CCS, CCSD, CCSDT, CCSDTQ, CCSDTQ5,... The quality of the calculation depends critically on the virtual space we employ atom-fixed Gaussian atomic orbitals virtual atomic orbitals are added in full shells at a time each new level introduces orbitals that recover the same amount of correlation energy the number of virtual AOs per atom increases rapidly: the correlation-consistent Gaussian basis sets SZ DZ TZ QZ 5Z 6Z 5 14 30 55 91 140 cc-pvdz, cc-pvtz, cc-pvqz, cc-pvtz, cc-pv6z,... Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 6 / 41

From Hartree Fock to coupled-cluster theory The two-dimensional chart of quantum chemistry The quality of ab initio calculations is determined by the description of 1 the N-electron space (wave-function model); 2 the one-electron space (basis set). Normal distributions of errors in AEs (kj/mol) CCSD(T) DZ CCSD(T) TZ CCSD(T) QZ CCSD(T) 5Z CCSD(T) 6Z -200 200-200 200-200 200-200 200-200 200 CCSD DZ CCSD TZ CCSD QZ CCSD 5Z CCSD 6Z -200 200-200 200-200 200-200 200-200 200 MP2 DZ MP2 TZ MP2 QZ MP2 5Z MP2 6Z -200 200-200 200-200 200-200 200-200 200 HF DZ HF TZ HF QZ HF 5Z HF 6Z -200 200-200 200-200 200-200 200-200 200 The errors are systematically reduced by going up in the hierarchies Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 7 / 41

Coupled-cluster convergence Convergence of the harmonic constant of N 2 371.9 84.6 4.1 13.8 23.5 4.7 0.8 HF CCSD FC CCSD T FC CCSD T CCSDT CCSDTQ CCSDTQ5 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 8 / 41

Coupled-cluster convergence Energy contributions to atomization energies (kj/mol) Contributions of each CC excitation level (left) and AO basis-set shell (right) 1000 Log Lin 1000 Log Log 100 100 10 10 1 1 HF CCSD CCSDT CCSDTQ DZ TZ QZ 5Z 6Z color code: HF, N 2, F 2, and CO The excitation-level convergence is approximately linear (log linear plot) each new excitation level reduces the error by about an order of magnitude the contributions from quintuples are negligible (about 0.1 kj/mol) The basis-set convergence is much slower (log log plot) each shell contributes an energy proportional to X 4 where X is the cardinal number a similarly small error (0.1 kj/mol) requires X > 10 clearly, we must choose our orbitals in the best possible manner Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 9 / 41

Coupled-cluster convergence The exhaustion of the Schrödinger equation Atomization energies (kj/mol) RHF SD T Q rel. vib. total experiment error HF 405.7 178.2 9.1 0.6 2.5 24.5 566.7 566.2±0.7 0.5 N 2 482.9 426.0 42.4 3.9 0.6 14.1 940.6 941.6±0.2 1.1 F 2 155.3 283.3 31.6 3.3 3.3 5.5 154.1 154.6±0.6 0.5 CO 730.1 322.2 32.1 2.3 2.0 12.9 1071.8 1071.8±0.5 0.0 Bond distances (pm) RHF SD T Q 5 rel. theory exp. err. HF 89.70 1.67 0.29 0.02 0.00 0.01 91.69 91.69 0.00 N 2 106.54 2.40 0.67 0.14 0.03 0.00 109.78 109.77 0.01 F 2 132.64 6.04 2.02 0.44 0.03 0.05 141.22 141.27 0.05 CO 110.18 1.87 0.75 0.04 0.00 0.00 112.84 112.84 0.00 Harmonic constants (cm 1 ) RHF SD T Q 5 rel. theory exp. err. HF 4473.8 277.4 50.2 4.1 0.1 3.5 4138.5 4138.3 0.2 N 2 2730.3 275.8 72.4 18.8 3.9 1.4 2358.0 2358.6 0.6 F 2 1266.9 236.1 95.3 15.3 0.8 0.5 918.9 916.6 2.3 CO 2426.7 177.4 71.7 7.2 0.0 1.3 2169.1 2169.8 0.7 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 10 / 41

Coupled-cluster convergence The electron cusp and the Coulomb hole The wave function has a cusp at coalescence 0.5 0.0-0.5 0.5 0.0-0.5 0.5 0.00 0.0-0.05-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0-0.10 It is difficult to describe by orbital expansions DZ TZ QZ 5Z -90 90-90 90-90 90-90 90 The error is inversely proportional to the number of virtual AOs E X N 1 T 1/4 Each new digit in the energy therefore costs 10000 times more CPU time! 1 minute 1 week 200 years Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 11 / 41

Coupled-cluster convergence Solutions to slow basis-set convergence 1 Use explicitly correlated methods! Include interelectronic distances r ij in the wave function: 50 100 150 200 250-2 Ψ R12 = K C K Φ K + C R r 12 Φ 0-4 CI -6 CI-R12-8 Hylleraas 2 Use basis-set extrapolation! Exploit the smooth convergence E = E X + AX 3 to extrapolate to basis-set limit: E = X 3 E X Y 3 E Y me h DZ TZ QZ 5Z 6Z R12 X 3 Y 3 plain 194.8 62.2 23.1 10.6 6.6 1.4 extr. 21.4 1.4 0.4 0.5 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 12 / 41

Density-functional theory The work horse of quantum chemistry The traditional wave-function methods of quantum chemistry are capable of high accuracy nevertheless, most calculations are performed using density-functional theory (DFT) Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 13 / 41

Density-functional theory The universal density functional The electronic energy is a functional E[v] of the external potential v(r) = K Z K rk Coulomb potential Traditionally, we determine E[v] by solving (approximately) the Schrödinger equation E[v] = inf Ψ Ψ Ĥ[v] Ψ variation principle However, the negative ground-state energy Ē[v] is a convex functional of the potential Ē(cv 1 + (1 c)v 2 ) cē(v 1 ) + (1 c)ē(v 2 ), 0 c 1 convexity f x1 cf x1 1 c f x2 f c x1 1 c x2 f x2 x1 c x1 1 c x2 x2 The energy may then be expressed in terms of its Legendre Fenchel transform ( ) F [ρ] = sup E[v] v(r)ρ(r) dr energy as a functional of density v ( ) E[v] = inf F [ρ] + v(r)ρ(r) dr ρ energy as a functional of potential the universal density functional F [ρ] is the central quantity in DFT Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 14 / 41

Density-functional theory Conjugate functionals As chemists we may choose to work in terms of E[v] or F [ρ]: ( ) F [ρ] = sup E[v] v(r)ρ(r) dr Lieb variation principle v ( ) E[v] = inf F [ρ] + v(r)ρ(r) dr ρ Hohenberg Kohn variation principle the relationship is analogous to that between Hamiltonian and Lagrangian mechanics The potential v(r) and the density ρ(r) are conjugate variables they belong to dual linear spaces such that v(r)ρ(r) dr is finite they satisfy the reciprocal relations δf [ρ] δρ(r) = v(r), δe[v] δv(r) = ρ(r) In molecular mechanics (MM), we work in terms of E[v] parameterization of energy as a function of bond distances, angles etc. widely used for large systems (in biochemistry) In density-functional theory (DFT), we work in terms of F [ρ] the exact functional is unknown but useful approximations exist more accurate the molecular mechanics, widely used in chemistry Neither method involves the direct solution of the Schrödinger equation Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 15 / 41

Kohn Sham theory The noninteracting reference system The Hohenberg Kohn variation principle is given by E[v] = min ρ ( F [ρ] + v(r)ρ(r) dr ) the functional form of F [ρ] is unknown the kinetic energy is most difficult A noninteracting system can be solved exactly, at low cost, by introducing orbitals F [ρ] = T s[ρ] + J[ρ] + E xc[ρ], ρ(r) = i φ i (r) φ i (r) where the contributions are T s[ρ] = 1 2 i φ i (r) 2 φ i (r)dr noninteracting kinetic energy J[ρ] = ρ(r 1 )ρ(r 2 )r 1 12 dr 1dr 2 Coulomb energy E xc[ρ] = F [ρ] T s[ρ] J[ρ] exchange correlation energy In Kohn Sham theory, we solve a noninteracting problem in an effective potential [ 1 2 2 + v eff (r) ] φ i (r) = ε i φ i (r), v eff (r) = v(r) + v J (r) + δexc[ρ] δρ(r) v eff (r) is adjusted such that the noninteracting density is equal to the true density it remains to specify the exchange correlation functional E xc[ρ] Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 16 / 41

Kohn Sham theory The exchange correlation functional The exact exchange correlation functional is unknown and we must rely on approximations Local-density approximation (LDA) XC functional modeled after the uniform electron gas (which is known exactly) E LDA xc [ρ] = f (ρ(r)) dr local dependence on density widely applied in condensed-matter physics not sufficiently accurate to compete with traditional methods of quantum chemistry Generalized-gradient approximation (GGA) introduce a dependence also on the density gradient E GGA xc [ρ] = f (ρ(r, ρ(r)) dr local dependence on density and its gradient Becke s gradient correction to exchange (1988) changed the situation the accuracy became sufficient to compete in chemistry indeed, surprisingly high accuracy for energetics Hybrid Kohn Sham theory include some proportion of exact exchange in the calculations (Becke, 1993) it is difficult to find a correlation functional that goes with exact exchange 20% is good for energetics; for other properties, 100% may be a good thing Progress has to a large extent been semi-empirical empirical and non-empirical functionals Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 17 / 41

Kohn Sham theory A plethora of exchange correlation functionals Table 1. MADs (all energies in ev) for various level of theory for the extended G2 set Method G2(MAD) Hf IP EA PA H-Ne, Etot TM E He2, E(Re) Ne2, E(Re) (H2O)2, De(RO...O) HF 6.47 1.036 1.158 0.15 4.49 1.09 Unbound Unbound 0.161 (3.048) G2 or best ab initio 0.07 a 0.053 b 0.057 b 0.05 b 1.59 c 0.19 d 0.0011 (2.993) e 0.0043 (3.125) e 0.218 (2.912) f LDA (SVWN) 3.94 a 0.665 0.749 0.27 6.67 0.54 g 0.0109 (2.377) 0.0231 (2.595) 0.391 (2.710) GGA BP86 0.88 a 0.175 0.212 0.05 0.19 0.46 Unbound Unbound 0.194 (2.889) BLYP 0.31 a 0.187 0.106 0.08 0.19 0.37 g Unbound Unbound 0.181 (2.952) BPW91 0.34 a 0.163 0.094 0.05 0.16 0.60 Unbound Unbound 0.156 (2.946) PW91PW91 0.77 0.164 0.141 0.06 0.35 0.52 0.0100 (2.645) 0.0137 (3.016) 0.235 (2.886) mpwpw h 0.65 0.161 0.122 0.05 0.16 0.38 0.0052 (2.823) 0.0076 (3.178) 0.194 (2.911) PBEPBE i 0.74 i 0.156 0.101 0.06 1.25 0.34 0.0032 (2.752) 0.0048 (3.097) 0.222 (2.899) XLYP j 0.33 0.186 0.117 0.09 0.95 0.24 0.0010 (2.805) 0.0030 (3.126) 0.192 (2.953) Hybrid methods BH & HLYP k 0.94 0.207 0.247 0.07 0.08 0.72 Unbound Unbound 0.214 (2.905) B3P86 l 0.78 a 0.636 0.593 0.03 2.80 0.34 Unbound Unbound 0.206 (2.878) B3LYP m 0.13 a 0.168 0.103 0.06 0.38 0.25 g Unbound Unbound 0.198 (2.926) B3PW91 n 0.15 a 0.161 0.100 0.03 0.24 0.38 Unbound Unbound 0.175 (2.923) PW1PW o 0.23 0.160 0.114 0.04 0.30 0.30 0.0066 (2.660) 0.0095 (3.003) 0.227 (2.884) mpw1pw p 0.17 0.160 0.118 0.04 0.16 0.31 0.0020 (3.052) 0.0023 (3.254) 0.199 (2.898) PBE1PBE q 0.21 i 0.162 0.126 0.04 1.09 0.30 0.0018 (2.818) 0.0026 (3.118) 0.216 (2.896) O3LYP r 0.18 0.139 0.107 0.05 0.06 0.49 0.0031 (2.860) 0.0047 (3.225) 0.139 (3.095) X3LYP s 0.12 0.154 0.087 0.07 0.11 0.22 0.0010 (2.726) 0.0028 (2.904) 0.216 (2.908) Experimental 0.0010 (2.970) t 0.0036 (3.091) t 0.236 u (2.948) v Hf, heat of formation at 298 K; PA, proton affinity; Etot, total energies (H-Ne); TM E, s to d excitation energy of nine first-row transition metal atoms and nine positive ions. Bonding properties [ E or De in ev and (Re) in Å] are given for He2, Ne2, and (H2O)2. The best DFT results are in boldface, as are the most accurate answers [experiment except for (H2O)2]. a Ref. 5. Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 18 / 41

Kohn Sham theory A comparison with coupled-cluster theory Reaction enthalpies (kj/mol) calculated using the DFT/B3LYP and CCSD(T) models B3LYP CCSD(T) exp. CH 2 + H 2 CH 4 543 1 543 1 544(2) C 2H 2 + H 2 C 2H 4 208 5 206 3 203(2) C 2H 2 + 3H 2 2CH 4 450 4 447 1 446(2) CO + H 2 CH 2O 34 13 23 2 21(1) N 2 + 3H 2 2NH 2 166 2 165 1 164(1) F 2 + H 2 2HF 540 23 564 1 563(1) O 3 + 3H 2 3H 2O 909 24 946 13 933(2) CH 2O + 2H 2 CH 4 + H 2O 234 17 250 1 251(1) H 2O 2 + H 2 2H 2O 346 19 362 3 365(2) CO + 3H 2 CH 4 + H 2O 268 4 273 1 272(1) HCN + 3H 2 CH 4 + NH 2 320 0 321 1 320(3) HNO + 2H 2 H 2O + NH 2 429 15 446 2 444(1) CO 2 + 4H 2 CH 4 + 2H 2O 211 33 244 0 244(1) 2CH 2 C 2H 4 845 1 845 1 844(3) Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 19 / 41

Kohn Sham excitation energies Excitation energies of CO HF (grey), CCSD (red), CC3 (black) LDA (yellow), BLYP (green), B3LYP (blue) 14 12 10 8 6 4 2 1 2 3 4 5 6 7 8 Statistics for errors for HF, CO, and H 2 O (%) HF CCSD LDA BLYP B3LYP 8.6 0.2 17.9 20.3 12.3 std 4.9 1.1 8.2 8.3 5.5 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 20 / 41

Kohn Sham excitation energies The asymptotic behaviour and importance of exact exchange DFT represents local excitations well excitations to outer valence and charge-transfer (CT) excitations less well described the potential falls off too fast the asymptotic behaviour should be lim r vxc(r) = 1 r This can be corrected by the inclusion of exact exchange exact exchange can be introduced in different manners 1.0 0.8 0.6 HF LC CAM B3LYP 0.4 0.2 B3LYP 2 4 6 8 the proportion of exact exchange as a function of r Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 21 / 41

Kohn Sham excitation energies Charge transfer excitations in tripeptide O H O N N N H O H excitation type PBE B3LYP CAM exp. n 2 π2 local 5.58 5.74 5.92 5.61 n 1 π1 local 5.36 5.57 5.72 5.74 n 3 π3 local 5.74 5.88 6.00 5.91 π 1 π2 CT 5.18 6.27 6.98 7.01 π 2 π3 CT 5.51 6.60 7.68 7.39 n 1 π2 CT 4.61 6.33 7.78 8.12 n 2 π3 CT 5.16 6.83 8.25 8.33 π 1 π3 CT 4.76 6.06 8.51 8.74 n 1 π3 CT 4.26 6.12 8.67 9.30 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 22 / 41

Kohn Sham excitation energies Diagnostic for TDDFT excitation energies The quality of excitation energies declines with increasing degree of charge transfer when can excitation energies be trusted? We have developed an inexpensive diagnostic Λ = ia κ2 ia φ i φ a /( ia κ2 ia ) the PBE functional (left) is erratic for Λ < 0.6 the CAMB3LYP functional (right) is uniformly reliable local excitations, Rydberg excitations, charge-transfer excitations Peach et al., J. Chem. Phys. 128, 044118 (2008) Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 23 / 41

Benchmarking by the Lieb variation principle The adiabatic connection Let E λ [v] be the ground-state electronic energy at interaction strength λ We may then calculate the energy by expansion about the noninteracting system λ = 0: λ F λ [ρ] = F 0 [ρ] + F λ [ρ] dλ = Ts[ρ] + λ (J[ρ] + Ex[ρ]) + E c,λ[ρ] 0 The adiabatic connection: F λ [ρ] plotted against the interaction strength λ F λ [ρ] = J[ρ] + Ex[ρ] + E c,λ [ρ] AC integrand 0.000 9.0 WXC,Λ a.u. 9.1 9.2 9.3 9.4 T c Ρ E c Ρ CCSD W CCSD T Ρ W CCSD Ρ a.u. 0.005 0.010 0.015 0.020 W CCSD T Ρ W CC 9.5 CCSD T 0.025 0.0 0.2 0.4 0.6 0.8 1.0 Λ 0.0 0.2 0.4 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 24 / 41

Benchmarking by the Lieb variation principle From dynamical to static correlation: dissociation of H 2 As H 2 dissociates, correlation changes from dynamical to static 0.05 1.4 bohr 0.10 0.15 0.20 5 bohr 0.25 10 bohr 0 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 25 / 41

Benchmarking by the Lieb variation principle BLYP cannot do static correlation The BLYP functional treats correlation as dynamical at all bond distances BLYP FCI BLYP FCI R 1.4 bohr R 3.0 bohr BLYP BLYP R 5.0 bohr FCI R 10.0 bohr FCI Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 26 / 41

Benchmarking by the Lieb variation principle BLYP manages by error cancellation The improved BLYP performance arises from an overestimation of exchange error cancellation between exchange and correlation reduces total error to about one third R 1.4 bohr R 3.0 bohr FCI exchange BLYP exchange BLYP correlation FCI correlation R 5.0 bohr R 10.0 bohr Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 27 / 41

Methods for large molecules Construction of Kohn Sham matrices and molecular gradients Quantum chemistry is well developed for small and medium-sized molecules well-established levels of theory of high accuracy and reliability wide variety of phenomena amenable to a rigorous treatment however, often a more realistic modeling requires studies on very large systems Our goal is to make molecular studies on thousands of atoms routine redesign algorithms to curb cost and utilize new computer architectures different requirements may be necessary for electronic-structure models Fast integral evaluation for Kohn Sham matrices and molecular gradients expansion of solid harmonics in Hermite rather than Cartesian Gaussians density fitting and fast-multipole methods: linear complexity and fast timings for BP86/6-31G** molecular gradients in linear polyene chains: 100 XC 80 FF-J Time (s) 60 40 NF-J 20 1el 0 0 50 100 150 200 250 Number of carbon atoms Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 28 / 41

Methods for large molecules Examples of energy optimizations and force evaluations Single Intel Xeon 2.66 GHz processor 392-atom titin fragment BP86/6-31G(*) energy in 50 min; gradient in 9 min (1.4 s per atom) 642-atom crambin protein BP86/6-31G energy in 3 h, gradient in 26 min (2.4 s per atom) Exact exchange is one to two orders of magnitude slower Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 29 / 41

Methods for large molecules Difficulties with large systems For more than 1000 atoms, standard optimization techniques become problematic Roothaan diagonalization combined with DIIS averaging may oscillate or diverge diagonalization is inherently of nonlinear complexity Large systems are typically more difficult to converge 0.4 alanine residue peptides 0.3 HF HOMO LUMO gap 0.2 lowest HF Hessian eigenvalue 0.1 0 B3LYP eigenvalue B3LYP HOMO LUMO gap 100 150 200 250 300 350 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 30 / 41

Methods for large molecules Density-matrix energy optimization We have developed a density-matrix minimization based on the parameterization D(X) = exp( XS)D 0 exp(sx), X T = X Avoids diagonalization (of cubic cost) and provides linear scaling by sparsity 10 000 8000 time in RH Newton equations against the number of atoms alanine residue peptides HF 6 31G dens 6000 4000 2000 sparse 200 400 600 800 1000 1200 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 31 / 41

Methods for large molecules Augmented Roothaan Hall (ARH) method Optimization based on D(X) may be carried out in many ways conjugate gradient, quasi-newton, Newton methods The full Newton equations are given by (F vv n F oo n ) X + X (F vv n F oo n ) + G ov ([D n, X]) G vo ([D n, X]) = F vo n }{{}}{{} large 2nd-order F term small 2nd-order G term dominant part of Hessian treated exactly, the remainder by update Comparison of ARH (left) and standard RH DIIS (right) F ov n }{{} 1st-order F term 51-molecule water cluster (full triangles), insulin (full squares), vitamin B12 (empty circles) 10 4 (a) 10 4 (b) Energy error / Hartree 10 2 10 0 10-2 10-4 10-6 10 2 10 0 10-2 10-4 10-6 0 10 20 30 40 Iterations 0 10 20 30 40 Iterations Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 32 / 41

Methods for large molecules Three-level ARH scheme I The ARH scheme has subsequently been implemented within a three-level (3L) scheme: I I I I grand-canonical atomic optimization: starting guess valence-basis molecular optimization: crude molecular optimization full-basis molecular optimization: final adjustments Comparison with other methods for 23 transition-metal complexes van Lenthe 100 I ARH-3L 76 QCP1 109 QCP2 157 A water droplet containing 736 water molecules (diameter 35 A ) I I I BP86/6-31G* level of theory (0+5+9 iterations) 2208 atoms, 7360 electrons, 25760 primitive and 13248 contracted Gaussians wall time 76 h, CPU time 270 h (4 IBM Power6 4.7 GHz cores) Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 33 / 41

Methods for large molecules Computers and hardware Requirements for our new linear-scaling code not only linear scaling but fast easily adaptable to new computational methods and computer architectures Hardware technology and platforms change rapidly we will never have routine access to largest and fastest computers this gives us some time to adapt code to emerging technologies early adaptation is difficult; late adaptation is dangerous Moderns computers combine many nodes (1000s) with many cores (4,8,16) the use of OpenMP for many cores is fairly straightforward the use of MPI for many nodes is much more difficult our current code uses OpenMP, we aim for a hybrid MPI/OpenMP solution Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 34 / 41

Centre for Theoretical and Computational Chemistry (CTCC) Theory and Modeling Centre of excellence established in 2007 for a period of 5 (10) years one of 21 Norwegian centres of excellence, the only one in chemistry shared between the Universities of Tromsø (UiT) and Oslo (UiO), with UiT as host institution Chemical biology Materials science Bioinorganic chem. Organic and organometallic chemistry Theory and modelling Solid-state systems Spectroscopy Heterogeneous and homogeneous catalysis Atmospheric chem. Experimentalists and theorists from chemistry, physics, and mathematics The vision of the CTCC is to become a leading international contributor to computational chemistry by carrying out cutting-edge research in theoretical and computational chemistry at the highest international level. Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 35 / 41

Centre for Theoretical and Computational Chemistry (CTCC) CTCC in numbers Financial support from the Norwegian Research Council (NRC) in 2009: 11.1 MNOK from home institutions in 2009: 7.9 MNOK Staff (total and UiT + UiO) 10 senior members (5+5) 3.5 researchers (2+1.5) 20 postdocs (11+9) 13 PhD students (6+7) 5 master students (2+3) 3 affiliates (1+2) 4 adjunct professors in 20% position (3+1) 1.6 administrative staff (1+0.6) Publications more than 220 papers more than 1200 citations Computer resources provided by NOTUR (the Norwegian Metacenter for Computational Science) 20 million CPU hours annually Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 36 / 41

Centre for Theoretical and Computational Chemistry (CTCC) Senior and affiliate members Principal investigators at the University of Tromsø Tor Flå, multiscale methods with wavelets Luca Frediani, properties and spectroscopy Abhik Ghosh, bioinorganic chemistry Kenneth Ruud, director Inge Røeggen, fragment approach for large systems Principal investigators at the University of Oslo Knut Fægri, clusters, surfaces and solids Trygve Helgaker, large periodic and nonperiodic systems Claus Jørgen Nielsen, gas-phase reactions and photochemistry Mats Tilset, catalysis and organometallic chemistry Einar Uggerud, dynamics and time development Affiliate members Bjørn Olav Brandsdali, University of Tromsø Harald Møllendal, University of Oslo Svein Samdal, University of Oslo Adjunct professors (20% positions) Sonia Coriani, University of Trieste Odile Eisenstein, University of Montpellier Benedetta Mennucci, University of Pisa Magdalena Pecul, University of Warsaw Trond Saue, University of Strasbourg Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 37 / 41

Centre for Theoretical and Computational Chemistry (CTCC) Organization and management Center Leadership Prof. Kenneth Ruud, Director, UiT Prof. Trygve Helgaker, Co-director, UiO Senior Researcher Forum all senior CTCC members Administrative staff Stig Eide, Head of Administration, UiT Anne Marie Øveraas, Office Manager, UiO (60% + 20% position) Board of Directors Prof. Fred Godtliebsen, chairman, vice dean of research, Faculty of Science, UiT Prof. Anne-Brit Kolstø, vice chairman, UiO Dr. Nina Aas, Statoil Prof. Knut J. Børve, University of Bergen Prof. Aslak Tveito, Simula Research Center Scientific Advisory Board Prof. Emily Carter, Princeton University Prof. Odile Eisenstein, University of Montpellier Prof. Kersti Hermansson, Uppsala University Prof. Mike Robb, Imperial College London Prof. Per-Olof Åstrand, Norwegian University of Science and Technology Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 38 / 41

Centre for Theoretical and Computational Chemistry (CTCC) CTCC seminars and workshops Biannual CTCC meetings joint Tromsø Oslo meetings of all members twice a year informal presentations of students and postdocs one meeting on conjunction with Norwegian Chemical Society (NKS) meeting CTCC group seminars 74 in Oslo since October 2007 101 in Tromsø since July 2007 6 workshops with UiO research groups Catalysis Seminar, with ingap 1 April 17 2008 Organic Quantum Chemistry, November 25 2008 From Ab Initio Methods To Density-Functional Theory, with CMA 2, January 13 2009 Mini-Seminar on Computational Inorganic Chemistry, with Fermio 3, April 29 2010 Mini-Seminar on Computational Materials Science, May 4 2010 Workshop On Computational Quantum Mechanics, with CMA, June 18 19 2010 1 Innovative Natural Gas Process and Products, a Centre for Research-based Innovation, UiO 2 Centre of Mathematics for Applications, a Centre of Excellence, UiO 3 Functional Energy Related Materials in Oslo, UiO Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 39 / 41

Centre for Theoretical and Computational Chemistry (CTCC) CTCC visitors CTCC visitors programs 12 months for visiting professors each year 24 months for graduate and postgraduate visitors each year Visiting professors Swapan Chakrabarti, University of Calcutta (3 months 2008, UiT) Daniel Crawford, Virginia Tech (6 months 2009, UiT + UiO) Pawe l Kozlowski, University of Louisville (3 months 2010, UiT) Ludwik Adamowicz, University of Arizona (5 weeks 2010, UiO) Taku Onishi, Mie University (10 months 2010 2011, UiO) Mark Hoffmann, University of North Dakota, (6 months 2010 2011, UiT + UiO) Wim Klopper, Universität Karlsruhe (6 months 2010 2011, UiO) Short visits 2007: 21 visits of 20 unique visitors from 11 countries 2008: 56 visits of 45 unique visitors from 21 countries 2009: 32 visits of 30 unique visitors from 17 countries 2010: 56 visits of 46 unique visitors from 18 countries Gropen Almlöf Lectures annual lecture series established by the CTCC in 2008 Björn Roos (2008), Tom Ziegler (2009), Michele Parrinello (2010) Division for Computational Chemistry of the Norwegian Chemical Society (NKS) established in 2008 following an initiative of the CTCC Kongsvinger 2008, Bergen 2009, Trondheim 2010, Lillstrøm 2011 Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 40 / 41

Centre for Theoretical and Computational Chemistry (CTCC) CTCC meetings and conferences Coastal Voyage of Current Density Functional Theory September 19 22 2007 Coastal Express between Tromsø and Trondheim 43 participants form 15 countries 22 talks and 11 posters Molecular Properties 2009 June 18 21 2009 Hotell Vettre, Asker (Oslo) satellite symposium to the 13th ICQC in Helsinki 117 participants from 21 countries 35 talks and 54 posters Quantum Chemistry beyond the Arctic Circle Promoting Female Excellence in Theoretical and Computational Chemistry June 23 26 2010 Sommarøy and Tromsø 75 participants from 20 countries 29 talks and 25 posters XVth European Seminar on Computational Methods in Quantum Chemistry June 16 19 2011 Oscarsborg, Drøbak (Oslo) about 100 participants Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7 8 2011 41 / 41