Local Approaches to the Simulation of Electron Correlation in complex systems

Transcription

1 Local Approaches to the Simulation of Electron Correlation in complex systems Martin Schütz Institut für Physikalische und Theoretische Chemie, Universität Regensburg Universitätsstraße 31, D Regensburg International School on Ab initio Modelling of Solids 2014 Regensburg, July 20-25, 2014

2 ab inito methods: Hartree-Fock Single Slater determinant Variational optimization of expectation value over (exact) Hamiltonian! w.r. to orbitals Stationary condition of Lagrangian yields effective one-electron Schrödinger equation for each orbital (effective potential: nucleii plus mean-potential of the other electrons). Reduction of the -particle problem to one-particle problems. Computational cost scales linear with # electrons N

3 Intrinsic Error of the Hartree-Fock Model Mean Field Approximation: Each electron moves in the field of the nucleii and the mean field of the other electrons The two-particle density factorizes as a product of one-particle density (matrices), P 2 (r 1, r 2 )=P 1 (r 1 )P 1 (r 2 ) P 1 (r 1 ; r 2 )P 1 (r 2 ; r 1 ) + P 1 (r 1 )P 1 (r 2 ) P 1 (r 1 ; r 2 )P 1 (r 2 ; r 1 ) + P 1 (r 1 )P 1 (r 2 ) + P 1 (r 1 )P 1 (r 2 ). In reality: Electrons experience the actual positions of the others, electronic motion is correlated The two-particle density is depleted near r 1 = r 2 (Coulomb hole). P 2 (r 1, r 2 ) 5.0 RHF UHF Exact Energy / ev At equilibrium distances: repulsion of electrons is overestimated 2.Qualitatively wrong description of the dissociation R / Angstrom

4 Long-range correlation dissociation RHF wavefunction for ground state of H2: X = Â g (1) g (2), with g = Z g ( A + B ), where A is an A s-like atomic orbital centered on atom A. For inifinite separation R between A and B, A s A,Z g 1/ 2 we have: X = 1 2Â s As B + s Bs A + s As A + s Bs B Covalent terms OK,but spurious ionic terms (H +... F - ) lead to unphysical the potential. dissociation limit is overestimated! R 1 behaviour of RHF failure due to neglect of correlation: Both electrons have same probability density on A and B, regardless of the position of the other electron ionic terms do contribute! Partial solution: spin-unrestricted HF (UHF),but beware of spin contamination! (UHF wavefunction ist no longer eigenfunction of operator! Ŝ 2

5 Long-range correlation dissociation + RHF wavefunction for doubly excited state of H2: where A is an A s-like atomic orbital centered on atom A. g E = Â u (1) u(2), with u = Z u ( A B ), For inifinite separation R between A and B, A s A,Z g 1/ 2 we have: E = 1 2Â s As B + s Bs A s A s A s B s B Again unphysical mixture of covalent and ionic terms, but the latter have opposite sign here. Linear combination of X and E,i.e., = c X X + c E E is flexible enough to describe the potential correctly over the whole range of. near equilibrium: at dissociation limit: ionic and covalent terms have similar weight ionic terms mutually cancel Simple example for configuration interaction (CI), wavefunction is a mixture of several Slater determinants. The CI-coefficients (here c X and c E ) are determined according to the variational principle.

6 Short-range correlation the interelectronic cusp Correlation also important even when Hartree-Fock model is reasonable (e.g. near equilibrium). Since Hamilton operator contains r 1 ij the behaviour of the electronic wavefunction near r ij =0 has strong effect on energy. Consider Hamilton operator for He atom in internal coordinates nucleus), and (interelectronic distance), i.e., r 12 Ĥ = i=1 r 1 r 1 2 r 2 i + 2 r i r i + 2Z r i r 12 r 12 r 1 + r 2 r 2 r 21 r 21 r 2 2 r 12 r 2 12 r 1,r r 12 r 12 r 12 (distances from Right-hand side of Schrödinger equation is well behaved, so the Coulomb singularities must cancel, leading to the nuclear and interelectronic cusps, r i r i =0 = Z (r i = 0), and r 12 r 12 =0 = 1 2 (r 12 = 0) Interelectronic cusp: increases r 12 (linearly) near r 12 =0!

7 Short-range correlation the interelectronic cusp The Hartree-Fock wavefunction, on the other hand, does not depend on r 12 near r 12 =0. It overestimates the possibility of finding two electrons close together, and thus overestimates the electron repulsion energy. One then has for the Hartree-Fock and the exact wavefunction HF =0, and r 12 r 12 =0 r 12 r 12 =0 = 1 2 (r 12 = 0), OLPRO (Due to the lack of Fermi exchange correlation) the (Coulomb) correlation effect is especially important for electron pairs of opposite spin. For electron pairs with same spin the Pauli exclusion principle (solidly built into the Slater determinant), keeps the electrons apart (Pauli correlation). (Coulomb) correlation is much weaker!

8 CI for short-range correlation? CI expansion for He atom in terms of Slater orbitals (STOs) nlm (r) =r n 1 exp( r)y lm (, ), n > l m 0 can be expressed in the following form [Helgaker, Jørgensen, Ohlsen, Molecular Electronic Structure Theory (2000)] CI = exp[ (r 1 + r 2 )] ijk (r i 1r j 2 + rj 1 ri 2)r 2k 12 i, j, k =0, 1,... Good news: CI expansion introduces r 12 dependence in wavefunction Bad news:...but only in even powers of r 12 (leading linear term of cusp is missing) CI describes the Coulomb hole, but not the correct Coulomb cusp behaviour ( CI / r 12 ) r12 =0 =0. Slow convergence of CI (and related methods) with respect to basis set (i, j, k in above example) Introduce explicit dependence on r 12 into the wavefunction (explicitely correlated wavefunctions like r12-ci, r12-mp2, f12-mp2, f12-cc (Kutzelnigg, Klopper, Manby, Ten-no, Werner,,Grüneis,Usvyat)

9 CI and the Coulomb hole Coulomb hole of the He ground state,ci with different basis sets 2s1p 3s2p1d s3p2d1f 5s4p3d2f1g Very slow convergence with basis set size, no cusp at the bottom, since ( CI / r 12 ) r12 =0 =0. Intrinsic problem shared by all wavefunction expansions in orbital products, i.e., Slater determinants!

10 Form of the exact N-electron wavefunction Full CI expansion based on complete one-particle basis spanning its Fock-space Impractical, cost of Full CI scales exponentially with molecular size, slow convergence w.r. to AO basis set (except for explicitely correlated methods) Hierarchy of correlation methods and basis sets, extrapolation FULL CI O(N!) CCSDTQ O(N 10 ) METHOD CCSD(T) MP2 O(N 7 ) O(N 5 ) HF O(N 3 ) WANTED: [3s2p1d] [4s3p2d1f] [5s4p3d2f1g] [6s5p4d3f2g1h] Complete Basis set faster convergence w.r. to AO basis (explicitely correlated methods, F12-MP2,F12-CCSD,...) size-extensive and compact wavefunctions (as close to FCI as possible with as few determinants as possible (Coupled Cluster Ansatz, Local Methods, i.e., LCC)

11 What about Density Functional Theory (DFT)? Hohenberg-Kohn theorem (1965), E = E[ (r)] Kohn-Sham 1-Det. Ansatz in principle exact (if functional would be known and could be computed in practice) provides rather good results in many cases computationally rather cheap applicable to extended systems BUT principal physical deficiencies in practice (self repulsion of electron, non-vanishing correlation energy for single electrons in many functionals) No proper description of long range van der Waals dispersion MP2 LDA PBE MP2 LDA PBE Different functionals may provide rather different results, but NO POSSIBILITY FOR SYSTEMATIC IMPROVEMENT (take the functional that suits you most approach)

12 Abandon The all Jacob s hope allladder ye whoofenter DFT here Hartree-Fock LDA GGA Meta-GGA RPA,DHyb, Hyper-GGA

13 Coupled Cluster Theory, CC>=exp(T) 0> Exponential ansatz: exponential excitation operator acting on reference determinant CC = exp (T ) 0, exp (T )=1+T + 1 2! TT + 1 3! TTT +..., T = µ t µ µ, µ 1 = a A a I Exponential ansatz ensures product separability, and with this, size extensivity, also for truncated CC wavefunctions, e.g., CCSD, with T = T 1 + T 2 = t I Aa A a I + 1 t IJ 4 ABa A a Ia B a J IJAB Truncated CC wavefunction contains all Determinants of FCI, e.g. 4-fold substitutions in CCSD as disconnected products T 2 T 2,T 1 T 1 T 2, etc. CC is much more compact than CI linked CC equations for Energy and amplitudes: E = 0 exp ( T )H exp (T ) 0, IA 0 = 0 µ exp ( T )H exp (T ) 0. Since H is a two-electron operator, the Hausdorff expansion terminates always (independently of the truncation on T ) after the fifth term, i.e., exp ( T )H exp (T ) 0 = H +[H, T]+ 1 2! [[H, T],T]+ 1 3! [[[H, T],T],T]+ 1 [[[[H, T],T],T,T] 4! CCSD(T), i.e., CCSD with a perturbative a posteriori treatment of connected triple substitutions is the Golden Standard in Quantum Chemistry (very high accuracy is achieved for closed-shell systems, if proper AO basis sets are applied)! Scaling of computational cost: O(N 6 ) CCSD + O(N 7 ) (T)

14 Simplified CC models, based on perturbation theory 1.CC2 model (cheapest model appropriate for time-dependent properties (excitation energies, oscillator strengths, etc.) in CC response) singles substitutions kept to all orders, doubles substitutions to first order, all higher omitted: E = 0 Ĥ +[Ĥ,T 2] 0, with Ĥ = exp ( T 1)H exp (T 1 ) 0 = 0 µ 1 (Ĥ +[Ĥ,T 2]) 0, 0 = 0 µ 2 (Ĥ +[F, T 2]) 0. Scaling of computational cost: O(N 5 ) 2.MP2 model (Møller-Plesset perturbation theory of second order, cheapest correlation method, treats dispersion energy a level corresponding to uncoupled HF polarizabilities) only doubles substitutions (accurate to first order): E = 0 H +[H, T 2 ] 0, 0 = 0 µ 2 (H +[F, T 2 ]) 0 Scaling of computational cost: O(N 5 ) 3.RPA model (ring-ccd) (random phase approximation) only doubles substitutions (accurate to first order, since many 2nd-order diagrams missing, but also terms up to infinite order included) E = h0 H +[H, T 2 ] 0i, 0 = h0 µ 2 (H +[H, T 2 ] ring +[[H, T 2 ],T 2 ] ring ) 0i

15 Basis set extrapolation Extrapolated vs. plain atomization energies/reaction enthalpies: Normal distribution of errors [kj/mol] CCSD(T) relative to experiment Atomization energies Reaction enthalpies Bak, Jørgensen, Olsen, Helgaker, Klopper, JCP 112, 9229 (2000) Two-point extrapolation with cc-pcv(x 1)Z calculation leads to a significant improvement over the cc-pcv(x)z calculation alone! High accuracy can be achieved!

16 The scaling problem of electron correlation methods H H O O O O C C C C H H H H N N Canonical Local Orbital Orbital

17 Local methods for electronic ground states (pair specific) local subspaces of virtual orbitals (exp(-r) decay, tensor factorization) Restricted LMO pair lists (r -6 decay) strong pairs: LCCSD weak pairs: LMP2, or rather: LCCD[S]-R -6 (JCP 140, , 2014). TpT CPD photo lesion

18 Scaling of CPU-time for Local Coupled Clusters System: Glycine-Polypeptide [Gly]n / VDZ Basis: ([Gly]16: 1160 BF/360 El.) Hardware: 1.2 GHz Athlon PC (T) (~N ) CCSD Iteration (~N ) CPU time / s MGS, J. Chem. Phys., 113, 9986 (2000). n LCCSD Iteration (~N) L(T) (~ N)

19 Reaction energies CCSD(T) and LCCSD(T) 10 Energie / kcal/mol CCSD(T) - Exp LCCSD(T) LMP2 - Exp - Exp Energie / kcal/mol BP - Exp B3LYP - Exp

20 Density Fitting (DF) for correlation methods Idea: Use DF to approximate the expensive transformed 2-el 4-index integrals, e.g., the 4- ext integrals (AB CD) in CCSD (AB CD)= dx 1 dx 2 A (x 1 ) B (x 1 )r 1 12 C(x 1 ) D (x 1 ) = dx 1 dx 2 AB (x 1 )r 1 12 CD(x 2 ) Expand orbital product densities Fitting error functional to minimize: AB(x 1 ), CD(x 2 ) AB(x 1 ) AB (x 1 ) = P in an auxiliary (fitting) basis, i.e., c AB P P (x 1 ) w 12 =( AB AB w 12 AB AB ) Differentiation w.r. to fitting coefficients c AB P yields linear equation system Q (P w 12 Q)c AB Q =(P w 12 AB) Robust fit (Dunlap), fit error of integral is always 2nd order w.r. to fir error in orbital product, (AB CD) ( AB CD )+( AB CD ) ( AB CD ) Factorization of 4-index integrals in terms of 3-index integrals, i.e., for w 12 = r 1 12 (AB CD) ( AB CD )=( AB CD )=( AB CD )= PQ (AB P )(P Q) 1 (Q CD) Since the fitted orbital products are local quantities, locality can be exploited also for the fitting basis Local Fitting Methods

21 Local Density Fitting CPU /s Calculation of 4-ext integrals (rs tu) Polyglycine chain AO basis: cc-pvdz Fit basis: 7s5p4d2f/3s2p1d LCCSD DF-LCCSD LDF-LCCSD LCCSD: O(N ) 0 scaling n DF-LCCSD: integral evaluation/transformation: O(N 2 ), Solve: O(N 3 ), Assembly: O(N 2 ) With local fitting (local AUX domains) O(N ) scaling restored for all steps!

22 Ex. LCCSD(T) calc. TpT vs. CPD lesion in DNA ATTA sequence of DNA double strand TpT CPD photo lesion Undamaged pyrimidine dimer (TpT) cyclobutane pyrimidine dimer (CPD) 90 atoms, 528 correlated electrons, 2636 basis functions in avtz basis LCCSD(T): CPD in ATTA sequence destabilized by 55.2 kj/mol Elapsed times Integrals: 272 min LMP2: 31min LCCSD: 252 min L(T): 63 min Total: 622 min

23 He scattering on the MgO (100) surface He Mg O He-MgO potential calculated employing periodic local MP2 method in AVTZ basis set (CRYSCOR) Higher order corrections (CCSD(T), CCSDT(Q), CBS limit) by scaling pair energies with factors obtained from corresponding finite cluster calculations Diffraction intensities by solving the scattering Schrödinger eq. in the Born-Oppernheimer approx. (close coupling method)

24 Scaled MP2 faithfully recovers CCSD(T) curve Intra Na 2 Mg 3 O 4 correlation Interaction energy, mev HF CCSD(T) Upscaled LMP2 LMP Intra He correlation Inter correlation He Mg distance, Angstrom

25 Diffraction intensities, PCCP (advance article) Intensity (a.u.) mev mev mev (a) (d) (b) CCSD(T) CCSDT(Q) Exper. LMP (e) (c) (f) Intensity (a.u.) Incident angle (degrees) Incident angle (degrees) Incident angle (degrees) mev mev mev Diffraction intensities extremely sensitive w.r. to the quality of the potential. CCSD(T) clearly insufficient and CCSDT(Q) required.

26 Which is the lowest bound state?, PRB 89, (2014) 0 V 0 (mev) Experiment 1 Periodic LMP2 LMP2->CCSD(T)/VTZ LMP2->CCSD(T)/VQZ LMP2->CCSD(T)/BSL 1. Finite cluster CCSDT(Q)-CCSD(T) 2. Periodic LMP2-F12 + finite cluster CCSD(T)-MP2-10 Experiment z(å) Calculations resolve the controversy in the experimental results. Bound state below -10 mev (J. Toennies et al.) can be ruled out!

27 The CRYSCOR Group Prof. Dr. C. Pisani, Dr. L. Maschio, Dr. S. Casassa, A. Erba (Torino) Prof. Dr. M. Schütz, Dr. D. Usvyat, M. Hinreiner (Regensburg)