Coupled Cluster Theory and Tensor Decomposition Techniques

Transcription

1 Coupled Cluster Theory and Tensor Decomposition Techniques Alexander A. Auer Atomistic Modelling Group Interface Chemistry and Surface Engineering

2 Hierarchy of quantum chemical methods

3 Analysis of orbitals and electron density

4 predicting 13 C nmr chemical shifts deviation from experiment [ppm] + vibr. corr. + large basis CCSD(T) CCSD MP2 DFT HF-SCF benchmark for 16 compounds : A.A.A., J.Gauss, J.F.Stanton J. Chem. Phys., 118, (2003)

5 problem from synthesis / spectroscopy NMR chemical shifts [ppm] : C 1 C 2 C 3 C S.G.D yachkova et al. Russ. Chem. Bull., Int. Ed., 4, 751 (2001)

6 the right testcase NMR chemical shifts [ppm] : C 1 C 2 C 3 C 4 C 1 C 2 C 3 C K.Banert in Science of Synthesis, Houben-Weyl, Methods of Molecular Transformations, Bd. 24, S. 1059, Thieme Verlag Stuttgart (2006)

7 benchmarks geometry: MP2 / pvtz - NMR: B3-LYP / TZVP C 1 C 2 C 2 C 3 C 4 C 3 C [ppm]

8 benchmarks geometry: MP2 / pvtz - NMR: B3-LYP / TZVP C 1 C 2 C 2 C 3 C 4 C 3 C geometry: CCSD(T) [ppm] / pvtz - NMR: MP2 / tz2p C 1 C 2 C 2 C 3 C 4 C 4 C

9 benchmarks geometry: MP2 / pvtz - NMR: B3-LYP / TZVP C 1 C 2 C 2 C 3 C 4 C 3 C geometry: CCSD(T) [ppm] / pvtz - NMR: MP2 / tz2p C 1 C 2 C 2 C 3 C 4 C 4 C geometry: CCSD(T) / pvtz - NMR: CCSD(T) / tz2p C 1 C 2 C 2 C 3 C 3 C

10 ZPV and solvent effects solvent effects: chem. shift [ppm] ε = 1 ε = 2 ε = 20 ε = 78 C C experimental observation: change in chemical shifts for EtOCCH in different solvents at most 1 ppm benchmark for ZVP effects for 13 C shifts: in the order of 1-2 ppm

11 CCSD(T) prediction for azide compound theory geometry: MP2 / pvtz experiment NMR: CCSD(T) / qz2p C 1 C 2 C 2 C 3 C 4 C 3 C [ppm]

12 E. Prochnow, A. A. Auer and K. Banert, J. Phys. Chem. A, 111, 9945 (2007) CCSD(T) prediction for azide compound theory geometry: MP2 / pvtz experiment NMR: CCSD(T) / qz2p C 1 C 2 C 2 C 3 C 4 C 3 C [ppm]

13 ab-initio methods : configuration interaction exact wavefunction is a many electron function!

14 ab-initio methods : configuration interaction exact wavefunction is a many electron function! idea : expand exact wavefunction in complete basis of many electron functions

15 ab-initio methods : configuration interaction exact wavefunction is a many electron function! idea : expand exact wavefunction in complete basis of many electron functions many electron function : HF-SCF slater determinant complete basis of many electron functions : all possible determinants from occupied and virtuals Configuration Interaction (CI) : linear expansion, one parameter per determinant, optimized variationally

16 ab-initio methods : configuration interaction wavefunction expression as linear CI expansion : Ψ CI = Ψ 0 + ci a Ψ a i + 1 ia 4 ijab c ab ij Ψ ab ij +...

17 ab-initio methods : configuration interaction wavefunction expression as linear CI expansion : Ψ CI = Ψ 0 + ci a Ψ a i + 1 ia 4 ijab c ab ij Ψ ab ij +... Ψ CI = Ψ 0 + Ĉ 1 Ψ 0 + Ĉ 2 Ψ diagonalization of Hamiltonian matrix yields energies for ground and excited states

18 ab-initio methods : configuration interaction use all determinants that can be constructed expansion in complete basis (number of determinants grows exponentially!!)

19 ab-initio methods : configuration interaction use all determinants that can be constructed expansion in complete basis (number of determinants grows exponentially!!) exact solution of electronic problem in given AO basis (Full Configuration Interaction FCI)

20 ab-initio methods : configuration interaction use all determinants that can be constructed expansion in complete basis (number of determinants grows exponentially!!) exact solution of electronic problem in given AO basis (Full Configuration Interaction FCI) in practice only truncated expansion feasible : CIS : only singly substituted determinants CISD : singly and double substituted determinants CISDT : single, double and triple substitutions...

21 configuration interaction and size constancy But : extensivity of energy not given!! (CI is not size-extensive) unphysical description leads to wrong asymptotic of energy for extended systems the larger the system, the smaller the fraction of correlation energy from a truncated CI

22 configuration interaction and size constancy But : extensivity of energy not given!! (CI is not size-extensive) unphysical description leads to wrong asymptotic of energy for extended systems the larger the system, the smaller the fraction of correlation energy from a truncated CI recommendation : if possible don t use CI!

23 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0

24 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0 cluster operator ˆT analogous to CI-Operator ˆT = ˆT 1 + ˆT 2 + ˆT

25 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0 cluster operator ˆT analogous to CI-Operator coefficients are called amplitudes: ˆT = ˆT 1 + ˆT 2 + ˆT e ˆT = 1 + ˆT + 1 2! ˆT ! ˆT

26 coupled cluster theory product terms resulting from exponential expansion Ψ CC = (1 + ˆT ! ˆT ! ˆT ˆT ! ˆT ! ˆT ˆT 1 ˆT ! ˆT 1 2 ˆT ! ˆT 1 ˆT ) Ψ 0

27 coupled cluster theory Energy equations : Amplitude equations : Ψ 0 e ˆT Ĥe ˆT Ψ 0 = E Ψ ab.. ij.. e ˆT Ĥe ˆT Ψ 0 = 0 energy as expectation value of similarity transformed Hamiltonian projection trick to derive amplitude equations iterative solution of nonlinear set of equations! state selective (ground state) quite expensive!

28 coupled cluster theory CI based method with inclusion of higher excitations strictly size consistent faster convergence towards FCI highly accurate! approximate methods : CCSD : Coupled Cluster Singles and Doubles CCSD(T):Singles, Doubles + perturbative approximation to Triples CCSDT : Singles, Doubles and Triples...

29 CCD T 2 equations T ab ij v ab ij fi m tmj ab f b teb ij t fa mnvef mn e tij ea tab tef ij t ab mnvij mn mnvef mn + t ea nj vie nb 1 2 tab mit ef nj vef mn tef ij vef ab tmit eb nj fa vef mn

30 typical problem: 50 occ, 400 virt T ab ij amplitudes (3 GB) W(abef) (200 GB) (less if stored in AO-Basis) T abc ijk (1.5 TB) one a<bc block (250 MB) memory requirements 10 GB typical nr. of iterations : typical calculation times : days, weeks (.. months) molecular properties: factor of >2 more expensive

31 parallel CCSD(T) implementation geometry optimization of Ferrocene (staggered and eclipsed), ae-ccsd(t)/cc-pwcvtz, 96 e, 672 Bf., 9 days per step on 14 nodes spin-rotation tensor of HC 7 N, ae-ccsd(t)/cc-pvqz, 470 Bf., 4 days on 7 nodes adamantyl cation (C 10 H + 15 ), ae-ccsd(t)/cc-pvtz, 510 Bf., 5 days per step on 9 nodes M. Harding, T. Metzroth, A. A. Auer and J. Gauss J. Chem. Theory Comput. 4, 64 (2008)

32 hierarchy of post-hf ab-initio methods MP2 / CC2 : non-iterative / iterative N 5 CCSD : iterative NoccN 2 vrt 4 (formal N 6 ) CC3, CCSD(T) : iterative / non-iterative NoccN 3 vrt 4 (formal N 7 ) CCSDT : iterative NoccN 3 vrt 5 (formal N 8 ) CCSDTQ : iterative NoccN 4 vrt 6 (formal N 10 )... T.U. Helgaker, W. Klopper and D.P. Tew Mol. Phys., 106, 2107 (2008)

33 task for future methods

34 tensor decomposition in quantum chemistry related techniques have been used for DFT since 70ies: resolution of the identity / density fitting Laplace-MP2 Cholesky decomposition various applications of SVD techniques

35 tensor decomposition SVD for n-dimensions is not well-defined but: any n-dimensional tensor can be approximated by decomposition find convenient format for decomposition construct trivial decomposition and then apply rank reduction rank reduction : algorithm for finding a low-rank approximation with control of error - used as back box here (one threshold!)

36 tensor decomposition format chosen for application in Coupled Cluster Theory: canonical format X ab ij k l =1 (x a ) l (x b ) l (x i ) l (x j ) l advantages: - most compact form - leads to most general contraction scheme - best reduction of scaling in contractions task: devise CC algorithm in decomposed format

37 notes on decomposition algorithm convert original tensor to canonical format by trivial decomposition find large entry in original tensor to use as new rank optimize rank to improve A-residual iterative procedure to adjust all ranks to minimize à A. if difference is larger ε, pick up more ranks U. Benedikt, A. A. Auer, M. Espig, W. Hackbusch, J. Chem. Phys., accepted

38 tensor decomposition for chemists decomposition of arbitrary tensor: a 1 a 2. a r br T r a 1 ( b 1 b 2... ) 1 + a 2. = A ( b 1 b 2... ) = 2 x 11 x x 21 x

39 tensor decomposition for chemists decomposition of arbitrary tensor: a 1 a 2. ( b 1 b 2... ) 1 + a 1 a 2. ( b 1 b 2... ) =

40 tensor decomposition for chemists decomposition of arbitrary tensor: ( ) = ( ) ( )

41 tensor decomposition for chemists rank reduction from 3 to 2 : 1 2 ( ) + 3 = ( )

42 notes on decomposition algorithm 1: Choose initial à 0 V R d and parameter ε R >0. Define iteration count k := 0, compute the gradient G 0 := J(à 0 ) and D 0 := G 0. 2: while the gradient G k > ε do 3: Compute the smallest root α k [0, 1] of the polynomial p(α) := { } α k := min α R 0 : p(α) := J(à k + αd k ), D k = 0. 4: Update the representation system of Ã, i.e. à k+1 := à k + α k D k. 5: Compute the gradient for the updated system, i.e. G k+1 := J(à k+1 ). 6: Compute β k := Gk+1 G k,g k+1, γ G k 2 k := max{0, β k }. 7: Update the new search direction, i.e. D k+1 := G k+1 + γ k D k. 8: k k : end while J(à k + αd k ), D k, i.e.

43 improved decomposition algorithm

44 coupled cluster in a nutshell calculation of HF yields MO coefficients C µ,p, energies ε i and integrals like µν σρ transform integrals to MO basis ab ij = µνσρ Cµ a Cν b Ci σ C ρ j µν σρ iteratively solve CC amplitude equations by performing contractions and updating residual obtain energy from amplitudes

45 two electron integrals decomposition of two electron integrals: µν σρ r l=1 quantity of 4 #AOs r benefit if rank r is smaller than #AO 3 (χ µ ) l (χ ν ) l (χ σ ) l (χ ρ ) l

46 integral transformation ab ij = Cµ a Cν b Ci σ C ρ j µν σρ µνσρ = = µνσρ r l=1 r l=1 Cµ a Cν b Ci σ C ρ j ( µ r l=1 C a µ (χ µ ) l ) ( ν (χ µ ) l (χ ν ) l (χ σ ) l (χ ρ ) l (v a ) l (v b ) l (v i ) l (v j ) l ) ( Cν b (χ ν ) l C σ σ i (χ σ ) l )( ρ C ρ j (χρ ) l )

47 integral transformation ab ij = Cµ a Cν b Ci σ C ρ j µν σρ µνσρ = r l=1 (v a ) l (v b ) l (v i ) l (v j ) l note : rank stays the same upon transformation rank is the same irrespective of basis used! computational effort for transformation : 4 #MOs #AOs r benefit if rank is smaller than #AO 3

48 perturbation theory E [2] = 1 4 ij ab 2 abij ε i + ε j ε a ε b use integrals in decomposed form factorizable denominator approximation

49 perturbation theory E [2] = 1 4 ij ab 2 abij ε i + ε j ε a ε b use integrals in decomposed form factorizable denominator approximation 1 ε i + ε j ε a ε b = N n=1 N ω n exp( α n (ε i + ε j ε a ε b )) n=1 ω n exp( α n ε i ) exp( α n ε j ) exp(α n ε a ) exp(α n ε b )

50 post HF-methods in spirit of coupled cluster - rewrite MP2 energy in terms of amplitudes : E [2] = 1 4 tij ab ij ab with tij ab = abij ij ab ε i + ε j ε a ε b if integrals and denominator are decomposed, amplitudes will always automatically be obtained in decomposed form t k l =1 (t a ) l (t b ) l (t i ) l (t j ) l

51 post HF-methods E [2] = 1 4 abij tij ab ij ab MP2 energy can be calculated as inner products ( k ) ( ) E [2] 1 4 r (t a ) l (t b ) l (t i ) l (t j ) l (v a ) l (v b ) l (v i ) l (v j ) l abij l=1 = 1 4 k r l =1 l=1 ( a=1 l =1 ) (t a ) l (v a ) l )( (t b ) l (v b ) l )( (t i ) l (v i ) l )( (t j ) l (v j ) l b=1 i=1 j=1

52 coupled cluster theory amplitude equations Ψ ab.. ij.. e ˆT Ĥe ˆT Ψ 0 = 0 example of T 2 equations for CCD: T ab ij v ab ij teb ij t fa fi m tmj ab f b mnvef mn e tij ea tef ij t ab tab mnvef mn mnvij mn + t ea 1 2 tab mit ef nj vie nb nj vef mn express all (!) quantities in decomposed form tef ij vef ab tmit eb nj fa vef mn

53 coupled cluster theory r ab ij tij ef vef ab = all tensors in decomposed representation t ef ij v ab = k r ef = l =1 l=1e=1 f =1 k r l =1 l=1 ( e=1 tij ef vef ab e=1 f =1 [ ] (t e ) l (t f ) l (t i ) l (t j ) l ][(v a ) l (v b ) l (v e ) l (v f ) l ) ( ) (t e ) l (v e ) l (t f ) l (v f ) l (v a ) l (v b ) l (t i ) l (t j ) l f =1 complexity is reduced from occ 2 virt 4 to k r N benefit if product of ranks is smaller than N 5

54 benchmarks and results decomposition of AO integrals H 2 O - different basis size (STO-3G, 6-31G, cc-pvdz, aug-cc-pvdz, cc-pvtz) initial rank eps = 10-2 eps = 10-4 eps = 10-6 slice = 250 slice = rank number of basis functions (N) scaling of rank with basis functions: n 2.5

55 benchmarks and results decomposition of AO integrals initial ranks final ranks (10-2 ) final ranks (10-4 ) final ranks (10-6 ) LiH-chain (6-31G) rank (LiH) n scaling of rank with system size: N 2

56 benchmarks and results decomposition of MO integrals scaling of v abcd -ranks in C n H 2n 6-31G scaling of v ijkl -ranks in C n H 2n 6-31G initial (n 2.7 ) eps=10-2 (n 1.3 ) eps=10-4 (n 2.1 ) eps=10-6 (n x ) initial (n 2.8 ) eps=10-2 (n 1.8 ) eps=10-4 (n 2.0 ) eps=10-6 (n 2.3 ) rank rank n n

57 benchmarks and results decomposition MP2 amplitudes 2000 initial eps=10-2 eps=10-4 eps=10-6 t(abij) in H 2 O using different basis size 1500 rank number of basis functions (N) scaling of rank with basis functions: n 1.2

58 benchmarks and results decomposition MP2 amplitudes t(abij) in (LiH) n 6-31G 2000 initial eps=10-2 eps=10-4 eps= rank (LiH) n scaling of rank with system size: N 1.2

59 benchmarks and results error in MP2 energy -1.0e e-01 E MP2 for different eps in C n H 2n 6-31G CH 4 C 2 H 6 C 3 H 8 C 4 H e-02 E MP2 [Hartree] -1.0e e e e e eps mh accuracy at thresholds around

60 CC algorithm - first tests test-lccd (not all terms yet...) only use decomposed integrals and amplitudes rank reduction after every contraction apply same threshold for all quantities yet no DIIS error in correlation energy for H 2 O using 6-31G correlation energy ε = 10-2 ε = error [Hartree] energy [Hartree] iteration

61 how compact is decomposed representation? rank does not change upon transformation no guarantee to find lowest possible rank does algorithms know about sparsity? Test : dissociate O 2 and investigate ranks MP2 energy in O 2 using 6 31G ε=10 4 interaction energy energy error rank E or E [µhartree] rank O O distance [Å]

62 improvement for CCSD(T) expected? Terms like : T abc ijk = ˆP can be cast into the form: ( ) R L (t m ) r (v m ) l r=1 m=1 l=1 Tim ab Vjk cm m + e Tij ae Vek bc (t a ) r (t b ) r (v c ) l (t i ) r (v j ) l (v k ) l current estimate : R N 1.5, L N 2.0 complexity : R L orb (with current estimate N 4.5 )

63 summary - tensor decomposition integral storage and transformation scales as r N coupled cluster iterations scale as r ampl. r int. N actual ranks scale as N 1.5 (ampl.) and N 2.0 (MO int.) decomposition scales with initial rank and quickly becomes main bottleneck! next steps: CCSD, CCSD(T) (N 5?) and CCSDT more efficient decomposition algorithm (trivial!) parallelization molecular properties

64 acknowledgments Udo Benedikt (MPIE) DFG : SPP 1145 AU 206/2-1 Eric Prochnow (TUC) AU 206/1-1 Prof. J. Gauss (Uni Mainz) BMBF M. Harding, T. Metzroth Fonds der Chemischen Industrie Prof. K. Banert (TUC) Prof. W. Hackbusch (MPI-MIS) M. Espig TU Chemnitz MPIE CES (RUB) M. Nooijen (Uni Waterloo) S. Hirata (QTP) TCE Group : D. Bernholt, R. Pitzer, P. Saddayappan, G. Baumgartner...

65 thanks for your attention!!