Coupled-Cluster Perturbative Triples for Bond Breaking

Transcription

1 Coupled-Cluster Perturbative Triples for Bond Breaking Andrew G. Taube and Rodney J. Bartlett Quantum Theory Project University of Florida INT CC Meeting Seattle July 8, 2008

2 Why does chemistry need triples? Method/ Molecule Hartree-Fock H2O N2 HF (99.7%) (99.7%) (99.9%) Total energies in hartree CCSD (99.995%) (99.987%) (99.998%) FCI hartree = 27.2 ev Scale (for these molecules) of total energies: 10 2 hartree = 10 0 kev Equilibrium structures. H2O in cc-pvdz, N2 in cc-pvdz, HF in DZ

3 Atomization Energies 16 Small Molecules, cc-pcvqz Bak, et. al., JCP 112, 9229 (2000) H2O 2H + O Scale of AE ~ kj/mol ~ 10 0 ev ev ;<=>5&:??@A??@A5>: 1 ev = 96.5 kj/mol Errors in AE ~ kj/mol ~ 10 1 mev mev * *!!""!#"!$"!%"!&" " &" %" $" #"!"" '(()(*+,*-.)/+01.+),*',2(34*5678/)9:

4 19 Rxns of those 16 small molecules, Ibid. Reactions <= BBCD BBCD6A; * Scale of ΔErxn ~ kj/mol ~ 10 2 mev ev Errors in ΔErxn ~ kj/mol ~ 10 0 mev mev *!!""!#"!$"!%"!&" " &" %" $" #"!"" '(()(*+,*-./01+),*',12/34+.5*6789:)3;

5 19 Rxns of those 16 small molecules, Ibid. Reactions <= BBCD BBCD6A; * Scale of ΔErxn ~ kj/mol ~ 10 2 mev ev Errors in ΔErxn ~ kj/mol ~ 10 0 mev mev Total Energy ~ 1 kev want errors in ΔErxn ~ 1 mev *!!""!#"!$"!%"!&" " &" %" $" #"!"" '(()(*+,*-./01+),*',12/34+.5*6789:)3;

6 What can we (reasonably) expect from high-level quantum chemistry Identification of and calculation of properties at stationary points of moderately-sized molecules Parameterization / reference for less accurate methods for larger molecules Driver for Molecular Dynamics / (Classical) Monte Carlo Simulations for small molecules

7 What can we (reasonably) expect from high-level quantum chemistry Identification of and calculation of properties at stationary points of moderately-sized molecules Parameterization / reference for less accurate methods for larger molecules Driver for Molecular Dynamics / (Classical) Monte Carlo Simulations for small molecules For the latter two, need data away from stationary points For MC: Energies For MD: Forces

8 Computational Scaling CCSD T=T1+T2 CCSD(T) T=T1+T2+T3 [2] CCSDT T=T1+T2+T3 O(N 6 ) ( o 3 v 3 + o 2 v 4) N it O(N 7 ) ( o 3 v 3 + o 2 v 4) N it + o 3 v 4 + o 4 v 3 O(N 8 ) ( o 4 v 4 + o 3 v 5) N it

9 The Gold Standard E CCSD = E ref + 0 F ov T W ( T ) 2 T E CCSD(T) = E CCSD + 0 ( ) T 1 + T 2 (F ov + W ) R 3 (W T 2 ) C 0 R 3 = ( ) 2 1 3! ijk abc abc ijk abc ijk ɛ abc ijk [2] = [2] [2] [2]

10 CCSD(T)

11 CCSD(T)

12 CCSD(T)

13 There are two different ways that CCSD(T) fails for bond breaking

14 Bond Length (n*re) There are two different ways that CCSD(T) fails for bond breaking Collapse of the spin-restricted (RHF) wavefunction RHF CCSD RHF CCSD(T) DMRG Full CI 376.5!E (kcal/mol)

15 There are two different ways that CCSD(T) fails for bond breaking Collapse of the spin-restricted (RHF) wavefunction Spin symmetry breaking RHF CCSD(T) UHF CCSD(T) Absolute Error from Full CI (kcal/mol) Bond Length (n*re)

16 F2 at 2Re Denominatorless T3 amplitudes δe t abc ijk RHF λ ijk abc [2] ɛ abc ijk δe t abc ijk UHF λ ijk abc ɛ abc ijk The T3 amplitudes themselves are bad - not denominators ( E CCSD = E ref + 0 F ov T W T ) 2 T O(T 2 ) ( ) E CCSD(T) = E CCSD + 0 T 1 + T 2 (F ov + W ) R 3 (W T 2 ) C 0 O(T2 2 ) If the CCSD wavefunction is bad then so is the CCSD(T) energy

17 Ways to fix this problem

18 Ways to fix this problem Symmetry-broken Reference Doesn t fully fix the problem

19 Ways to fix this problem Symmetry-broken Reference Multireference Techniques Not Black-box, Extensivity, Computational Cost

20 Ways to fix this problem Symmetry-broken Reference Multireference Techniques Higher excitations Computational Cost

21 Ways to fix this problem Symmetry-broken Reference Multireference Techniques Higher excitations Use CI Extensivity, Accuracy

22 Ways to fix this problem Symmetry-broken Reference Multireference Techniques Higher excitations Use CI Improved Functionals Our Choice

23 Properties of CCSD(T) that are important to keep Extensivity Invariance to OO and VV rotations Availability of analytical gradients O[(o 3 v 3 + o 2 v 4 )N it + o 3 v 4 + o 4 v 3 ] Applicability to arbitrary reference functions

24 Löwdin Partitioning H P P c P + H P Q c Q H QP c P + H QQ c Q = Ec P = Ec Q c Q = ˆQ ( E H QQ ) 1 ˆQ HQP c P H P P + H P Q ˆQ ( E HQQ ) 1 ˆQ HQP c P = Ec P H is a similarity transformation of the Hamiltonian same eigenvalue spectrum

25 Perturbative Expansion E = 0 L P HP P R P L P HP Q ˆQ ( E HQQ ) 1 ˆQ HQP R P 0 Want corrections to CCSD Expand perturbatively around CCSD solution H [0] P P = ˆP H CCSD ˆP E [0] = E CCSD H[0] P Q L [0] P = (1 + Λ CCSD) R [0] P = 1 = H [0] QP = 0 CCSD(T) Choice ) L [0] P (1 + T CCSD

26 Partitioning Choices H [0] Λ(T) = ˆP H CCSD ˆP + ˆQ H [0] (2) T = ˆP H CCSD ˆP + ˆQ ( E CCSD + H (0)) ˆQ ( E CCSD + H (0)) ˆQ H [0] CR(2,3) = ˆP H CCSD ˆP + ˆQ ( ECCSD + H CCSD ) ˆQ H [1] Λ(T) = H (1) CCSD H [1] (2) T = H CCSD H [1] CR(2,3) = H CCSD H [0] Λ(T) H [0] (2) T H [0] CR(2,3) H (0) = F oo + F vv H (1) = F ov + F vo + W

27 (T)-Like Functionals ( ) 2 E [3] 1 CCSD(T) = 3! ijk E [3] ΛCCSD(T) = ( 1 3! E [2] CCSD(2) T = E [2] CR(2,3) = ( 1 3! ( 1 3! abc ) 2 ijk abc ) 2 ijk abc ) 2 ijk abc 0 T (F ov + W ) abc ijk 0 Λ (F ov + W ) abc ijk 0 Λ H SD abc ijk 0 Λ H SD abc ijk 1 ɛ abc ijk 1 ɛ abc ijk 1 ɛ abc ijk 1 ɛ abc ijk abc ijk (W T 2 ) C 0 abc ijk (W T 2 ) C 0 abc ijk H SD 0 abc ijk H SD 0 Formally should have quadruples CCSD(T): Many CCSD(2)T: Hirata ΛCCSD(T): Bartlett CR(2,3) : Piecuch

28 Non-iterative O(N 7 ) Triples ΛCCSD(T) is the minimal extension to CCSD(T), yet... ΛCCSD(T) CCSD(T) CCSD(2)T CR(2,3) Improved RHF bond breaking Orbital Invariance Non-HF Reference Analytical Gradients * CR-CCSD(T) is not extensive

29 ΛCCSD(T) and CCSD(T) E ΛCCSD(T) = E CCSD + 0 (Λ 1 + Λ 2 ) (F ov + W ) R 3 (W T 2 ) C 0 [2] [2] [2] E CCSD(T) = E CCSD + 0 ( ) T 1 + T 2 (F ov + W ) R 3 (W T 2 ) C 0 [2] [2] [2]

30 Stationary Formulation of ΛCCSD(T) E ΛCCSD(T) = 0 (1 + Λ + Π) H (1 + Λ) ( HΣ )C Λ 1 W T [2] Λ 2 (F ov + W ) T [2] Λ [2] 3 (F oo + F vv ) T [2] Λ [2] 3 (W T 2) C 0 Allows for (Generalized) Hellman-Feynman Theorem to be used for gradients

31 Stationary Formulation of ΛCCSD(T) E ΛCCSD(T) = 0 (1 + Λ + Π) H (1 + Λ) ( HΣ )C 0 Combination of the Stat. Functional for CCSD(T) and EOM-CCSD + 0 Λ 1 W T [2] Λ 2 (F ov + W ) T [2] Λ [2] 3 (F oo + F vv ) T [2] Λ [2] 3 (W T 2) C 0 Allows for (Generalized) Hellman-Feynman Theorem to be used for gradients

32 Reaction Energies CCSD(T)!CCSD(T) Error in Reaction Enthalpy (kcal/mol) ~40 well-characterized reactions from Lynch, Zhao, Truhlar data set

33 Lithium Fluoride RHF CCSD RHF CCSD(T) RHF "CCSD(T) DMRG Full CI!E (kcal/mol) double-ζ Bond Length (n*re) FCI: Bauschlicher, Langoff, JCP 89, 4246 (1988). DMRG: Legeza, Roder, Hess, Mol. Phys. 101, 2019 (2003).

34 Carbon Monoxide RHF CCSD RHF CCSD(T) RHF "CCSD(T) cc-pvtz 284.7!E (kcal/mol) De ~ 8 kcal/mol too high Bond Length (n*re)

35 Carbon Monoxide RHF CCSD RHF CCSD(T) RHF "CCSD(T) cc-pvtz 284.7!E (kcal/mol) De ~ 8 kcal/mol too high Bond Length (n*re)

36 H2O RHF 0 cc-pvdz Error from Full CI (kcal/mol) RHF CCSD(T) RHF!CCSD(T) RHF CR-CCSD(T) RHF CCSD(2)T RHF CR-CCSD(T)L Bond Length (n*re) FCI: Olsen, et. al., JCP 104, 8007 (1996). CR-CCSD(T)/CCSD(2)T: Hirata, et. al., JCP 121, (2004 ). CR-CCSD(T) L: Włoch, et. al., MP 104, 2149 (2006)

37 Average Non-parallelity Error CCSD CCSD(T)!CCSD(T) Weighted Non-Parallelity Error (millihartree) HF: Dutta, Sherrill, JCP 118, 1610 (2003). H2O: Olsen, et al., JCP 104, 8007 (1996). C2: Sherrill, Piecuch, JCP 122, (2005). N2: Chan, et al., JCP 121, 6110 (2004).

38 HF Force Curve RHF CCSD(T) RHF!CCSD(T) 6-31G** Force (E h /a 0 ) Bond Length (n*r e )

39 FCI: Olsen, et. al., JCP 104, 8007 (1996). CR-CCSD(T)/CCSD(2)T: Hirata, et. al., JCP 121, (2004 ). H2O UHF 7.53 cc-pvdz UHF CCSD(T) UHF!CCSD(T) UHF CCSD(2)T Error from Full CI (kcal/mol) Bond Length (n*re)

40 [2] Denominatorless T3 amplitudes δe t abc ijk UHF λ ijk abc T3 amplitudes not source of the UHF problem ɛ abc ijk Error from RHF CCSDT (kcal/mol) F2 at 2Re RHF CCSD(T) RHF!CCSD(T) UHF CCSD(T) UHF!CCSD(T) Bond Length (n*r e )

41 RHF: The ΛCCSD(T) Story So Far Improvements for intermediate bond lengths Asymptotics not quantitative UHF: No Improvements for intermediate bond lengths Asymptotics still ok ΛCCSD(T) fixes spin-restricted PT failure Not spin-symmetry breaking problem AGT, RJB. JCP 128, & (2008)

42 Can spin-restricted refs be good longer? Yes... Brueckner Orbitals Best single-determinant approximation to the exact wavefunction 0 B min 0 0 Ψ 2 equivalent to: 0 B max 0 0 Ψ * For normalized wfns and determinants Brueckner orbitals should maintain symmetry better than Hartree-Fock orbitals

43 CC Condition What about the case using a CC wfn? Intermediate normalization: Ω = e κ 1 0 Ψ Ψ 1 κ 1 = ia κ a i ( {a i} {i a} ) Define f(κ 1 ) = Ω e κ 1 Ψ Ψ Ψ 1/2 0 B max κ 1 f(κ 1 )

44 CC Condition, continued f(κ 1 ) κ a i Expanding κ1 =0 = Ω ( i a a i ) e κ 1 Ψ Ψ Ψ 1/2 = Ωa i Ψ Ψ Ψ 1/2 = 0 κ1 =0 Ψ = e T Ω = Ω + ia t a i Ω a i +

45 CC Condition, continued f(κ 1 ) κ a i Expanding κ1 =0 = Ω ( i a a i ) e κ 1 Ψ Ψ Ψ 1/2 = Ωa i Ψ Ψ Ψ 1/2 = 0 κ1 =0 Ψ = e T Ω = Ω + ia t a i Ω a i + t a i = 0

46 Brueckner Instability Condition 2 f(κ 1 ) κ a i κb j κ1 =0 = Ω ( i a a i ) ( j b b j ) e κ 1 Ψ Ψ Ψ 1/2 Ω ab ij = Ψ δ ij δ ab Ω Ψ Ψ Ψ 1/2 = Ψ Ψ 1/2 ( t ab ) ij δ ij δ ab κ1 =0 Want this to be negative definite (maximum) Diagonalize A = I T 2 Any negative eigenvalues are instabilities

47 Onset of Instability for BCCD Molecule N2 F2 C2 HF RHF UHF 1.1Re Re 0.5Re 1.5Re RB UB 1.9Re >5Re 2.1Re >5Re

48 Computational Procedure Brueckner Instability Analysis Brueckner DIIS (Padé Approximants for matrix exponentials and logarithms) Regularization of CC equations Initial guess using prior converged brueckner result

49 An interlude on solving the CC equations [n] t ab ij = 1 ɛ ab ij F ( [n 1] t, [n 1] t 2, [n 1] t 3, [n 1] t 4) Note the possibility of a (near-)singularity due to the denominator [n] t ab ij = ( ɛ ab ij + ɛ ab ij ) 2 + δ 2 F ( [n 1] t, [n 1] t 2, [n 1] t 3, [n 1] t 4) ( ɛ ab ij δ 2 ) 2 + δ 2 [n 1] t ab ij Based on an approximate Tikhonov regularization. Will converge to the same result as unregularized methods

50 t ab ij = At convergence of CC equations... [ (ɛ ab ij ɛ ab ij ( ( ) ɛ ab F t, t 2, t 3, t 4) δ 2 ij ( ɛ ab ij δ 2 ) 2 + δ 2 tab ) ] 2 + δ 2 t ab ij = ɛ ab ( ij F t, t 2, t 3, t 4) + δ 2 t ab ij ( ) ɛ ab 2 ij t ab ij = ɛ ab ( ij F t, t 2, t 3, t 4) t ab ij = 1 ɛ ab ij F ( t, t 2, t 3, t 4) ij

51 Brueckner Functional E BCC = 0 (1 + Λ) [ (He κ 1 ) e T ] C C 0 T = n=2 T n κ 1 = κ 1 1 Λ = n=1 Λ n BΛCCSD(T) same as ΛCCSD(T) with modified Λ Singles are needed for Λ, not for T equations Reformulation of Handy, Pople, Head-Gordon, Raghavachari, Trucks. CPL 164, 185 (1989)

52 HF RHF / UHF / RB G** Error from Full CI (kcal/mol) RHF CCSD(T) RHF!CCSD(T) RB CCD(T) RB!CCSD(T) UHF CCSD(T) UHF!CCSD(T) Bond Length (n*re) FCI: Dutta, Sherrill, JCP 118, 1610 (2003).

53 H2O UHF / UB Error from Full CI (kcal/mol) cc-pvdz UHF CCSD(T) UHF!CCSD(T) UB CCD(T) UB!CCSD(T) Bond Length (n*re) FCI:Olsen, et. al., JCP 104, 8007 (1996).

54 F2 RHF / UHF / RB 5 cc-pvdz Error from RHF CCSDT (kcal/mol) RHF CCSD(T) RHF!CCSD(T) RB CCD(T) RB!CCSD(T) UHF CCSD(T) UHF!CCSD(T) Bond Length (n*r e ) CCSDT: Kowalski, Piecuch, CPL 344, 165 (2001).

55 N2 UHF / UB 15 cc-pvdz 10 Error from FCI (kcal/mol) UHF CCSD(T) UHF!CCSD(T) UB CCD(T) UB!CCSD(T) Bond Length (n*r e ) FCI: Chan, et al., JCP 121, 6110 (2004).

56 Conclusions about BΛCCSD(T) Turned Spin-Symmetry Breaking (T) Problem into Spin-Restricted (T) Problem Allows BΛCCSD(T) PES to be Substantially improved over HF CCSD(T) or BCCD(T) Brueckner orbitals improve perturbative approximations; no evidence this is true for iterative CC methods

57 Acknowledgments Symposium Organizers and Staff Prof. Rod Bartlett and the Bartlett Group All of you Funding: National Defense Science and Engineering Fellowship AFOSR, ARL