Coupled cluster theory: Fundamentals Wim Klopper and David P. Tew Lehrstuhl für Theoretische Chemie Institut für Physikalische Chemie Universität Karlsruhe (TH) C4 Tutorial, Zürich, 2–4 October 2006 Outline The coupled cluster wave functions • The coupled cluster Schrödinger equation • Evaluating the CCSD energy • Optimizing the coupled cluster wave function • The diagnostic • T1 Size consistency • Coupled cluster perturbation theory • The coupled cluster Lagrangian • The 2n + 1 rule • The CC2 and CC3 hybrid models • The CCSD(T) noniterative method • Coulomb correlation We take Hartree-Fock as the zeroth order reference. • This takes care of Fermi correlation. • The correlation energy is then defined as the remaining • deviation from the exact energy. E = EHF + Ecorr When the HF reference is a good approximation to the • wave function, coupled-cluster methods accurately describe the correction due to dynamic Coulomb correlation. We use the language of second quantization to discuss • coupled-cluster methods. Pair Clusters The wave function for the motion of non-interacting • fermions is described by a slater determinant of occupied spin-orbitals I = 1, N. N HF = aI† vac | ! ! # | ! I"=1 Allowing for instantaneous Coulomb repulsion, the motion • of two electrons in spin orbitals I and J is disturbed AB a† a† a† a† + t a† a† . I J → I J IJ A B A>B$ The disturbance is represented by excitations into virtual • AB spin orbitals A and B with probabilities tIJ . This is known as a pair cluster or two-electron cluster. • Let us introduce an operator that describes this correlation • process AB τˆIJ = aA† aI aB† aJ We may write the pair cluster IJ as • AB AB AB (1 + tIJ τˆIJ ) aI† aJ† vac = aI† aJ† vac + tIJ aA† aB† vac . ! # | ! | ! | ! A>B" A>B$ Allowing each pair of electrons to interact, we arrive at a • coupled cluster wave function CCD = (1 + tABτˆAB) HF . | ! IJ IJ | ! A>B",I>J Note that all τˆAB commute with each other. • IJ General coupled clusters Pair clusters account for the (dominant) two-electron • interactions. The simultaneous interaction of three or more electrons • also occurs, and in addition each orbital relaxes. We introduce a generalized excitation operator • ABC... τˆµ = τˆIJK... = aA† aI aB† aJ aC† aK . The generalized coupled cluster wave function is • CC = (1 + t τˆ ) HF . | ! µ µ | ! ! µ # " Note the product form, in contrast to the CI wave function. • CI = 1 + C τˆ HF . | ! µ µ | ! % µ & $ The exponential ansatz Since the spin-orbital excitation operators satisfy • 2 τˆµ = 0, then • (1 + tµτˆµ) = exp(tµτˆµ). and • CC = (1 + t τˆ ) HF = exp(Tˆ) HF , | ! µ µ | ! | ! ! µ # " with • Tˆ = tµτˆµ. µ $ The order of the operators does not matter, [τˆ , τˆ ] = 0 • µ ν This exponential formulation affords significant • simplifications later on. The hierarchy of excitation levels We may classify the excitation operators according to the • level of excitation Tˆ = Tˆ + Tˆ + + Tˆ 1 2 · · · N where Tˆ contains excitations involving one electron only • 1 ˆ A A A T1 = tI aA† aI = tI τˆI $AI $AI and Tˆ contains excitations involving two electrons only etc. • 2 ˆ AB T2 = tIJ aA† aI aB† aJ A>B I$>J 1 AB 1 AB AB = 4 tIJ aA† aI aB† aJ = 4 tIJ τˆIJ AI$BJ AI$BJ The truncation of the excitation operator Tˆ defines the • coupled cluster method Tˆ = Tˆ + Tˆ + + Tˆ 1 2 · · · N CCS Tˆ1 CCSD Tˆ1 + Tˆ2 CCSDT Tˆ1 + Tˆ2 + Tˆ3 CCSDTQ Tˆ1 + Tˆ2 + Tˆ3 + Tˆ4 . Excitations Tˆ are able to account for the simultaneous • N interaction of N electrons. This series rapidly converges if the HF wave function is a • good zeroth order reference. Coupled cluster convergence towards full CI The difference between the cc-pVDZ energies of H O from • 2 the full CI limit of various coupled-cluster methods (Eh). o The HOH angle is 110.565 and Rref = 1.84345 a0. method R = Rref R = 2Rref RHF 0.217822 0.363954 CCSD 0.003744 0.022032 CCSDT 0.000493 -0.001405 CCSDTQ 0.000019 -0.000446 CCSDTQ5 0.000003 The series converges very fast at equilibrium and slower at • stretched geometry where there a higher multi-reference character. Connected and disconnected clusters Expanding CC , we obtain • | ! exp(Tˆ) HF = 1 + t τˆ + t t τˆ τˆ + HF | ! µ µ µ ν µ ν · · · | ! % µ µ>ν & $ $ = HF + t µ + t t + µν . | ! µ| ! µ ν · · · | ! µ µ>ν $ $ Note that each excitation τˆ occurs only once in Tˆ. • µ Collecting determinants we see that a given determinant is • generated in several ways Cˆ0 = 1 Cˆ = Tˆ ˆ ˆ 1 1 exp(T ) HF = Ci HF 1 2 | ! | ! Cˆ2 = Tˆ2 + Tˆ i 2 1 $ ˆ ˆ ˆ ˆ 1 ˆ3 C3 = T3 + T1T2 + 6 T1 E.g., the determinant with orbitals IJ replaced by AB • (tAB)total AB = tABτˆAB HF + tAtBτˆAτˆB HF tBtAτˆBτˆA HF IJ |IJ ! IJ IJ | ! I J I J | ! − I J I J | ! tAtB and tBtA are tAB are connected • I J I J • IJ disconnected cluster cluster amplitudes amplitudes The maximum excitation in Tˆ determines the maximum • connected amplitude. All possible excitations are included in CC , the • | ! amplitudes of all higher excitations are determined by products of connected amplitudes. Whereas each determinant is parameterized in CI theory, • in CC theory the excitation process is parameterized. We shall see that this leads to size consistency. The coupled cluster Schrödinger equation We wish to solve Hˆ CC = E CC . • | ! | ! What happens if we apply the variational principle? • CC Hˆ CC Emin = min $ | | ! tµ CC CC $ | ! CC depends on t in a non-linear way: • | ! µ ∂ CC = (1 + t τˆ ) µ ∂t | ! ν ν | ! µ ! ν # " The variational condition results in an intractable set of • nonlinear equations for the amplitudes. µ (1 + t τˆ†) Hˆ CC = E µ (1 + t τˆ†) CC $ | ν ν | ! min$ | ν ν | ! ! ν # ! ν # " " The projected coupled cluster equations In a given orbital basis, the full coupled cluster wave • function that includes all M 1 excitations satisfies N − Hˆ exp(Tˆ) HF+ =,E exp(Tˆ) HF | ! | ! We may set up M equations to find the energy and • N amplitudes by left projecting by each determinant. + , HF Hˆ exp(Tˆ) HF = E $ | | ! µ Hˆ exp(Tˆ) HF = E µ exp(Tˆ) HF $ | | ! $ | | ! The are the projected coupled cluster equations and must • be solved self consistently. For truncated cluster operator Tˆ the Schrödinger • N equation is never satisfied. Hˆ exp(Tˆ ) HF = exp(Tˆ!) HF = E exp(Tˆ ) HF N | ! | ! % N N | ! For a cluster operator that includes n-tuple excitations we • project the Schrödinger equation onto the manifold of all n-tuple excited determinants. HF Hˆ exp(Tˆ) HF = E $ | | ! µ Hˆ exp(Tˆ) HF = E µ exp(Tˆ) HF $ n| | ! $ n| | ! Solving self consistently yields approximate amplitudes • and an approximate energy, which may be above or below the exact energy. The error in the amplitudes is due to the missing • interactions with the excitations absent from TˆN . The similarity transformed Hamiltonian Since HF exp( Tˆ) = HF , we may left multiply by • $ | − $ | exp( Tˆ). − Hˆ exp(Tˆ) HF = E exp(Tˆ) HF | ! | ! exp( Tˆ)Hˆ exp(Tˆ) HF = E HF − | ! | ! The projected energy and amplitude equations are then • decoupled. HF exp( Tˆ)Hˆ exp(Tˆ) HF = E $ | − | ! µ exp( Tˆ)Hˆ exp(Tˆ) HF = 0 $ | − | ! We define the similarity transformed Hamiltonian: • Hˆ T = exp( Tˆ)Hˆ exp(Tˆ) − Hˆ T may been seen as an effective Hamiltonian and is • non-Hermitian. The BCH expansion The similarity transformed Hamiltonian may be simplified • through the Baker-Campbell-Hausdorff (BCH) expansion. We Taylor expand Hˆ λT around λ = 0 • Hˆ λT = exp( λˆT )Hˆ exp(λˆT ) − ˆ λT 2 ˆ λT ˆ λT dH 1 2 d H = H + λ + 2! λ + λ=0 dλ λ=0 dλ2 λ=0 · · · - - - dHˆ λT - - - = exp( - λˆT )[Hˆ , Tˆ] exp(- λˆT ) - dλ − d2Hˆ λT = exp( λˆT )[[Hˆ , Tˆ], Tˆ] exp(λˆT ) dλ2 − Setting λ = 1 we obtain the BCH expansion • Hˆ T = Hˆ + [Hˆ , Tˆ] + 1 [[Hˆ , Tˆ], Tˆ] + 1 [[[Hˆ , Tˆ], Tˆ], Tˆ] + 2! 3! · · · In fact, the BCH expansion of Hˆ T terminates after only four • nested commutators. ˆ T ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ H = H + [H, T ] + 2! [[H, T ], T ] + 3! [[[H, T ], T ], T ] 1 ˆ ˆ ˆ ˆ ˆ + 4! [[[[H, T ], T ], T ], T ] We recall that a commutator [A,ˆ Bˆ] has a rank one less • than the operator AˆBˆ, reducing the rank of the Aˆ and Bˆ contributions by one half. e.g. [a† a , a† a ] = δ a† a δ a† a P Q A I QA P I − P I A Q Since the Hamiltonian is a rank 2 operator, four successive • commutations removes the contribution from Hˆ leaving only the excitation operator contributions, which always commute, [aA† aI , aB† aJ ] = 0. The coupled cluster energy Applying BCH to the projected energy expression, • E = HF Hˆ HF + HF [Hˆ , Tˆ] HF + 1 HF [[Hˆ , Tˆ], Tˆ] HF $ | | ! $ | | ! 2! $ | | ! + 1 HF [[[Hˆ , Tˆ], Tˆ], Tˆ] HF + 1 HF [[[[Hˆ , Tˆ], Tˆ], Tˆ], Tˆ] HF 3! $ | | ! 4! $ | | ! This simplifies considerably, since HF Tˆ = 0 • $ | and since Hˆ is a two particle operator, it can de-excite a • maximum of 2 electrons: HF Hˆ Tˆ HF = 0 for i > 2 $ | i| ! also HF Hˆ Tˆ HF = 0 due to the Brillouin theorem. • $ | 1| ! The projected energy expression is simply • E = E + HF [Hˆ , Tˆ ] HF + 1 HF [[Hˆ , Tˆ ], Tˆ ] HF HF $ | 2 | ! 2 $ | 1 1 | ! Only singles and doubles amplitudes contribute directly to • the energy, but the singles and doubles amplitudes depend on all the other amplitudes. The coupled cluster amplitude equations Applying the BCH expansion to the projected amplitude • equations 0 = µ exp( Tˆ)Hˆ exp(Tˆ) HF $ | − | ! = µ Hˆ HF + µ [Hˆ , Tˆ] HF + 1 µ [[Hˆ , Tˆ], Tˆ] HF $ | | ! $ | | ! 2! $ | | ! + 1 µ [[[Hˆ , Tˆ], Tˆ], Tˆ] HF + 1 µ [[[[Hˆ , Tˆ], Tˆ], Tˆ], Tˆ] HF 3! $ | | ! 4! $ | | ! These form coupled a set of nonlinear equations, at most • quartic in the amplitudes.