Coupled-Cluster Theory Lecture Series in Electronic Structure Theory, Summer 2010 K. Sahan Thanthiriwatte, July 8, 2010 cc theory – p. 1/41 References Crawford, T. D. and Schaefer, H. F. “An Introduction to Coupled Cluster Theory for Computational Chemists”, In Reviews of Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: New York, 2000; Vol. 14, Chapter 2, pp 33-136, Shavitt, I and Bartlett, R. J. “Many-body Methods in Chemistry and Physics”, Cambridge University Press, Cambridge, 2009. Bartlett, R. J. and Stanton, J. F. “Applications of Post-Hartree-Fock Methods: A Tutorial”, In Reviews of Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: New York, 1994; Vol. 5, Chapter 2, pp 65-169, Bartlett, R. J. Coupled-Cluster Theory: An Overview of Recent Developments. In Modern Electronic Structure Theory, Part II; Yarkony, D. R., Ed.; World Scientific Publishing Co.: Singapore, 1995; Chapter 16, pp 1047-1131. cc theory – p. 2/41 Introduction The purpose of all many-body methods is to describe electron correlation. Löwdin definition of correlation energy ∆E = Eexact − EHF . (1) Since the variationally optimized Hartree-Fock energy is an upper bound to the exact energy, the correlation energy must be a negative value. There are three main methods for calculating electron correlation: Configuration Interaction (CI) Many-Body Perturbation Theory (MBPT) or (MPPT) Coupled-Cluster Theory (CC) HF theory recovers 99% of overall energy. But, lots of important chemistry happen in the remaining 1%. cc theory – p. 3/41 Conventions and Notation Level Symbol Name Alternative 1 S singles mono-substitutions 2 D doubles di–substitutions 3 T triples tri–substitutions 4 Q quadraples tetra–substitutions 5 P/5 pentuples penta–substitutions 6 H/6 hextupels hexa–substitutions i,j,k,... occupied orbitals or holes. a,b,c,... virtual, unoccupied orbitals or particles. p,q,r,... generic orbitals ˆi, a,ˆ F,ˆ H,...ˆ or H , F , V , W ,... or Λ, Ω,... φ or ϕ Φ0 or Ψ0 or |0i cc theory – p. 4/41 Extensivity and Consistencivity Size-extensive The energy of a non-interacting system computed with this model scales correctly with the size of the system, which satisfy E(NHe)= NE(He) . Size-consistency The energies of two systems A and B and of the combined system AB with A and B very far apart, computed in equivalent ways, satisfy E(AB)= E(A)+ E(B) . cc theory – p. 5/41 The Exponential Ansatz The essential idea in CC theory is the ground state wave function |Ψ0i can be given by the exponential ansatz Tˆ |ΨCC i = e |Φ0i, (2) Tˆ2 Tˆ3 = 1+ Tˆ + + + .... |Φ0i. (3) 2! 3! ! Tˆ = Tˆ1 + Tˆ2 + Tˆ3 + ..., (4) ˆ a a T1|Φ0i = ti Φi , (5) i,a X ˆ ab ab T2|Φ0i = tij Φij , (6) i>j a>bX ˆ2 ab cd abcd T2 |Φ0i = tij tkl Φijkl (7) i>j k>l a>bX c>dX cc theory – p. 6/41 Basic Considerations in CC Theory The non-relativistic time-independent Schrödinger equation Hˆ |Ψi = E|Ψi, (8) and can be written as Tˆ Tˆ Heˆ |Φ0i = Ee |Φ0i, −Tˆ Tˆ −Tˆ Tˆ e Heˆ |Φ0i = Ee e |Φ0i −Tˆ Tˆ e Heˆ |Φ0i = E|Φ0i. (9) The energy and amplitude expressions can be obtained from Eq. 9 left-multiplying by the reference and an excited state determinant, respectively, and integrating over all space −Tˆ ˆ Tˆ hΦ0|e He |Φ0i = E, (10) ab... −Tˆ ˆ Tˆ hΦij... |e He |Φ0i = 0. (11) cc theory – p. 7/41 CCD Equations If we consider, Tˆ2 ΨCCD = e |Φ0i (12) and substitute to Hˆ Ψ= EΨ, we can get Tˆ2 Tˆ2 Heˆ |Φ0i = ECCDe |Φ0i (13) Tˆ2 hΦ0|Hˆ − ECCD|e Φ0i =0 (14) 1 2 1 3 hΦ0|Hˆ − ECCD| 1+ Tˆ2 + Tˆ2 + Tˆ2 + .... Φ0i =0 (15) 2! 3! ECCD = hΦ0|Hˆ |Φ0i + hΦ0|Hˆ Tˆ2|Φ0i, (16) ˆ ab ab = EHF + hΦ0|H|Φij itij , (17) i>j a>bX ab = EHF + hij||abitij . (18) i>j a>bX cc theory – p. 8/41 CCD Equations ˆ ab ˆ T2 hΦij |H − ECCD|e Φ0i =0 (19) ab ab ab 1 2 hΦ |Hˆ |Φ0i + hΦ |Hˆ − ECCDTˆ2|Φ0i + hΦ |Hˆ − ECCD Tˆ |Φ0i =0 (20) ij ij ij 2 2 ab ab cd cd 1 ab cdef cd ef hΦ |Hˆ |Φ0i + hΦ |Hˆ − ECCD|Φ it + hΦ |Hˆ − ECCD|Φ it t =0 ij ij kl kl 2 ij klmn kl mn k>l>m>n c>d>e>fX (21) Tˆ = Tˆ2 −→ CCD Tˆ = Tˆ1 + Tˆ2 −→ CCSD Tˆ = Tˆ1 + Tˆ2 + Tˆ3 −→ CCSDT cc theory – p. 9/41 Full CI vs. Full CC Tˆ ΨCI = 1+ Cˆ |Φ0i ΨCC = e |Φ0i Cˆ = Cˆ1 + Cˆ2 + Cˆ3+ Cˆ4 + ··· Tˆ = Tˆ1 + Tˆ2 + Tˆ3 + ··· Tˆ 1 2 1 3 e = 1+ Tˆ2 + Tˆ2 + Tˆ2 + ··· 2! 3! 1 2 = 1+ Tˆ1 + Tˆ2 + ··· + Tˆ1 + Tˆ2 + ··· + ··· 2 1 2 1 3 = 1+ Tˆ1 + Tˆ2 + Tˆ + Tˆ3 + Tˆ1Tˆ2 + Tˆ + (22) 2 1 3! 1 1 2 1 2 1 4 Tˆ4 + Tˆ1Tˆ3 + Tˆ + Tˆ Tˆ2 + Tˆ + ··· (23) 2! 2 2! 1 4! 1 |Ψi = |Ψ0i + cS|Si + cD|Di + cT|T i + cQ|Qi + ··· XS XD XT XQ cc theory – p. 10/41 Second Quantization Second-quantization techniques evolved primarily for problems in which the number of particles is not fixed or known and in the context of independent-particle models. We assumes the existence of an unspecified number of functions in a given fixed one-particle basis, say {φi} = {φ1,φ2,φ3,...} , (24) In general, the functions ψi are spinorbitals, and we assume that the basis is orthonormal hφi|φj i = δij (25) The given one-particle basis generates Hilbert spaces for N =1, 2,... particles, for which the basis functions are products of N one-particle basis functions. However, since we are dealing with fermions, we will restrict the many-particle functions to be antisymmetric and thus assume a many-particle basis constructed of Slater determinants made up of the one-particle basis functions. cc theory – p. 11/41 Creation and Annihilation Operators The representation of a normalized Slater determinant Φ=Φijk... ≡A(ϕiϕj ϕk ··· ) ≡|ϕiϕj ϕk ···i≡|ijk ···i (26) where A is the antisymmetrizer and each ϕ is a spinorbital in our one-particle basis. The Slater determinant (SD) Φ is represented in second-quantized form by specifying the occupancies (or occupation numbers) n1,n2,... of the basis spinorbitals ϕ1,ϕ2,... in the determinant. 0 if φi is empty ni (Φ) = , (i = i, 2,...) (27) 1 if φ is occupied j The determinant itself (and various operators on it) are represented in terms of a set of creation and annihilation operators. ˆ † † ˆ† † Creation operator for ϕi, Xi , cˆi , ii or aˆi ˆ ˆ Annihilation operator for ϕi, Xi, c,ˆ ii or aˆi † Here we shall use aˆi and aˆi. cc theory – p. 12/41 Creation and Annihilation Operators An annihilation operator which is the adjoin of the creation operator † † aˆi =a ˆi (28) Creation and annihilation operators defined in terms of their action on SD † aˆi |jk ···i = |ijk ···i (29) aˆi|ijk ···i = |jk ···i (30) A given SD may be written as a chain of creation operators acting on the true vacuum † † † aˆpaˆq ··· aˆs|i = |φpφq ··· φsi. (31) An annihilation operator acting on the vacuum state gives a zero result, aˆp|i =0. (32) † aˆi |Φi =0 if ni(Φ) = 1, (33) aˆi|Φi =0 if ni(Φ) = 0, (34) cc theory – p. 13/41 Permutation Operator and Anticommutation Relations It is convenient to arrange the spinorbitals in an SD in lexical order as |ijk ···i, where i>j>k> ··· and therefore it is necessary define permutation operator ˆ Pˆ|ijk ···i =(−1)σ(P )|ijk ···i, (35) Pairwise permutations of the operators introduce changes in the sign of the resulting determinant, † † † † aˆqaˆp|i = |φqφpi = −|φpφqi = −aˆpaˆq|i. (36) The anticommutation relation for a pair of creation operators is simply † † † † † † aˆp, aˆq =a ˆpaˆq +ˆaqaˆp = 0ˆ. (37) + h i The analogous relation for a pair of annihilation operators is aˆ , aˆ =a ˆ aˆ +ˆa aˆ = 0ˆ. (38) p q + p q q p The anticommutation relation for the “mixed” product is † † † † ˆ aˆp, aˆq = aˆp, aˆq =a ˆpaˆq +ˆaqaˆp = δpq, (39) + + h i h i cc theory – p. 14/41 Normal Order for Second Quantized Operators † “all annihilation operators (ˆai and aˆa) standing to the right of all creation operators † (ˆai and aˆa) .” ˆ † † A =a ˆpaˆqaˆraˆs † † † = δpqaˆraˆs − aˆqaˆpaˆraˆs † † † † = δpqδrs − δpqaˆsaˆr − δrsaˆqaˆp +ˆaqaˆpaˆsaˆr † † † † † = δpqδrs − δpqaˆsaˆr − δrsaˆqaˆp + δpsaˆqaˆr − aˆqaˆsaˆpaˆr. (40) The quantum mechanical expectation value of this operator in the true vacuum state, |i, ˆ † † † † † h|A|i = h|δpqδrs|i−h|δpqaˆsaˆr|i−h|δrsaˆqaˆp|i + h|δpsaˆqaˆr|i−h|aˆqaˆsaˆpaˆr|i = δpqδrs, (41) The matrix element of Aˆ between the single-particle states, hφt| and |φui, ˆ ˆ † ˆ ˆ † hφt|A|φui = h|aˆtAaˆu|i and a new operator, B ≡ aˆtAaˆu, we can express hφt|Aˆ|φui = h|Bˆ|i = δtuδpqδrs + δtqδpsδru − δtqδpuδrs − δtsδpqδru. (42) cc theory – p. 15/41 Contraction A contraction between two arbitrary annihilation/creation operators, A and B, is defined as AB ≡ AB −{AB} (43) where the notation {AB} indicates the normal-ordered form of the pair. Case I aˆpaˆq =a ˆpaˆq −{aˆpaˆq} =a ˆpaˆq − aˆpaˆq =0 . (44) Case II † † † † † † † † † † aˆpaˆq =a ˆpaˆq −{aˆpaˆq} =a ˆpaˆq − aˆpaˆq =0 . (45) Case III † † † † † aˆpaˆq =a ˆpaˆq −{aˆpaˆq} =a ˆpaˆq − aˆpaˆq =0 . (46) Case IV † † † † † aˆpaˆq =a ˆpaˆq −{aˆpaˆq} =a ˆpaˆq +ˆaqaˆp = δpq . (47) cc theory – p. 16/41 Wick’s theorem An arbitrary string of annihilation and creation operators, ABCDE ··· , may be written as a linear combination of normal-ordered strings ABCDE ··· ={ABCDE ···} + {ABCDE ···}{ABCDE ···}+ singlesX {ABCDE ···}{ABCDE ···}+ doublesX {ABCDEF ···}{ABCDEF ···} + ··· + {···} (48) triples fully X contractedX terms hΦ0 |{ABCD ···}| Φ0i =0 (49) hΦ0 |{ABCD ···}| Φ0i = {ABC...} (50) fully contractedX cc theory – p.