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

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

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

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 , cî , ii or aî ˆ ˆ Annihilation operator for ϕi, Xi, c,ˆ ii or aî † Here we shall use aî and aî. cc theory – p. 12/41 Creation and Annihilation Operators

An annihilation operator which is the adjoin of the creation operator

† † aî =a î (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 (âi and aâ) standing to the right of all creation operators † (âi and aâ) .”

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)

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. 17/41 terms Wick’s theorem - Example

hΦ0|{ABCD}|Φ0i =h————————-Φ0|{ABCD}|Φ0i + h————————-Φ0|{ABCD}|Φ0i+

h————————-Φ0|{ABCD}|Φ0i + h————————-Φ0|{ABCD}|Φ0i+

h————————-Φ0|{ABCD}|Φ0i + hΦ0|{ABCD}|Φ0i+

hΦ0|{ABCD}|Φ0i + hΦ0|{ABCD}|Φ0i (51)

cc theory – p. 18/41 The Fermi Vacuum and the Particle-Hole Formalism

A complete set of operators required to generate |Φ0i from the true vacuum

† † † |Φ0i =a ˆi aˆj aˆk ... |i (52)

Quasiparticle (q-particle) operators. † aî and aâ annihilate holes and particles. † aî and aâ create holes and particles

† † † † † aî aˆj =a î aˆj −{aî aˆj } =a î aˆj +âj aî = δij (53) and † † † † † aâaˆb =a âaˆb −{aâaˆb} =a âaˆb +âbaâ = δab . (54)

† † aˆaaˆb =a ˆiaˆj =0. (55)

† † † † † † † † aâaî = aîaâ = aâaî = aî aâ = aâaî = aî aâ = aâaî = aîaâ =0 . (56)

cc theory – p. 19/41 The Normal-Ordered Electronic Hamiltonian

One-electron operator

† † † ˆ Using Wick’s theorem, aˆpaˆq = {aˆpaˆq} + aˆpaˆq, we can write one-electron operator F as

ˆ ˆ † F = hp|f|qiaˆpaˆq pq X ˆ † ˆ = hp|f|qi{aˆpaˆq} + hp|f|qiδpq pq pq X X ˆ † ˆ = hp|f|qi{aˆpaˆq} + hi|f|ii pq i X X = FˆN + hi|fˆ|ii (57) i X where FN is the normal-product form of the operator.

Fˆ ˆ † N = hp|f|qi{aˆpaˆq} (58) pq X

cc theory – p. 20/41 The Normal-Ordered Electronic Hamiltonian

Fˆ ˆ † N = hp|f|qi{aˆpaˆq}, (59) Fˆ pq Since h0| N |0i =0, we have X h0|FˆN |0i = hi|fˆ|ii, Fˆ = FˆN + h0|Fˆ|0i, (60) i X and FˆN represents the difference between F and its Fermi-vacuum expectation value

FˆN = Fˆ −h0|Fˆ|0i. (61) FˆN contains hole-hole, particle-particle and hole-particle terms, Fˆ ˆ † ˆ † ˆ † ˆ † N = fij {aî aˆj } + fab{aâaˆb} + fia{aî aâ} + fai{aâaî} ij ia ia X Xab X X ˆ † ˆ † ˆ † ˆ † = − fij aˆj aî + fabaâaˆb + fiaaî aâ + faiaâaî (62) ij ia ia X Xab X X FˆN can be separated into diagonal and off-diagonal parts, Fˆ Fˆd Fô N = N + N , Fˆd ˆ † Fˆd ˆd N = fppaˆpaˆp = −h0|F |0i, p X Fô ˆ † Fô ô Fô N = fpqaˆpaˆq = −h0|F |0i = . (63) pq X

cc theory – p. 21/41 The Normal-Ordered Electronic Hamiltonian Two-electron operator

Wˆ 1 † † N = hpq||rsiaˆpaˆqaˆsaˆr (64) 4 pqrs X † † † † aˆpaˆq = aˆpaˆq =0, aˆi aˆj = δij, aˆaaˆb =0 (65) We obtain,

† † † † † † † † † † † † aˆpaˆqaˆsaˆr ={aˆpaˆqaˆsaˆr} + {—————-aˆpaˆqaˆsaˆr} + {aˆpaˆqaˆsaˆr} + {—————-aˆpaˆqaˆsaˆr} + {aˆpaˆqaˆsaˆr}+

† † † † † † † † † † {aˆpaˆqaˆsaˆr} + {aˆpaˆqaˆsaˆr} + {—————-aˆpaˆqaˆsaˆr} + {aˆpaˆqaˆsaˆr} + {aˆpaˆqaˆsaˆr} † † † † ={aˆpaˆqaˆsaˆr} + δqs{aˆpaˆr}− δps{aˆqaˆr}+ † † − δqr{aˆpaˆs} + δpr{aˆqaˆs} + −δpsδqr + δprδqs (66)

Wˆ 1 † † 1 † 1 † = hpq||rsiaˆpaˆqaˆsaˆr + hpi||riiaˆpaˆr − hiq||riiaˆqaˆr 4 pqrs 4 pqi 4 qri X X X 1 † 1 † 1 1 − hpi||isiaˆpaˆs + hiq||isiaˆqaˆs − hij||jii + hij||iji (67) 4 psi 4 qsi 4 ij 4 ij X X X X cc theory – p. 22/41 The Normal-Ordered Electronic Hamiltonian

Using antisymmetrized two-electron integrals in Dirac’s notation, hpq||rsi = −hpq||sri = −hqp||rsi = hqp||sri, we re-index sums and combine terms where appropriate to obtain

Wˆ 1 † † † 1 = hpq||rsi{aˆpaˆqaˆsaˆr} + hpi||rii{aˆpaˆr} + hij||iji. (68) 4 pqrs pri 2 ij X X X The complete Hamiltonian is therefore

ˆ † † 1 † † H = hp|h|qi{aˆpaˆq} + hpi||rii{aˆpaˆr} + hpq||rsi{aˆpaˆqaˆsaˆr} pq pri 4 pqrs X X X 1 + hi|h|ii + hij||iji. (69) i 2 ij X X

ˆ † 1 † † ˆ H = fpq{aˆpaˆq} + hpq||rsi{aˆpaˆqaˆsaˆr} + hΦ0|H|Φ0i (70) pq 4 pqrs X X or

Hˆ = FˆN + VˆN + hΦ0|Hˆ |Φ0i, (71)

cc theory – p. 23/41 The Normal-Ordered Electronic Hamiltonian

The normal-ordered Hamiltonian (canonical HF) is

HˆN = Hˆ −hΦ0|Hˆ |Φ0i. (72) where, HˆN = FˆN + VˆN But, more generally,

Vˆ Fˆo Wˆ N = N + N (73)

The total normal-product Hamiltonian

Hˆ Fˆ Wˆ Fˆd Fˆo Wˆ Fˆd Vˆ N = N + N = N + N + N = N + N (74)

cc theory – p. 24/41 Simpliﬁcation of the Coupled Cluster Hamiltonian

Tˆ |ΨCC i = e |Φ0i, (75) Tˆ2 Tˆ3 = 1+ Tˆ + + + .... |Φ0i. (76) 2! 3! !

Tˆ Tˆ Heˆ |Φ0i = Ee |Φ0i, Tˆ Tˆ Hˆ −hΦ0|Hˆ |Φ0i e |Φ0i = E −hΦ0|Hˆ |Φ0i e |Φ0i,

−Tˆ Tˆ hΦ0|e HˆN e |Φ0i = ∆E, (78) ab... −TˆHˆ Tˆ hΦij... |e N e |Φ0i = 0. (79)

cc theory – p. 25/41 Simpliﬁcation of the Coupled Cluster Hamiltonian

ˆ a † T1 ≡ tî = ti aâaî, (80) i ia X X and 1 1 ˆ ˆ ab † † T2 ≡ tij = tij aâaˆbaˆj aî, (81) 2 ij 4 X ijabX More generally, an n-orbital cluster operator may be defined as

2 n 1 ab... † † Tˆn = t aˆ aˆ ... aˆ aˆ . (82) n! ij... a b j i ij...ab...X

cc theory – p. 26/41 Similarity-Transformed Normal-Ordered Hamiltonian

Hˆ Ψk = EΨk (83)

−1 Hˆ ΩΩ Ψk = EΨk (84)

−1 −1 −1 Ω Hˆ Ω Ω Ψk = EΩ Ψk (85)

H¯ Ψ¯k = EΨ¯k (86) This change eigenvectors but not the eigenvalues of the operator. −Tˆ Tˆ The similarity-transformed normal-ordered Hamiltonian, H¯ ≡ e HˆN e , we obtain 1 1 H¯ = Hˆ + H,ˆ Tˆ + H,ˆ Tˆ , Tˆ + H,ˆ Tˆ , Tˆ , Tˆ + 2! 3! 1 h i hh i i hhh i i i H,ˆ Tˆ , Tˆ , Tˆ , Tˆ + ··· . (87) 4! hhhh i i i i

cc theory – p. 27/41 Campbell-Baker-Hausdorff Relationship

−Tˆ Tˆ Tˆ e Heˆ = HˆN e (88) C Tˆ hΦ0| HˆN e |Φ0i = ∆E, (89) C ab... ˆ Tˆ hΦij... | HN e |Φ0i = 0. (90) C The energy and T amplitudes can be obtained by solving equations 89 and 90, respectively. In order to satisfy the generalized Hellmann–Feynman theorem and to calculate molecular properties, CC equations can be reformulated while introducing an antisymmetrized, perturbation-independent, deexcitation operator Λ

Λ = Λ1 +Λ2 + ... +Λn, (91) 2 n 1 ij... † † Λn = λ i j ...ba . (92) n! ab... ij...ab...X n o

Tˆ h0| (1+Λ) HˆN e |0i =∆E, (93) C ˆ ˆ T ab... cc theory – p. 28/41 h0| (1+Λ) HN e − ∆E|Φij... i =0. (94) C The CCSD Energy Equation

(Tˆ1+Tˆ2) hΦ0| HˆN e |Φ0i =∆E, (95) C

1 2 1 2 1 3 1 2 H¯ = HˆN 1+ Tˆ1 + Tˆ2 + Tˆ + Tˆ + Tˆ1Tˆ2+ Tˆ + Tˆ Tˆ2+ 2 1 2 2 6 1 2 1 1 2 1 3 1 4 1 3 1 2 2 1 3 1 4 Tˆ1Tˆ + Tˆ + Tˆ + Tˆ Tˆ2 + Tˆ Tˆ + Tˆ1Tˆ + Tˆ (96) 2 2 6 2 24 1 6 1 4 1 2 6 2 24 2 c but only survives, 1 2 HˆN 1+ Tˆ1 + Tˆ2 + Tˆ (97) 2 1 C

1 2 (FˆN + VˆN ) 1+ Tˆ1 + Tˆ2 + Tˆ (98) 2 1 C

1 2 1 2 FˆN + VˆN + FˆN Tˆ1 + FˆN Tˆ2 + FˆN Tˆ + VˆN Tˆ1 + VˆN Tˆ2 + VˆN Tˆ (99) 2 1 2 1 C

cc theory – p. 29/41 The CCSD Energy Equation

We have,

ˆ † FN = fpq{aˆpaˆq} (100) pq X ˆ 1 † † VN = hpq||rsi{aˆpaˆqaˆsaˆr} (101) 4 pqrs X ˆ 1 a † T1 = ti {aˆaaˆi} (102) 2 ai X 1 ab † † Tˆ2 = t {aˆ aˆ aˆ aˆ } (103) 4 ij a b j i abijX 1 † Tˆ2 = tatb{aˆ† aˆ }{aˆ aˆ } (104) 1 4 i j a i b j aibjX

cc theory – p. 30/41 The CCSD Energy Equation

1 2 1 2 FˆN + VˆN + FˆN Tˆ1 + FˆN Tˆ2 + FˆN Tˆ + VˆN Tˆ1 + VˆN Tˆ2 + VˆN Tˆ (105) 2 1 2 1 C

ˆ ˆ 1 a † † FN T1 = fpqti {aˆpaˆq}{aˆaaˆi} (106) 2 pq ai X X

† † † † {aâaî}{aˆpaˆq} = {aâaîaˆpaˆq}

† † † † † † † † = {aˆpaˆqaâaî} + {aˆpaˆqaâaî} + {aˆpaˆqaâaî} + {aˆpaˆqaâaî} † † † † = {aˆpaˆqaâaî} + δpi{aˆqaâ} + δqa{aˆpaî} + δpiδqa. (107)

ˆ ˆ a FN T1 = fiati , (108) ai X

cc theory – p. 31/41 The CCSD Energy Equation

cc theory – p. 32/41 The CCSD Energy Equation

cc theory – p. 33/41 The CCSD Energy Equation

cc theory – p. 34/41 The CCSD Energy Equation

1 2 1 2 FˆN + VˆN + FˆN Tˆ1 + FˆN Tˆ2 + FˆN Tˆ + VˆN Tˆ1 + VˆN Tˆ2 + VˆN Tˆ (109) 2 1 2 1 C ˆ † FN = fpq{aˆpaˆq} =0 (110) pq X ˆ 1 † † VN = hpq||rsi{aˆpaˆqaˆsaˆr} =0 (111) 4 pqrs X ˆ ˆ 1 a † † a FN T1 = fpqti {aˆpaˆq}{aâaî} = fiati , (112) 2 pq ai ai X X X ˆ ˆ 1 ab † † † FN T2 = fpqtij {aˆpaˆq}{aâaˆbaˆj aî} =0, (113) 4 pq X abijX 1 ˆ ˆ2 1 a b † † † FN T1 = fpqti tj {aˆpaˆq}{aâaî}{aˆbaˆj } =0, (114) 2 4 pq X aibjX ˆ ˆ 1 † † 1 a † VN T1 = hpq||rsi{aˆpaˆqaˆsaˆr} ti {aâaî} =0, (115) 4 pqrs 2 ai X X

a 1 ab 1 a b ECCSD − E0 = fiat + hij||abit + hij||abit t . (116) i 4 ij 2 i j ia aibj aibj X X X cc theory – p. 35/41 The CCSD Amplitude Equations

ˆ ˆ a ˆ (T1+T2) hΦi | HN e |Φ0i =0 (117) C ˆ ˆ ab ˆ (T1+T2) hΦij | HN e |Φ0i =0 (118) C a ˆ ˆ hΦi | FN + VN |Φ0i = (119) † † 1 † † † fpqhΦ0|{aî aâ}{aˆpaˆq}|Φ0i + hpq||rsihΦ0|{aî aâ}{aˆpaˆqaˆsaˆr}|Φ0i. pq 4 pqrs X X

a ˆ † † hΦi |FN |Φ0i = fpqhΦ0|{aî aâ}{aˆpaˆq}|Φ0i pq X † † = fpq{aî aâaˆpaˆq} pq X = fpqδiqδap pq X = fai. (120) cc theory – p. 36/41 The CCSD Amplitude Equations

ab ˆ ˆ hΦij | FN + VN |Φ0i = (121) † † † 1 † † † † fpqhΦ0|{aî aˆj aˆbaâ}{aˆpaˆq}|Φ0i + hpq||rsihΦ0|{aî aˆj aˆbaâ}{aˆpaˆqaˆsaˆr}|Φ0i. pq 4 pqrs X X

cc theory – p. 37/41 The CCSD Amplitude Equations

a ˆ ˆ ˆ b † † † hΦi | FN + VN T1 |Φ0i = fpqtj hΦ0|{aî aâ} {aˆpaˆq}{aˆ aˆj } |Φ0i + c b c pq h i X Xjb 1 b † † † † hpq||rsitj hΦ0|{aî aâ} {aˆpaˆqaˆsaˆr}{aˆ aˆj } |Φ0i, b c 4 pqrs X Xjb

cc theory – p. 38/41 The CCSD Amplitude Equations

c c a b = hab||cjiti −hab||ciitj + hij||bkitk −hij||akitk c k X X cc theory – p. 39/41 The CCSD Amplitude Equations

ˆ T1 Contribution,

c a c ac 1 cd 0 = fai + facti − fkitk + hka||ciitk + fkctik + hka||cditki − c 2 X Xk Xkc Xkc Xkcd 1 ca c a c a c d hkl||ciit − fkct t − hkl||ciit t + hka||cdit t − (122) 2 kl i k k l k i Xklc Xkc Xklc Xkcd 1 1 hkl||cditc tdta + hkl||cditc tda − hkl||cditcdta − hkl||cditcatd, k i l k li 2 ki l 2 kl i klcdX klcdX klcdX klcdX

cc theory – p. 40/41 The CCSD Amplitude Equations

ac bc ab ab 1 ab 0 = hab||iji + f t − f t − f t − f t + hkl||ijit (123) X bc ij ac ij X kj ik ki jk X kl c k 2 kl

1 cd ac c a + hab||cdit + P (ij)P (ab) hkb||cjit + P (ij) hab||cjit − P (ab) hkb||ijit X ij X ik X i X k 2 cd kc c k

1 ac db 1 cd ab 1 ac bd + P (ij)P (ab) hkl||cdit t + hkl||cdit t − P (ab) hkl||cdit t X ik lj X ij kl X ij kl 2 klcd 4 klcd 2 klcd

1 ab cd 1 a b 1 c d −P (ij) hkl||cdit t + P (ab) hkl||ijit t + P (ij) hab||cdit t X ik jl X k l X i j 2 klcd 2 kl 2 cd a bc c ab c ab c db P (ab) f t t + P (ij) f t t P (ij) hkl||ciit t + P (ab) hka||cdit t + X kc k ij X kc i jk X k lj X k ij kc kc klc kcd d bc a bc a c P (ij)P (ab) hak||dcit t + P (ij)P (ab) hkl||icit t − P (ij)P (ab) hkb||icit t + X i jk X l jk X k j kcd klc kc

1 c ab 1 a cd P (ij) hkl||cjit t − P (ab) hkb||cdit t − X i kl X k ij 2 klc 2 kcd

1 c a d 1 c a b P (ij)P (ab) hkb||cdit t t + P (ij)P (ab) hkl||cjit t t − X i k j X i k l 2 kcd 2 klc c d ab c a db P (ij) hkl||cdit t t − P (ab) hkl||cdit t t + X k i lj X k l ij klcd klcd

1 c d ab 1 a b cd P (ij) hkl||cdit t t + P (ab) hkl||cdit t t + X i j kl X k l ij 4 klcd 4 klcd

c b ad 1 c a d b P (ij)P (ab) hkl||cdit t t + P (ij)P (ab) hkl||cdit t t t . X i l kj X i k j l klcd 4 klcd cc theory – p. 41/41