Introduction to perturbation theory and coupled-cluster theory for electron correlation

Transcription

1 Introduction to perturbtion theory nd coupled-cluster theory for electron correltion Julien Toulouse Lortoire de Chimie Théorique Université Pierre et Mrie Curie et CNRS, Pris, Frnce October 11, Contents 1 Review of the mny-body problem The Hmiltonin nd the mny-body wve function The Hrtree-Fock pproximtion Strightforwrd configurtion-interction methods Perturbtion theory Generl Ryleigh Schrödinger perturbtion theory Møller-Plesset perturbtion theory Generl spin-unrestricted theory in terms of spin orbitls Spin-restricted theory in terms of sptil orbitls for closed-shell systems Digrmmtic representtion of perturbtion theory Coupled-cluster theory The exponentil nstz Trunction of the cluster opertor The coupled-cluster energy nd the coupled-cluster equtions Exmple: coupled-cluster doubles References 18 1

2 These notes provide n introduction to perturbtion theory nd coupled-cluster theory for ground-stte electron correltion. For generl reference on this subject, see e.g. Refs. [1, 2]. 1 Review of the mny-body problem 1.1 The Hmiltonin nd the mny-body wve function We consider N-electron system (tom, molecule, solid) in the Born-Oppenheimer nd nonreltivistic pproximtions. The electronic Hmiltonin in the position representtion is, in tomic units, H(r 1,r 2,...,r N ) = N h(r i )+ 1 2 i N i N i j 1 r i r j, (1) whereh(r i ) = (1/2) 2 r i +v ne (r i )istheone-electroncontributioncomposedofthekinetic-energy opertor nd of the nuclei-electron interction v ne (r i ) = α Z α/ r i R α (where R α nd Z α re the positions nd chrges of the nuclei). The sttionry electronic sttes re determined by the time-independent Schrödinger eqution H(r 1,r 2,...,r N )Ψ(x 1,x 2,...,x N ) = EΨ(x 1,x 2,...,x N ), (2) where Ψ(x 1,x 2,...,x N ) is wve function written with spce-spin coordintes x i = (r i,σ i ) (with r i R 3 nd σ i = or ) which is ntisymmetric with respect to the exchnge of two coordintes, nd E is the ssocited energy. Using Dirc nottions, the Schrödinger eqution (2) cn be rewritten in convenient representtion-independent formlism Ĥ Ψ = E Ψ. (3) We re interested in clculting n pproximtion for the wve function Ψ nd the ssocited energy E of specific stte, most often the ground-stte wve function Ψ 0 nd the ground-stte energy E The Hrtree-Fock pproximtion The Hrtree-Fock (HF) method 1 consists in pproximting the ground-stte wve function s single Slter determinnt, Ψ 0 Φ 0, χ 1 (x 1 ) χ 2 (x 1 ) χ N (x 1 ) Φ 0 (x 1,x 2,...,x N ) = 1 χ 1 (x 2 ) χ 2 (x 2 ) χ N (x 2 ) N!....., (4). χ 1 (x N ) χ 2 (x N ) χ N (x N ) where χ i (x) re orthonorml spin orbitls. The HF totl electronic energy, E HF = Φ 0 Ĥ Φ 0, cn be expressed in terms of integrls over these spin orbitls, using Slter s rules for clculting expecttion vlues over Slter determinnts, E HF = Φ 0 Ĥ Φ 0 = h + 1, (5) 2 1 We present here the unrestricted Hrtree-Fock (UHF) method, in which the sptil prt of the - nd -spin orbitls re llowed to be different, which generlly leds to the breking of Ŝ 2 symmetry. By contrst, the restricted Hrtree-Fock (RHF) method imposes Ŝ2 symmetry by constrining the sptil prt of the - nd -spin orbitls to be the sme. 2

3 where the sums over nd b re over upied spin orbitls. In this expression, h re the one-electron integrls h = dx χ (x)h(r)χ (x), (6) nd = b re the ntisymmetrized two-electron integrls (in physicists nottion) where χ i ij kl = dx 1 dx (x 1)χ j (x 2)χ k (x 1 )χ l (x 2 ) 2. (7) r 2 r 1 The spin orbitls re determined by minimizing the HF energy subject to the normliztion constrints { } E HF [{χ }] ε χ χ, (8) min {χ } where ε re the orbitl energies plying the role of Lgrnge multipliers for the normliztion constrints. This minimiztion leds to the HF eigenvlue equtions f(x)χ i (x) = ε i χ i (x), (9) whichdetermineboththeupiedndvirtulspinorbitlsχ i (x)ndssocitedorbitlenergies ε i. In these equtions, f(x) is the one-electron HF Hmiltonin (or often clled simply Fock Hmiltonin) f(x) = h(r)+v HF (x), (10) where v HF (x) is the one-electron HF potentil opertor v HF (x) = J (x) K (x), (11) composed of Coulomb (or Hrtree) opertor written s χ J (x 1 ) = dx (x 2 )χ (x 2 ) 2, (12) r 2 r 1 nd n exchnge (or Fock) opertor whose ction on spin orbitl χ i (x 1 ) is given by χ K (x 1 )χ i (x 1 ) = dx (x 2 )χ i (x 2 ) 2 χ (x 1 ). (13) r 2 r 1 The HF potentil v HF (x) is one-electron men-field potentil pproximting the effect of the two-electron interction (1/2) N N i i j 1/ r i r j. In other words, the HF pproximtion only ccounts for the electron-electron interction in n verged, men-field wy. The effect of the electron-electron interction beyond the HF pproximtion is clled electron correltion. The difference between the exct ground-stte totl energy E 0 nd the HF totl energy E HF is clled the correltion energy E c = E 0 E HF. (14) Even though E c is usully smll percentge of the totl energy, it very often mkes lrge nd crucil contribution to energy differences (such s rection energies, rection brrier heights,...) which re the quntities of chemicl interest. It is therefore importnt to go beyond the HF pproximtion nd clculte the vlue of the correltion energy, which is the gol of the post-hf methods. 3

4 1.3 Strightforwrd configurtion-interction methods The most strightforwrd post-hf method is the configurtion-interction (CI) method. In this method, the wve function is expnded in the bsis of the HF determinnt Φ 0, the singleexcited determinnts Φ r, the double-excited determinnts Φ rs, nd so on Ψ CI = c 0 Φ 0 + c r Φ r + r <b r<s c rs Φrs +. (15) The coefficients c = (c 0,c r,c rs,...) corresponding to the ground-stte wve function re found by minimizing the totl CI energy Ψ CI Ĥ Ψ CI with the constrint of the normliztion of the wve function { } min Ψ CI Ĥ Ψ c CI E Ψ CI Ψ CI, (16) leding to the following eigenvlue eqution Φ 0 Ĥ Φ 0 Φ 0 Ĥ Φr Φ 0 Ĥ Φrs Φ r Ĥ Φ 0 Φ r Ĥ Φr Φ r Ĥ Φrs Φ r s b Ĥ Φ 0 Φ r s b Ĥ Φr Φ r s b Ĥ Φrs c 0 c r c rs. = E c 0 c r c rs.. (17) The eigenvector ssocited to the lowest eigenvlue corresponds to the ground stte, wheres the other eigenvectors correspond to excited sttes. If ll levels of excittions re included (i.e., up to N-fold excittions for N-electron system), then exct wve functions re obtined within the underlying one-electron bsis set used for expnding the orbitls. This is referred to s the full configurtion-interction (FCI) method. In prctice, FCI cn only be performed for very smll systems with smll bsis sets. For most systems, one hs to truncte the CI expnsion in Eq. (15) t given level of excittions for mngele clcultions. This is referred to s truncted CI. Often, only the single nd double excittions re included in the expnsion, leding to the configurtion-interction singles doubles (CISD) method. The truncted CI method hs serious shortcoming. Consider the size consistency property tht the totl energy of system composed of two non-intercting frgments A nd B must be the sum of the totl energies of the seprte frgments E(A B) = E(A)+E(B). (18) This property is prticulrly importnt in chemistry since it is often concerned with systems composed of frgments (toms, molecules). It is of course stisfied for the exct totl energy, but not necessrily with pproximte methods. A method which gives totl energies stisfying this property is sid to be size-consistent. For exmple, the unrestricted HF method is sizeconsistent but the restricted HF method is generlly not. The FCI method is size-consistent, but the truncted CI method hs the importnt drwbck of being generlly not size-consistent. We will now see two other post-hf pproches which hve the dvntge of being sizeconsistent: perturbtion theory nd coupled-cluster theory. 4

5 2 Perturbtion theory This section provides bsic introduction to Møller-Plesset perturbtion theory. For historicl perspective nd recent reserch developments, see e.g. Ref. [3]. 2.1 Generl Ryleigh Schrödinger perturbtion theory We strt by reviewing the generl expressions of Ryleigh Schrödinger perturbtion theory. Consider Hmiltonin Ĥλ depending on coupling constnt λ Ĥ λ = Ĥ(0) +λˆv, (19) where Ĥ(0) is zeroth-order Hmiltonin opertors nd ˆV is perturbtion opertor. These two opertors re chosen so tht the physicl Hmiltonin of interest corresponds to λ = 1, i.e. Ĥ = Ĥλ=1 = Ĥ(0) + ˆV. By vrying λ from 0 to 1, we cn thus go from the zeroth-order Hmiltonin, Ĥ λ=0 = Ĥ(0), to the physicl Hmiltonin Ĥλ=1 = Ĥ. We will be ultimtely interested in the vlue λ = 1. The zeroth-order Hmiltonin Ĥ(0) is chosen such tht its eigensttes Φ n nd ssocited eigenvlues E (0) n re known. They of course stisfy the eigenvlue eqution Ĥ (0) Φ n = E (0) n Φ n, (20) nd the eigensttes re chosen to be orthonorml, i.e. Φ n Φ m = δ n,m. We would like to determine the eigensttes Ψ λ n nd ssocited eigenvlues E λ n of the Hmiltonin Ĥλ Ĥ λ Ψ λ n = E λ n Ψ λ n. (21) In the following, we will only consider the specific cse of the determintion of the ground stte Ψ λ 0 nd its energy Eλ 0. We ssume tht the energy cn be expnded in powers of λ nd, similrly, for the wve function Ψ λ 0 E λ 0 = E (0) 0 +λe (1) 0 +λ 2 E (2) 0 +, (22) Ψ λ 0 = Ψ (0) 0 +λ Ψ(1) 0 +λ2 Ψ (2) 0 +. (23) Note tht the zeroth-order wve function is just Ψ (0) 0 = Φ 0. We re free to choose the normliztion of Ψ λ 0. A convenient choice is the so-clled intermedite normliztion, i.e. Φ 0 Ψ λ 0 = 1 for ll λ. Since the zeroth-order wve function is normlized s Φ 0 Φ 0 = 1, it implies tht Φ 0 Ψ (i) 0 = 0 for ll i 1, i.e. the wve-function correction t ech order is orthogonl to the zeroth-order wve function. Inserting Eqs. (22) nd (23) into Eq. (21) gives (Ĥ(0) +λˆv)( + ) Φ 0 +λ Ψ (1) 0 +λ2 Ψ (2) 0 ( )( + ) = E (0) 0 +λe (1) 0 +λ 2 E (2) 0 + Φ 0 +λ Ψ (1) 0 +λ2 Ψ (2) 0. (24) Looking t this eqution order by order in λ, we obtin t zeroth order which is just Eq. (20) for the ground stte. Ĥ (0) Φ 0 = E (0) 0 Φ 0, (25) 5

6 At first order, we obtin Ĥ (0) Ψ (1) 0 + ˆV Φ 0 = E (0) 0 Ψ(1) 0 +E(1) 0 Φ 0. (26) Projecting this eqution on the br stte Φ 0, nd using Φ 0 Ĥ(0) = E (0) 0 Φ 0 nd Φ 0 Ψ (1) 0 = 0, gives the first-order energy correction E (1) 0 = Φ 0 ˆV Φ 0. (27) Projecting now Eq. (26) on the br sttes Φ n for n 0, nd using Φ n Ĥ(0) = E n (0) Φ n nd Φ n Φ 0 = 0, gives E n (0) Φ n Ψ (1) 0 + Φ n ˆV Φ 0 = E (0) 0 Φ n Ψ (1) 0, (28) leding to the projection coefficients of the Ψ (1) 0 in the bsis of Φ n Φ n Ψ (1) 0 = Φ n ˆV Φ 0, (29) E n (0) E (0) 0 nd, since Φ 0 Ψ (1) 0 = 0, it leds to the first-order wve-function correction Ψ (1) 0 = n 0 Similrly, t second order, we obtin Φ n ˆV Φ 0 E n (0) E (0) Φ n. (30) 0 Ĥ (0) Ψ (2) (1) 0 + ˆV Ψ 0 = E(0) 0 Ψ(2) 0 +E(1) 0 Ψ(1) 0 +E(2) 0 Φ 0. (31) Projecting this eqution on the br stte Φ 0, nd using Φ 0 Ĥ(0) = E (0) 0 Φ 0 nd Φ 0 Ψ (1) 0 = Φ 0 Ψ (2) 0 = 0, gives the second-order energy correction or, fter using Eq. (30), 0 = Φ 0 ˆV Ψ (1) 0, (32) E (2) E (2) 0 = Φ 0 ˆV Φ n 2 n 0 E n (0) E (0). (33) 0 Note tht E (2) 0 diverges if there is stte Φ n (with n 0) of energy E n (0) equls to E (0) 0, i.e. if the zeroth-order Hmiltonin Ĥ(0) hs degenerte ground stte. In this cse, the expnsions in Eqs. (22) nd (23) re not vlid, nd one must insted digonlize the Hmiltonin in the degenerte spce before pplying perturbtion theory, which is known s degenerte perturbtion theory. Exercise 1 : Prove tht the third-order energy correction hs the following expression E (3) 0 = Φ 0 ˆV Ψ (2) 0 = Φ 0 ˆV Φ n Φ n ˆV Φ m Φ m ˆV Φ 0 n,m 0 (E n (0) E (0) 0 )(E(0) m E (0) 0 ) E (1) 0 n 0 Φ 0 ˆV Φ n 2 (E n (0) E (0) 0 )2. (34) 6

7 2.2 Møller-Plesset perturbtion theory Generl spin-unrestricted theory in terms of spin orbitls Møller-Plesset (MP) perturbtion theory is prticulr cse of Ryleigh Schrödinger perturbtion theory for which the zeroth-order Hmiltonin is chosen to be the (mny-electron) Hrtree-Fock (sometimes lso simply clled Fock) Hmiltonin where the expression of ˆF in the position representtion is Ĥ (0) = ˆF, (35) F(x 1,x 2,...,x N ) = N f(x i ). (36) i The corresponding perturbtion opertor ˆV is thus the difference between the electron-electron Coulomb interction nd the HF potentil V(x 1,x 2,...,x N ) = 1 2 N N i i j 1 N r i r j v HF (x i ). (37) i Zeroth order The zeroth-order ground-stte wve function is the HF single determinnt Φ 0, nd the zerothorderexcited-sttewvefunctionsrethesingle,double,... exciteddeterminntsφ n = Φ r,φ rs,... According to Eq. (9), the zeroth-order ground-stte energy E (0) 0 is given by the sum of upied orbitl energies E (0) 0 = ε. (38) First order The first-order energy correction is the expecttion vlue of the HF determinnt over the perturbtion opertor, which is clculted ccording to Slter s rules, E (1) 0 = Φ 0 ˆV Φ 0 = 1 ˆv HF 2 = 1, (39) 2 where, ccording to the definition of the HF potentil in Eq. (11), we hve used the fct tht ˆv HF =. Therefore, the sum of the zeroth-order energy nd first-order 7

8 energy correction just gives bck the HF energy E (0) 0 +E (1) 0 = = = ε 1, 2 ( ) h b h where we hve used the fct tht ε = h + b. = E HF, (40) Second order The second-order energy correction, which is clled in this context the second-order Møller- Plesset (MP2) correltion energy, is E (2) 0 = Ec MP2 = Φ 0 ˆV Φ n 2 n 0 E n (0) E (0), (41) 0 where Φ n cn be priori single, double, triple,... excited determinnts. In fct, since ˆV is two-body opertor, ccording to Slter s rules, triple nd higher excittions with respect to Φ 0 give vnishing mtrix elements Φ 0 ˆV Φ n. In ddition, it turns out tht single excittions only give vnishing contribution Φ 0 ˆV Φ r = = rb ˆv HF r b rb rb b b = 0. (42) It thus remins only the double excittions, Φ n = Φ rs. Only the two-body prt of the perturbtion opertor gives non-zero mtrix element, Φ 0 ˆV Φ rs = rs. (43) Besides, the zeroth-order energy corresponding to the doubly-excited determinnts Φ rs is E (0) n = E rs,(0) = E (0) 0 +. (44) UsingEqs.(43)nd(44)inEq.(41), werrivetthefollowingexpressionforthemp2correltion energy Ec MP2 rs 2 =. (45) <b r<s Using the ntisymmetry property of the integrls, i.e. rs = sr = b rs, nd the fct tht rs = 0 if = b or r = s, the MP2 correltion energy cn lso be written without constrints in the sums E MP2 c = rs 2. (46)

9 Note the MP2 correltion energy is lwys negtive. It diverges to if one energy denomintor is zero. This hppens for systems with zero HF HOMO-LUMO gp, in which cse MP perturbtion theory cnnot be pplied. The MP2 totl energy is simply defined s E MP2 = E HF +Ec MP2. Since it is not vritionl theory, the MP2 totl energy is not necessrily ove the exct ground-stte totl energy. The MP2 method cn be considered s the computtionlly chepest post-hf method, nd is thus widely used. Third order Similrly, strting from Eq. (34), it cn be shown, fter much work, tht the third-order Møller- Plesset (MP3) energy correction hs the following expression E (3) 0 = ,c,d +,c,t,u,t rs rs cd cd ( )(ε r +ε s ε c ε d ) rs rs tu tu ( )(ε t +ε u ε ε b ) rs cs tb rt c ( )(ε r +ε t ε ε c ). (47) The clcultion of the third- or higher-order terms is often considered s not worthwhile in comprison with coupled-cluster methods for exmple. When strting from n unrestricted HF clcultion, MP perturbtion theory is correctly size consistent t ech order. This is consequence of the fct tht the energy correction t ech order cnnot be fctorized in uncoupled sums. This coupling between ll the orbitl indices is n expression of the link-cluster theorem Spin-restricted theory in terms of sptil orbitls for closed-shell systems For closed-shell systems, with spin-singlet symmetry, the MP2 correltion energy expression cn be simplified by summing over the spin coordintes. One cn first rewrite Eq. (46) s E MP2 c = 1 4 = 1 4 = 1 2 rs sr 2 ( rs sr )( rs sr ) rs rs rs rs b, (48) where the lst line hs been obtined by expnding, using the permuttion symmetry property of the integrls such s sr = rs b, nd exchnging the dummy indices such s r nd s in the summtion. We cn now perform the summtions over the spin coordintes. After pying 9

10 ttention to whether the spin integrtion in the two-electron integrls gives 0 or 1, we rrive t E MP2 c spt = spt = spt spt 2 rs rs rs rs b, rs [2 rs rs b ], (49) where nd refer now to sptil upied nd virtul orbitls, respectively. This lst expression is lso frequently given using chemists nottion for the two-electron integrls(ij kl) = ik jl E MP2 c spt = spt (r bs)[2(r sb) (rb s)]. (50) Exercise 2 : Write down the MP2 correltion energy expression for the cse of the H 2 molecule in miniml bsis (orbitls: 1 = σ g, 2 = σ u ). Wht hppens in the dissocition limit? Digrmmtic representtion of perturbtion theory The vrious terms ppering in perturbtion theory cn be conveniently represented by Feynmn digrms, used in mny res of mny-body theory. In quntum chemistry, the prticulr kind of digrms most often used re clled Goldstone digrms. For exmple, the spin-orbitl expression of the MP2 correltion energy written in Eq. (48) s the sum of direct nd exchnge term E MP2 c = 1 2 rs rs is represented by the corresponding two digrms: rs rs b, (51) E MP2 c = r s b + s r b Ech digrm is mde of only three types of lines: full line with downwrd rrow representing upied spin orbitl (, b,...), nd is clled hole line; full line with upwrd rrow representing virtul spin orbitl (r, s,...), nd is clled prticle line; horizontl dsh line representing the Coulomb interction between four spin orbitls, nd is clled n interction line. 10

11 Ech interction line hs two extremities clled interction vertices. Ech such interction vertex is connected with one full line coming in nd one full line coming out of the vertex. The mthemticl expression corresponding to ech digrm is obtined by pplying the following rules: 1. ech interction line contributes two-electron integrl fctor with spin-orbitl indices s orb-left-in, orb-right-in orb-left-out, orb-right-out ; 2. ech pir of djcent interction lines contributes fctor 1/( ε prticle ε hole ) where the sums re over indices of ll prticle nd hole lines crossing n imginry line seprting the two djcent interction lines; 3. sum over ll prticle nd hole indices; 4. if the digrm is left/right symmetric, there is n overll fctor of 1/2; 5. the overll sign is given by ( 1) h+l where h nd l re the number of hole lines nd closed full-line loops, respectively. The digrms do not only provide visully ppeling representtion of the perturbtion expnsion tht one cn drw fter hving derived the mthemticl expressions of the perturbtion terms. They cn lso be used to void the mthemticl derivtions. Indeed, one cn first drw ll possible digrms t given perturbtion order, nd then trnslte them into mthemticl expressions. In the context of digrmmtic perturbtion theory, the linked-cluster theorem cn be esily formulted: ech digrm contributing to the perturbtive expnsion of the energy is mde of single connected piece. Unconnected digrms do not contribute. This ensures the size consistency of the perturbtive expnsion t ech order. 11

12 3 Coupled-cluster theory This section provides bsic introduction to coupled-cluster theory. For more detils, its extension for excited-stte clcultions, nd recent developments, see e.g. Ref. [4]. 3.1 The exponentil nstz In coupled-cluster (CC) theory, one strts by n exponentil nstz for the CC wve function Ψ CC = eˆt Φ 0, (52) where Φ 0 is the HF wve function, nd ˆT is the cluster opertor which is the sum of cluster opertors of different excittion levels ˆT = ˆT 1 + ˆT ˆT N. (53) In this expression, ˆT 1 is the cluster opertor for the single excittions, which cn be written in second-quntiztion formlism s ˆT 1 = t r â râ, (54) where t r re the single-excittion cluster mplitudes to be determined, nd â nd â r re nnihiltion nd cretion opertors for the spin-orbitls nd r, respectively. When the opertor â râ cts on the HF single determinnt Φ 0, it genertes the single-excited determinnt Φ r = â râ Φ 0. Similrly, ˆT 2 is the cluster opertor for the double excittions, which is written s ˆT 2 = <b r<s t rs â râ sâ b â = 1 4 r t rs â râ sâ b â, (55) where t rs re the double-excittion cluster mplitudes to be determined. When the opertor â râ sâ b â cts on the HF single determinnt Φ 0, it genertes the double-excited determinnt Φ rs = â râ sâ b â Φ 0. The second equlity in Eq. (55) comes from imposing to the mplitudes to be ntisymmetric with respect to the exchnge of two indices, i.e. trs nd from the nticommuttion property of two nnihiltion opertors, i.e. â b â = â â b, or two cretion opertors, i.e. â râ s = â sâ r. And so on up to the ˆT N cluster opertor for N-fold excittions. t rs = trs b = tsr = tsr b, To understnd the ction of the opertor eˆt on the HF wve function Φ 0, one cn expnd the exponentil nd rerrnge the opertors in terms of excittion levels eˆt = ˆ1+ ˆT + ˆT 2 2! + ˆT 3 3! + = ˆ1+Ĉ1 +Ĉ2 + +ĈN, (56) where the opertor Ĉ1 genertes single excittions, Ĉ2 genertes double excittions, etc. Noting tht the cluster opertors ˆT 1, ˆT 2,..., ˆT N commute with ech other, we find for exmple for the first four excittion opertors Ĉ 1 = ˆT 1, (57) Ĉ 2 = ˆT ˆT 2 1, (58) 12

13 Ĉ 3 = ˆT 3 + ˆT 1ˆT ˆT 3 1, (59) Ĉ 4 = ˆT 4 + ˆT 1ˆT ˆT ˆT 2 1 ˆT ˆT 4 1, (60) nd so on. The single excittions re generted only by ˆT 1. The double excittions cn be generted in two wys: by ˆT 2 describing simultneous excittions of two electrons, or by ˆT 1 2 describing two independent single excittions. The triple excittions cn be generted in three wys: by ˆT 3 describing simultneous excittions of three electrons, by ˆT 1ˆT2 describing independent single nd double excittion, or by ˆT 1 3 three independent single excittions. And so on. The CC wve function cn thus be written s Ψ CC = Φ 0 + c r Φ r + c rs Φrs + <b<c r r<s<t <b r<s c rst c Φrst c + with coefficients relted to the cluster mplitudes by <b<c<d c rstu cd Φrstu cd r<s<t<u +, (61) c r = t r, (62) c rs = trs + tr t s b, (63) c rst c = t rst c +tr t st bc +tr t s b tt c, (64) c rstu cd = t rstu cd +tr t stu bcd +trs ttu cd +tr t s b ttu cd +tr t s b tt c t u d, (65) nd so on. In these expressions, mens n ntisymmetric product mking the resulting coefficients properly ntisymmetric with respect to ny exchnge of two upied spin orbitls or two virtul spin orbitls. For exmple, we hve t r t s b = tr t s b tr b ts, (66) t r t st bc = tr t st bc tr b tst c +t r ct st ts t rt bc +ts b trt c t s ct rt +tt t rs bc tt b trs c +t t ct rs, (67) t r t s b tt c = t r t s b tt c t r t s ct t b tr b ts t t c t r ct s b tt +t r ct s t t b +tr b ts ct t, (68) etc. These expressions cn be obtined by strting from Eqs. (57)-(60), introducing the definitionsoftheclusteropertors ˆT 1, ˆT 2,..., ndimposinglltheconstrintsofthetype < b < c < d or r < s < t < u in the sums by using the nticommuttion property of the nnihiltion nd cretion opertors. Exercise 3 : Check Eqs. (62)-(68), nd find the expressions of t r t stu bcd, trs ttu cd, tr t s b ttu cd, nd t r t s b tt c t u d. Wrning: the time required to do this exercise is inversely proportionl to your ese with combintorics. Thus, the CC wve function contins ll excited determinnts, just s the FCI wve function. Iftheclusteropertor ˆT isnottruncted, theccwvefunctionisjustnonlinerreprmetriztion of the FCI wve function: optimizing the cluster mplitudes t = (t r,t rs,trst c,...) so s to minimize the totl energy would led to the FCI wve function. The interest of the CC pproch only ppers when the cluster opertor is truncted. 13

14 3.2 Trunction of the cluster opertor Let us now consider the trunction of the cluster opertor t given excittion level. For exmple, it is frequent to keep only ˆT 1 nd ˆT 2 ˆT = ˆT 1 + ˆT 2, (69) which is known s coupled-cluster singles doubles (CCSD). The expnsion of eˆt gives ( eˆt = ˆ1+ ˆT 1 + ˆT ) ( 2 ˆT ˆT 1ˆT2 + 1 ) ( 6 ˆT ˆT ˆT 1 2ˆT ) 24 ˆT (70) Applying this expnsion to the HF wve function Φ 0, we see tht the CCSD wve function hs the sme form s in Eq. (61), i.e. it contins ll excited determinnts with coefficients now given by c r = t r, (71) c rs = trs +tr t s b, (72) c rst c = t r t st bc +tr t s b tt c, (73) c rstu cd = t rs ttu cd +tr t s b ttu cd +tr t s b tt c t u d, (74) nd so on. In comprison to the untruncted CC cse, the coefficients of triple excittions c rst c re fully determined by only products of single- nd double-excittion mplitudes t r nd t st bc, nd similrly for the coefficients of qudruple excittions c rstu cd, nd ll higher-level excittions. Presumly, the triple-excittion mplitudes t rst c, qudruple-excittion mplitudes trstu cd, nd higher-level-excittion mplitudes re smller thn the double-excittion mplitude t rs, nd this is thus resonle pproximtion. The CCSD wve function contins much more excited determinnts thn the CISD wve function, while keeping the sme number of free prmeters t = (t r,t rs ) to optimize. One big dvntge of truncted CC over truncted CI is tht truncted CC is size-consistent. This directly stems from the exponentil form of the wve function. Consider system composed of two infinitely seprted (nd thus non-intercting) frgments A nd B. Becuse the orbitls of ech frgments do not overlp, the cluster opertor of the system is dditively seprle, i.e. ˆTA B = ˆT A + ˆT B, where ˆT A nd ˆT B re the clusters opertors of frgments A nd B, respectively. Moreover, in the cse where the HF clcultion is size-consistent, the strting HF wve function is multiplictively seprle Φ0 A B = Φ A 0 ΦB 0. We cn then write Ψ A B CC = eˆt A B Φ A B 0 = eˆt A +ˆT B Φ A 0 Φ B 0 = eˆt A Φ A 0 eˆt B Φ B 0 = Ψ A CC Ψ B CC, (75) i.e. the CC wve function is multiplictively seprle. This implies in turn tht the CC totl energy is dditively seprle, i.e. the method is size-consistent. 14

15 3.3 The coupled-cluster energy nd the coupled-cluster equtions Let us consider n rbitrry trunction level of the cluster opertor. The most nturl wy to clculte the cluster mplitude t would seem to be by using the vritionl method, i.e. minimizing the CC totl energy with the normliztion constrint, just like in CI, { } min Ψ CC Ĥ Ψ t CC E Ψ CC Ψ CC. (76) However, the CC wve function includes ll excited determinnts up to N-fold excittions which contribute to these expecttion vlues, giving too complex equtions to be efficiently solved. A more convenient pproch for obtining the CC energy nd mplitudes is the projection method. In this method, we require tht the CC wve function stisfies the Schrödinger eqution (Ĥ E) Ψ CC = 0, (77) projected onto the spce spnned by the HF determinnt Φ 0, nd the excited determinnts Φ r, Φ rs, Φrst c,... Φ 0 (Ĥ E) Ψ CC = 0, (78) Φ r (Ĥ E) Ψ CC = 0, (79) Φ rs (Ĥ E) Ψ CC = 0, (80) Φ rst c (Ĥ E) Ψ CC = 0, (81) nd so on. Using the expnsion of Ψ CC in terms of determinnts giving in Eq. (61), we see tht Φ 0 Ψ CC = 1 due to the orthonormlity of the determinnts, nd thus Eq. (78) directly gives the CC totl energy E = Φ 0 Ĥ Ψ CC = Φ 0 Ĥ Φ 0 + c r Φ 0 Ĥ Φr + r <b r<s c rs Φ0 Ĥ Φrs, (82) in which ccording to Slter s rules triple nd higher excited determinnts do not contribute. SincethefirstterminEq.(82)isjusttheHFtotlenergyE HF = Φ 0 Ĥ Φ 0, ndsincethesecond term vnishes by virtue of Brillouin s theorem Φ 0 Ĥ Φr = 0, we obtin the CC correltion energy E c = E E HF E c = = = 1 4 c rs Φ0 Ĥ Φrs <b r<s <b r<s (t rs +tr t s b ) rs (t rs +2 tr t s b ) rs, (83) where the ntisymmetry property of the mplitudes nd of the integrls hs been used. Thus, t ny trunction level, the expression of the CC correltion energy obtined with the projection method is quite simple. It only involves mtrix elements over double-excited determinnts nd only single- nd double-excittion mplitudes t r nd t rs enter the expression. Except in the uninteresting cse where the cluster opertor ˆT is not truncted, the totl CC energy obtined with the projection method is not identicl to the one tht would hve been obtined with the 15

16 vrition method of Eq. (76). Consequently, the totl CC energy is not necessrily ove the exct ground-stte totl energy, just s in perturbtion theory. The other equtions (79), (80), (81),... determine the CC mplitudes. They re often more explicitly written s Φ r (Ĥ E)eˆT Φ 0 = 0, (84) Φ rs (Ĥ E)eˆT Φ 0 = 0, (85) Φ rst c (Ĥ E)eˆT Φ 0 = 0, (86) nd so on, which re known s the unlinked CC mplitude equtions. They represent system of coupled nonliner equtions for the mplitudes t r, t rs, trst c, etc. To hve the sme number of equtions s the number of unknown mplitudes, the projection spce must correspond to the trunction level of the cluster opertor. For exmple, for determining the CCSD mplitudes, one needs to consider projection on to single nd double-excited determinnts only. In prctice, it is often more convenient to write the CC mplitude equtions is different wy, by first multiplying from the left by the opertor e ˆT in the Schrödinger eqution (77) before projecting on the excited determinnts Φ r e ˆT ĤeˆT Φ 0 = 0, (87) Φ rs e ˆT ĤeˆT Φ 0 = 0, (88) Φ rst c e ˆT ĤeˆT Φ 0 = 0, (89) nd so on, which re known s the linked CC mplitude equtions. Although equivlent to the unlinked equtions, the linked equtions hve the dvntge of leding to more compct expressions which re mnifestly size-consistent nd t most qurtic in the mplitudes (t ny trunction level). This lst feture comes from the fct tht the Bker-Cmpbell-Husdorff (BCH) expnsion of e ˆT ĤeˆT exctly termintes t fourth order becuse Ĥ contins t most two-electron opertor e ˆT ĤeˆT = Ĥ +[Ĥ, ˆT]+ 1 2! [[Ĥ, ˆT], ˆT]+ 1 3! [[[Ĥ, ˆT], ˆT], ˆT]+ 1 4! [[[[Ĥ, ˆT], ˆT], ˆT], ˆT]. (90) 3.4 Exmple: coupled-cluster doubles As n exmple, we now write down the complete equtions in the simple cse of coupledcluster doubles (CCD). In this cse, the cluster opertor only contins double excittions ˆT = ˆT 2, (91) nd the CCD wve function thus contins double-excited determinnts, qudruple-excited determinnts,... Ψ CCD = Φ 0 + t rs Φrs + <b r<s <b<c<d (t rs ttu cd ) Φrstu cd r<s<t<u +, (92) where the coefficients of the qudruple excittions re given by the ntisymmetrized product of the coefficients of the double excittions, nd so on. The CCD correltion energy is given by E CCD c = t rs rs. (93) <b r<s 16

17 The double-excittion mplitudes t rs cn be determined from the unlinked CC mplitude equtions Φ rs (Ĥ E) Ψ CCD = 0, (94) leding to Φ rs Ĥ Φ 0 + c<d t<u Φ rs Ĥ E Φtu cd ttu cd + c<d t<u Φ rs Ĥ Φrstu cd (trs ttu cd ) = 0, (95) where the qudruple-excittion term hs been simplified by tking into ccount tht mtrix elements of Ĥ over Slter determinnts differing by more thn 2 spin orbitls re zero. Using now Φ rs Ĥ Φ 0 = rs nd Φ rs Ĥ Φrstu cd = cd tu, nd inserting E = E HF + Ec CCD nd replcing Ec CCD by its expression in Eq. (93) gives rs + Φ rs Ĥ E HF Φ tu cd ttu cd + c<d t<u c<d t<u cd tu (t rs ttu cd trs ttu cd ) = 0. (96) The remining mtrix element is more complicted to clculte. After considering the different possibilities of equlity between the indices nd c,d, nd between nd t,u, it cn be found Φ rs Ĥ E HF Φ tu cd = ( )δ,cd δ rs,tu + rs tu δ,cd + cd δ rs,tu The finl CCD mplitude equtions re + ds ub t ru d d u vir rs +( )t rs + rs tu t tu c u cs ub t ru + ds ub δ,c δ r,t cs ub δ,d δ r,t ds tb δ,c δ r,u + cs tb δ,d δ r,u. (97) c d t t<u + ds tb t tr d + + which re qudrtic equtions to be solved itertively. c<d t<u c c<d t cd t rs cd cs tb t tr c cd tu (t rs ttu cd trs ttu cd ) = 0,(98) It is interesting to consider the expnsion of the mplitudes in powers of the electron-electron interction: t rs = trs,(1) +t rs,(2) +. The first-order mplitudes re given by tht is rs +( )t rs,(1) = 0, (99) t rs,(1) rs =. (100) By inserting this expression of t rs,(1) in the expression of the correltion energy in Eq. (93), we then recover the MP2 correltion energy E MP2 c = <b r<s t rs,(1) rs = <b r<s rs 2. (101) Thus, CCD correctly reduces to MP2 t second order in the electron-electron interction. 17

18 References [1] A. Szo nd N. S. Ostlund, Modern Quntum Chemistry: Introduction to Advnced Electronic Structure Theory (Dover, New York, 1996). [2] T. Helgker, P. Jørgensen nd J. Olsen, Moleculr Electronic-Structure Theory (Wiley, Chichester, 2002). [3] D. Cremer, WIREs Comput. Mol. Sci. 1, 509 (2011), doi: /wcms.58. [4] R. J. Brtlett, WIREs Comput. Mol. Sci. 2, 126 (2012), doi: /wcms