60 3 THE CONTINUOUS COUPLED CLUSTER METHOD
to the (canonical, projected) Coupled Cluster method, e.g. termed CISD resp. CCSD if only basis functions corresponding to single and double excitations are included. The analysis in [190] now examines the approximation properties of the projected Coupled Cluster method to the “full CI” solution, and thus provides an analysis of the second discretisation step. On the other hand this first approach, taken mainly to circumvent the problems associated with the formulation of the Coupled Cluster method for the orig-inal, infinite-dimensional problem, does not allow for estimates with respect to the true solution Ψ ∈ H1k, and thus a priori excludes approaches where the size of the underly-ing one-particle basis Bdisc is varied. The latter are of interest in the context of error estimation though, especially in view of the fact that convergence of different CC models towards the limit within the full CI-space usually is rather fast, while the convergence of the full-CI solutions Ψdisc ∈ H1,disck to the continuous limit Ψ ∈ H1k is often rather slow with respect to the size of the underlying one-particle basis set.
In this part of this work, we will therefore formulate the Coupled Cluster equations in a coefficient space reflecting the continuous (“complete CI”) space H1 := H1k, and the re-sulting method will be termed “the continuous Coupled Cluster method”. First of all, the continuity properties of cluster operators in the respective function spacesH1, H−1 have to be established (Theorem 3.6), and indeed, this poses the main obstacle in the analysis of the continuous CC method. Once this is done, we will formulate the continuous CC equations and define the continuous CC function f. We prove thatf possesses the prop-erty of being locally strongly monotone in a neighborhood of the solution t∗ (Theorem 3.18); then, techniques from operator theory partly already used in [190] apply to obtain existence/uniqueness and convergence results (Theorem 3.21), and we will prove a goal oriented error estimator [22] for convergence of the energyE∗ (Theorem 3.24). Finally, we will indicate how the CC equations can be simplified to obtain computable expressions (Section 3.5) and show convergence of a quasi-Newton method (also formulated in the continuous space) when applied to the CC function.
3.1 Notations, basic assumptions and definitions 61
of spin ζk = −N/2 +k, which will be fixed in this section. A spin orbital χI contained Ψ0 in will be called occupied orbital, and this situation will be abbreviated by I ∈ occ.
Iff A /∈ occ, a spin orbital χA is called virtual orbital, denoted by A ∈ virt. It is a notational convention that in summations etc., occupied orbitals are labeled by letters I, J, K, . . .∈occ, virtual orbitals by letters A, B, C, . . .∈virt, and unspecified orbitals by lettersP, Q, R, . . . ∈ I, and we will also use this convention here.
For the discrete (“projected”) Coupled Cluster method in its simplest form, a (discrete) basis BΣ of so-called canonical orbitals is in practice provided by diagonalization of the final (discrete) Fock operator FHF = FHF,ΨHF, so that BΣ is an eigenbasis of the Fock operator, and this discrete setting was analysed in [190]. In the infinite dimensional setting,FΨHF does not allow for a complete eigensystem anymore, so that the formulation of the Coupled Cluster method and also the analysis from [190] do not extend straight-forwardly to the continuous setting. Also many of the more sophisticated CC schemes are not based on canonical orbitals (i.e. the eigenfunctions of the Fock operator), but use certain localization criteria to rotate the occupied orbitals (to e.g. Foster-Boys-type orbitals [33], Pipek-Mazay-type orbitals [172] or enveloped localized orbitals [13]), use non-orthogonal bases for the virtual orbitals (e.g. the projected atomic orbitals (PAOs) in the LCCSD approach [100, 194]), or enhance the virtual space by specialized basis functions taking the electron-electron cusp in account (as e.g. the recent powerful r1,2 -andf1,2- methods [119]). Nevertheless, if a HF ground state exists (see 2.1(i)), the infinite dimensional Fock operator FHF possesses an invariant subspace belonging to N lowest eigenvalues, and the L2− and FHF-orthogonality between virtual and occupied orbitals is maintained in all of the aforementioned methods. Motivated by this, we will base our analysis on the following mild assumptions covering all of the above cases.
Assumption 3.1. We have a symmetric mapping F :H1(R3× {±1
2})→H−1(R3× {±1 2})
at hand that induces a norm spectrally equivalent to the H1(R3× {±12})-norm, i.e. there areγ,Γ>0 such that
γ hϕ, ϕi1 ≤ hF ϕ, ϕi ≤ Γ hϕ, ϕi1 for all ϕ∈H1(R3× {±1
2}). (3.3) For the spin basis (3.1), we will suppose that χP are eigenfunctions of the z-spin operator SNz, see Section 1.2(ii). We also demand that {χI|I ∈ occ} is a basis of an invariant subspace of F, that is, there holds
hF χI, χAi = hχI, χAi = 0 for all I ∈occ, A∈virt. (3.4)
62 3 THE CONTINUOUS COUPLED CLUSTER METHOD
By Lemma 1.25, the above mapping F induces a norm on the tensor space L2R that is equivalent to the H1-norm. Note that the condition (3.3) is in particular fulfilled by the (continuous or discrete) shifted Fock operatorF =FHF −µI and also by suitable Kohn-Sham Hamiltonians, see Lemma 1.23 and Remark 2.8.
We will in this section abbreviate by H1 := H1k, L2 := L2k the corresponding spaces of real-valued, antisymmetric functions of fixed spin number 0 ≤ k ≤ N/2 introduced in Section 1.2, and rewrite the weak Schr¨odinger equation with the dual argument to the right,
hΨ,( ˆH−E∗)Ψi = 0 for all Ψ∈H1 :=H1k, (3.5) to stay consistent with other literature on the Coupled Cluster method. For convenience, we will impose to the solution of the weak Schr¨odinger equation theintermediate normal-ization condition, i.e. we drop the normalnormal-ization condition (1.11) and instead we look for eigenfunctions Ψ = Ψ0+ Ψ∗ for which
hΨ0,Ψ∗i= 0, i.e. hΨ,Ψ0i= 1 (3.6) is fulfilled. This poses no additional restriction if the reference solution is sufficiently good so that hΨ,Ψ0i 6= 0, and we assume this latter condition from now on. The eigenfunction Ψ is thus fixed by its component Ψ∗ ∈ span{Ψ0}⊥, and we will now, as a first step towards the CC formulation, rewrite Ψ∗ in terms of so-called excitations of the reference determinant Ψ0. We start with some definitions.
Definition 3.2. (Operator strings and excitation operators)
Let b1, . . . , bn be any canonical creation or annihilation operators. An operator of the formS :F→F, SΨ =b1. . . bnΨ, will be called operator string.
An operator string
XIA1,...,Ar
1,...,Ir =a†A
1. . . a†AraI1. . . aIr (3.7) is called excitation operator if I1 < . . . < Ir ∈ occ, A1 < . . . < Ar ∈ virt, and if in {I1, . . . , Ir} and {A1, . . . , Ar}, the numbers of contained “spin up” indices coincide. The number r = r(XIA1,...,Ar
1,...,Ir ) ≤ N of annihilators (resp. creators) contained in XIA1,...,Ar
1,...,Ir is called the (excitation) rank of XIA11,...,I,...,Arr.
Definition 3.3. (Indices and index operations)
(i) We denote by µ0 := (1, . . . , k,1, . . . , N −k) the index belonging to the reference determinant, and let
M∗ =M\{µ0}, M∗k =Mk\{µ0} with the index sets M,Mk from Definition 1.18 .
3.1 Notations, basic assumptions and definitions 63
(ii) For a multi-indexµ∈ M∗k, corresponding to an excitation operatorXµ=XIA11,...,I,...,Arr = a†A
1. . . a†A
raI1. . . aIr and a determinant Ψµ= ΨAI1,...,Ar
1,...,Ir ,we define its rank as r(µ) :=
r(Xµ).
IffP ∈ {I1, . . . , Ir, A1, . . . , Ar} we say thatP is contained in µ, P ∈µin short. For µ0, we define thatP /∈µ0 for all P ∈ I.
(iii) For two multi-indices ν, µ ∈ M, we write µ ⊆ ν iff for all indices P ∈ I, P ∈ µ implies P ∈ν.
(iv) Obviously, for each pair µ⊆ν ∈ Mk, there is exactly one multi-index α⊆ν ∈ Mk determined by the condition that P ∈ α iff P ∈ ν, P /∈ µ, and we will denote the relation between these indices by ν=µ⊕α, α=ν µ.
Additionally, we define for the situations where ⊕, is not defined by the above that µ⊕α=−1 if{P|P ∈µ} ∩ {P|P ∈α} 6=∅, and ν µ=−1 for the caseµ6⊆ν.
(v) Finally, we declare for convenience thatXµ0 =I, define that for coefficients turning up in summations etc. c−1, t−1, . . .= 0, and also let Ψ−1 = 0, X−1 = 0.
Remarks 3.4. (Properties of determinants and excitation operators)
(i) An excitation operator XIA11,...,I,...,Arr maps the reference determinant Ψ0 ∈Bk (of fixed spin number k) by definition to a Slater determinant Ψµ ∈ Bk by replacing the occupied orbitals I1, . . . , Ir by the virtual orbitals A1, . . . , Ar. More precisely, we have a one-to-one correspondence between the basis functions Ψµ ∈ Bk and the excitation operators XIA1,...,Ar
1,...,Ir , and because both notations will be convenient in some situations, we will identify the index sets and therefore write Ψµ= ΨAI1,...,Ar
1,...,Ir :=
XIA1,...,Ar
1,...,Ir Ψ0. Also, we will denote the excitation operator taking Ψ0 to Ψµ by Xµ; further, we will call Ψµ= ΨAI1,...,Ar
1,...,Ir anr−fold excited determinant or determinant of excitation rankr. Note also that by Lemma 1.20,(XIA11,...,I,...,Arr)†=a†I1. . . a†IraA1. . . aAr, so that
(XIA1,...,Ar
1,...,Ir )† XIA1,...,Ar
1,...,Ir Ψ0 = (XIA1,...,Ar
1,...,Ir )† ΨAI1,...,Ar
1,...,Ir = Ψ0, (3.8) and the adjoints of excitation operators are therefore sometimes termed decitation operators.32
(ii) For two determinants Ψr,Ψs of excitation ranks r 6=s,
hΨr,Ψsi=hΨr,ΨsiF = 0 (3.9) due to (3.4).
32Note that (XIA1,...,Ar
1,...,Ir )† is not the inverse ofXIA1,...,Ar
1,...,Ir though.
64 3 THE CONTINUOUS COUPLED CLUSTER METHOD
(iii) It follows from the anticommutator relations 1.20(v) that all operators contained in any excitation operators anticommute. Therefore, Definition 3.3 implies that for all indices α, β ∈ Mk, Xα⊕β also defines an excitation operator, and that
XαXβ = Xα⊕β = Xβ⊕α = XβXα. (3.10) The same holds for products of decitation operators Xα†Xβ† =Xα⊕β† . Also,
Xα†Xβ =Xβ α, XαΨβ = Ψβ⊕α, Xα†Ψβ = Ψβ α. (3.11) Observation/Definition 3.5. (Cluster operator)
Due to Remark 3.4(i), every intermediately normed Ψ∈L2 can be expanded in the tensor basis Bk as
Ψ = Ψ0+ Ψ∗ = Ψ0+ X
µ∈M∗k
tµXµΨ0 =: (I +TΨ∗)Ψ0 (3.12) of at most N-fold spin-k-excitations XµΨ0 of the reference determinant Ψ0 ∈ Bk. The operator TΨ∗ introduced in the above remark will be called cluster operator of Ψ∈L2.