In various methods used in quantum chemistry, including the Coupled Cluster method to be treated in Section 3, the use of the formalism of Second Quantization [27] greatly simplifies the derivation of implementable equations. In Second Quantization, operators defined on the antisymmetric tensor space L2 are written in terms of annihilation and creation operators belonging to a fixed one particle spin basis of L2(R3 ×Σ), inducing a tensor basis of L2 as constructed in the last section. Operators are then completely determined by a corresponding set of coefficients, see [206] for results on the related concept of “matrix operators”. In this section, we will introduce annihilation and creation operators in part (i), leading in part (ii) to a mathematically rigorous definition of the (weak) Second Quantization Hamiltonian that will be used later.
(i) Annihilation and creation operators. We will in this part (i) have to utilize the antisymmetric, real valued space L2 = L2N for a varying number N of electrons.
Therefore, the spaces, operators etc. under consideration will be equipped with an index N indicating the number of particles where needed. Because notations used are intuitive and only needed in this part, we will not introduce them at all length. From part (ii) on, the particle number N will be fixed again; consequently, the indices will be omitted again. Let us introduce the (fermion) Fock space [77]
F :=
∞
M
N=0
L2N,
where the symbol L
denotes the direct orthogonal sum of the antisymmetric N-fold tensor product Hilbert spaces L2N. In F, we may embed any N-electron state vector ΨN ∈ L2N by writing it as (δk,NΨN)k∈N = (0,0, . . . ,0,ΨN,0, . . .). Note that the case N = 0 is also included in the above definition of the space F. For this case, L20 is (by definition of the tensor product) the underlying field of the complex numbers. This is a one-dimensional vector space, thus containing up to a phase factor only one state vector called the vacuum state |i. This state is in some sense the starting point for the formalism of second quantization, as any state vector may be created from it by the use of the creation operators introduced below.
Motivated by our application in Section 3, the following definition of those operators also allows for non-orthogonal basis sets and functions f not contained in the basis BΣ; the naming of the operators introduced will be motivated in the remarks given afterwards.
1.5 Second Quantization 23
Definition 1.19. (Creation and annihilation operators)
(i) For 1≤N ∈N, f ∈L2(R3× {±12}) and Ψµ∈BN, we at first define a†f,NΨµ := QN+1 f ⊗Ψµ
, (1.54)
where
QN+1 :L2R,N+1 →L2N+1
is the mapping from Definition 1.16.
By linear continuation of the above definition to linear combinations, and by closing [206] the operator in L2N, we extend16 eacha†f,N to a linear map
a†f,N :L2N →L2N+1.
For N = 0, we let a†f,0|i = f ∈ L21. The creation operator or creator of f is now defined by
a†f :F→F, a†f :=
∞
M
N=0
a†f,N. (1.55)
In particular, if f = χP from the fixed basis set B, we will denote a†P := a†χ
P for convenience.
(ii) We define theannihilation operator orannihilator af :F→F off as the adjoint of the creation operator a†f :F → F of f. The annihilator of a basis function χP ∈ B is denoted by aP.
We remark that for any normed finite linear combination Ψ =PM
n=1αµΨµ of basis func-tions, it is easy to show ka†f,NΨkL2 ≤ kfkL2, so (as was already asserted above,) the closure [206] ofa†f,N is an operatorL2N →L2N−1.
Additionally, because the creation operatora†f is closed, the adjoint of the adjoint of a†f isa†f, so that the adjoint of the annihilatoraf is indeed a†f, as indicated by the notation.
16See the remarks after this definition.
24 1 A FRAMEWORK FOR ELECTRONIC STRUCTURE CALCULATION
Later on, we will need the properties of the annihilation and creation operators compiled in the following lemma. The proofs can - given in the so-called “ket notation”17- be found in [103, 201] or are generalized from them straightforwardly, so they are omitted here.
Lemma 1.20. (Properties of the creation and annihilation operators) (i) For f ∈span{χP1, . . . χPN}, we have
a†f ⊗ˆNn=1χPn
= 0.
(ii) The action of af on an N-electron elementary tensor Ψ =⊗Ni=1χPi is given by
˜
afΨ :=
N
X
n=1
(−1)n−1hf, χPni Q
⊗n−1i=1 χPi
⊗ ⊗Ni=n+1χPi
. (1.56) (iii) In particular, there holds for Ψµ=⊗Ni=1χPi ∈B and Pi ∈ {P1, . . . , PN} that
aPi,N ⊗ˆNn=1χPn
= (−1)i−1Q
⊗i−1n=1χPn
⊗ ⊗Nn=i+1χPn
∈ L2N−1, so that aPi “annihilates” the basis function χPi and adds a corresponding sign.
(iv) For J /∈ {P1. . . PN},
aJ ⊗ˆNn=1χPn
= 0,
where 0 is the zero vector 0∈F (not to be confused with the vacuum state).
(v) Using the anticommutator [A, B]+ = AB +BA, there hold the anticommutator relations
[af, ag]+= 0, [a†f, a†g]+ = 0, (1.57) and if f, g∈L2(R3× {±12}) are orthogonal,
[af, a†g]+ = [a†f, ag]+= 0. (1.58) If B is an orthogonal one-electron basis,
[aP, a†Q]+= [a†P, aQ]+ =δP,Q (1.59) for allP, Q∈ I, whereδP,Q = 1only ifP =QandδP,Q= 0otherwise. Furthermore, all creation and annihilation operators are nilpotent,
afaf =a†fa†f = 0. (1.60) The importance of creation and annihilation operators is rooted in the fact that any linear operator onFmay be written as a sum of polynomials in creation and annihilation operatorsa†I, aI [44]. In particular, this of course includes the Hamiltonian, and its second quantization form will be introduced in the next section.
17In quantum chemistry, Slater determinants are usually denoted in the ket notation |P1, . . . , PNi, related to the above by|P1, . . . , PNi:=QΨµ for any (not necessarily sorted) indexµ= (P1, . . . , PN).
1.5 Second Quantization 25
(ii) The weak Hamiltonian of Second Quantization. For numerical treatment of the Schr¨odinger equation, one usually fixes a basisBk of H1k as constructed in Definition 1.16. For this basis (or rather for a finite selection from Bk in practice), the matrix elements h(Ψµ,Ψµ) of the bilinear form h then have to be evaluated. By definition of h, this task involves for each pair Ψµ,Ψν of Slater determinants with coinciding spin a high-dimensional integration overR3N, which would in view of the size of the tensor basis and additionally the dimension of the integration domain quickly become an infeasible task even for very small N. It is therefore an essential fact that in an orthonormal basis set, this task reduces due to the structure of the Hamiltonian to the computation ofO(|D|4) integrals, where|D|is the size of the used discretised one particle basis set{χp|p∈D⊆I}.
Additionally, those integrals are now involving at most 2 spatial variablesxi,xj, i.e. they are integrals over R6. We now introduce notations for those integrals, and afterwards derive the weak Hamiltonian of Second Quantization.
Definition 1.21. (Antisymmetric integrals of quantum chemistry)
ForχP, χQ, χR, χS ∈BΣ, we introduce the single electron interaction integrals
hP,Q := 1
2h∇χP,∇χQi+
K
X
ν=1
hχP, Zν
|xi−Rν|χQi (1.61) and theelectron pair interaction integrals18,19
hP Q|RSi := X
s,s0∈{±12}
Z
R6
χP(x, s)χQ(y, s0) 1
|x−y|χR(x, s)χS(y, s0) dxdy (1.62)
as well as theantisymmetrized integrals
hP QkRSi := hP Q|RSi − hP Q|SRi. (1.63)
18The notation for electron pair interaction integrals introduced here the is the standard physicist’s notation for the Coulomb integrals, which may be read as abbreviation for the inner product in (1.62).
Note though that concurrently to this, the so-called Mullikan notation (P RkQS) is preferred by most chemists, related to the above by (P RkQS) = hP QkRSi. To avoid confusion, we will stick to the physicist’s notation in this work.
19Note that (except for the case of closed shell calculations, i.e. k =N/2) the integrals depend not only on the indicesp, q, r, sfor the spin free basis functions, but on the spin orbital indicesP, Q, R, S, i.e.
e.g. hpQkRSi 6=hpQkRSiin general.
26 1 A FRAMEWORK FOR ELECTRONIC STRUCTURE CALCULATION
With these definitions at hand, we can now introduce the Second Quantization Hamilto-nian.
Lemma 1.22. (Second Quantization Hamiltonian)
By standard functional analysis [206], the bilinear formh:H1k×H1k defines a corresponding bounded linear operator Hˆ : H1k→H−1k , which maps Ψ∈H1k to a functional
HΨ :ˆ H1k →R, Ψ0 7→h(Ψ,Ψ0). (1.64) If B from (1.40) is anL2-orthonormal basis set, this operator is in terms of annihilation and creation operators given by
Hˆ = X
P,Q∈I
hP,Qa†PaQ + 1 2
X
P,Q,R,S∈I
hP QkRSia†Pa†QaSaR. (1.65)
Proof. Because of the linearity and continuity of h on H1k, it suffices to show the claim for all Slater basis function Ψµ =⊗Nn=1χQn,Ψν =⊗Nn=1χPn ∈ H1k. The conjecture thus is a consequence of the following equalities, see below for some comments.
h(Ψµ,Ψν) =
N
X
i=1
hPi,Qi Y
`6=i
hχQ`, χP`i +
N
X
i,j=1
hPiPjkQiQji Y
`6=i,j
hχQ`, χP`i
=
N
X
i=1
hPi,QihaQiΨµ, aPiΨνi+
N
X
i,j=1
hPiPjkQiQjihaQjaQiΨµ, aPjaPiΨνi
= X
P,Q∈I
hP,QhaQΨµ, aPΨνi+ 1 2
X
P,Q,R,S∈I
hP QkRSihaSaRΨµ, aQaPΨνi
= X
P,Q∈I
hP,Qha†PaQΨµ,Ψνi+ 1 2
X
P,Q,R,S∈I
hP QkRSiha†Pa†QaSaRΨµ,Ψνi
= hHΨˆ µ,Ψνi.
In the preceding, the representation of h in the first line follows from evaluation of h(Ψµ,Ψν) for the antisymmetric Ψ,Ψ0. As this is rather straightforward, we do not prove it here for sake of brevity; see [201] for the related Slater-Condon rules. The transition from the first to the second third line is due to (iii) of Lemma 1.20, while the third follows from (iv) of Lemma 1.20 and the fourth from the adjoint relation between aI and a†I. Additionally, symmetry of the coefficients and orthogonality of the basis functions were used.
1.6 Ellipticity results 27