Introduction
One of the most important problems of quantum chemistry is solving the notorious Schrödinger equation of a many-body quantum system. This equation is often unsolvable, and one therefore has to resort to approximations. Traditional and more hands-on approximation schemes1 are often imprecise, tedious, or numerically inconvenient. Therefore there are a lot of different approaches to this matter, one of which is coupled-cluster theory. Coupled-cluster methods transform the Schrödinger equation from a problem of finding an eigenfunction into a problem of finding an operator, which maps a zero-order eigenfunction ansatz into the actual eigenfunction of the corresponding Hamiltonian. This operator belongs to an algebra of bounded operators, and therefore it is natural to approach this theory using C*-algebra theory.
1: such as e.g. many different perturbative approaches.
Fock space
Let \(\mathcal{H}\) be the single-particle Hilbert space, i.e. a Hilbert space where its elements correspond to single-particle quantum states. Using this, we can construct the full Fock space as the following direct sum
\(\mathcal{F(\mathcal{H})} := \bigoplus_{0 \leq n} \mathcal{H}^{\otimes n} = \mathbb{C} \oplus \mathcal{H} \oplus (\mathcal{H}\otimes \mathcal{H}) \oplus (\mathcal{H}\otimes \mathcal{H}\otimes \mathcal{H})\cdots,\)
where \(\mathcal{H}^{\otimes 0} \cong \mathbb{C}\) by convention. In this article, we want to focus on the fermionic subspace of this Fock space, and in order to do so, we define the following unitary representation
\( \pi \colon S_n \to \mathcal{U}(\mathcal{H}^{\otimes n}) \)
\( \xi_1 \otimes \cdots \otimes \xi_n \mapsto \xi_{\sigma(1)} \otimes \ \cdots \otimes \xi_{\sigma(n)} \)
That is, the operators \(\pi(\sigma):= u_\sigma\) shuffle the components of the elementary tensor \(\xi_1 \otimes \cdots \otimes \xi_n\) in accordance with the permutation \(\sigma \in S_n\). Using this representation, we can construct the following projections
\(P_n: \mathcal{H}^{\otimes n} \to P_n(\mathcal{H}^{\otimes n}) := \bigwedge^n \mathcal{H} \)
\(\xi_1 \otimes \cdots \otimes \xi_n \mapsto \frac{1}{n!}\sum_{\sigma \in S_n} (-1)^{|\sigma|} u_\sigma (\xi_1 \otimes \ \cdots \otimes \xi_n),\)
where \(|\sigma|\) is the sign of the permutation \(\sigma\), i.e. its number of transpositions. This projection maps the elementary tensor to its "anti-symmetric counterpart", i.e. the corresponding elementary tensor where if you interchange two of its components, it's equal to the initial vector, but with an additional minus sign. The image of this operator is most often referred to as the (\(n\)-fold) exterior product of \(\mathcal{H}\), and its anti-symmetric properties reflect the symmetry of fermions and the Pauli exclusion principle of quantum mechanics2. We also denote the elements of this subspace using the following wedge-product notation
\(\xi_1 \wedge \cdots \wedge \xi_n := \sqrt{n!}P_n(\xi_1 \otimes \cdots \otimes \xi_n) \in \bigwedge^n \mathcal{H},\)
where the prefactor in front of \(P_n\) is chosen such that the inner-product in the \(n\)-fold exterior product is normalized. Using these projections, we can define the Fermi-Fock space according to
\(\mathcal{F}_{-}(\mathcal{H}) := \bigoplus_{0 \leq n} P_n(\mathcal{H}^{\otimes n}) = \bigoplus_{0 \leq n} \bigwedge^n \mathcal{H}\)
which is the subspace in which our fermionic many-body states reside.
2: c.f. any standard textbook on elementary quantum mechanics.
The CAR-algebra
Creation- and annihilation operators
For each component of the Fermi-Fock space, we define the following class of operators
\(c_n^\dagger(\xi) \colon \bigwedge^n \mathcal{H} \longrightarrow \bigwedge^{n+1} \mathcal{H} \)
\(\eta_1 \wedge \cdots \wedge \eta_n \longmapsto \xi \wedge \eta_1 \wedge \cdots \wedge \eta_n \)
with corresponding adjoint operators
\(c_n(\xi) \colon \bigwedge^{n + 1} \mathcal{H} \longrightarrow \bigwedge^{n} \mathcal{H}\)
\(\eta_1 \wedge \cdots \wedge \eta_{n + 1} \longmapsto \sum_{j = 1}^{n + 1} (-1)^{j + 1} \langle \eta_j, \xi \rangle \eta_1 \wedge \cdots \hat{\eta_j} \cdots \wedge \eta_{n + 1}\) (9)
where \(\hat{\eta}_j\) indicates that the \(\eta_j\)-component has been removed from the wedge-product, making it an element of the \(n\)-fold rather than the \(n+1\)-fold exterior product. If \(\xi\) and the family \(\{\eta_j\}_{j = 1}^{n + 1}\) form an orthonormal set3, then eq. (9) collapse into a single term which is equal to zero if \(\xi \notin \{\eta_j\}_{j = 1}^{n + 1}\), otherwise it's equal to the wedge-product where the \(\xi\)-component has been removed, modulo a possible minus sign. In this case, \(c_n^\dagger(\xi)\) "creates" a fermion in state \(\xi\) and \(c_n(\xi)\) "annihilates" the same state. These operators are therefore known as creation and annihilation operators in chemistry and physics literature.
3: Which most often is the case in physics and chemistry.
Defining the CAR-algebra
The operators defined in the previous section are bounded and in particular independent of the index \(n\), and we can therefore define the following creation- and annihilation operators acting on the whole Fermi-Fock space
\(\{c^\dagger(\xi), c^\dagger(\eta)\} = c^\dagger(\xi)c^\dagger(\eta) + c^\dagger(\eta)c^\dagger(\xi) = 0 \)
\(\{c(\xi),c(\eta)\} = 0\)
\(\{c^\dagger(\xi),c(\eta)\} = \langle \xi , \eta \rangle I_{\mathcal{F}(\mathcal{H})}\)
which are the so-called canonical anti-commutation relations (CAR) [1]. These operators are anti-linear and linear mappings from \(\mathcal{H} \to \mathcal{B}(\mathcal{F}_-(\mathcal{H}))\) respectively, and we define their the image as follows4
\(c^\dagger(\mathcal{H}) = \{c^\dagger (\xi) \colon \xi \in \mathcal{H}\}\)
\(c(\mathcal{H}) = \{c(\xi) \colon \xi \in \mathcal{H}\}.\)
Then we can define the following C*-algebra action on Fermi-Fock space
Definition. (The CAR-algebra) The algebra of canonical anti-commutation relations (CAR-algebra) is the C*-algebra generated by the image of the creation- and annihilation operators, i.e.
\(\mathcal{CAR}(\mathcal{H}) := C^*(c^\dagger(\mathcal{H}), c(\mathcal{H})),\)
which is a unital C*-subalgebra of \(\mathcal{B}(\mathcal{F_-}(\mathcal{H}))\). Furthermore, for a given Hilbert space, this algebra is unique up to *-isomorphism [3].
4: Here \(\mathcal{H}\) can be viewed as a discrete group with addition. We could also simply use a basis for \(\mathcal{H}\) rather than the whole Hilbert space.
Irreducibility of the CAR-algebra and second quantization
Using Shur's lemma, one can quite easily prove that
Theorem. ([3]) \(\mathcal{CAR}(\mathcal{H}) \subset \mathcal{B}(\mathcal{F_-}(\mathcal{H}))\) is irreducible.
The same lemma implies the following consequences:
Lemma. (Cyclicity of the CAR-algebra) \(\forall \eta \in \mathcal{F}_-(\mathcal{H})\), \(\mathcal{CAR}(\mathcal{H})\) is cyclic, i.e.
\(\big[\mathcal{CAR}(\mathcal{H})\eta\big] := \overline{\mathrm{span}\{T\eta \colon T \in \mathcal{CAR}(\mathcal{H})\}}^{\|\cdot\|} = \mathcal{F}_-(\mathcal{H}) \quad \forall \eta \in \mathcal{F}_-(\mathcal{H}).\)
This statement means that we can obtain any element of our Fermi-Fock space by acting with an element of the CAR-algebra on an arbitrary vector. In many-body theory, this vector is most often chosen to be the unital so-called \say{vacuum state} \(\Omega \in \bigwedge^0 \mathcal{H} \cong \mathbb{C}\). In this case, we can formulate this lemma using more physical terminology: we can obtain any many-body state by repeatedly acting with creation operators on the vacuum state.
Lemma. (Density of the CAR-algebra) \(\mathcal{CAR}(\mathcal{H}) \subset \mathcal{B}(\mathcal{F}_-(\mathcal{H}))\) is dense in the strong operator topology (SOT), i.e. \(\forall U\) SOT-open neighborhoods of \(T \in \mathcal{B}(\mathcal{F}_-(\mathcal{H}))\), there exists \(T' \in \mathcal{CAR}(\mathcal{H}) \cap U\) s.t. \(\|T\eta - T'\eta\| < \epsilon\), \(\forall \eta \in V \subset \mathcal{F}_-(\mathcal{H})\) and \(\epsilon > 0\) (which depends on \(U\)).
So if we are given a bounded operator acting on Fermi-Fock space, in particular a self-adjoint operator corresponding to a quantum observable, we can approximate this operator arbitrarily well by replacing it with a corresponding element of the CAR-algebra5. This, combined with the previous statement, makes up the mathematical foundation of second quantization6 and therefore many-body physics in general.
5: Although this statement guarantees its existence, it doesn't show us how to explicitly obtain such an operator. If this is of interest, c.f. any standard text on many-body physics.
6: i.e. the procedure of writing states and operators using elements of Fermi-Fock space and the CAR-algebra (in the fermionic case).
Finite-dimensional coupled-cluster theory
In this section, we make our single-particle Hilbert space somewhat more concrete by considering an orthonormal family of states which is also a subset of square-integrable functions, i.e.
\( \{\phi_p\}_{p = 1}^D \subset L^2(X),\)
where \(D\) is a positive integer and \(X\) is some arbitrary physical configuration space7. Then, for a many-body system consisting of \(N \leq D\) fermions, we get the following
\( \mathrm{dim}(\bigwedge^N \mathcal{H}) = \binom{D}{N} \\ \mathrm{dim} (\mathcal{F_-(\mathcal{H})}) = \sum_{i = 0}^N \binom{D}{i} = 2^D. \)
Note in particular that the direct sum of the Fermi-Fock space terminates when \(n \geq D\), because if we form a wedge-product of more than \(D\) elements of our orthonormal family, then there must be two equal components, which due to the anti-symmetric properties of the exterior products of \(\mathcal{H}\) implies that this vector must be zero.
If we are given a Schödinger equation, we can use its Hamiltonian operator to find such a family by e.g. using the Hartree-Fock method8. In this case, each state has a corresponding energy eigenvalue, and we can therefore order this set accordingly. We can therefore also express \(\bigwedge^N \mathcal{H}\) as follows:
\( \bigwedge^N \mathcal{H} = \mathrm{span}(\Phi_0) + \{\Phi_0\}^\perp\)
where \(\Phi_0:= \phi_1 \wedge \cdots \wedge \phi_N\) is the wedge-product of the \(N\) first elements of this ordered family, which is our zero-order approximation for the solution to our given Schrödinger equation.
7: Typically \(\mathbb{R} \times \{\uparrow, \downarrow \}\), where the former represents space and the latter represent spin.
8: The details of this procedure are not of importance in this discussion, c.f. any standard textbook on quantum chemistry
Excitation operators
Before proceeding, we will need some notation. Define
- \(p = 1, 2, \cdots, N\): occupied indices (labelled by \(i, j, k, \cdots \in \mathcal{O}\)).
- \(p > N\): virtual indices (labelled by \(a, b, c, \cdots \in \mathcal{V}\)).
The reason for the terminology "occupied" and "virtual" us as follows: let there be \(N\) fermions in a quantum system where there is \(D \geq N\) available quantum states. If temperatures are low, these fermions will occupy the \(N\) first states (ordered according to energy) and the rest will be virtual states for which these fermions can be excited into. In this notation, we can define single excitation operators:
\(X_i^a \colon \mathrm{span}(\Phi_0) \to \{\Phi_0\}^\perp \nonumber \\ \phi_1 \wedge \cdots \wedge \phi_i \wedge \cdots \wedge \phi_N \mapsto \phi_1 \wedge \cdots \wedge \phi_a \wedge \cdots \wedge \phi_N, \)
That is: \(X_i^a\) replaces \(\phi_i\) by \(\phi_a\), i.e. it takes the \(i\)'th occupied quantum state and excites it into the \(a\)'th virtual state. Since our family of states is orthonormal, we can write
\(X_i^a = c^\dagger (\phi_a) c(\phi_i).\) (24)
Using eq. (24) and our canonical anti-commutation relations, its easy to verify that these operators commute and that they are nilpotent, i.e.
\([X_i^a,X_j^b] = 0 \\ (X_i^a)^2 = 0.\)
We can also define multiple (double, triple, etc.) excitation operators in a similar manner
\(X_{ij\cdots}^{ab\cdots} = X_i^a X_j^b \cdots.\)
Furthermore, since we can at most act with \(N\) annihilation operators on \(\Phi_0\), the multiple excitation operators are of order at most \(N\).
The Cluster algebra
An alternative (and more shorthand) way of labelling multiple excitation operators is using the following matrix notation
\(\mu = \begin{pmatrix} a_1 & a_2 & \cdots & a_N\\ i_1 & i_2 & \cdots & i_N\end{pmatrix},\)
where the first row corresponds to the virtual indices and the second row corresponds to the occupied indices of its corresponding operator. Furthermore, we assume that the rows are individually ordered according to energy. In this notation, the product of two multiple-excitation operators (containing \(L\) and \(M\) single-excitation operators respectively) can be written as
\(X_\mu X_\nu = X_{i_1}^{a_1} \cdots X_{i_L}^{a_L} X_{j_1}^{b_1} \cdots X_{j_M}^{b_M} = (-1)^{s(\mu, \nu)} X_{k_1}^{c_1} \cdots X_{k_{L + M}}^{c_{L + M}}\)
\( = (-1)^{s(\mu, \nu)} X_\eta \qquad \eta = \begin{pmatrix} c_1 & c_2 & \cdots & c_{L + M}\\ k_1 & k_2 & \cdots & k_{L + M}\end{pmatrix},\)
for unique \(\eta\) and where the index sets \(\{k_n\}, \{c_n\}\) are ordered according to energy. Furthermore, this product is non-zero if and only if \(L + M \leq N\) and the indices of the rows in \(\eta\) are all distinct.
Definition. (Cluster algebra) The cluster algebra \(\mathcal{A}\) is the commutative and nilpotent algebra generated by the set of single excitation operators \(\{X_i^a \colon (i,a) \in \mathcal{O}\times\mathcal{V} \} := \{X_i^a\}\).
Furthermore, since there are finitely many single-excitation operators when \(\mathcal{H}\) is finite-dimensional, the cluster algebra is a finite-dimensional subalgebra of \(\mathcal{B}(\bigwedge^N \mathcal{H}) \subset \mathcal{B}(\mathcal{F_-(\mathcal{H}))}\), and since \(\mathcal{F}_-(\mathcal{H})\) is finite-dimensional, we immediately get that \(\mathcal{A}\) is a Banach algebra.
Schrödinger equation
Given the Schrödinger equation9
\(H|\Psi\rangle = E|\Psi\rangle\) (30)
we make the following assumption
\(|\Psi\rangle = |\Phi_0\rangle + |\Phi^*\rangle \qquad \langle\Psi|\Phi_0\rangle = 1,\)
where \(|\Phi^*\rangle\) is often assumed to be small in norm compared to \(|\Phi_0\rangle\) for computational convenience. By construction of \(\mathcal{A}\) and cyclicity of the CAR-algebra, we get that \(\bigwedge^N \mathcal{H}\) is generated by the cluster operators, i.e.
Lemma. (Exponential ansatz) The solution to eq. (30) can be written as
\(|\Psi\rangle = (I + S)|\Phi_0\rangle = \exp\{T\} |\Phi_0\rangle\) (32)
for unique cluster operators \(S, T \in \mathcal{A}\).
The exponential ansatz in eq. (32) is well defined since \(\mathcal{A}\) is a Banach algebra10. Its one-to-one because the series expansion of \(\exp{\cdot}\) terminates at order \(N\), and since \(\mathcal{A}\) is finite dimensional, we get finitely many terms. By ordering these according to excitation order, we can compare them with the corresponding terms in \(S\) and thereby establish injectivity. Using this lemma, we can rewrite the Schrödinger equation accordingly
Theorem. (Coupled-cluster theory) Let \(\Psi, \Phi_0\) be given as in the previous statements. Then we can find cluster operator \(T \in \mathcal{A}\) such that
\(H \exp\{T\} |\Phi_0\rangle = E \exp\{T\} |\Phi_0\rangle.\) (33)
Consequently, the problem of solving the Schrödinger equation has been transformed into a problem of finding an element \(T \in \mathcal{A}\) given a zero-order approximation \(\Phi_0\).
9: Using Diracs braket notation and assuming the Hamiltonian is written in its second quantized form.
10: This is an almost trivial example of the holomorphic functional calculus.
Baker-Campbell-Hausdorff (BCH) expansion
The above coupled-cluster equation can be rewritten as
\(\exp\{-T\}H\exp\{T\}|\Phi_0\rangle = E |\Phi_0\rangle.\)
Using the BCH-expansion formula [2], we can rewrite this as follows
\(\exp\{-T\}H\exp\{T\} = H +[H,T] + \frac{1}{2}[[H,T],T] + \cdots\)
which terminates to fourth order in \(T\) if \(H\) is a two-body operator, i.e. a forth-order polynomial in terms of creation- and annihilation operators [2]11. Thus the coupled-cluster equation of (33) is polynomial in \(T\). This makes coupled-cluster theory more numerically convenient [2].
11: This generalizes to \(n\)-body operators, but as far as chemistry is concerned, the world is described by a two-body operator.
Approximations
Coupled-cluster theory as written in eq. (33) is exact and therefore equivalent to the original Schrödinger equation of eq. (30). However, by transforming it from a problem of finding a wave function into a problem of finding an element in the cluster algebra, we can use a different arsenal of (non-perturbative)12 approximation schemes. A common way to approximate (33) is by restricting to a subspace of the cluster algebra, e.g. \(\mathcal{A}_S = \mathrm{span}(\{X_i^a\})\), \(\mathcal{A}_{SD} = \mathrm{span}(\{X_{i}^{a}, X_{jk}^{bc}\})\), etc.
12: i.e. it does not require the physical coupling terms to be small in energy compared to free (e.g. kinetic) energy.
References
- P. de la Harpe & V. Jones. An introduction to C*-algebras - The algebra of canonical anticommutation relations. Université de Genevé: section de mathematiques, 1995.
- S. Kvaal. Coupled cluster theory - an abstract development. 2022.
- Ola Bratteli & D. W. Robinson. Operator algebras and quantum statistical mechanics : 2 : Equilibrium states : models in quantum statistical mechanics. International series of monographs on physics. Springer, 1997. isbn: 3540614435.