An Introduction to Coupled Cluster Diagrams

In this section, we present a simple diagrammatic formalism popularized by Kucharski and Bartlett[20] by which one may construct the coupled cluster energy and amplitude equations far more quickly than by direct application of Wick's theorem.^4.6 We begin by describing some of the general features of the diagrams, including their relationship to the particle-hole formalism and how they may be used to represent normal-ordered dynamical operators. Next we describe how the operator diagrams may be connected together to form operator products in a manner analogous to Wick's theorem. We then construct the diagrammatic form of the CCSD energy and amplitude equations, and, as each new diagram is presented, we provide rules for its algebraic interpretation. The diagrams described here may be used to represent either wavefunctions, operators, or matrix elements, depending on the context of the mathematical analysis. However, the set of rules we will present for interpreting the diagrams algebraically will apply only to the matrix element representation, since that is the most appropriate context for the coupled cluster energy and amplitude equations.^4.7

We make use of the particle-hole formalism in diagrammatic analyses by drawing upward and downward directed lines that identify those orbitals which differ from those in the reference determinant, $\Phi_0$, as shown in Figure 1. Downward directed lines represent hole states (orbitals occupied in the reference) and upward directed lines represent particle states (orbitals unoccupied in the reference). Hence, one may interpret the fourth diagram of the figure as a single-determinant wavefunction that differs from the reference by a single excitation from orbital $\phi_i$ to orbital $\phi_a$. Furthermore, this convention implies that the reference wavefunction itself is represented by empty space as indicated in Figure 1(c).

(a)	(b)	(c)	(d)

Figure 1: Some basic components of coupled cluster diagrams: (a) hole lines; (b) particle lines; (c) the reference wavefunction, $\Phi_0$, represented by empty space; (d) a single-determinant wavefunction, $\Phi_i^a$, which differs from the reference by a single excitation.

Diagrams representing dynamical operators (such as the one- and two-electron components of the normal-ordered Hamiltonian, $\hat{H}_N$) are depicted by horizontal ``interaction lines'' with vertical directed lines like those in Figure 1 representing the annihilation and creation operator strings. We will choose different interaction lines to represent different types of operators (e.g., a dashed line to indicate components of the electronic Hamiltonian, a solid line for cluster operators, $\hat{T}_1$, $\hat{T}_2$, etc.). The directed lines emanate from ``vertices'' on the interaction line; each vertex represents the action of the operator on individual electrons. Thus, one-electron diagrams have one vertex, two-electron diagrams have two vertices, etc. Each vertex has two directed lines attached to it, one incoming and one outgoing, associated with the annihilation and creation operators of the operator's normal-ordered string. Since one-electron operators contain two second-quantized components (see, for example, Eq. [3.12]), their diagrammatic representations contain two directed lines. Similarly, diagrams representing two-electron operators contain four directed lines, three-electron operators contain six directed lines, etc. The upward and downward directions of these lines are dependent on the orbital subspaces in which the second-quantized operators act: q-creation operators^4.8 lie above the interaction line, whereas q-annihilation lines lie below the interaction line.

For example, we denote the one-electron component of the Hamiltonian, $\hat{F}_N$, by a dashed interaction line capped by an ``X''. This operator may be written in four fragments as shown in Figure 2. The first fragment, which involves only operators in the particle (unoccupied) space, has one q-creation line above the interaction line corresponding to the $a_{a}^{\dagger}$ component of its operator string, and one q-annihilation line below the interaction line corresponding to the $a_{b}^{\ }$ component. Similarly, the second fragment in the figure, which involves only operators in the hole (occupied) space, has one q-creation line above the interaction line corresponding to the $a_{j}^{\ }$ component of the operator string, and one q-annihilation line below the interaction line corresponding to the $a_{i}^{\dagger}$ component. The third $\hat{F}_N$ fragment contains only q-annihilation lines below the interaction line since the operator string consists only of $a_{i}^{\dagger}$ and $a_{a}^{\ }$ components. Finally, the fourth fragment contains only q-creation lines above the interaction line representing the $a_{a}^{\dagger}$ and $a_{i}^{\ }$ components of the operator string.

$\hat{F}_N$	=	${\displaystyle \sum_{ab}}f_{ab}\{a_{a}^{\dagger}a_{b}^{\ }\}$	+	${\displaystyle \sum_{ij}}f_{ij}\{a_{i}^{\dagger}a_{j}^{\ }\}$	+	${\displaystyle \sum_{ia}}f_{ia}\{a_{i}^{\dagger}a_{a}^{\ }\}$	+	${\displaystyle \sum_{ia}}f_{ai}\{a_{a}^{\dagger}a_{i}^{\ }\}$
	=		+		+		+
		0		0		-1		+1

Figure 2: Diagrammatic representation of each fragment of the one-particle component of the Hamiltonian operator, $\hat{F}_N$. The excitation level of each diagram is indicated beneath it. The interaction line is indicated by the dashed horizontal line capped by the ``X''.

$\hat{V}_N$	=	$\frac{1}{4}{\displaystyle \sum_{abcd}}\langle ab \vert\vert cd \rangle\{a_{a}^{\dagger}a_{b}^{\dagger}a_{d}^{\ }a_{c}^{\ }\}$	+	$\frac{1}{4}{\displaystyle \sum_{ijkl}}\langle ij \vert\vert kl \rangle\{a_{i}^{\dagger}a_{j}^{\dagger}a_{l}^{\ }a_{k}^{\ }\}$	+	${\displaystyle \sum_{iabj}}\langle ia \vert\vert bj \rangle\{a_{i}^{\dagger}a_{a}^{\dagger}a_{j}^{\ }a_{b}^{\ }\}$
	+	$\frac{1}{2}{\displaystyle \sum_{aibc}}\langle ai \vert\vert bc \rangle\{a_{a}^{\dagger}a_{i}^{\dagger}a_{c}^{\ }a_{b}^{\ }\}$	+	$\frac{1}{2}{\displaystyle \sum_{ijka}}\langle ij \vert\vert ka \rangle\{a_{i}^{\dagger}a_{j}^{\dagger}a_{a}^{\ }a_{k}^{\ }\}$	+	$\frac{1}{2}{\displaystyle \sum_{abci}}\langle ab \vert\vert ci \rangle\{a_{a}^{\dagger}a_{b}^{\dagger}a_{i}^{\ }a_{c}^{\ }\}$
	+	$\frac{1}{2}{\displaystyle \sum_{iajk}}\langle ia \vert\vert jk \rangle\{a_{i}^{\dagger}a_{a}^{\dagger}a_{k}^{\ }a_{j}^{\ }\}$	+	$\frac{1}{4}{\displaystyle \sum_{abij}}\langle ab \vert\vert ij \rangle\{a_{a}^{\dagger}a_{b}^{\dagger}a_{j}^{\ }a_{i}^{\ }\}$	+	$\frac{1}{4}{\displaystyle \sum_{ijab}}\langle ij \vert\vert ab \rangle\{a_{i}^{\dagger}a_{j}^{\dagger}a_{b}^{\ }a_{a}^{\ }\}$
	=		+		+
		0		0		0
	+		+		+
		-1		-1		+1
	+		+		+
		+1		+2		-2

Figure 3: Diagrammatic representation of each fragment of the two-particle component of the Hamiltonian operator, $\hat{V}_N$. The excitation level of each diagram is indicated beneath it. The interaction line is indicated by the dashed horizontal line.

The two-electron fragment of the Hamiltonian may be partitioned in a similar manner as shown in Figure 3, with a dashed horizontal interaction line and with implicit antisymmetry with respect to permutation of the lines leaving or entering the left and right vertices. For example, in the third diagram, corresponding to a sum over the operator components, $\langle ia \vert\vert bj \rangle\{a_{i}^{\dagger}a_{a}^{\dagger}a_{j}^{\ }a_{b}^{\ }\}$, the diagram as shown may be written in four equivalent ways (differing only by a sign), each formed by permuting either the two outgoing lines or the two incoming lines from the left and right vertices:

$$\begin{tabular}{m{1.0in}} \\ \epsfig{file=diagrams/V/Viabj_1... ...ile=diagrams/V/Viabj_4.eps,height=3.0cm}\\ \\ \end{tabular}.$$

In addition, diagrammatic representations of the cluster operators, $\hat{T}$, are shown in Figure 4, with solid horizontal interaction lines. Since the cluster operators contain only q-creation strings (and thereby generate excited determinants from the reference wavefunction), they contain no lines below the horizontal bar. Furthermore, these representations are also fully antisymmetric in that exchange of any pair of outgoing or incoming lines introduces a change in the sign of the diagram. We will discuss this point in greater detail later when we introduce rules for interpreting the diagrams algebraically.

$\hat{T}_1$	=	${\displaystyle \sum_{ia}} t_i^a \{ a_{a}^{\dagger} a_{i}^{\ } \}$	=		+1
$\hat{T}_2$	=	${\displaystyle \frac{1}{4} \sum_{ijab}} t_{ij}^{ab} \{ a_{a}^{\dagger} a_{b}^{\dagger} a_{j}^{\ } a_{i}^{\ } \}$	=		+2
$\hat{T}_3$	=	${\displaystyle \frac{1}{36} \sum_{ijkabc}} t_{ijk}^{abc} \{ a_{a}^{\dagger} a_{b}^{\dagger} a_{c}^{\dagger} a_{k}^{\ } a_{j}^{\ } a_{i}^{\ } \}$	=		+3

Figure 4: Diagrammatic representation of the $\hat{T}_1$, $\hat{T}_2$, and $\hat{T}_3$ excitation operators. The excitation level of each diagram is indicated to its right. The interaction line is indicated by a solid horizontal bar.

Other than the operator representation above, we will interpret the diagrams in this chapter from bottom to top as matrix elements of operators (or operator products) between determinants. For the coupled cluster energy and amplitude equations shown in Eqs. [3.9] and [3.10], the pertinent matrix elements always contain the reference determinant, $\Phi_0$, on the right and either $\Phi_0$ or excited determinants such as $\Phi_{ij}^{ab}$ on the left. Diagrams are particularly convenient for constructing such matrix elements since they provide a straightforward method for evaluating the types of determinants to which individual operator fragments in Figures 2-4 may be applied or what determinants they produce. As an example, consider the fourth $\hat{F}_N$ fragment in Figure 2, which contains no lines below and two lines above the horizontal operator line. Since the reference wavefunction, $\Phi_0$, is represented by empty space, and a singly excited determinant, $\Phi_i^a$, by a pair of directed lines such as those in Figure 1(c), we may interpret the diagram from bottom to top to obtain the matrix element

$$\langle \Phi_i^a \vert \hat{F}_N \vert \Phi_0 \rangle = \begi... ...g{file=diagrams/F/Fai_labelled.eps,height=1.7cm}\end{tabular}.$$ (4.79)

A similar analysis may be applied to the two-electron operator in the third diagram in Figure 3, which contains particle-hole pairs of lines both above and below the interaction line. Each of these pairs may be interpreted as singly excited determinants to obtain the general matrix element

$$\langle \Phi_i^a \vert \hat{V}_N \vert \Phi_j^b \rangle = \b... ...psfig{file=diagrams/V/Viabj_4a.eps,height=3.0cm}\end{tabular}.$$ (4.80)

The cluster operator diagrams are particularly simple to interpret as matrix elements; the diagrams always involve the reference determinant on the right (because they contain no lines below the interaction line) and an excited determinant on the left, e.g.,

$$\langle \Phi_{ij}^{ab} \vert \hat{T}_2 \vert \Phi_0 \rangle =... ...g{file=diagrams/T2/T2_labelled.eps,height=1.5cm}\end{tabular}.$$ (4.81)

We also make use of a simple bookkeeping system[20] which indicates the ``excitation level'' a particular operator fragment produces. This value is determined by subtracting the number of q-annihilation lines from the number of q-creation lines and dividing the result by two. For example, the first and second one-electron Hamiltonian fragments shown in Figure 2 are assigned an excitation level of 0, since both the wavefunction to which they are applied (at the bottom of the diagram) and the wavefunction they produce (at the top of the diagram) differ from the reference by a single orbital; no net excitation is produced. The fourth one-electron fragment, however, has an excitation level of +1since it effectively produces a single excitation from the reference wavefunction. Two-electron Hamiltonian fragments have excitation levels ranging from +2 to -2, as indicated in Figure 3, and the $\hat{T}$ operators have the obvious excitation levels indicated in Figure 4.