... procedure.^2.1

We will denote those functions that are part of the occupied space with the subscripts i, j, k, $\ldots,$ those within the virtual space with a, b, c, $\ldots$and arbitrary functions which may lie in either space with p, q, r, $\ldots$

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... function.^2.2

The annihilation operator $a_{p}^{\ }$ is simply the Hermitian conjugate of the creation operator $a_{p}^{\dagger}$. An equivalent perspective on Eq. [2.14], therefore, is the annihilation operator $a_{p}^{\ }$ acting to the left on the bra-state, $\langle \Phi_0 \vert$, to give

$$\langle \phi_q \ldots \phi_s \vert a_{p}^{\ } = \langle \phi_... ...\dagger} \vert \phi_q \ldots \phi_s \rangle \right)^{\dagger}.$$

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... commute.^2.3

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue.

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...respectively.^2.4

The factors of 1/2 and 1/4 are included here to correct for the ``double counting'' resulting from the now unrestricted summations over i, j, a, and b.

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... contributions.^2.5

This terminology should not be confused with so-called disconnected diagrammatic contributions, which are discussed later in the chapter.

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... fragments.^2.6

It should be noted that the localized orbital requirement is used here strictly for ease of analysis, and the property of multiplicative separability of the coupled cluster wavefunction does not strictly depend on this computational requirement, as discussed in Ref. 88.

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... etc.^3.1

In second-quantization terminology, $\vert \Phi_{ij\ldots}^{ab\ldots} \rangle = a_{a}^{\dagger} a_{b}^{\dagger} \ldots a_{j}^{\ } a_{i}^{\ } \vert \Phi_0 \rangle$.

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...$\hat{T}$:^3.2

The inequality with the final term in this expression relies on the fact that the Hermitian adjoint of an excitation (cluster) operator, $\hat{T}$, is a de-excitation operator as, can be seen from the properties of its component annihilation and creation operators. For example, we note that

$$\hat{T}_1 = \sum_{ia} t_i^a a_{a}^{\dagger} a_{i}^{\ } \neq \... ...} = \sum_{ia} \left(t_i^a\right)^* a_{i}^{\dagger} a_{a}^{\ }.$$

On the other hand, the inverse of the exponentiated excitation operator, $e^{-\hat{T}}$, is also an excitation operator, as can be seen from its power series expansion,

$$e^{-\hat{T}} = 1 - \hat{T} + \frac{1}{2} \hat{T}^2 - \frac{1}{3!} \hat{T}^3 + \ldots.$$

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... obtain^4.1

The vacuum, $\vert \ \rangle$, is a state containing no electrons.

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...]:^4.2

Note that the use of the brackets, $\{\ \}_v$, around a string implies that the operators contained therein, except for any pair being contracted, exactly anticommute. Hence, a general term such as $\{ABC\ldots XYZ\}_v$ may be written exactly as $-\{BAC\dots XYZ\}_v$, without concern for the anticommutation relations.

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... crossing.^4.3

This sign rule only applies to fully contracted terms and assumes that one places all the contraction lines above the expression.

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...$a_{a}^{\dagger}$).^4.4

Note that this q-particle definition of annihilation and creation simply reverses the roles of second-quantized operators acting in the occupied (hole) space, but leaves the those acting in the unoccupied (particle) space untouched.

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... earlier.^4.5

Since the cluster operators commute, we have

$$\left[\left[\hat{H}_N,\hat{T}_1\right],\hat{T}_2\right] = \fr... ...{1}{2}\left[\left[\hat{H}_N,\hat{T}_2\right],\hat{T}_1\right].$$

Therefore, a factor of 1/2 does not appear in front of this term in the above expansion.

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... theorem.^4.6

Many varieties of diagrams have used throughout the chemical physics literature for many years (e.g., see Refs. 1, 2, 80, 117, and 119). The diagrammatic formalism we have chosen here has been frequently used in work by the Bartlett group among others[120] and is particularly straightforward for ``conventional'' coupled cluster and many-body perturbation theories.

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... equations.^4.7

The algebraic rules for interpreting the diagrams as operators or wavefunctions differ only slightly from the matrix element approach discussed here. We recommend Refs. 80 and 88 for additional information.

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... operators^4.8

See the earlier discussion of the particle-hole formalism for an explanation of q-creation and q-annihilation operators.

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... expression.^4.9

It is possible for groups of three or more lines to be identified as equivalent, though this can happen only in many-body perturbation theory, expectation-value coupled cluster theory, or unitary coupled cluster theory. For such diagrams, a prefactor of $\frac{1}{n!}$, where n is the number of electron lines, must be included.

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... lines.^4.10

The ``disconnected'' diagram of Eq. [4.119] is not unlinked since the inclusion of an additional $\hat{V}_N$fragment can connect its two components -- Harris et al.[80] have recommended that such terms should be called ``linkable.'' With terms such as ``disconnected,'' ``connected,'' ``linked,'' and ``unlinked'' used to describe diagrams, it is not surprising that these techniques have caused much confusion in the past.

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... wavefunction.^5.1

The choice of $\hat{F}_N$ as the zeroth-order Hamiltonian requires the use of either a spin-restricted (closed-shell) Hartree-Fock (RHF) or spin-unrestricted Hartree-Fock (UHF) determinant as the zeroth-order (reference) wavefunction. Since spin-restricted open-shell Hartree-Fock (ROHF) reference functions are not eigenfunctions of the spin-orbital $\hat{F}_N$, other partitionings are required.[127,128,129,130,,132,133,134]

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....^5.2

It should be noted that the ``procedure'' outlined here for computing E_T⁽⁴⁾ is certainly not the most efficient approach. As discussed more than two decades ago[139,136], the expression for E_T⁽⁴⁾ may be cast into the form

$$E_T^{(4)} = \frac{1}{36} \sum_{ijkabc} t_{ijk}^{abc(2)} D_{ijk}^{abc} t_{ijk}^{abc(2)},$$

where D_ijk^abc is the three-electron counterpart of D_ij^ab. Instead of storing individual triple excitation amplitudes, however, each contribution to the summation above is computed separately using equations involving only two-electron integrals and energy denominators.

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... example,^6.1

Of course, other partitionings of the four indices i, j, a, and b are equally valid. For example, the following equality also holds based on Eq. [6.14]:

$$\Gamma_i = \Gamma_{jab} \equiv \Gamma_j \otimes \Gamma_a \otimes \Gamma_b.$$

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... orbitals.^6.2

Although a spin-orbital formulation is conceptually simple, desirable properties such as spin-adaptation may be lost when the electronic state of interest is open shell, for example. A rigorously spin adapted theory must include spin-free definitions of the cluster operators, $\hat{T}$, and an appropriate (perhaps multi-determinant) reference wavefunction[156,157,39,158,41,42]. Such general coupled cluster derivations are beyond the scope of this chapter, though some of the issues associated with difficult open-shell problems are discussed in the next section.

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... species,^7.1

We wish to emphasize that the present discussion focuses only on high-spin open-shell systems to which a single-determinant reference wavefunction is applicable. Coupled cluster techniques for low spin cases, such as open-shell singlets, have been pursued in the literature for many years, however, and provide a fertile area of research[167,168,169,170,158].

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... energy,^7.2

The ROHF-CCSD energy is indeed completely spin projected as discussed in Refs. 35, 27, and 37, but is still different from that computed using a spin-adapted coupled cluster wavefunction.

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... orbitals.^7.3

This diagonalization affects neither the ROHF determinant itself nor the ROHF or CCSD energies due to the well-known invariance of those methods with respect to certain classes of orbital rotations[134].

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....^7.4

It should be noted that the goal of true spectroscopic accuracy may be unattainabledue to the implicit errors associated with the use of a Born-Oppenheimer, non-relativistic Hamiltonian to describe molecular systems.[225]

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