Review of biorthogonal coupled cluster representations for electronic excitation

Abstract

Single-reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states, the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison to a full (and hermitian) ISR method.

Appendices

Appendix 1: Order relations of bCC representations

A general proof of the canonical order relations in the lower left (LL) triangle of the bCC secular matrix can be found in Ref. [18]. A brief review of the derivation of these order relations is given in the following.

Let us first consider the simpler case of a one-particle operator $\hat{D},$ reading in second-quantized notation

$$ \hat{D}=\sum d_{pq} c^{\dagger}_p c_q $$

(143)

where $d_{pq}=\langle {\phi_p}|{\hat{d}}|{\phi_q}\rangle$ denote the one-particle matrix elements associated with $\hat{D}.$ The bCC representation of $\hat{D},$

$$ \begin{aligned} D_{IJ}&=\langle {\overline{\Upphi}_I}|{\hat{D}}|{\Uppsi^{0}_J}\rangle\\ &=\langle {\Upphi_I}|{e^{-\hat{T}} \hat{D} e^{\hat{T}}}|{\Upphi_J}\rangle \end{aligned}$$

(144)

was encountered in the treatment of transition moments and excited-state properties, as discussed in Sects. 4.2 and 4.5 (see Eq. 71).

The bCC representation matrix D has an order structure associated with the partitioning according to excitation classes, as shown in Fig. 5. In the upper right (UR) triangle one finds the familiar CI structure for a one-particle operator. This result follows along the lines of the first paragraph in Sect. 4.1. In the lower left (LL) triangle, the canonical order relations

$$ D_{IJ}=O([I]- [J] - 1),\quad [I] > [J] $$

(145)

apply, which is to be shown in the following.

The operator in the bCC matrix element (144) has a finite Baker–Hausdorff (BH) expansion,

$$ e^{-\hat{T}} \hat{D} e^{\hat{T}}=\hat{D}+[\hat{D}, \hat{T}] + \frac{1}{2}[[\hat{D}, \hat{T}],\hat{T}]$$

(146)

terminating here already after the double commutator term, because

$$ \hat{T}=\sum t_I \hat{C}_I $$

(147)

consists of physical excitation operators only, and $\hat{D}$ has at most two unphysical operators. Let us now write the $\hat{T}$ operator according to

$$ \hat{T}=\sum \hat{T}_{\mu} $$

(148)

in terms of individual class operators $\hat{T}_{\mu}, \mu = 1,2,\ldots.$ The T-amplitudes, being themselves subject of a well-defined (diagrammatic) PT, exhibit the order relations (see Hubbard [49])

$$ \hat{T}_{\mu} \sim O(\mu-1),\quad\mu > 1$$

(149)

This means, for example, that the PT expansions of the T ₂ amplitudes

$$ \hat{T}_{2}=\sum t_{abij} \hat{C}_{abij} $$

(150)

begin in first order. The T ₁ amplitudes (μ = 1), being of second order, are an exception reflecting Brioullin’s theorem.

What are the consequences of the expansion (146) and the order relations (149)? Since the BH expansion (146) begins with $\hat{D},$ there will be non-vanishing zeroth-order contributions to D _IJ for [I] = [J] and [I] = [J] + 1. Now suppose that I and J differ by more than one class, that is, [I] ≥ [J] + 2. In that case non-vanishing contributions in D _IJ will arise only if there are terms in the BH expansion that are at least of rank r =  [I] − [J]. Here, the rank of an operator is the number of its $c^{\dagger}$ (or c) factors. For example, $\hat{D}$ is of rank 1 and the $\hat{T}_{\mu}$ operators are of rank μ. Now it is readily established that the commutators $[\hat{D}, \hat{T}_{\mu}]$ and $[[\hat{D}, \hat{T}_{\mu}],\hat{T}_{\nu}]$ are of rank μ and μ + ν − 1, respectively (a commutator of two operators $\hat{A}$ and $\hat{B}$ with definite ranks, a and b, respectively, is of rank a + b−1). To determine the lowest (non-vanishing) PT contribution to the D _IJ matrix elements, one has to inspect the terms of the BH expansion (146) having rank r =  [I] − [J] (which is the lowest rank allowing for non-vanishing matrix elements) and find the lowest PT order of those terms. For example, $[\hat{D}, \hat{T}_{2}]$ is of rank 2 and PT order 1, which means that for [I] = [J] + 2 the PT order of D _IJ is 1. In the general case, [I] = [J] + μ, μ ≥ 3, terms with the required rank r = μ and lowest PT order are due to the $[\hat{D}, \hat{T}_{\mu}]$ commutators, being of rank μ and PT order μ − 1. Likewise, also the double commutator $[[\hat{D}, \hat{T}_{2}],\hat{T}_{\mu-1}]$ gives rise to terms with rank μ and order μ − 1, but there are no rank μ terms with PT order lower than μ − 1. This proves the order relations (145).

Let us note that the order relations $\langle {\overline{\Upphi}_I}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle \sim O([I] - 1)$ for the left basis state transition moments (Eq. 65) follow as special case ([J] = 0).

In a similar way, the canonical order relations

$$ M_{IJ}=O([I]-[J]),\quad [I] \geq [J] $$

(151)

for the bCC secular matrix (LL triangle) elements

$$ \begin{aligned} M_{IJ}&=\langle {\overline\Upphi_I}|{\hat{H} - E_0}|{\Uppsi^0_J}\rangle\\ &=\langle {\Upphi_I}|{e^{-\hat{T}} [\hat{H}, \hat{C}_J] e^{\hat{T}}}|{\Upphi_0}\rangle\\ \end{aligned} $$

(152)

can be established. Now, we have to consider the BH expansion involving the commutator $\hat{K}_J = [\hat{H},\hat{C}_J]$ and check the emerging transition matrix elements of the type $\langle {\Upphi_I}|{\hat{O}}|{\Upphi_0}\rangle.$ In contrast to the case of the transition operator considered above, $\hat{K}_J$ is itself of PT order 1 and of rank [J] + 1 (regarding here only the relevant two-particle part of the Hamiltonian). The BH expansion

$$ e^{-\hat{T}} \hat{K}_J e^{\hat{T}} = \hat{K}_J + [\hat{K}_J, \hat{T}]+\frac{1}{2} [[\hat{K}_J, \hat{T}],\hat{T}] +\frac{1}{6}[[[\hat{K}_J, \hat{T}],\hat{T}],\hat{T}] $$

(153)

terminates after the triple commutator, since K _J has not more than three unphysical $c^{\dagger}$ (c) operators. Let us consider a secular matrix element M _IJ , where [I] = [J] + μ, μ ≥ 1. Obviously, the terms of the BH expansion (153) do not contribute to M _IJ if their rank is smaller than [I]. As above, we may analyze the terms of rank r = [J] + μ with respect to their PT order. For μ = 1, the first term $\hat{K}_J$ on the RHS of Eq. 153 is of rank [J] + 1 and order 1, thus giving rise to a first-order contribution to M _IJ . For higher values of μ, it suffices to consider the commutators $[\hat{K}_J, \hat{T}_\mu],$ being of the required rank r = [J] + μ and PT order μ. Again, it is readily established that there are no rank r = [J] + μ terms of lower PT order.

Appendix 2: Order relations of CI and bCC eigenvector matrices

The order structure of the CI and bCC secular matrices give rise to specific order relations for the eigenvector matrices, which, in turn, imply the respective truncation errors in the excitation energies and transition moments. In the following, we will first consider the CI eigenvector matrix, and then turn to the left and right eigenvector matrices associated with the bCC representation. Finally, we shall show how order relations established only for a triangular part of a matrix can be extended to the entire matrix as a consequence of unitarity.

2.1 CI eigenvector matrix

The order relations of the CI eigenvector matrix rely on PT for the exact states. Let us first consider the familiar case of the ground state, where the well-known Rayleigh–Schrödinger PT can be cast in the compact form

$$ |{\Uppsi_0}\rangle= |{\Upphi_0}\rangle+\sum_{\nu=1}^{\infty} \left[\frac{\hat{Q}_0}{E^{(0)}_0-\hat{H}_0}\left(\hat{H}_I - E_0 +E^{(0)}_0\right)\right]^{\nu} |{\Upphi_0}\rangle$$

(154)

Here, the usual Møller–Plesset decomposition of the Hamiltonian

$$ \hat{H}=\hat{H}_0 + \hat{H}_I $$

(155)

into an unperturbed (HF) part $\hat{H}_0$ and an interaction part $\hat{H}_I$ is supposed; $|{\Upphi_0}\rangle$ is the (HF) ground state of $\hat{H}_0$ with the energy $E_{0}^{(0)},$ and $\hat{Q}_0 = \hat{1} - |{\Upphi_0}\rangle\langle {\Upphi_0}|.$ To determine the lowest (nonvanishing) PT order for a specific eigenvalue component,

$$ X_{J0}=\langle {\Upphi_J}|{\Uppsi_0}\rangle$$

(156)

one has to analyze the contributions arising from the expansion on the RHS of Eq. 154. In the $\nu$th order, the leading operator term is $\hat{H}_I^{\nu}.$ Due to the two-electron (Coulomb repulsion) part of $\hat{H}_I,$ the matrix element $\langle {\Upphi_J}|{\hat{H}_I^{\nu}}|{\Upphi_0}\rangle$ vanishes if the excitation class of J exceeds the value 2ν. For the excitation classes [J] = 2ν and [J] = 2ν − 1, on the other hand, the matrix element gives rise to a non-vanishing $\nu$th order contribution. Obviously, there is no lower-order coupling between the HF ground state and excitations of class 2ν and 2ν − 1. This means that X _J0 is of PT order ν for [J] = 2ν and [J] = 2ν − 1.

This result can also be written in the form

$$ O[\underline{X}_{\mu0}]= \left\{ \begin{array}{ll} \frac{1}{2} \mu, & \,\mu\;\hbox{even}\\ \frac{1}{2} (\mu + 1), &\, \mu\;\hbox{odd}, > 1 \end{array} \right. $$

(157)

where μ denotes collectively the configurations of class μ. The p–h excitation class (μ = 1) is an exception, as here X _J0 ∼ O(2) due to Brioullin’s theorem. The ground-state component X ₀₀ is of course of zeroth order. In Fig. 1, the order structure of $\underline{X}_0$ is depicted.

Now we turn to the order relations of excited states $|{\Uppsi_n}\rangle.$ Rather than using individual PT expansions, the following analysis will be based directly on the order structure of the CI secular matrix (Fig. 1). However, a remark concerning the significance of excited-state PT is appropriate. As is well known, PT expansions for excited states and excited-state energies are of little practical use because the possibility of small or vanishing denominators (“dangerous denominators”) in the PT expansions prevents meaningful computational results. In a formal sense, however, excited-state PT expansions can be generated analogously to the ground-state case, which then can be used to analyze, e.g., truncation errors of excited-state energies and transition moments. Underlying such a formal PT is the concept that each excited state is related to a specific CI state,

$$ |{\Uppsi_n}\rangle \leftarrow |{\Upphi_J}\rangle,$$

from which it emerges when the scaled interaction, e.g., in the form $\lambda \hat{H}_I,$ is gradually increased from λ = 0 to 1. For our purpose, we do not need the individual PT descent of an exact excited state. It suffices to suppose that the exact states can be classified according to their derivation from the (unperturbed) CI excitation classes, p–h, 2p–2h, ..., etc. Analogously to the notation used for the CI excitation classes, we will denote by [n] the class of the exact state $|{\Uppsi_n}\rangle,$ that is, [n] = μ if the excited state n derives from the μp–μh class of CI states. The classification of both the CI and the exact states allows one to partition the CI eigenvector matrix X into sub-blocks X _μν , where μ and ν refer to the component and state classes, respectively. Figure 9 shows the partitioning and the associated order structure of X. A general expression for the order structure is as follows:

$$ O[\user2{X}_{\mu \nu}]= \left\{ \begin{array}{ll} \frac{1}{2} |\mu - \nu|, & \mu - \nu\;\hbox{even}\\ \frac{1}{2} |\mu - \nu|+\frac{1}{2}, & \mu - \nu\;\hbox{odd}\\ \end{array} \right.$$

(158)

Fig. 9

Order relations of the CI eigenvector matrix X. Block structure and entries as in Figs. 1 and 3

The order relations of the eigenvector matrix reflect the underlying order structure of the CI secular matrix. We begin by considering the class of singly excited states ([n] = 1). Rather than dealing with individual eigenvectors, we can treat the entire set of class-1 eigenvectors at the same time. Let us therefore denote by $\underline{{\user2 X}}_{1}$ the rectangular matrix formed by all (column) eigenvectors $\underline{X}_n$ with [n] = 1. The eigenvalue equations for the eigenvectors of class 1 can be written compactly as

$$ \user2{H}\underline{\user2{X}}_{1} = \underline{\user2{X}}_{1} \varvec{\Upomega}_{1} $$

(159)

where $\varvec{\Upomega}_{1}$ denotes the diagonal matrix of the p–h energy eigenvalues. Since any eigenvalue ω _n has an orbital energy (zeroth order) contribution, that is, $\varvec{\Upomega}_{1} \sim O(0),$ Eq. 159 leads to the following order equation

$$ O[\user2{H} \underline{\user2{X}}_{1}]= O[\underline{\user2{X}}_{1}]$$

(160)

This equation can be used to establish successively the individual orders of the component blocks, X _k,1. Obviously, the starting point is given by X _1,1 ∼ 1 + O(1), which merely reflects the fact that the singly excited states derive from the p–h CI configurations. To proceed, we inspect the matrix–vector block products ∑ _j H _k,j X _j,1 for successive values of the (row) index k. (to visualize these products, it is helpful to write the order structure of H (Fig. 1) alongside the $\underline{\user2{X}}_{1}$ column matrix and fill in the successively determined order entries here, starting with the entry 0 in the X _1,1 sub-block). For k = 2 we may readily conclude that

$$ \user2{X}_{2,1} \sim \sum_{j \geq 1} \user2{H}_{2,j} \user2{X}_{j,1} \sim O(1) $$

(161)

where the first-order behavior comes from the first term in the sum, H _2,1 X _1,1 ∼ H _2,1 ∼ O(1). [Note that the diagonal eigenvector block behaves as X _1,1 = 1 + O(1)]. In a similar way, we may establish that also the next sub-block is of first order, $\underline{\user2{X}}_{3,1} \sim O(1).$ For the fourth sub-block, the situation changes since the $\user2{H}_{4,1}$ matrix block vanishes so that here and beyond the zeroth-order X _1,1 block drops out. One here obtains

$$ \underline{\user2{X}}_{4,1} \sim \sum_{j \geq2} \user2{H}_{4,j} \user2{X}_{j,1} \sim O(2) $$

(162)

where the second-order behavior of X _4,1 derives from the first two summands involving the two first-order sub-blocks X _2,1 and X _3,1. In the next step, one first-order block drops out of the matrix–vector product, but the second one, X _3,1, combined with the H _5,3 block of the secular matrix, again leads to second-order behavior of X _5,1. Only for k = 6 the order jumps to 3, since now due to the structure of the secular matrix, the first-order blocks (and the zeroth-order block) no longer contribute to the matrix–vector product. Continuing in this way, the order relations of successively higher-class sub-blocks can be obtained. The general pattern is that for each even k, the order of the X _k,1 block increases by 1. Let us note that the procedure can readily be cast in the formally correct form of induction.

Now, we may proceed to the next higher class of 2p–2h states. Let $\underline{\user2{X}}_{2}$ denote the 2p–2h eigenvector matrix with the (sub)blocks X _k,2. In accord with the PT origin of the 2p–2h states, here the k = 2 (diagonal) block is of zeroth order, more specifically, X _2,2 ∼ 1 + O(1). But the order of the first block, X _1,2 is fixed as well. As will be demonstrated below (Order relations for biorthogonal matrices), the orthogonality of the p–h and 2p–2h eigenvectors requires that X _1,2 is of first order. In a similar way as above, the orders of the higher k blocks can now be derived successively from the matrix–vector products

$$ \user2{X}_{k,2} \sim \sum_{j \geq1} \user2{H}_{k,j} \user2{X}_{j,2}$$

(163)

It is readily established that the X _3,2 and X _4,2 blocks are of first order, followed by two blocks of second order and so forth.

The procedure outlined here for the p–h and 2p–2h states can easily be extended to higher excitation classes, μ. The diagonal block, X _μ,μ ∼ 1 + O(1) is always of zeroth order, while the order relations for blocks above the diagonal, k <  μ, are determined by the orthogonality between the eigenvectors of class μ and those of the lower classes [see below: Order relations for biorthogonal matrices]. In the 3p–3h states, for example, the orthogonality constraint with respect to the p–h and 2p–2h states requires the X _1,3 and X _2,3 blocks to be of first order. The full procedure for establishing the order relations of X, ascending both to higher excitation classes and higher blocks within a given excitation class, can, of course, be reformulated in a formally satisfactory way making use of induction.

Once the order structure of the eigenvectors has been established, it is straightforward to analyze the truncation errors of the excitation energies. For this purpose, one has to express an eigenvalue according to

$$ E_n=\underline{X}^{\dagger}_n \user2{H} \underline{X}_n $$

(164)

as an energy expectation value, involving the full secular matrix H and the exact eigenvector $\underline{X}_n.$ This expectation value can be written more explicitly as

$$ E_n = \sum _{\kappa,\lambda} \underline{X}^{\dagger}_{\kappa n}\user2{H}_{\kappa,\lambda} \underline{X}_{\lambda n} $$

(165)

where the Greek subscripts refer to excitation classes rather than to individual configurations. To specify the error arising from truncating the CI manifold after class μ, one has to inspect the PT order of the terms with κ = μ + 1, λ ≤ μ + 1 (or λ = μ + 1, κ ≤ μ + 1). Due to the structure of H, it suffices to consider the diagonal contribution $\underline{X}^{\dagger}_{\mu+1,n} \user2{H}_{\mu + 1,\mu + 1} \underline{X}_{\mu + 1, n}$ Since H _μ+1,μ+1 ∼O(0), the latter term is of the order

$$ O[\underline{X}^{\dagger}_{\mu + 1,n} \underline{X}_{\mu + 1,n}] = 2 O[\underline{X}_{\mu +1,n}]$$

(166)

Using Eq. 158, this translates into the general truncation error formula

$$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \mu - [n] + 2, & \mu-[n]\;\hbox{even}\\ \mu -[n] + 1, & \mu - [n]\;\hbox{odd} \end{array} \right. $$

(167)

for the CI excitation energies of class [n], where, of course, μ ≥ [n] is supposed. In a similar way, one may analyze the expression (9) for the transition moments, yielding the following truncation error formula:

$$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \mu -\frac{1}{2}[n] + 1, &[n]\;\hbox{even}\\ \mu -\frac{1}{2}[n] +\frac{1}{2}, & [n]\;\hbox{odd}\\ \end{array} \right. $$

(168)

Note that for the case of single excitations, [n] = 1, the general formulas (167, 168) reduce to the simple expressions (7) and (11), respectively, given in Sect. 2.

For completeness, let us note that the simple expression

$$ O^{[n]}_{\rm TE}(\mu) = \mu -[n] + 1 $$

(169)

applies to the truncation error of property matrix elements

$$ T_{nn} = \underline{X}^{\dagger}_n \user2{D} \underline{X}_n$$

(170)

Here, D is the CI representation (10) of a one-particle operator.

2.2 bCC eigenvector matrices

The order structures of the right and left bCC eigenvector matrices X and Y, respectively, are shown in Fig. 10. Note that X now denotes the right bCC eigenvector matrix rather than the CI eigenvector matrix as above (CI eigenvector matrix). The bCC eigenvector matrices display both canonical and CI-type order structures. The LL part of X and the UR part of Y are canonical,

$$ O[\user2{X}_{\mu\nu}]=O[\user2{Y}_{\nu\mu}]=\mu-\nu,\quad\mu \geq \nu$$

(171)

while the UR part of X and the LL part of Y are of CI-type,

$$ O[\user2{X}_{\nu\mu}]=O[\user2{Y}_{\mu\nu}]= \left\{ \begin{array}{ll} \frac{1}{2} (\mu - \nu), & \mu - \nu\;\hbox{even}, \geq 0\\ \frac{1}{2} (\mu - \nu) +\frac{1}{2}, & \mu - \nu\;\hbox{odd}, \geq 0 \end{array} \right.$$

(172)

The order structures of the bCC eigenvector matrices can be deduced from the order structure of the bCC secular matrix by an obvious generalization of the procedure used above for the CI eigenvector matrix. Here, it is important that at each successive step associated with an excitation class μ, both the right and left eigenvector matrices $\underline{\user2{X}}_{\mu}$ and $\underline{\user2{Y}}_{\mu}$ must be treated in parallel, while the biorthogonality to the respective eigenvectors of the lower excitation classes, $1, \ldots, \mu -1,$ must successively be taken into account. A formally correct way of translating the structure of the bCC secular matrix into the order structures of the eigenvector matrices can readily be spelled out.

Fig. 10

Order relations of the right and left bCC eigenvector matrices X and Y, respectively. Block structure and entries as in Figs. 1 and 3

However, one may take also an alternative route, in which the order structures of the UR parts of both X and Y are determined directly. The order relations of the complementary LL parts are then obtained as a result of the biorthogonality, $\user2{Y}^{\dagger}\user2{X}={\mathbf{1}}.$ Let us first consider an eigenvector component

$$ X_{Jn}=\langle {\overline{\Upphi}_J}|{\Uppsi^{(r)}_n}\rangle,\quad [n] \geq [J] $$

(173)

from a UR block of X. Since $|{\Uppsi^{(r)}_n}\rangle$ differs from the (normalized) exact eigenstate $|{\Uppsi_n}\rangle$ only by a normalization constant (N _n ∼1 + O(2)), we may consider $\langle {\overline{\Upphi}_J}|{\Uppsi_n}\rangle$ rather than X _Jn , that is,

$$ X_{Jn} \sim \langle {\overline{\Upphi}_J}|{\Uppsi_n}\rangle$$

(174)

Using the expansion (30) for $\langle{\overline{\Upphi}_J}|,$ it is apparent that for [n] ≥ [J] the bCC eigenvecor component is of the same order as the corresponding CI eigenvector component:

$$ O[\langle {\overline{\Upphi}_J}|{\Uppsi_n}\rangle] = O[\langle{\Upphi_J}|{\Uppsi_n}\rangle]$$

(175)

This establishes the CI-type order relations for the UR part of X.

In analogy to Eq. 174, the left bCC eigenvector components,

$$ Y_{Jn}=\langle {\Uppsi^{(l)}_n}|{\Uppsi^0_J}\rangle $$

(176)

can be related to the normalized eigenstates,

$$ Y_{Jn} \sim \langle {\Uppsi_n}|{\Uppsi^0_J}\rangle= \langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi^{\rm cc}_0}\rangle \sim \langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle $$

(177)

where we have used the fact that the CC and the normalized ground state differ by a normalization constant of the order 1 +  O(2). So it remains to show the order relations

$$ \langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle \sim O([n] - [J]), \quad [n] \geq [J] $$

(178)

for the “generalized transition moments” (GTM) $\langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle.$ Let us first note that these GTM order relations are non-trivial. For a triply excited state ([n] = 3) and a p–h excitation operator ([J] = 1), for example, the (canonical) order is 2, rather than 1, as one might expect in view of two individual first-order contributions associated with $|{\Uppsi^{(1)}_n}\rangle$ and $|{\Uppsi^{(1)}_0}\rangle,$ respectively. It is instructive to verify that these two first-order contributions, in fact, cancel each other.

To prove the GTM order relations we may rely on the canonical order relations

$$ \langle {\Uppsi_n}|{\tilde{\Uppsi}_J}\rangle \sim O(|[n] - [J]|) $$

(179)

as established for the ISR eigenvector components (see Eq. 132) discussed in Sect. 5. Since the intermediate states $|{\tilde{\Uppsi}_K}\rangle$ (including $|{\Uppsi_0}\rangle$) form a complete and orthonormal set of states, they can be used to expand the GTM $\langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle$ as follows:

$$ \begin{aligned} \langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle=&\sum_K \langle {\Uppsi_n}|{\tilde{\Uppsi}_K}\rangle \langle {\tilde{\Uppsi}_K}|{\hat{C}_J}|{\Uppsi_0}\rangle\\ =&\sum_{[K] \leq [J]} \langle {\Uppsi_n}|{\tilde{\Uppsi}_K}\rangle \langle {\tilde{\Uppsi}_K}|{\hat{C}_J}|{\Uppsi_0}\rangle \end{aligned} $$

(180)

where the second line is due to the fact that by construction the intermediate states $|{\tilde{\Uppsi}_K}\rangle$ are orthogonal to all states $\hat{C}_J|{\Uppsi_0}\rangle$ with [J] < [K], that is, $\langle {\tilde{\Uppsi}_K}|{\hat{C}_J}|{\Uppsi_0}\rangle= 0$ for [J] <  [K]. Now for [n] > [J], the order of $\langle {\Uppsi_n}|{\tilde{\Uppsi}_K}\rangle$ decreases with increasing [K] and so does the order of $\langle {\tilde{\Uppsi}_K}|{\hat{C}_J}|{\Uppsi_0}\rangle.$ This means that the minimal order contributions in the sum on the RHS of Eq. 180 are due to those where [K] = [J]:

$$ \langle {\Uppsi_n}|{\hat{C}_J}|{\Uppsi_0}\rangle \sim O[\langle {\Uppsi_n}|{\tilde{\Uppsi}_J}\rangle \langle {\tilde{\Uppsi}_J}|{\hat{C}_J}|{\Uppsi_0}\rangle]=O[\langle {\Uppsi_n}|{\tilde{\Uppsi}_J}\rangle]$$

(181)

The last equality follows from the observation that $\langle {\tilde{\Uppsi}_J}|{\hat{C}_J}|{\Uppsi_0}\rangle$ is of zeroth order. This completes the proof of the canonical order relations for the eigenvector components in the UR blocks of Y.

The biorthogonality of the left and right bCC eigenvector matrices exacts the order relations of the LL blocks of the X and Y matrices, as will be shown below (Order relations for biorthogonal matrices).

Like in the last paragraph of the first subsection (CI eigenvector matrix), the truncation error of the bCC excitation energies for general excitation classes [n] can be deduced from the eigenstate order relations. The starting point is the expression

$$ E_n=\underline{Y}^{\dagger}_n \user2{M} \underline{X}_n = \sum _{\kappa,\lambda} \underline{Y}^{\dagger}_{\kappa n} \user2{M}_{\kappa,\lambda} \underline{X}_{\lambda n} $$

(182)

where, analogously to Eq. 165, the second equation reflects the partitioning of the energy expectation value with respect to excitation classes. To determine the TEO at a given truncation level μ, one has to analyze the contributions where κ = μ + 1, [n] ≤ λ ≤ μ + 1 (set S ₁) and λ = μ + 1, [n] ≤ κ ≤ μ (set S ₂). Here, μ ≥ [n] is supposed. The former set of contributions is given by

$$ S_1 = \underline{Y}^{\dagger}_{\mu + 1,n} \sum^{\mu +1}_{\lambda = [n]} \user2{M}_{\mu + 1,\lambda} \underline{X}_{\lambda n}$$

(183)

Due to the order relations in the LL parts of X (Eq. 171) and M (Eq. 52), we find

$$ O[\user2{M}_{\mu + 1,\lambda} \underline{X}_{\lambda n}] = \mu - [n] +1 $$

(184)

irrespective of λ. Together with the OR of $\underline{Y}_{\mu + 1,n}$ (Eq. 172), this leads to the following TEO formula

$$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \frac{3}{2} (\mu -[n]) + 2, & \mu - [n]\;\hbox{even}\\ \frac{3}{2}(\mu - [n])+\frac{3}{2}, & \mu - [n]\;\hbox{odd}\\ \end{array} \right.$$

(185)

The S ₂ set consists only of two contributions,

$$ S_2=\underline{Y}^{\dagger}_{\mu,n} \user2{M}_{\mu,\mu + 1} \underline{X}_{\mu + 1, n}+ \underline{Y}^{\dagger}_{\mu -1 ,n} \user2{M} _{\mu -1,\mu + 1} \underline{X}_{\mu + 1, n} $$

(186)

because M _κ,μ+1  = 0 for κ < μ − 1. The two involved sub-blocks of M are of first order, $O[\user2{M}_{\mu,\mu+1}] =O[\user2{M}_{{\mu}-1,{\mu}+1}] =1,$ and the TEOs of the S ₂ contributions are seen to exceed those of S ₁. This means that Eq. 185 is the final expression for the truncation errors in the bCC excitation energies.

In a similar way, one can derive general TEO formulas for the left and right transition moments and the excited-state properties. In case of the right transition moments, the dual ground-state eigenvector $\underline{Y}_0$ comes into play. The CI-type order relations of $\underline{Y}_0$ (see Fig. 3b) can readily be established by analyzing the eigenvalue equation $\underline{Y}^{\dagger}_{0} \user2{M} =- \underline{v}^{\rm t}$ (Eq. 34) in a similar way as in the first subsection (CI eigenvector matrix), but assuming that only the p–h and 2p–2h components of $\underline{v}^{\rm t}$ are non-vanishing, the latter being of first order. As expected, both the dual and the CI ground-state have the same order relations, as specified by Eq. 157.

The resulting TEO formulas are as follows.

• (i) left transition moments:

  $$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \frac{3}{2}\mu -\frac{1}{2}[n], & \mu - [n]\;\hbox{even}\\ \frac{3}{2}\mu-\frac{1}{2}[n]+\frac{1}{2}, & \mu - [n]\;\hbox{odd}\\ \end{array} \right. $$

  (187)

• (ii) right transition moments:

  $$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \frac{3}{2}\mu - [n] + 1, & \mu\;\hbox{even}\\ \frac{3}{2}\mu - [n]+\frac{1}{2}, & \mu\;\hbox{odd}\\ \end{array} \right. $$

  (188)

• (iii) property matrix elements:

  $$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \frac{3}{2}(\mu -[n]) +1, & \mu - [n]\;\hbox{even}\\ \frac{3}{2}(\mu - [n])+\frac{1}{2}, & \mu-[n]\;\hbox{odd}\\ \end{array} \right. $$

  (189)

  As above, [n] and μ denote the final state excitation class and the truncation level, respectively, where of course μ ≥ [n].

The first expression (i) is obtained by generalizing the derivation of Eq. 67 in Sect. 4.2. Here the error associated with the truncation of the excited state manifold follows from Eq. 64, using the order relations (66) of the left basis set transition moments together with the CI-type order relations (Eq. 172) in the LL block of the left eigenvector matrix. In addition, one has to account for the error arising from the truncation of the ground-state CC expansion, assuming here the same truncation levels in the ground and excited states. Supposing a sufficiently large or even complete ground-state CC expansion, the TEO increases by 1 for even values of μ − [n] (first line on the RHS of Eq. 187).

In the case of the right transition moments (ii), the TEOs are determined by the second term on the RHS of Eq. 70. This term can be analyzed in an analogous way as the bCC eigenvalues by using the bCC order relations of the transition operator matrix D given in Fig. 5a, rather than those of the bCC secular matrix. It should be noted that apart from the case of single excitations, [n] = 1, the right transition moments have larger truncation errors (lower orders) than the left ones.

In (iii), finally, one has to analyze the vector × matrix × vector product in Eq. 100. The order relations that come into play are those of D (Fig. 5a) and the LL parts of the left and right eigenvector matrices. It should be noted that the expression (iii) applies also to inter-state transition moments for excited states of the same class, [n] =  [m].

In the latter two cases, there is each an additional contribution, arising from the admixture of the ground state in the right eigenstate expansion (Eq. 41), specified by the respective (extended) eigenvector component $x_n =-\underline{Y}^{\dagger}_0 \underline{X}_n$ (Eq. 40). Using the order relations for $\underline{Y}^{\dagger}_0$ and the LL part of the right eigenvector matrix, one can readily derive the following TEO formula for the x _n components:

$$ O^{[n]}_{\rm TE}(\mu)= \left\{ \begin{array}{ll} \frac{3}{2}\mu - [n] +2, & \mu\;\hbox{even}\\ \frac{3}{2}\mu-[n]+\frac{3}{2}, & \mu\;\hbox{odd} \end{array} \right. $$

(190)

As discussed in Sects. 4.2 and 4.5, the TEOs in the additional (ground-state admixture) terms exceed those of the respective main contribution.

2.3 Order relations for biorthogonal matrices

In the preceding subsection (bCC eigenvector matrices), the order relations for the bCC eigenvector matrices have been established only for the respective UR blocks, being of CI-type in X and canonical in Y. Now we will show that the biorthogonality of X and Y requires canonical and CI-type behavior in the LL blocks of X and Y, respectively.

Let us first note that the biorthogonality relation

$$ \user2{Y}^{\dagger} \user2{X}={\mathbf{1}} $$

(191)

also implies that

$$ \user2{XY}^{\dagger}={\mathbf{1}}$$

(192)

which will be our starting point here. In this form, the UR order relations of the first factor X (CI-type) match the LL order relations of the second factor ${\user2 Y}^{\dagger}$ (canonical), which for brevity will be denoted by Y′ henceforth. For a graphical notion of the following procedure, we recommend to place the X and Y′ partitioning schemes next to each other and fill in successively the emerging order relation entries.

For the first row of X-blocks, X _1,k , k = 1,2,..., and the first column of Y′-blocks, Y ^′_k,1 , k = 1,2,..., the order relations are already given. In the second row of X-blocks and the second column of Y′-blocks, there is one undetermined block each, namely, X _2,1 and Y ^′_1,2 , respectively. The orthogonality of the second X row and the first Y′ column can be expressed as follows:

$$ \sum_{k} \user2{X}_{2,k} \user2{Y}'_{k, 1}={\bf 0}$$

(193)

What can be concluded from this with respect to the order of X _2,1? Let us focus on the first two terms in the sum, the remainder being (at least) of the order 2:

$$ \user2{X}_{2,1} \user2{Y}'_{1,1} +\user2{X}_{2,2} \user2{Y}'_{2,1}+O(2)={\bf 0}$$

(194)

Since the diagonal blocks of the eigenvector matrices behave as Y ^′_1,1  = 1 + O(1) and X _2,2 = 1 + O(1), respectively, and Y ^′_2,1  ∼ O(1), the following relation holds through first order:

$$ \user2{X}_{2,1} + O(1)={\mathbf{0}}$$

(195)

This means that X _2,1 must cancel a (non-vanishing) first-order contribution and, thus, is itself of the first order (more accurately, the lowest non-vanishing contribution in the PT expansion of X _2,1 is of first order). In a similar way, we may conclude Y ^′_1,2  ∼ O(1), being a consequence of the orthogonality of the second column of Y′-blocks and the first row of X-blocks.

After having completed the order relations in the second row and column of X and Y′, respectively, we may proceed to the third row of X-blocks. Here, the orders of the first two blocks, X _3,1 and X _3,2, have to be derived, which in turn can be achieved by exploiting that this row is orthogonal to both the first and second column of Y′-blocks. Expanded explicitely through the first three terms, these order relations are:

$$ \user2{X}_{3,1} \user2{Y}'_{1,1} + \user2{X}_{3,2} \user2{Y}'_{2,1} +\user2{X}_{3,3} \user2{Y}'_{3,1} + O(4)={\mathbf{0}} $$

(196)

$$ \user2{X}_{3,1} \user2{Y}'_{1,2} + \user2{X}_{3,2} \user2{Y}'_{2,2} +\user2{X}_{3,3} \user2{Y}'_{3,2} + O(3)={\mathbf{0}}$$

(197)

The second equation allows us to determine the order of X _3,2. Using Y ^′_1,2  ∼ O(1), Y ^′_3,2  ∼ O(1), and Y ^′_2,2  = 1 + O(1), X _3,3 =  1 + O(1) (as diagonal eigenvector blocks), we may conclude that the relation

$$ \user2{X}_{3,2} + O(1)={\mathbf{0}} $$

(198)

holds through first order, which, as above, implies that X _3,2 is of first order. Using this result in the first orthogonality equation, together with Y ^′_1,1 = 1 + O(1), Y ^′_2,1  ∼ O(1), and Y ^′_3,1  ∼ O(2), yields through second order

$$ \user2{X}_{3,1} + O(2)={\mathbf{0}}$$

(199)

which means that X _3,1 ∼O(2), consistent with the canonical order relations. In a completely analogous way, the first two blocks in the third column of Y′-blocks can be treated, yielding the expected CI-type order results, Y′_1,3 ∼ O(1), Y′_2,3 ∼O(1).

This brief demonstration shows how the given CI-type UR order relations of X and the canonical LL order relations of Y′ ($={\user2 Y}^{\dagger}$) impose canonical order relations in the LL part of X and CI-type order relations in the UR part ${\user2 Y}^{\dagger}$ as a result of the biorthogonality of right and left bCC eigenvector matrices. Of course, this derivation can readily be cast into a formally correct proof by induction (see Ref. [18]).

In a related way, the unitarity of the CI eigenvector matrix (denoted X in the first subsection, CI eigenvector matrix) can be used to extend the CI order relations (157), established in the first subsection (CI eigenvector matrix) only for the LL part, to the entire matrix X. Writing the unitarity relation of the CI eigenvector matrix in the form ${\user2 X}^{\dagger}{\user2 X} = {\bf 1},$ the order relations of the UR part of ${\user2 X}^{\dagger}$ combine with those of the LL part of X, similar to the product (192) of the bCC eigenvector matrices. The successive construction of the CI order relations in the UR part of X can be performed essentially as in the bCC case above.

Appendix 3: Equivalence of CCLR and ordinary bCC transition moments

3.1 Right transition moments

In the exact (full) bCC treatment, the ordinary right bCC transition moment (Eq. 46)

$$ T^{(r)}_n=\langle {\overline{\Uppsi}_0}|{\hat{D}\hat{C}_n}|{\Uppsi^{\rm cc}_0}\rangle+ x_n \langle{\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(200)

and the separable CCLR expression (Eq. 97)

$$ T^{(r)}_n=\langle {\overline{\Uppsi}_0}|{[\hat{D}, \hat{C}_n]}|{\Uppsi^{\rm cc}_0}\rangle-\sum_{I,J} \langle {\overline{\Uppsi}_0}|{[[\hat{H}, \hat{C}_I],\hat{C}_n]}|{\Uppsi^{\rm cc}_0}\rangle (\user2{M} + \omega_n)^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(201)

are equivalent. Here, as in Sect. 4.4,

$$ \hat{C}_n = \sum X_{In} \hat{C}_I $$

(202)

denotes the (right) excitation operator associated with the nth excited state, $|{\Uppsi^{(r)}_n}\rangle= x_n |{\Uppsi^{\rm cc}_0}\rangle+ \hat{C}_n |{\Uppsi^{\rm cc}_0}\rangle.$ This by no means obvious result was shown explicitly by Koch et al. [19]. The following is essentially a reformulation and slight extension of the original proof, using the more transparent wave function notations adopted here.

Let us start from the the ordinary bCC expression (200) and transform it successively into the CCLR expression (201). As a first step we make use of the commutator relation $\hat{D} \hat{C}_n = [\hat{D},\hat{C}_n] + \hat{C}_n \hat{D},$ yielding

$$ T^{(r)}_n = \langle {\overline{\Uppsi}_0}|{[\hat{D},\hat{C}_n]}|{\Uppsi^{\rm cc}_0}\rangle+ \langle {\overline{\Uppsi}_0}|{\hat{C}_n \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle+ x_n \langle {\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle$$

(203)

To proceed, let us consider the (trivial) identity

$$ \langle {\overline{\Uppsi}_0}|{\hat{C}_n \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=\langle{\overline{\Uppsi}_0}|{\hat{C}_n (\hat{H} -E_0 + \omega_n)(\hat{H} -E_0 + \omega_n)^{-1} \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(204)

and replace the inverse matrix operator on the RHS by its bCC representation. Noting that M′ + ω _n is the bCC representation of $\hat{H} -E_0+\omega_n,$ where M′ is the extended bCC secular matrix given by Eq. 27, the bCC representation of the inverse operator reads

$$ (\user2{M}' + \omega_n)^{-1}=\left( \begin{array}{ll} \omega_n^{-1} & \underline{w}^{t}_n\\ \underline{0} & (\user2{M}+\omega_n)^{-1}\\ \end{array} \right) $$

(205)

where

$$ \underline{w}^{t}_n = - \omega_n^{-1} \underline{v}^{\rm t}(\user2{M} + \omega_n)^{-1}$$

(206)

This means that we can express $(\hat{H} -E_0 + \omega_n)^{-1}$ as follows:

$$ (\hat{H} -E_0 + \omega_n)^{-1} = \sum_{I,J} \left|{\Uppsi^0_I}\right\rangle(\user2{M} + \omega_n)^{-1}_{IJ} \left\langle{\overline{\Upphi}_J} \right| + \sum_J w_{nJ} |{\Uppsi^{\rm cc}_0}\rangle \left\langle{\overline{\Upphi}_J} \right| + \omega^{-1}_n \left|{\Uppsi^{\rm cc}_0}\right\rangle \left\langle{\Upphi_0} \right|$$

Inserting this expansion into Eq. 204 yields

$$ \begin{aligned} \langle {\overline{\Uppsi}_0}|{\hat{C}_n \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=&\langle {\overline{\Uppsi}_0}|{\hat{C}_n (\hat{H} -E_0 + \omega_n)}|{\Uppsi^0_I}\rangle (\user2{M} + \omega_n)^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle\\ -&x_n \left\{\langle {\Upphi_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle+\sum_J\omega_n w_{nJ}\langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle\right\} \end{aligned}$$

(207)

In deriving this result we have used that $(\hat{H} -E_0 + \omega_n) \left|{\Uppsi^{\rm cc}_0}\right\rangle = \omega_n \left|{\Uppsi^{\rm cc}_0}\right\rangle$ and $\langle {\overline{\Uppsi}_0}|{\hat{C}_n}|{\Uppsi^{\rm cc}_0}\rangle=- x_n.$ Note that at this point, another x _n -term comes into play, augmenting the third term on the RHS of Eq. 203. The reformulation of Eq. 204 is still not complete. To proceed, the matrix elements $\langle {\overline{\Uppsi}_0}|{\hat{C}_n (\hat{H} -E_0 + \omega_n)}|{\Uppsi^0_I}\rangle$ in the first term on the RHS of Eq. 207 can be expressed according to

$$ \langle {\overline{\Uppsi}_0}|{\hat{C}_n (\hat{H} -E_0 + \omega_n) \hat{C}_I}|{\Uppsi^{\rm cc}_0}\rangle=-\langle {\overline{\Uppsi}_0}|{[[\hat{H}, \hat{C}_I],\hat{C}_n]}|{\Uppsi^{\rm cc}_0}\rangle-x_n \omega_n Y^*_{I0} $$

(208)

in terms of the double commutator $[[\hat{H}, \hat{C}_I],\hat{C}_n].$ Here, we have used the eigenvalue equation for the nth excited state in the form

$$ (\hat{H} -E_0) \hat{C}_n \left|{\Uppsi^{\rm cc}_0}\right\rangle=\omega_n \left (\hat{C}_n \left|{\Uppsi^{\rm cc}_0}\right\rangle + x_n \left|{\Uppsi^{\rm cc}_0}\right\rangle \right)$$

(209)

Moreover, recall that $\langle {\overline{\Uppsi}_0}|{\hat{C}_I}|{\Uppsi^{\rm cc}_0}\rangle = Y^*_{I0}$ and the operators $\hat{C}_n$ and $\hat{C}_I$ commute. Inserting Eq. 208 on the RHS of Eq. 207 constitutes the final step in the reformulation of $\langle {\overline{\Uppsi}_0}|{\hat{C}_n \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle.$ Using this result in Eqs. 207 and 203, respectively, validates the second term in the CCLR expression (201), but also introduces a third x _n -term,

$$ -x_n \omega_n \sum_{I,J} Y^*_{I0}(\user2{M} + \omega_n)^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle$$

(210)

due to the second term on the RHS of Eq. 208. It remains to show that the three x _n terms cancel each other, that is,

$$ \langle {\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle-\langle {\Upphi_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle- \sum_J\omega_n w_{nJ}\langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle- \omega_n \sum_{I,J} Y^*_{I0}(\user2{M} + \omega_n)^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle= 0 $$

(211)

where the contributions 1, 2 + 3 and 4 on the LHS arise from Eqs. 203, 207 and 210, respectively. The contributions 3 and 4 can be combined and further evaluated in a compact matrix notation as follows:

$$ \omega_n (\underline{w}^{t} + \underline{Y}^{\dagger}_0) (\user2{M} + \omega_n)^{-1}=\omega_n (-\omega^{-1}_n \underline{v}^{\rm t}{(\user2{M}+ \omega_n)^{-1}} - {{\underline{v}}^{\rm t} \user2{M}^{-1}(\user2{M}+\omega_n)^{-1})} $$

(212)

$$ =-\omega_n \underline{v}^{\rm t} (\omega^{-1}_n + \user2{M}^{-1})(\user2{M}+\omega_n)^{-1} $$

(213)

$$ =-\omega_n \underline{v}^{\rm t} \omega^{-1}_n \user2{M}^{-1}(\user2{M}+ \omega_n)(\user2{M}+\omega_n)^{-1} $$

(214)

$$ =-\underline{v}^{\rm t} \user2{M}^{-1}=\underline{Y}^{\dagger}_0$$

(215)

As a result, the sum of the three x _n terms becomes

$$ x_n \{\langle {\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle- \langle {\Upphi_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle-\sum Y^*_{J0} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle\}=0 $$

(216)

where the cancellation now is obvious, as $\langle {\overline{\Uppsi}_0}| = \langle {\Upphi_0}|+ \sum Y^*_{J0} \langle {\overline{\Upphi}_J}|.$

3.2 Excited-state transition moments

In a similar way, one may show the equivalence of the ordinary bCC and the CCLR expressions for excited-state transition moments and properties, the latter reading (Eq. 121)

$$ T_{nm} = \langle {\overline{\Uppsi}^{(l)}_n}|{[\hat{D},\hat{C}_m]}|{\Uppsi^{\rm cc}_0}\rangle - \sum_{I,J} \langle {\overline{\Uppsi}^{(l)}_n}|{[[\hat{H}, \hat{C}_I],\hat{C}_m]}|{\Uppsi^{\rm cc}_0}\rangle (\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle+\delta_{nm} \langle {\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(217)

where ω _mn = ω _m −ω _n . As above, we start from the ordinary bCC expression (100)

$$ T_{nm}=\langle {\overline{\Uppsi}^{(l)}_n}|{\hat{D} \hat{C}_m}|{\Uppsi^{\rm cc}_0}\rangle+x_m \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(218)

and use the commutator relation $\hat{D}\hat{C}_m=[\hat{D},\hat{C}_m] + \hat{C}_m \hat{D}$ yielding

$$ T_{nm}=\langle {\overline{\Uppsi}^{(l)}_n}|{[\hat{D},\hat{C}_m]}|{\Uppsi^{\rm cc}_0}\rangle +\langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle+x_m \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle$$

(219)

Analogously to Eq. 204, we consider the identity

$$ \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=\langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m (\hat{H} -E_0 + \omega_{mn})(\hat{H} -E_0 + \omega_{mn})^{-1} \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle $$

(220)

and use the the bCC representation of $(\hat{H} - E_0 + \omega_{mn})^{-1}$ to further evaluate the RHS. Note that the only difference to Eqs. 205 and 206 is the replacement of ω _n by ω _mn . This leads to

$$ \begin{aligned} \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=&\sum_{I,J} \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m (\hat{H} -E_0 + \omega_{mn})}|{\Uppsi^0_I}\rangle(\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle\\ +&\delta_{mn} (\langle {\Upphi_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle- \sum_{I,J} v_I (\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle) \end{aligned} $$

(221)

using here $\langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m}|{\Uppsi^{\rm cc}_0}\rangle= \delta_{mn}.$ Let us consider the second term on the RHS, which is non-vanishing only for n = m due to the Kronecker symbol. Since ω _mn = 0 for m = n, and

$$ \underline{v}^t\user2{M}^{-1} =-\underline{Y}^{\dagger}_0$$

(222)

the latter term becomes $\delta_{mn} \langle {\overline{\Uppsi}_0}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle,$ thus reproducing the third term in the CCLR expression (217). As in Eq. 208, we now may introduce a double commutator according to

$$ \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m (\hat{H} - E_0 + \omega_{mn}) \hat{C}_I}|{\Uppsi^{\rm cc}_0}\rangle=- \langle {\overline{\Uppsi}^{(l)}_n}|{[[\hat{H}, \hat{C}_I],\hat{C}_m]}|{\Uppsi^{\rm cc}_0}\rangle-x_m \omega_m Y^*_{In}$$

(223)

Here, we have used the eigenvector equation for the mth excited state in the form

$$ (\hat{H} -E_0) \hat{C}_m \left|{\Uppsi^{\rm cc}_0}\right\rangle = \omega_m (\hat{C}_m \left|{\Uppsi^{\rm cc}_0}\right\rangle + x_m \left|{\Uppsi^{\rm cc}_0}\right\rangle ) $$

(224)

and the relations $\langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_I}|{\Uppsi^{\rm cc}_0}\rangle= Y^*_{In}$ for the left eigenvector components. Using this result in Eq. 221 gives

$$ \begin{aligned} \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{C}_m \hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=&-\sum_{I,J} \langle {\overline{\Uppsi}^{(l)}_n}|{[[\hat{H}, \hat{C}_I],\hat{C}_m]}|{\Uppsi^{\rm cc}_0}\rangle (\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle\\ -&x_m \omega_m \sum_{I,J} Y^*_{In} (\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle \end{aligned}$$

(225)

The first term on the RHS is seen to reproduce, via Eq. 219, the second term of the CCLR expression (217). It remains to inspect the second term on the RHS, containing the factor x _m . Since $\underline {Y}_n$ is a left eigenvector of M, it follows that

$$ \underline{Y}^{\dagger}_n (\user2{M} + \omega_{mn})^{-1} = (\omega_{n} + \omega_{mn})^{-1} \underline{Y}^{\dagger}_n = \omega^{-1}_m \underline{Y}^{\dagger}_n$$

(226)

Thus,

$$ -x_m \omega_m \sum_{I,J} Y^*_{In} (\user2{M} + \omega_{mn})^{-1}_{IJ} \langle {\overline{\Upphi}_J}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle=- x_m \langle {\overline{\Uppsi}^{(l)}_n}|{\hat{D}}|{\Uppsi^{\rm cc}_0}\rangle$$

(227)

which cancels the original x _m term on the RHS of Eq. 219. This concludes our proof.