# On the cluster structure of linear-chain fermionic wave functions

## Abstract

Using the model of cyclic polyenes \(\hbox {C}_N\hbox {H}_N\) with a nondegenerate ground state, \(N = 4 \nu + 2 \; (\nu = 1, 2, \ldots )\), as a prototype of extended linear metallic-like systems we explore the cluster structure of the relevant wave functions. Based on the existing configuration interaction and coupled cluster (CC) results, as obtained with the Hubbard and Pariser–Parr–Pople Hamiltonians in the entire range of the coupling constant extending from the uncorrelated Hückel limit to the fully correlated limit, we recall the breakdown of the CCD or CCSD methods as the size of the system increases and the strongly correlated regime is approached. We introduce the concept of the indecomposable quadruply-excited clusters which arise for \(\nu > 1\) and represent those connected quadruples that do not possess any corresponding disconnected cluster component. It is shown via explicit enumeration that the ratio of the number of these indecomposables relative to that of the decomposables depends linearly on the size of the polyene \(N\), so that the limit of the ratio of the number of indecomposables relative to the total number of quadruples approaches unity as \(N \rightarrow \infty \). We then briefly outline the implications of these results for the applicability of CC approaches to extended systems and provide a qualitative argument for an even more extreme behavior of hexa-excited, octa-excited, etc., clusters as \(N \rightarrow \infty \).

### Keywords

Linear chain models Cyclic polyenes Coupled cluster approach (CCA) Cluster analysis Decomposable and indecomposable connected quadruples CCA to extended linear fermionic chains### Mathematics Subject Classification

81Q05 81Q80 81V55 81V70 92E10## 1 Introduction

Quantum chemical methods that are based on the coupled cluster (CC) Ansatz for the wave function represent nowadays often used, highly accurate and reliable approaches to the electronic structure of molecular systems (see, e.g., [1, 2, 3, 4, 5, 6, 7]; for a historical overview, see [8, 9]). This is particularly the case for closed-shell (CS), non-degenerate ground states where the single reference (SR) CC methods have been successfully applied, even though much progress was also made in handling of quasi-degenerate and open-shell (OS) systems, including excited states, by relying on multi-reference (MR) CC approaches (see, e.g., [10]). Nonetheless, the exploitation of CC Ansätze for quasi-degenerate or highly-degenerate states is far from being settled.

The correlation problem is particularly challenging for one-dimensional (1D) extended systems, such as found in metallic-like linear chains. The restriction to one dimension was initially employed for the sake of simplicity, as in the case of the pioneering work of Bethe [11] on the Heisenberg model [12] of a linear chain of spin-\(\frac{1}{2}\) particles, which introduced what is nowadays referred to as the Bethe Ansatz. Seven years later this effort was extended by Hulthén [13], who considered the ground state of an antiferromagnetic case of the Heisenberg model, while relying on Bethe’s Ansatz. However, this Ansatz (also referred to as the Bethe-Hulthén scheme; for a nice brief overview of these developments see, e.g., [14]) was not revived until a quarter of a century later for the 1D model of interacting spinless bosons [15] and five years later for the fermionic case, where one has to account for the spin-degree of freedom, requiring a generalization to the nested Bethe Ansatz [16]. The latter was also crucial for the handling of the Hubbard lattice model of electrons with short-range on-site interaction via Lieb–Wu equations [17, 18]. The 1D fermionic problem then led to a formulation of the Yang–Baxter equations [16, 19], which opened the way to the wealth of applications in statistical mechanics and solid-state physics, ranging from superconductivity and neutron scattering to quantum entanglement, quantum field theory, as well as to the introduction of quantum groups and string theory. Very recently it is also providing a testing ground for a realization of various models for the trapping of atoms in optical lattices [20]. There exists nowadays a very rich literature on these topics, including various review papers [14, 21] and monographs (see, e.g., [22, 23, 24, 25, 26, 27, 28, 29, 30]).

An excellent model example of such 1D extended systems is provided by the so-called cyclic polyenes \(\hbox {C}_N\hbox {H}_N, N=2n=4\nu +2,\quad \nu =1, 2, 3, \ldots \), when \(\nu \rightarrow \infty \). These systems have a nondegenerate CS ground state and may also be regarded as linear metals with Born-von Kármán cyclic boundary conditions. When described by semi-empirical Hamiltonians, such as the Hubbard or Pariser–Parr–Pople (PPP) Hamiltonian, one can in fact vary the degree of quasi-degeneracy by varying the coupling constant via the scaling of the resonance (or hopping) integral \(\beta \) characterizing its one-electron component (see below). In this way we can achieve a continuous transition from a non-degenerate, uncorrelated case \((\beta \rightarrow \infty )\) to a degenerate, completely correlated state \((\beta = 0)\). Moreover, when relying on the Hubbard Hamiltonian, the geometry of these systems is irrelevant. Another great advantage of the Hubbard Hamiltonian description is the possibility to compare the CC energies with the exact ones obtained by solving the pertinent Lieb-Wu equations [31, 32] (see also the Appendix to [33]). Unfortunately, the corresponding wave functions that are based on the Bethe Ansatz are not easily accessible to analysis. Yet, the dependence of the Hubbard correlation energies on the coupling constant is qualitatively the same as that obtained with the more realistic PPP Hamiltonian. For these reasons the cyclic polyene model was widely studied using various approaches accounting for the correlation effects (see, e.g., [34, 35, 36, 37, 38, 39, 40, 41, 42] and references therein).

The SR CC approaches—even when truncated to only doubles (i.e., CCD, which is in this case equivalent to CCSD)—work reasonably well for small finite polyenes away from the fully correlated limit (i.e., for small coupling constants or large \(\beta \) values), while their performance rapidly deteriorates when approaching the fully correlated limit. For example, for the PPP model of the smallest \(N=6\) polyene (i.e., the benzene molecule) the error in the CCD energy^{1} for the spectroscopic value of the resonance integral \((\beta \approx -2.4\hbox { eV})\) amounts only to about 3 %, yet it rapidly increases to about 80 % in the fully correlated limit \((\beta = 0)\). For the next \(N=10\) polyene, this error at \(\beta =0\) amounts already to about 1,000 % and for the \(N=14\) ring CCD breaks down completely at about \(\beta \approx -1.75\,\hbox {eV}\) (\(-1.37\,\hbox {eV}\) for the Hubbard model) before reaching the fully correlated limit [34]. Extending CCD to CCDTQ (\(\equiv \) CCSDTQ) helps to some extent, yet the same difficulties as found with CCD remain and are only shifted to larger cycles or larger coupling constants [43].

As is well known, the main reason for the success of the SR CC methods is the fact that for non-degenerate systems the quadruply-excited configurations are reasonably well approximated by their disconnected components given by the products of corresponding pair clusters. In the language of the CC formalism, the exact wave function is represented in the exponential form, \(| \Psi \rangle = \exp (T) | \Phi \rangle \), with \(| \Phi \rangle \) representing an independent particle model (IPM) wave function, most often realized by a Hartree-Fock (HF) wave function. The cluster operator \(T\) is then given by the sum of one-, two-, three-, etc., up to the \(N\)-body components, \(T= \sum _{i=1}^N T_i\), so that, symbolically, we can describe the above mentioned property as \(T_4 \ll \frac{1}{2}T_2^2\). Nonetheless, the importance of connected quadruply-excited amplitudes constituting \(T_4\) increases with the increasing quasi-degeneracy.

Now, already for the \(N=10\) polyene we find an interesting phenomenon, namely the existence of connected quadruples that have no corresponding disconnected counterpart, as first pointed out in the context of the cluster analysis of its full configuration interaction (FCI) wave function [44]. We shall refer to such quadruples as *indecomposable* ones in order to distinguish them from their *decomposable* counterparts that possess the corresponding disconnected quadruply-excited component given by the product of relevant pair clusters. We shall see that this phenomenon arises thanks to a high symmetry of our models and cannot arise in asymmetric systems.

## 2 Basic notation

## 3 Semiempirical \(\pi \)-eletron Hamiltonians

The one-electron part of \(H_{\pi }\) is thus proportional to the resonance integral \(\beta \) whose reciprocal value can be viewed as a coupling constant, so that by varying \(\beta \) from zero to large negative values (in practice \(-5\) or \(-10\hbox { eV}\)) we can explore the whole range of the correlation effects from the fully or strongly correlated limit to the weakly or uncorrelated limit, respectively. We note that the strongly or fully correlated limit \((\beta \rightarrow 0)\) corresponds to the strong coupling or a low density regime \((r_s \rightarrow \infty )\) in the parlance of the electron-gas model and, similarly, the weakly correlated limit \((|\beta |\rightarrow \infty )\) corresponds to the high-density limit \((r_s\rightarrow 0)\). The physical (spectroscopic) value of \(\beta \) for the standard Coulomb integral approximations amounts to about \(-2.4\) or \(-2.5\hbox { eV}\).

In the Hubbard Hamiltonian only the on-site interactions are allowed, so that \(\gamma _{\mu \nu }=\gamma \delta _{\mu \nu }\), with \(\gamma = 5\hbox { eV}\), roughly corresponding to the difference \((\gamma _{00}-\gamma _{01})\) in the PPP model [49]. In solid state texts the resonance integral \(\beta \) is referred to as the hopping integral \(t = |\beta |\) and the on-site repulsion integral \(\gamma _{11} \equiv \gamma \) is designated by \(U\). The relevant parameter defining the correlation strength is thus \(U/\beta \). As already mentioned, the actual geometry of the model is irrelevant in this case since only the on-site Coulomb integral \(\gamma \) and the nearest-neighbor hopping term \(\beta \) play the role.

## 4 Cyclic polyene model

As already pointed out above the \(\pi \)-electron cyclic polyene model is useful when exploring the correlation effects in one-dimensional systems. The limiting case of infinitely large polyenes may then be regarded as a model of polyenic chains and the same model may also be thought of as a model of a linear metal with Born-von Kármán cyclic boundary conditions. In fact, this model is essentially equivalent to the model of an electron in a box with suitable boundary conditions [50, 51, 52].

## 5 Coupled cluster method

The main benefit of the exponential cluster Ansatz (19) stems from the possibility of an efficient truncation of the wave function expansion based on the linked cluster theorem of the many-body perturbation theory (MBPT) [53]. In this way the higher-excited components may be effectively approximated—at least for nonmetallic systems—by their disconnected components that arise automatically in the CC formalism via the exponential Ansatz. Specifically, since the singly-excited clusters \(T_1\) usually play a secondary role (contributing only in higher orders of MBPT) and can be completely eliminated by relying on the maximum overlap (or Brueckner) MOs, the most important role is played by the pair cluster amplitudes \(t_j^{(2)}\) constituting the doubly-excited component \(T_2\) (doubles for short). Thus, setting \(T \approx T_2\) we automatically account for quadruples by their disconnected component \(\frac{1}{2}T_2^2\), yielding the so-called CCD method, which generally provides a very good description of the correlation effects. A small, yet often important connected triple \((T_3)\) contribution is then usually accounted for perturbatively via the CCSD(T) approximation [54, 55].

We must emphasize here that the actual values of the cluster amplitudes \(t_j^{(i)}\) and thus of the operator \(T_i\) are well defined only in the context of the exact (i.e., FCI) wave function \(| \Psi \rangle \). Otherwise, their values will depend on the approximation (i.e., the degree of truncation) employed. In fact, while they are always well defined relative to the exact \(| \Psi \rangle \), they may be undefined in cases when the CC equations do not possess any real solution, as will be seen below.

The spin orbital form of the Hamiltonian, Eq. (23), is primarily employed in handling of OS systems (yet, often also for CS systems) when relying on the unrestricted HF (UHF) reference \(| \Phi \rangle \), thus ignoring the spin-free nature of the molecular Hamiltonian and unnecessarily increasing the dimension of the problem. In the CS case, when \(| \Phi \rangle \) is represented by a single Slater determinant with doubly occupied MOs, one can simply achieve the spin adaptation by assigning a factor of two to each closed loop of oriented lines in the resulting diagrams [56, 57]. Nonetheless, such a formalism is based on nonorthogonal CSFs and once going beyond the doubles will involve more cluster amplitudes than necessary (see the discussion in [59]). The orthogonally spin-adapted version of the CCD method (referred to at the time as the coupled-pair many-electron theory CPMET) was presented in [62] and further developed in [63, 64] (see also [65, 66, 67] and references therein). Yet another, a totally different spin-adapted version, may be obtained via the cluster Ansatz based on the unitary group approach (UGA-CC) [68, 69].

## 6 Nature of many-body wave functions

As stated by Thouless [70] “the wave function of a many-body system is so complicated that any approximation to it will be almost orthogonal to it”. In order to get some idea of these complexities we shall rely on earlier studies of cyclic polyenes \(\hbox {C}_N\hbox {H}_N\) representing typical extended systems when \(N\rightarrow \infty \). In fact, some of the characteristics of the relevant wave functions may be gleaned by examining those describing finite polyenes or chains while gradually increasing their size.^{2} Moreover, the parametrization of the semi-empirical Hamiltonians defining these models enables us to vary the quasidegeneracy over the whole range of the coupling constant, as pointed out above. Since even here the corresponding wave functions are rather complex, most of the studies focussed on the correlation energies. Yet, we can draw some conclusions about the importance of various wave function components from limited CI and CC results.

The FCI and limited CI results and their CC structure were investigated in a considerable detail for the \(N=6\) and \(N=10\) rings using both the Hubbard and the PPP Hamiltonians [44]. The PPP FCI results were later extended to the \(N=14\) and \(N=18\) rings by Bendazzoli and Evangelisti [39] and Bendazzoli et al. [40] (see also [41, 42]). However, already for the \(N=18\) ring, both the dimension of the FCI problem \((\approx 73\times 10^6)\) and the increasing quasidegeneracy did not allow the authors to reach the fully-correlated limit and proceed beyond \(\beta = -2.5\,\hbox {eV}\).

Contribution of doubles (D), triples (T), quadruples (Q), and higher than quadruples (\(>\)Q) to the FCI correlation energy of the PPP \(N=10\) cyclic polyene model (in %) and the difference \(R\) of the CCD and CIDQ energies relative to the FCI energy, \(R \equiv (\Delta \epsilon ^\mathrm{(CCD)} - \Delta \epsilon ^\mathrm{(CIDQ)})/\Delta \epsilon ^\mathrm{(FCI)}\) (in %), as a function of the resonance integral \(\beta \) (in eV)

\(-\beta \) | D | T | Q | \(>\)Q | \(R\) |
---|---|---|---|---|---|

5.0 | 94.8 | 1.2 | 3.9 | 0.1 | 0.2 |

4.0 | 92.4 | 1.7 | 5.8 | 0.2 | 0.4 |

3.0 | 87.5 | 2.6 | 9.3 | 0.6 | 1.3 |

2.5 | 83.4 | 3.3 | 12.2 | 1.1 | 2.9 |

2.0 | 77.7 | 3.9 | 16.1 | 2.4 | 7.2 |

1.5 | 69.7 | 3.9 | 21.5 | 5.0 | 26.5 |

1.0 | 59.5 | 2.8 | 26.6 | 11.1 | 608.6 |

0.5 | 51.0 | 0.8 | 31.2 | 17.0 | 897.0 |

0.0 | 42.8 | 0.1 | 32.3 | 24.8 | 837.9 |

In this regard it is of interest to note the role played by the reference configuration \(| \Phi \rangle \) in the FCI wave function. As can be seen from the results given in Table IX of [44] and Table I of [40], listing the coefficients (or their squares) of the reference configuration \(| \Phi \rangle \) in the normalized FCI wave function \(| \Psi \rangle (\langle \Psi |\Psi \rangle = 1)\), the HF (or Brueckner) IPM reference \(| \Phi \rangle \) provides the exact result in the uncorrelated limit when \(|\beta | \rightarrow \infty \), while its weight almost vanishes in the fully correlated limit. In fact, we see that with the increasing \(N\) this weight rapidly approaches zero, amounting to 0.0686, 0.0094, and 0.0013 for \(N = 6, 10\), and 14, respectively [40]. Clearly, as \(N \rightarrow \infty \) this weight will vanish making the exact wave function orthogonal to \(| \Phi \rangle \). In fact, this will also be the case when we choose a better reference than the IPM one (e.g., CID or CCD wave function), as the results of Table III of [40] clearly indicate. We recall here that since in the fully correlated limit the one-electron component of the Hamiltonian is absent, all excited CSFs become degenerate with the reference configuration. The above described behavior thus clearly indicates that in the fully correlated limit the cyclic polyene model represents a highly complex degenerate system.

Another aspect of the difficulties that we encounter when handling cyclic polyenes is revealed by the fact that with the increasing quasidegeneracy, the truncated CC equations may not possess any real valued, physically meaningful solution [34]. Thus, while at the CCD level we find real solutions in the whole range of the coupling constant for \(N=6\) and \(N=10\) (even though we encounter, respectively, a 100 and 1,000 % error in the correlation energy for \(\beta = 0\)), no real solution exists for \(N=14\) below certain critical value of \(|\beta |\) (see [34] for details). This behavior persists even when one employs higher-level approximations, such as CC(S)DTQ [43]. This is clearly related with the growing importance of higher-excited cluster components as documented by the analysis given in [44]. As already alluded to in the Introduction, one interesting aspect that arises for larger and larger cycles is the existence of connected quadruples that have no corresponding disconnected conterparts, which we defined as the indecomposable quadruples. Clearly, such quadruples cannot be accounted for via their disconnected components, which is essential for the success of the CCD or CCSD approaches. We shall now turn our attention to this aspect.

## 7 Role of indecomposable quadruples

The cluster analysis of the FCI wave function for the first two cyclic polyenes \(N=6\) and \(N=10\) clearly indicates an increasing importance of connected quadruples as we approach the fully correlated limit (cf. Table 1 above and Table X and Figs. 8–11 of [44]). We again recall here a parallel behavior for the Hubbard and the PPP Hamiltonians, as well as the fact that the actual geometry of the model is irrelevant in the former case, where only topology plays a role (i.e., the adjacency in the tight-binding model). This parallel behavior is also revealed by the results for the correlation energy in larger polyenes (see, e.g., Fig. 5 of [34]).

*indecomposable*ones in contrast to the

*decomposable*quadruples, which possess at least one disconnected counterpart. Of course, a similar situation will arise for even higher than quadruply-excited configurations, such as sextuples, octuples, etc. While no such indecomposable quadruple appears in the \(N=6\) case, we find already 22 of them for the \(N=10\) ring; they are listed in Table 2 (for simplicity’s sake we ignore here the alternancy and hole-particle symmetries). Note that all these configurations correspond to one singlet CSF except for configurations 4 and 19 which involve four singly-occupied MOs and thus generate two singlet CSFs. Thus, the number of indecomposable CSFs for \(N=10\) equals 24, which will also be the number of corresponding indecomposable, connected, quadruply-excited cluster amplitudes.

Indecomposable quadruples for \(N=10\) cyclic polyene

\(i\) | \(|\Phi _i^{{\mathrm{occ} \rightarrow \mathrm{unocc}}} \rangle \) | \(i\) | \(|\Phi _i^{{\mathrm{occ} \rightarrow \mathrm{unocc}}} \rangle \) |
---|---|---|---|

1 | \(0^2 \, 2^2 \rightarrow 3^2 \, 4^2\) | 12 | \(2^2 \, 8^2 \rightarrow 3 \; 5 \; 6^2\) |

2 | \(0 \; 1^2 \, 2 \rightarrow 3^2 \, 4^2\) | 13 | \(2^2 \, 9^2 \rightarrow 3 \; 6^2 \, 7\) |

3 | \(1 \; 2^2 \, 9 \rightarrow 3^2 \, 4^2\) | 14 | \(2^2 \, 8^2 \rightarrow 4^2 \, 5 \; 7\) |

4 | \(0 \; 1 \; 2^2 \rightarrow 3^2 \, 4 \; 5\) | 15 | \(1^2 \, 2 \; 8 \rightarrow 4^2 \, 7^2\) |

5 | \(1^2 \, 2^2 \rightarrow 3^2 \, 4 \; 6\) | 16 | \(8^2 \, 9^2 \rightarrow 4 \; 6 \; 7^2\) |

6 | \(1^2 \, 2^2 \rightarrow 3^2 \, 5^2\) | 17 | \(8^2 \, 9^2 \rightarrow 5^2 \, 7^2\) |

7 | \(2 \; 8 \; 9^2 \rightarrow 3^2 \, 6^2\) | 18 | \(8^2 \, 9^2 \rightarrow 5 \; 6^2 \, 7\) |

8 | \(0 \; 1^2 \, 8 \rightarrow 3^2 \, 7^2\) | 19 | \(0 \; 8^2 \, 9 \rightarrow 5 \; 6 \; 7^2\) |

9 | \(0 \; 2 \; 9^2 \rightarrow 3^2 \, 7^2\) | 20 | \(0^2 \, 8^2 \rightarrow 6^2 \, 7^2\) |

10 | \(1^2 \, 2^2 \rightarrow 3 \; 4^2 \, 5\) | 21 | \(0 \; 8 \; 9^2 \rightarrow 6^2 \, 7^2\) |

11 | \(1^2 \, 8^2 \rightarrow 3 \; 4^2 \, 7\) | 22 | \(1 \; 8^2 \, 9 \rightarrow 6^2 \, 7^2\) |

Possible disconnected quadruples for the 17th configuration \(8^2 \, 9^2 \rightarrow 5^2 \, 7^2\) of Table 2

\(i\) | \(|\Phi _i^{{\mathrm{occ} \rightarrow \mathrm{unocc}}} \rangle \) | \(k\) | \(|\Phi _i^{{\mathrm{occ} \rightarrow \mathrm{unocc}}} \rangle \) | \(k\) |
---|---|---|---|---|

1 | \(8^2 \rightarrow 7^2\) | \((-2)\) | \(9^2 \rightarrow 5^2\) | \((2)\) |

2 | \(8 \; 9 \rightarrow 5^2\) | \((-3)\) | \(8 \; 9 \rightarrow 7^2\) | \((3)\) |

3 | \(9^2 \rightarrow 7^2\) | \((-4)\) | \(8^2 \rightarrow 5^2\) | \((4)\) |

4 | \(8^2 \rightarrow 5 \; 7\) | \((-4)\) | \(9^2 \rightarrow 5 \; 7\) | \((4)\) |

5 | \(8 \; 9 \rightarrow 5 \; 7\) | \((\pm 5)\) | \(8 \; 9 \rightarrow 5 \; 7\) | \((\pm 5)\) |

Number of all \([Q \equiv Q(N) = Q^\mathrm{dec}(N)+Q^\mathrm{indec}(N)]\), decomposable \([Q^\mathrm{dec}(N) \equiv D]\), and indecomposable \([Q^\mathrm{indec}(N) \equiv I]\) quadruple configurations and their ratios for cyclic polyenes \(\hbox {C}_N\hbox {H}_N, N=4\nu + 2\)

\(\nu \) | \(N\) | Orbital configurations | Ratios | |||
---|---|---|---|---|---|---|

\(Q(N)\) | \(Q^\mathrm{dec}(N)\) | \(Q^\mathrm{indec}(N)\) | \(Q^\mathrm{indec}/Q\) | \(Q^\mathrm{indec}/Q^\mathrm{dec}\) | ||

1 | 6 | 8 | 8 | 0 | 0.0000 | 0.0000 |

2 | 10 | 217 | 195 | 22 | 0.1014 | 0.1128 |

3 | 14 | 1927 | 1523 | 404 | 0.2096 | 0.2653 |

4 | 18 | 9852 | 6976 | 2876 | 0.2919 | 0.4123 |

5 | 22 | 36380 | 23432 | 12948 | 0.3559 | 0.5526 |

6 | 26 | 108361 | 64151 | 44210 | 0.4080 | 0.6892 |

7 | 30 | 277063 | 151935 | 125128 | 0.4516 | 0.8236 |

8 | 34 | 631840 | 322904 | 308936 | 0.4889 | 0.9567 |

9 | 38 | 1318088 | 630920 | 687168 | 0.5213 | 1.0891 |

10 | 42 | 2560161 | 1152651 | 1407510 | 0.5498 | 1.2211 |

11 | 46 | 4689759 | 1993283 | 2696476 | 0.5750 | 1.3528 |

12 | 50 | 8180364 | 3292848 | 4887516 | 0.5975 | 1.4843 |

13 | 54 | 13688428 | 5233224 | 8455204 | 0.6177 | 1.6157 |

14 | 58 | 22101793 | 8045751 | 14056042 | 0.6360 | 1.7470 |

15 | 62 | 34595919 | 12019495 | 22576424 | 0.6526 | 1.8783 |

16 | 66 | 52698656 | 17510152 | 35188504 | 0.6677 | 2.0096 |

17 | 70 | 78364008 | 24949600 | 53414408 | 0.6816 | 2.1409 |

18 | 74 | 114055465 | 34856067 | 79199398 | 0.6944 | 2.2722 |

19 | 78 | 162839671 | 47844971 | 114994700 | 0.7062 | 2.4035 |

20 | 82 | 228490844 | 64640376 | 163850468 | 0.7171 | 2.5348 |

21 | 86 | 315606524 | 86087096 | 229519428 | 0.7272 | 2.6661 |

22 | 90 | 429735449 | 113163439 | 316572010 | 0.7367 | 2.7975 |

23 | 94 | 577517943 | 146994599 | 430523344 | 0.7455 | 2.9288 |

24 | 98 | 766839392 | 188866664 | 577972728 | 0.7537 | 3.0602 |

25 | 102 | 1006997640 | 240241296 | 766756344 | 0.7614 | 3.1916 |

26 | 106 | 1308884657 | 302771027 | 1006113630 | 0.7687 | 3.3230 |

27 | 110 | 1685183055 | 378315203 | 1306867852 | 0.7755 | 3.4544 |

28 | 114 | 2150578316 | 468956568 | 1681621748 | 0.7819 | 3.5859 |

29 | 118 | 2721987052 | 577018496 | 2144968556 | 0.7880 | 3.7173 |

30 | 122 | 3418801873 | 705082839 | 2713719034 | 0.7938 | 3.8488 |

31 | 126 | 4263153759 | 856008447 | 3407145312 | 0.7992 | 3.9803 |

32 | 130 | 5280192224 | 1032950304 | 4247241920 | 0.8044 | 4.1118 |

33 | 134 | 6498383848 | 1239379312 | 5259004536 | 0.8093 | 4.2433 |

34 | 138 | 7949830105 | 1479102715 | 6470727390 | 0.8139 | 4.3748 |

35 | 142 | 9670604743 | 1756285171 | 7914319572 | 0.8184 | 4.5063 |

36 | 146 | 11701111292 | 2075470440 | 9625640852 | 0.8226 | 4.6378 |

37 | 150 | 14086461660 | 2441603744 | 11644857916 | 0.8267 | 4.7693 |

38 | 154 | 16876876041 | 2860054743 | 14016821298 | 0.8305 | 4.9009 |

39 | 158 | 20128104711 | 3336641159 | 16791463552 | 0.8342 | 5.0324 |

40 | 162 | 23901872704 | 3877653040 | 20024219664 | 0.8378 | 5.1640 |

41 | 166 | 28266347560 | 4489877672 | 23776469888 | 0.8412 | 5.2956 |

42 | 170 | 33296630721 | 5180625107 | 28116005614 | 0.8444 | 5.4271 |

43 | 174 | 39075273599 | 5957754363 | 33117519236 | 0.8475 | 5.5587 |

44 | 178 | 45692818476 | 6829700240 | 38863118236 | 0.8505 | 5.6903 |

45 | 182 | 53248364812 | 7805500784 | 45442864028 | 0.8534 | 5.8219 |

46 | 186 | 61850162017 | 8894825391 | 52955336626 | 0.8562 | 5.9535 |

47 | 190 | 71616228815 | 10108003559 | 61508225256 | 0.8589 | 6.0851 |

Number of all \([Q \equiv Q_0(N) = Q_0^\mathrm{dec}(N)+Q_0^\mathrm{indec}(N)]\), decomposable \([Q_0^\mathrm{dec}(N) \equiv D]\), and indecomposable \([Q_0^\mathrm{indec}(N) \equiv I]\) singlet quadruple configurations and their ratios for cyclic polyenes \(\hbox {C}_N\hbox {H}_N, N=4\nu + 2\)

\(\nu \) | \(N\) | SINGLET CONFIGURATIONS | RATIOS | |||
---|---|---|---|---|---|---|

\(Q_0(N)\) | \(Q_0^\mathrm{dec}(N)\) | \(Q_0^\mathrm{indec}(N)\) | \(Q_0^\mathrm{indec}/Q_0\) | \(Q_0^\mathrm{indec}/Q_0^\mathrm{dec}\) | ||

1 | 6 | 11 | 11 | 0 | 0.0000 | 0.0000 |

2 | 10 | 522 | 498 | 24 | 0.0460 | 0.0482 |

3 | 14 | 6587 | 5847 | 740 | 0.1123 | 0.1266 |

4 | 18 | 42472 | 35000 | 7472 | 0.1759 | 0.2135 |

5 | 22 | 185039 | 140727 | 44312 | 0.2395 | 0.3149 |

6 | 26 | 624122 | 438014 | 186108 | 0.2982 | 0.4249 |

7 | 30 | 1758183 | 1141235 | 616948 | 0.3509 | 0.5406 |

8 | 34 | 4333488 | 2610272 | 1723216 | 0.3976 | 0.6602 |

9 | 38 | 9635059 | 5405959 | 4229100 | 0.4389 | 0.7823 |

10 | 42 | 19737706 | 10353958 | 9383748 | 0.4754 | 0.9063 |

11 | 46 | 37825363 | 18617791 | 19207572 | 0.5078 | 1.0317 |

12 | 50 | 68586984 | 31781276 | 36805708 | 0.5366 | 1.1581 |

13 | 54 | 118697319 | 51939171 | 66758148 | 0.5624 | 1.2853 |

14 | 58 | 197390778 | 81797054 | 115593724 | 0.5856 | 1.4132 |

15 | 62 | 317136639 | 124780815 | 192355824 | 0.6065 | 1.5415 |

16 | 66 | 494423936 | 185154004 | 309269932 | 0.6255 | 1.6703 |

17 | 70 | 750664219 | 268144623 | 482519596 | 0.6428 | 1.7995 |

18 | 74 | 1113220442 | 380081610 | 733138832 | 0.6586 | 1.9289 |

19 | 78 | 1616570331 | 528538955 | 1088031376 | 0.6730 | 2.0586 |

20 | 82 | 2303612408 | 722489340 | 1581123068 | 0.6864 | 2.1884 |

21 | 86 | 3227122927 | 972467679 | 2254655248 | 0.6987 | 2.3185 |

22 | 90 | 4451372090 | 1290741938 | 3160630152 | 0.7100 | 2.4487 |

23 | 94 | 6053907703 | 1691493687 | 4362414016 | 0.7206 | 2.5790 |

24 | 98 | 8127514528 | 2191008632 | 5936505896 | 0.7304 | 2.7095 |

25 | 102 | 10782357715 | 2807874203 | 7974483512 | 0.7396 | 2.8400 |

26 | 106 | 14148318458 | 3563186954 | 10585131504 | 0.7482 | 2.9707 |

27 | 110 | 18377530131 | 4480770151 | 13896759980 | 0.7562 | 3.1014 |

28 | 114 | 23647123304 | 5587398064 | 18059725240 | 0.7637 | 3.2322 |

29 | 118 | 30162187767 | 6913030279 | 23249157488 | 0.7708 | 3.3631 |

30 | 122 | 38158959818 | 8491056278 | 29667903540 | 0.7775 | 3.4940 |

31 | 126 | 47908243231 | 10358546499 | 37549696732 | 0.7838 | 3.6250 |

32 | 130 | 59719072016 | 12556513496 | 47162558520 | 0.7897 | 3.7560 |

33 | 134 | 73942623227 | 15130183575 | 58812439652 | 0.7954 | 3.8871 |

34 | 138 | 90976388250 | 18129274558 | 72847113692 | 0.8007 | 4.0182 |

35 | 142 | 111268610667 | 21608283855 | 89660326812 | 0.8058 | 4.1493 |

36 | 146 | 135322998952 | 25626787092 | 109696211860 | 0.8106 | 4.2805 |

37 | 150 | 163703722447 | 30249742643 | 133453979804 | 0.8152 | 4.4117 |

38 | 154 | 197040698698 | 35547806550 | 161492892148 | 0.8196 | 4.5430 |

39 | 158 | 236035180407 | 41597658207 | 194437522200 | 0.8238 | 4.6742 |

40 | 162 | 281465650464 | 48482331596 | 232983318868 | 0.8277 | 4.8055 |

41 | 166 | 334194033123 | 56291557119 | 277902476004 | 0.8316 | 4.9368 |

42 | 170 | 395172229578 | 65122114274 | 330050115304 | 0.8352 | 5.0682 |

43 | 174 | 465448986419 | 75078189659 | 390370796760 | 0.8387 | 5.1995 |

44 | 178 | 546177105016 | 86271745652 | 459905359364 | 0.8420 | 5.3309 |

45 | 182 | 638621000087 | 98822900143 | 539798099944 | 0.8453 | 5.4623 |

46 | 186 | 744164615946 | 112860311242 | 631304304704 | 0.8483 | 5.5937 |

47 | 190 | 864319708463 | 128521572871 | 735798135592 | 0.8513 | 5.7251 |

## 8 Conclusions

Having established the linear dependence of the ratio \(I/D\) of the number of indecomposable \(I\) to the number of decomposable \(D\) singlet quadruples as a function of the size of the polyene \(N = 2n = 4 \nu + 2\) via direct evaluation of the number of respective clusters or CSFs, we recall [34] a qualitative reasoning that anticipates this result. Having \(n\) occupied and \(n\) unoccupied MOs, the number of singly-excited configurations scales as \(n^2\) and the number of doubly-excited as \(n^4\). The requirement of a zero overall quasimomentum reduces the \(n^4\) dependence to \(n^3\) [note that the available symmetries can only modify an overall numerical factor—e.g., when all the symmetries including the alternancy symmetry, are accounted for, the dependence is \(n^3/6\)—see also Eq. (29)]. Similarly, the number of quadruply-excited configurations will scale as \(n^8\), and as \(n^7\) when we impose the requirement of a zero quasimomentum. At the same time, the number of \(\frac{1}{2}T_2^2\)-type disconnected quadruples will vary only as \((n^3)^2 = n^6\), ignoring again the numerical factors due to various symmetries. Consequently, the dependence of the ratio of all quadruples to the number of decomposable ones, i.e., \((I+D)/D = I/D + 1\), on \(N = 2n\) as \(N \rightarrow \infty \) will be linear, since \(n^7/n^6 = n\). This simple dimensional reasoning thus corroborates our results obtained by actual evaluation of the number of pertinent clusters. Clearly, the number of indecomposable quadruples will dominate the quadruply-excited manifold as \(N \rightarrow \infty \). Yet, the actual role played by these clusters in the calculation of the exact correlation energy is not presently known and would deserve to be investigated.

The just outlined dimensional argument can now be extended to higher than quadruples. The number of \(p(=2t)\)-times excited configurations scaling as \(n^{2p}\) is reduces to the \(n^{2p-1}=n^{4t-1}\) dependence when we take into account the zero quasimomentum requirement. (Note that for the odd-number-of-times excited clusters, such as \(T_3\), the role of connected and disconnected clusters is reversed, the latter ones being much less important than the former ones, not to mention the much smaller importance of even connected clusters of this type for the correlation energy, cf., e.g., Table 1; moreover, the \(T_1\) clusters vanish entirely in view of the Brueckner character of the MOs). Next, the number of corresponding disconnected clusters constituting \((1/t!)T_2^t\) scales as \((n^3)^t\), so that the relevant ratio yields \(n^{4t-1}/n^{3t} = n^{t-1} = n^{\frac{1}{2}p - 1}\). Thus, for quadruples \((p=4)\), hexuples \((p=6)\), octuples \((p=8)\), etc., we find that the pertinent ratio \(I/D\) scales as \(n, n^2, n^3\), etc., as \(n \rightarrow \infty \), making the role of indecomposables even more important as the excitation order increases. This at least partially explains the difficulties of calculating the correlation energies for 1D extended systems as well as the complexity of the Bethe Ansatz wave functions. We recall that as one approaches the fully correlated limit \((\beta =0)\), all orbital energies become degenerate and thus all the configurations involved.

## 9 Summary

Relying on semi-empirical models of cyclic polyenes we have shown that within the CC description there exist two distinct types of connected, quadruply-excited clusters, which we designated as the *decomposable* and the *indecomposable* ones. The indecomposable quadruples, in contrast to the ubiquitous decomposable ones, do not possess the corresponding disconnected counterparts given by the product of pair clusters. As we have pointed out, this phenomenon is associated with the high spatial symmetry of our model and cannot arise in asymmetric species. However, it will generally arise when handling low-dimensional extended systems, such as the one-dimensional (1D) or quasi-1D metals, 1D photonic crystals or polymeric chains with cyclic boundary conditions.

An account of quadruples is essential for a proper description of correlation effects. For most molecular systems, this may be efficiently achieved by exploiting the CCD (or CCSD) formalism (with an eventual perturbative correction for triples), in which case the quadruples are satisfactorily approximated by their disconnected component \(\frac{1}{2}T_2^2\). However, there is no such possibility for indecomposable quadruples which have no disconnected counterparts. Moreover, as we have seen, their preponderance increases with the size of the linear chain. It would thus be useful to find out the role played by this type of clusters when accounting for the correlation effects in strongly correlated systems. In particular, it would be important to find out to what extent we can handle quadruples via the ACPQ-type approaches [34, 64, 66] that provide excellent results even for medium-sized systems (such as \(\hbox {C}_{22}\hbox {H}_{22}\)) in the entire range of the coupling constant and faithfully reproduce the exact correlation energy in the strongly correlated limit for the PPP and Hubbard models. Moreover, the role of even higher-than-quadruply-excited clusters may be of concern including their indecomposables. These and similar factors may render the standard CC-type approaches ineffectual in handling of degenerate strongly-correlated systems unless suitably modified as in the ACPQ case. A better understanding of the structure of the Bethe Ansatz wave functions could be certainly helpful in these efforts.

## Footnotes

## Notes

### Acknowledgments

Two of the authors (J.P. and T.S.) are greatly indebted to the Alexander von Humboldt Foundation for its kind support that enabled their stay at the Max-Planck-Institute for Astrophysics at Garching bei München in Germany and they thank the latter Institute for its hospitality during their stay. Their heartfelt thanks are also due to their host, Prof. Dr. Geerd H. F. Diercksen, for his kind advice and collaboration and for making their stay as pleasant and productive as possible.

### References

- 1.R.J. Bartlett, in
*Modern Electronic Structure Theory*, vol. 1, ed. by D.R. Yarkony (World Scientific, Singapore, 1995), pp. 1047–1131.Google Scholar - 2.J. Paldus, X. Li, Adv. Chem. Phys.
**110**, 1 (1999)Google Scholar - 3.T.D. Crawford, H.F. Schaefer III, in
*Reviews of Computational Chemistry*, vol. 14, ed. by K.B. Lipkowitz, D.B. Boyd (Wiley, New York, 2000), pp. 33–136CrossRefGoogle Scholar - 4.J. Paldus, in
*Handbook of Molecular Physics and Quantum Chemistry, Part 3, Chap. 19*, vol. 2, ed. by S. Wilson (Wiley, Chichester, 2003), pp. 272–313.Google Scholar - 5.R.J. Bartlett, M. Musiał, Rev. Mod. Phys.
**79**, 291 (2007)CrossRefGoogle Scholar - 6.I. Shavitt, R.J. Bartlett,
*Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory*(Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar - 7.P. Čársky, J. Paldus, J. Pittner (eds.),
*Recent Progress in Coupled Cluster Methods: Theory and Applications*(Springer, Berlin, 2010)Google Scholar - 8.J. Paldus, in
*Theory and Applications of Computational Chemistry: The First Forty Years, Chap. 7*, ed. by C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria (Elsevier, Amsterdam, 2005), pp. 115–147.Google Scholar - 9.R.J. Bartlett, in
*Theory and Applications of Computational Chemistry: The First Forty Years, Chap. 42*, ed. by C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria (Elsevier, Amsterdam, 2005), pp. 1191–1221.Google Scholar - 10.J. Paldus, J. Pittner, P. Čársky, in
*Recent Progress in Coupled Cluster Methods: Theory and Applications, Chap. 17*, ed. by P. Čársky, J. Paldus, J. Pittner (Springer, Berlin, 2010), pp. 455–489Google Scholar - 11.H.A. Bethe, Z. Phys.
**71**, 205 (1931)CrossRefGoogle Scholar - 12.W. Heisenberg, Z. Phys.
**49**, 619 (1928)CrossRefGoogle Scholar - 13.L. Hulthén, Arkiv. Mat. Astron. Fys. A
**26**, 1 (1938)Google Scholar - 14.M.T. Batchelor, Phys. Today
**60**, 36 (2007)CrossRefGoogle Scholar - 15.E.H. Lieb, W. Liniger, Phys. Rev.
**130**, 1605 (1963)CrossRefGoogle Scholar - 16.C.N. Yang, Phys. Rev. Lett.
**19**, 1312 (1967)CrossRefGoogle Scholar - 17.E.H. Lieb, F.Y. Wu, Phys. Rev. Lett.
**20**, 1445 (1968)CrossRefGoogle Scholar - 18.E.H. Lieb, F.Y. Wu, Phys. A
**321**, 1 (2003)CrossRefGoogle Scholar - 19.R.J. Baxter, Ann. Phys. (N.Y.) 70, 193 (1972).Google Scholar
- 20.I. Bloch, Nat. Phys.
**1**, 23 (2005)CrossRefGoogle Scholar - 21.Z. Bajnok, L. Šamaj, Acta Phys. Slov.
**61**, 129 (2011). and references thereinGoogle Scholar - 22.E.H. Lieb, D.C. Mattis (eds.),
*Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles*(Academic Press, New York, 1966)Google Scholar - 23.R.J. Baxter,
*Exactly Solved Models in Statistical Mechanics*(Academic Press, New york, 1982)Google Scholar - 24.M. Gaudin,
*La Fonction d’Onde de Bethe (Masson, Paris, 1983); English translation: The Bethe Wavefunction*(Cambridge University Press, Cambridge, 2014)Google Scholar - 25.A. Montorsi (ed.),
*The Hubbard Model: A Reprint Volume*(World Scientific, Singapore, 1992)Google Scholar - 26.V.E. Korepin, N.M. Bogoliubov, A.G. Izergin,
*Quantum Inverse Scattering Method and Correlation Functions*(Cambridge University Press, Cambridge, 1993)CrossRefGoogle Scholar - 27.D. Baeriswyl, D.K. Campbell, J.M.P. Carmelo, F. Guinea, E. Louis,
*The Hubbard Model: Its Physics and Mathematical Physics*(Plenum Press, New York, 1995)CrossRefGoogle Scholar - 28.Z.N.C. Ha,
*Quantum Many-Body Systems in One Dimension*(World Scientific, Singapore, 1996)CrossRefGoogle Scholar - 29.B. Sutherland,
*Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems*(World Scientific, Singapore, 2004)CrossRefGoogle Scholar - 30.F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper, V.E. Korepin,
*The One-Dimensionsl Hubbard Model*(Cambridge University Press, Cambridge, 2005)CrossRefGoogle Scholar - 31.K. Hashimoto, Int. J. Quantum Chem.
**28**, 581 (1985)CrossRefGoogle Scholar - 32.P. Piecuch, J. Čížek, J. Paldus, Int. J. Quantum Chem.
**42**, 165 (1992)CrossRefGoogle Scholar - 33.K. Hashimoto, J. Čížek, J. Paldus, Int. J. Quantum Chem.
**34**, 407 (1988)CrossRefGoogle Scholar - 34.J. Paldus, M. Takahashi, R.W.H. Cho, Phys. Rev. B
**30**, 4267 (1984)CrossRefGoogle Scholar - 35.R. Pauncz, J. de Heer, P.-O. Löwdin, J. Chem. Phys.
**36**, 2247 (1962)CrossRefGoogle Scholar - 36.R. Pauncz, J. de Heer, P.-O. Löwdin, J. Chem. Phys.
**36**, 2257 (1962)CrossRefGoogle Scholar - 37.J. de Heer, R. Pauncz, J. Mol. Spectr.
**5**, 326 (1960)CrossRefGoogle Scholar - 38.R. Pauncz,
*Alternant Molecular Orbital Method*(W. B. Saunders, Philadelphia, 1967)Google Scholar - 39.G.L. Bendazzoli, S. Evangelisti, Chem. Phys. Lett.
**185**, 125 (1991)CrossRefGoogle Scholar - 40.G.L. Bendazzoli, S. Evangelisti, L. Gagliardi, Int. J. Quantum Chem.
**51**, 13 (1994)CrossRefGoogle Scholar - 41.S. Evangelisti, G.L. Bendazzoli, Chem. Phys. Lett.
**196**, 511 (1992)CrossRefGoogle Scholar - 42.G.L. Bendazzoli, S. Evangelisti, Int. J. Quantum Chem.
**66**, 397 (1998)CrossRefGoogle Scholar - 43.R. Podeszwa, S.A. Kucharski, L.Z. Stolarczyk, J. Chem. Phys. 116, 480 (2002)Google Scholar
- 44.J. Paldus, M.J. Boyle, Int. J. Quantum Chem.
**22**, 1281 (1982)CrossRefGoogle Scholar - 45.J. Paldus, in
*Theoretical Chemistry: Advances and Perspectives*, vol. 2, ed. by H. Eyring, D.J. Henderson (Academic Press, New York, 1976), pp. 131–290.Google Scholar - 46.J. Paldus, in
*Mathematical Frontiers in Computational Chemical Physics, IMA Series*, vol. 15, ed. by D.G. Truhlar (Springer, Berlin, 1988), pp. 262–299Google Scholar - 47.R.G. Parr,
*The Quantum Theory of Molecular Electronic Structure*(Benjamin, New York, 1963)Google Scholar - 48.N. Mataga, K. Nishimoto, Z. Phys, Chem.
**13**, 140 (1957)Google Scholar - 49.J. Čížek, J. Paldus, I. Hubač, Int. J. Quantum Chem. Symp.
**8**, 293 (1974)CrossRefGoogle Scholar - 50.Y. Ooshika, J. Phys. Soc. Jpn.
**12**, 1246 (1957)CrossRefGoogle Scholar - 51.T. Murai, Progr. Theor. Phys.
**27**, 899 (1962)CrossRefGoogle Scholar - 52.D. Cazes, L. Salem, C. Tric, J. Polymer Sci.:Part C, No. 29, pp. 109–118 (1970).Google Scholar
- 53.J. Hubbard, Proc. R. Soc. A
**244**, 199 (1958)CrossRefGoogle Scholar - 54.M. Urban, J. Noga, S.J. Cole, R.J. Bartlett, J. Chem. Phys.
**83**, 4041 (1985)CrossRefGoogle Scholar - 55.K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett.
**157**, 479 (1989)CrossRefGoogle Scholar - 56.J. Čížek, J. Chem. Phys.
**45**, 4256 (1966)CrossRefGoogle Scholar - 57.J. Čížek, Adv. Chem. Phys.
**14**, 35 (1969)Google Scholar - 58.J. Čížek, J. Paldus, Int. J. Quantum Chem.
**5**, 359 (1971)CrossRefGoogle Scholar - 59.J. Paldus, J. Čížek, I. Shavitt, Phys. Rev. A
**5**, 50 (1972)CrossRefGoogle Scholar - 60.J. Paldus, J. Čížek, Adv. Quantum Chem.
**9**, 105 (1975)CrossRefGoogle Scholar - 61.P.E.S. Wormer, J. Paldus, Adv. Quantum Chem.
**51**, 59 (2006)CrossRefGoogle Scholar - 62.J. Paldus, J. Chem. Phys.
**67**, 303 (1977)CrossRefGoogle Scholar - 63.P. Piecuch, J. Paldus, Theor. Chim. Acta
**78**, 65 (1990)CrossRefGoogle Scholar - 64.P. Piecuch, R. Toboła, J. Paldus, Phys. Rev. A
**54**, 1210 (1996)CrossRefGoogle Scholar - 65.P. Piecuch, J. Paldus, Theor. Chim. Acta
**83**, 69 (1992)CrossRefGoogle Scholar - 66.P. Piecuch, R. Toboła, J. Paldus, Int. J. Quantum Chem.
**55**, 133 (1995)CrossRefGoogle Scholar - 67.A.E. Kondo, P. Piecuch, J. Paldus, J. Chem. Phys.
**104**, 8566 (1996)CrossRefGoogle Scholar - 68.X. Li, J. Paldus, J. Chem. Phys.
**101**, 8812 (1994)CrossRefGoogle Scholar - 69.B. Jeziorski, J. Paldus, P. Jankowski, Int. J. Quantum Chem.
**56**, 129 (1995)CrossRefGoogle Scholar - 70.D.J. Thouless, in
*The Quantum Mechanics of Many-Body Systems,*2nd edn. (Academic, New York, 1972), p. 57 (p. 35 in the 1961, 1st edn.)Google Scholar - 71.J.-M. Maillet, in
*Quantum Spaces: Poincaré Seminar 2007, Progress in Mathematical Physics*, vol. 53, ed. by B. Duplantier, V. Rivasseau (Birkhäuser Verlag, Basel, 2007), pp. 161–201Google Scholar