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 $N \rightarrow \infty$.

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., [1–7]; 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., [22–30]).

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., [34–42] 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 energy¹ 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.

In view of the fact that cyclic polyenes may be regarded as a model of linear metals, it is of interest to explore the relative importance of indecomposable and decomposable clusters, particularly in the $N \rightarrow \infty$ limit, as manifested by the trend in their relative occurrences. For example, designating the number of quadruples for the $N=4\nu +2$ polyene by $Q(N)$ and, similarly, the number of decomposable and indecomposable quadruples by $Q^\mathrm{dec}(N)$ and $Q^\mathrm{indec}(N)$, respectively, it would be of interest to find out how these quantities depend on the size of the polyene $N$ and, in particular, to determine the limits like

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{Q^\mathrm{indec}(N)}{Q(N)} \qquad \mathrm{or} \qquad \lim _{N \rightarrow \infty } \frac{Q^\mathrm{indec}(N)}{Q^\mathrm{dec}(N)}. \end{aligned}$$

(1)

This information should help us to better understand the cluster structure of the corresponding wave functions and enable us to assess the capability of CC approaches to describe this type of systems. For large values of $N$ we can also expect the importance of higher than quadruples and a similar behavior of their decomposable and indecomposable components.

2 Basic notation

Unlike the ab initio approaches that rely on nonorthogonal atomic orbitals (AOs) to define a truncated finite-dimensional Hilbert space employed, the semi-empirical approaches utilize a minimum basis set of hypothetical orthonormal AOs. We can thus employ the standard second quantization formalism associated with these basis sets as in the case of molecular orbitals (MOs) given as linear combinations of AOs. We shall label AOs by lower case Greek letters, e.g., $|\mu \rangle , \, |\nu \rangle$, etc., and MOs by the lower case letters of the Latin alphabet, e.g., $|a \rangle , \, |b \rangle$, etc., designating generic MOs by $|i \rangle , \, |j \rangle$, etc. The corresponding spin-orbitals, given by a simple product of an orbital and a spin function $|\sigma \rangle , \, \sigma = \pm \frac{1}{2}$, will then be designated by the corresponding capitals, e.g., $|M \rangle = |\mu \rangle |\sigma \rangle$ or $|A \rangle = |a \rangle |\sigma \rangle$. The corresponding creation and annihilation operators are then designated as $X^{\dagger }_A \equiv X^{\dagger }_{a \sigma }$ and $X_A \equiv X_{a \sigma }$, respectively. We also define the orbital unitary group, $\hbox {U}(N)$, generators $E_{i j}$ as (see, e.g., [45, 46])

$$\begin{aligned} E_{i j} = \sum _{\sigma } X^{\dagger }_{i \sigma } X_{j \sigma }, \end{aligned}$$

(2)

so that we can express the relevant spin-independent electronic Hamiltonian involving at most two-body interactions

$$\begin{aligned} H=\sum _i z_i + \sum _{i<j} v_{ij}, \end{aligned}$$

(3)

in the following second-quantized form

$$\begin{aligned} H&= \sum _{i,j} \langle i |z| j \rangle \sum _\sigma X^{\dagger }_{i \sigma } X_{j \sigma } + \frac{1}{2}\sum _{i,j,k,l} \langle i j |v| k l \rangle \sum _{\sigma , \tau } X^{\dagger }_{i \sigma } X^{\dagger }_{j \tau } X_{l \tau } X_{k \sigma } \nonumber \\&= \sum _{i,j} \langle i |z| j \rangle E_{ij} + \frac{1}{2}\sum _{i,j,k,l} \langle ij |v| kl \rangle \left( E_{ik} E_{jl} - \delta _{jk} E_{il}\right) , \end{aligned}$$

(4)

with $i,j,k,l$ representing either the effective, hypothetical AOs $\mu , \nu$, etc., or MOs $a, b, c$, etc., either set being orthonormal.

3 Semiempirical $\pi$-eletron Hamiltonians

The $\pi$-eletron systems can be conveniently described by the semi-empirical Hamiltonian $H_\pi$ of the PPP type (see, e.g., Sec. VI of [45])

$$\begin{aligned} H_\pi = \sum _{\mu ,\nu } z_{\mu \nu } E_{\mu \nu } + \frac{1}{2}\sum _{\mu ,\nu } \gamma _{\mu \nu } \left( E_{\mu \mu } E_{\nu \nu } - \delta _{\mu \nu } E_{\mu \nu }\right) , \end{aligned}$$

(5)

with indices $\mu , \nu = 1, 2, \ldots , N$ labeling the atomic sites and the AO unitary group generators $E_{\mu \nu }$ defined as in Eq. (2). Like any $\pi$-electron semi-empirical Hamiltonian, it is defined directly by specifying the one- and two-electron parameters $z_{\mu \nu }$ and $\gamma _{\mu \nu }$, respectively, rather than by selecting some actual spin orbital basis set as is the case of ab initio model Hamiltonians. For the two-center Coulomb-type two-electron integrals one invokes the zero differential overlap approximation [47], so that only the Coulomb-type integrals $\gamma _{\mu \nu } \equiv \langle \mu (1) \nu (2) |r_{12}^{-1}|\mu (1) \nu (2) \rangle$ are required.

The diagonal one-electron parameters $z_{\mu \mu }$ are approximated by relying on the Goeppert-Mayer and Sklar approximation (see, e.g., [47])

$$\begin{aligned} z_{\mu \mu } = \alpha _\mu - \sum _{\nu \ne \mu } Z_\nu \gamma _{\mu \nu }, \end{aligned}$$

(6)

where $\alpha _\mu$ is the so-called Coulomb integral (usually approximated by the valence state ionization potential) and $Z_\mu$ designates the number of $\pi$-electrons contributed by the $\mu$-th atomic site. For the one-electron off-diagonal parameters $z_{\mu \nu }\,(\mu \ne \nu )$ one invokes the tight-binding approximation, so that in view of the $D_{Nh}$ (or simply $C_N$) point group symmetry of our models (see Sec. 4 below) we can write

$$\begin{aligned} z_{\mu ,\mu \pm 1} = \beta \quad \mathrm{and} \quad z_{\mu \nu }=0 \;\; \mathrm{otherwise}, \end{aligned}$$

(7)

with all indices taken modulo $N$.

When we include the internuclear repulsion term $\sum _{\mu <\nu } Z_\mu Z_\nu \gamma _{\mu \nu }$, we can thus rewrite the Hamiltonian (5) as follows

$$\begin{aligned} H_\pi&= \sum _\mu \alpha _\mu + {\sum _{\mu \nu }}^\prime \beta E_{\mu \nu } + \sum _{\mu < \nu } \gamma _{\mu \nu } \left( E_{\mu \mu } - Z_\mu \right) \left( E_{\nu \nu } - Z_\nu \right) \nonumber \\&\quad + \frac{1}{2}\sum _\mu \gamma _{\mu \mu } E_{\mu \mu } \left( E_{\mu \mu } - 1\right) , \end{aligned}$$

(8)

where the prime on the second summation symbol implies the sum over nearest neighbors only. For cyclic polyenes each atomic site contributes only one electron $(Z_\mu = 1)$ and all one-center Coulomb integrals are equivalent, so that

$$\begin{aligned} \alpha _\mu = \alpha _1 = 0 \quad \mathrm{and} \quad \gamma _{\mu \mu } = \gamma _{11}, \end{aligned}$$

(9)

with $\alpha _1 = 0$ defining the origin of the energy scale. Thus, finally, for the case of cyclic polyenes we can write the Hamiltonian (5) or (8) in a simple form (see, e.g., Eq. (6.19) of [45])

$$\begin{aligned} H_\pi = \beta \sum _\mu \left( E_{\mu ,\mu +1} + E^\dagger _{\mu ,\mu +1}\right) + \frac{1}{2}\sum _{\mu ,\nu } \gamma _{\mu \nu } (n_\mu -1)(n_\nu -1), \end{aligned}$$

(10)

where we designated the $\mu$th site occupation number operator by $n_\mu \equiv E_{\mu \mu }$.

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 $-2.5\hbox { eV}$.

Various approximations are used for the two-electron Coulomb repulsion parameters $\gamma _{\mu \nu }$, the most common one for the PPP Hamiltonian being the Mataga–Nishimoto parametrization [48], i.e., a modified point charge approximation

$$\begin{aligned} \gamma _{\mu \nu } = e^2/(R_{\mu \nu }+c), \quad c=e^2/\gamma _{00}, \end{aligned}$$

(11)

with one center integral $\gamma _{00} \equiv \gamma _{\mu \mu }$ given by the so-called $I$–$A$ approximation [47] (for the PPP Hamiltonian $\gamma _{00}=10.84\hbox { eV}$) and $R_{\mu \nu }$ designating the internuclear separation between the atomic sites $\mu$ and $\nu$ (with the nearest neighbor site separation taken as 1.4 Å). The geometry of the model implies that $\gamma _{\mu \nu } = \gamma _{\mu +\kappa ,\nu +\kappa } = \gamma _{0,\mu -\nu }$.

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 [50–52].

Our hypothetical systems thus represent a $\pi$-electron model of cyclic polyenes $\hbox {C}_N\hbox {H}_N$ having a non-degenerate ground state $(N=2n=4\nu +2, \, \nu =1,2,\ldots )$ with carbon atomic sites located at the vertices of a regular $N$-gon. In view of the Bloch theorem the MOs $|a \rangle ,\, a = 0, 1, 2,\ldots ,(N-1)$, are completely determined by the symmetry of the system

$$\begin{aligned} |a \rangle = \left( 1/\sqrt{N}\right) \sum _{\mu = 0}^{N-1} \exp {(i\omega a \mu )} |\mu \rangle , \qquad \omega = 2\pi /N. \end{aligned}$$

(12)

The one- and two-electron matrix elements of the Hamiltonian take a very simple form in the MO basis, namely (cf. [34])

$$\begin{aligned} \langle a |z| b \rangle&= 2 \beta \delta _{ab} \cos (\omega a), \nonumber \\ \langle a b |v| c d \rangle&= K(a-c) \delta _{a+b,c+d}, \end{aligned}$$

(13)

where

$$\begin{aligned} K(a) = K(-a)&= N^{-1} \sum _{\mu =0}^{N-1} \gamma _{0 \mu } \exp (i \omega a \mu ) \nonumber \\&= N^{-1} \left[ \gamma _{00} + (-1)^a \gamma _{0n} + 2 \sum _{\mu =1}^{n-1} \gamma _{0 \mu } \cos (\mu a \omega )\right] , \end{aligned}$$

(14)

with all indices or arguments taken modulo $N$.

In order to distinguish MOs that are defining a CS ground-state IPM wave function, referred to as the occupied MOs, from the virtual or unoccupied ones, we shall label the former ones by the subscripted and the latter ones by the superscripted indices, i.e., as $a_1, a_2$, etc., and $a^1, a^2$, etc., respectively. The Hartree-Fock orbital energies $\epsilon _a$ are then

$$\begin{aligned} \epsilon _a = \langle a |f| b \rangle \delta _{ab} = 2 \beta \cos (\omega a) + N K(0) - \sum _{a_1} K(a - a_1), \end{aligned}$$

(15)

with the summation extending over the occupied (or hole) MOs $a_1\, (-\nu \le a_1 \le \nu )$ defining the HF sea. We note that the Bloch MOs (12) are pairwise degenerate, $\epsilon _a = \epsilon _{-a} \equiv \epsilon _{N-a}$, except when $a=0$ or $a=n$.

The expressions for the one-and two-electron matrix elements of the Hamiltonian $H_\pi$ and for the orbital energies take on a particularly simple form when we rely on the Hubbard Hamiltonian. Since only on-site Coulomb integrals are involved, the actual geometry of the molecular framework is irrelevant, just as in the case of the Hückel Hamiltonian. Thus, the quantities $K(a)$ are independent of the orbital label,

$$\begin{aligned} K(a) \equiv K = \gamma /N, \end{aligned}$$

(16)

and the orbital energies are essentially those of the Hückel model, i.e.,

$$\begin{aligned} \epsilon _a = 2 \beta \cos (\omega a) + \frac{1}{2}\gamma \end{aligned}$$

(17)

5 Coupled cluster method

In contrast to the linear SR configuration interaction (CI) Ansatz for the exact wave function $| \Psi \rangle$,

$$\begin{aligned} | \Psi \rangle = \sum _{i=0}^N C_i |\Phi \rangle , \quad C_0 \equiv \hat{1}, \end{aligned}$$

(18)

the SR CC method relies on the exponential Ansatz

$$\begin{aligned} | \Psi \rangle = \exp \Big ( \sum _{i=0}^N T_i \Big ) |\Phi \rangle , \end{aligned}$$

(19)

where $| \Phi \rangle$ designates a single-determinantal IPM (usually HF) wave function. We employ the intermediate normalization $\langle \Psi | \Phi \rangle = 1$ in both cases. The CI and CC $i$-body operators $C_i$ and $T_i (i=1, 2, \ldots , N)$, respectively, are expressed as linear combinations of $i$-body excitation operators $G_i(\Omega )$ with coefficients representing the corresponding CI and CC amplitudes $c_j^{(i)}$ and $t_j^{(i)}$,

$$\begin{aligned} C_i = \sum _j c_j^{(i)} G_i(\Omega _j) \quad \mathrm{and} \quad T_i = \sum _j t_j^{(i)} G_i(\Omega _j). \end{aligned}$$

(20)

Labeling the spin orbitals $| A \rangle$ that are occupied and unoccupied in the reference IPM wave function $| \Phi \rangle$ by the subscripted and superscripted labels $A_k$ and $A^k$, respectively, the excitation operators $G_i(\Omega _j)$ are given by

$$\begin{aligned} G_i(\Omega _j) = \prod _{k=1}^i (X^\dagger _{A^k} X_{A_k}), \quad \Omega _j \equiv \bigg \{A_k, A^k|k=1, \ldots , i\bigg \}. \end{aligned}$$

(21)

The CI and CC operators $C_i$ and $T_i$ are then related as follows

$$\begin{aligned} C_i = T_i + \sum _{\lambda \vdash i} \prod _j \big (n_j!\big )^{-1} T_j^{n_j}, \end{aligned}$$

(22)

where the summation extends over all nontrivial partitions $\lambda$ of $i, \lambda \vdash i$, as given by the solutions of the Diophantine equation $i = \sum _{j=1}^{i-1} j n_j$. Here $T_i$’s represent the connected cluster components, while the product terms on the right hand side of (22) represent the disconnected components. When Brueckner MOs are employed, the $C_1 = T_1$ components vanish, so that the first disconnected component appears in the quadruply-excited $C_4, C_4 = T_4 + \frac{1}{2}T_2^2$.

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].

When deriving the explicit form of CC equations [56–59] (for a historical account see [8] and references therein) it is convenient to employ the normal product ($N$-product) form of the Hamiltonian (cf. [60]) relative to the IPM wave function $| \Phi \rangle$ as a Fermi vacuum for the hole–particle (h–p) second-quantized formalism, i.e.,

$$\begin{aligned} H_N&\equiv H - \langle \Phi | H | \Phi \rangle \nonumber \\&= \sum _{A,B} \langle A|f|B \rangle N\left[ X_A^\dagger X_B\right] + \frac{1}{4}\sum _{A,B,C,D} \langle A B |v| C D \rangle _{\mathcal {A}} N\left[ X_A^\dagger X_B^\dagger X_D X_C\right] , \qquad \quad \end{aligned}$$

(23)

where $\langle A B |v| C D \rangle _{\mathcal {A}}$ designates the antisymmetrized two-body matrix element

$$\begin{aligned} \langle A B |v| C D \rangle _{\mathcal {A}} \equiv \langle A B |v| C D \rangle - \langle A B |v| D C \rangle , \end{aligned}$$

(24)

and $\langle A |f| B \rangle$ are the one-body IPM (usually HF) matrix elements

$$\begin{aligned} \langle A |f| B \rangle \equiv \langle A |z| B \rangle + \sum _{C_1} \langle A C_1 |v| B C_1 \rangle _{\mathcal {A}}. \end{aligned}$$

(25)

This facilitates the exploitation of Wick’s theorem and of Feynman-like diagrams while relying on the h-p formalism (see, e.g., [60]) and of the spin-angular-momentum diagrams (see, e.g., [61]) when deriving the spin-adapted version of the CC formalism (see, e.g., [62]). Note that the excitation operators $G_i$ are automatically in the $N$-product form since they involve only creation h-p operators.

Since the variational determination of cluster amplitudes leads to a very cumbersome formalism one relies on the Schrödinger equation $H_N |\Psi \rangle = \Delta \epsilon | \Psi \rangle$ projected onto the manifold of relevant excited configuration state functions (CSFs). Here $\Delta \epsilon \equiv E - \langle \Phi |H| \Phi \rangle$ designates the exact energy relative to the IPM model (i.e., the correlation energy when we use a HF reference). Thus, substituting the Ansatz (19) into the Schrödinger equation and exploiting the connected cluster theorem, i.e., $H_N \exp (T) |\Phi \rangle = \exp (T) [H_N \exp (T)]_{\mathcal {C}} | \Phi \rangle$, we obtain (see, e.g., [56–59])

$$\begin{aligned} \left( \left[ H_N \exp (T)\right] _{\mathcal {C}} - \Delta \epsilon \right) | \Phi \rangle = 0, \end{aligned}$$

(26)

where the subscript $\mathcal {C}$ indicates the connected part of the expression. By relying on the diagrammatic technique (or symbol manipulation codes) one can then obtain an explicit form of the above equation as a set of nonlinear algebraic equations for the unknown cluster amplitudes.

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 [65–67] 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 $\beta = -2.5\,\hbox {eV}$.

The $N=6$ and $N=10$ results clearly revealed an overwhelming importance of doubles and a rapidly increasing role of quadruples as the fully correlated limit is approached (cf. Figs. 4 and 5 and Tables IV–VII of [44]). Moreover, already the $N=10$ results (Table 1) indicate an increasing importance of higher than quadruples in this limit. Contribution of triples is not only very small relative to that of doubly- and quadruply-excited CSFs but—except for the highly correlated region—is also approximately additive (cf. Table VIII of [44]).

Table 1 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)

Interestingly enough, while the difficulty of obtaining the FCI results rapidly increases when approaching the fully correlated limit and the CC methods break down completely (even when accounting for connected quadruples [43]) for large enough $N$ values, the handling of the fully correlated limit $(\beta = 0)$ becomes easy in view of the fact that only the electrostatic component of the Hamiltonian plays a role here. Indeed, it is not difficult to show (see, e.g., [44]) that the exact correlation energy at $\beta = 0$ becomes

$$\begin{aligned} -\Delta \epsilon ^\mathrm{PPP} (\beta = 0) = n(2n - 1) K(0) - 2 \sum _{i<j}^\mathrm{(occ)} K(i-j) - \sum _{\mu <\nu } \gamma _{\mu \nu }, \end{aligned}$$

(27)

for the PPP Hamiltonian, while an almost trivial result is obtained for the Hubbard Hamiltonian

$$\begin{aligned} -\Delta \epsilon ^\mathrm{H} (\beta = 0) = n^2 K(0) = \left( \nu + \frac{1}{2}\right) \gamma _{11} = \frac{1}{4}N \gamma . \end{aligned}$$

(28)

Of course, in the latter case we can generate the exact FCI energies for any value of the coupling constant and any $N$ using Lieb–Wu equations [31, 33]. Unfortunately, this approach (like any based on the Bethe Ansatz) does not provide us with any information about the cluster structure of the corresponding wave functions. It is, however, an important realization that the Hubbard and PPP results exhibit a parallel behavior, both as far as the role played by various excitation orders in their contribution to the correlation energy and their weight in the resulting wave functions.

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]).

Now, starting with the $N=10$ ring we observe an interesting new phenomenon, which is clearly associated with the high symmetry of our models, namely that some of the connected quadruples do not possess corresponding disconnected counterparts given by the product of two doubles. We can thus refer to such quadruples as the 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.

Table 2 Indecomposable quadruples for $N=10$ cyclic polyene

We recall that we label the MOs by their quasimomentum $k$ (or $a$), $k=0, 1, \ldots , (N-1)$, Eq. (12), (see, e.g., Fig. 1 of [34] for a general schematic representation of Bloch orbital labeling and orbital energy distribution for $\hbox {C}_N\hbox {H}_N$). The ground state configuration $(0^2 \, 1^2 \, \ldots \, \nu ^2 \, (N-\nu )^2 \, (N-\nu +1)^2 \, \ldots \, (N-1)^2)$ or the corresponding CSF is then associated with a zero quasimomentum $k=0\,(\mathrm{mod}\,N)$, so that only CSFs belonging to the same irrep of $C_{N}$ (i.e., having the same quasimomentum $k=0$) can interact with the ground state and thus appear in the pertinent wave function or CC formalism. Clearly, all singles are associated with a nonvanishing quasimomentum $k > 1$ and can be ignored. The number of singlet and triplet coupled doubles (the latter involving four distinct MO labels), $N^\mathrm{(bi)}_{S=0}$ and $N^\mathrm{(bi)}_{S=1}$, respectively, that are associated with a zero quasimomentum transfer (mod $N$) are given by

$$\begin{aligned} N^\mathrm{(bi)}_{S=0} = \frac{4}{3}\nu (\nu ^2 + 2) + 3 \nu ^2 + 1 \quad \mathrm{and} \quad N^\mathrm{(bi)}_{S=1} = \frac{2}{3}\nu (2 \nu ^2 + 1) + \nu ^2 \, . \end{aligned}$$

(29)

Note that such configurations (CSFs or CC amplitudes) are simply generated by considering all possible pairs of, say, virtual MOs $a^1,\, a^2$ with occupied MOs given by $a_1 = a^1 - k, \, a_2 = a^2 + k$ yielding a zero quasimomentum transfer. Of course, here we require that $-\nu \le a_i \le \nu , i=1,2$ (all labels modulo $N$). We can then similarly generate possible zero quasimomentum transfer quadruples and check if there exists at least one possible pair of $k=0$ doubles. In this way we can determine the number of decomposable and indecomposable quadruples for each polyene.

To provide an example of an indecomposable quadruple in the case of $N=10$ polyene let us consider, for example, the excitation $8^2 \, 9^2 \rightarrow 5^2 \, 7^2$ (i.e., $\Phi _{17}$ in Table 2), shown schematically in Fig. 1. The ground state configuration $(0^2 \, 1^2 \, 2^2 \, 8^2 \, 9^2)$ being associated with a zero quasimomentum $[k=0\,(\mathrm{mod} N)]$, only CSFs belonging to the same irrep of $C_{10}$ (i.e., having the same quasimomentum $k=0$) can interact with it and appear in the pertinent wave function or CC formalism. Considering now possible disconnected quadruples given by pairs of doubles, we find five possible candidates listed in Table 3, where we also parenthetically indicate the corresponding quasimomenta $k$ for each double. We see that even though these pairs of doubles have zero quasimomentum, none of the individual doubles is associated with a zero quasimomentum. Consequently, none of these doubles can appear as a component of the ground state wave function and thus these disconnected quadruples will not play any role in the CC (in this case CCD) formalism.

Table 3 Possible disconnected quadruples for the 17th configuration $8^2 \, 9^2 \rightarrow 5^2 \, 7^2$ of Table 2

Fig. 1

A schematic representation of quadruple excitation $8^2 \, 9^2 \rightarrow 5^2 \, 7^2$ (or $1^2 \, 2^2 \rightarrow 3^2 \, 5^2$)

We have developed appropriate codes that generate all relevant quadruples and analyze them for their decomposability. The resulting counts for the decomposable $[Q^\mathrm{dec}(N) \equiv D]$ and indecomposable $[Q^\mathrm{indec}(N) \equiv I]$ quadruples, as well as the number of all quadruples $[Q \equiv Q(N) = Q^\mathrm{dec}(N)+Q^\mathrm{indec}(N) = D+I]$ are given in Table 4 for cyclic polyenes $\hbox {C}_N\hbox {H}_N, N=4\nu + 2$ with $1 \le \nu \le 47$, where we also included the ratios $I/Q [\equiv Q^\mathrm{indec}(N)/Q(N)]$ and $I/D [\equiv Q^\mathrm{indec}(N)/Q^\mathrm{dec}(N)]$. The corresponding results for CSFs (accounting for a number of distinct singlets for configurations involving four or more distinct MOs) are then listed in Table 5. In each case we observe an initial rapid increase in the number of indecomposables $Q^\mathrm{indec}(N)$ relative to the total number of quadruples $Q(N)$, as graphically illustrated in Fig. 2a. The observed trend seems to indicate that the ratio $Q^\mathrm{indec}(N)/Q(N) \equiv I/Q$ will approach unity as $N \rightarrow \infty$. This fact is then confirmed by considering the ratio of indecomposables relative to the decomposables $Q^\mathrm{indec}(N)/Q^\mathrm{dec}(N) \equiv I/D$ shown in Fig. 2b. This figure clearly implies a linear dependence of this ratio for large enough values of $N$, so that we can write that $I/D \sim N$ or, more precisely, that $I/D = cN - d$, with a small negative intercept $d > 0$. We can thus conjecture with great confidence that the sought after limits, Eq. (1), are given by

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{Q^\mathrm{indec}(N)}{Q^\mathrm{dec}(N)} = \lim _{N \rightarrow \infty } \frac{I}{D}&\rightarrow \infty , \nonumber \\ \lim _{N \rightarrow \infty } \frac{Q^\mathrm{indec}(N)}{Q(N)} = \lim _{N \rightarrow \infty } \frac{I}{Q}&= \lim _{N \rightarrow \infty } \frac{N-d/c}{N-(d+1)/c} = 1. \end{aligned}$$

(30)

Fig. 2

a Dependence of the ratio $I/(D+I)$ of the number of indecomposable quadruples $I$ to the total number of quadruples $(D+I)$, both decomposable $D$ and indecomposable ones, and b dependence of the ratio $I/D$ of the number of indecomposable $I$ to the number of decomposable $D$ quadruples, both as a function of the size of the polyene $N = 4 \nu + 2$

Table 4 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 $\hbox {C}_N\hbox {H}_N, N=4\nu + 2$

Table 5 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 $\hbox {C}_N\hbox {H}_N, N=4\nu + 2$

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.