Brad Ganoe and
James Shee*
Department of Chemistry, Rice University, Houston, TX 77005, USA. E-mail: james.shee@rice.edu
First published on 10th April 2024
Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples – hydrogen systems (H_{n}, n = 2, 4, 6), Be_{2}, H–He–H, and benzene – and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of “correlation energy” and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird’s rule.
1. The wavefunction cannot qualitatively be described by a single Slater determinant, and is “multiconfigurational”. A chemical example is when two states of different character – e.g., ionic and covalent configurations that can be non-orthogonal along much of a dissociation coordinate – are of competing significance (energetically nearly degenerate).^{5} Another example from physics of such an intermediate regime involves fluctuations in materials such as transition metal oxides close to a Mott metal–insulator transition^{6} or f-block compounds,^{7} in which electrons “hesitate” between delocalized (wave-like) and localized (particle-like) character.^{4}
2. The two-body interaction is large relative to one-body terms in the Hamiltonian. This has its origin in the Hubbard model:
(1) |
3. Electrons are substantially “entangled”.^{9} From an information theory perspective, this can refer to entanglement entropy, e.g. Shannon or Renyi formulations.^{10,11} Concepts such as 1- and 2-orbital entanglement entropies have also been proposed.^{12,13}
4. The overlap of the Hartree–Fock (HF) state or the largest-weighted configuration in a multi-determinant expansion with the true ground-state is small.
5. The eigenvalues of the one-electron reduced density matrix – the natural orbital occupation numbers (NOONs) – are non-integer, deviating substantially from 0 and 1 (or 2 for restricted models).^{14}
6. The amplitudes of Coupled Cluster (CC) theory, especially T_{1} and T_{2} of CCSD,^{15–17} become large. As the connected portions of triples amplitudes and higher become significant, the corresponding residuals remain large such that no finite truncation is well posed.^{18}
7. Two-body spin or particle correlation functions, e.g. 〈S_{0}·S_{i}〉 or 〈_{0}_{i}〉 in a real-space site basis^{19} that do not decay exponentially. This idea of long correlation lengths is familiar to statistical physics.
8. The energies from Møller–Plesset perturbation theory through low orders with the generalized Hartree–Fock (GHF) reference state qualitatively differ from that of Full Configuration Interaction (FCI).^{20} The causal relationship between strong correlation and the breakdown of many-body perturbation theory is relatively well recognized in the physics literature.^{4}
The failure of approximate density functional theory (DFT) is another common characterization. Alternatively, a colleague recently exclaimed of strong correlation that “you know it when you see it”!
While Small and Head-Gordon have defined strong correlation in terms of the breakdown of a specific perturbation theory, there is general agreement that a consequence of strong correlation is that feasibly-scaling electronic structure models such as low orders of single-reference perturbation or coupled cluster theories and approximate density functionals (especially those from the higher rungs of Jacob’s ladder) break down.^{4,21,22} Thus it is of practical value to computational scientists to reliably diagnose when a system shows symptoms of strong correlation. For example, a recent work has delineated correlation regimes inside of which CCSD(T) is expected to produce ionization potentials of transition metal complexes to within kcal mol^{−1} accuracy, and outside of which CCSD(T) is expected to qualitatively fail (no matter which orbitals are used in the reference determinant).^{23} Such predictive classification relied on a composite diagnostic involving both spin-symmetry breaking in unrestricted CCSD wavefunctions and the largest CI coefficient in selected CI wavefunctions (in the basis of natural orbitals).
In contrast, the present work is motivated by a more fundamental question concerning the physical nature/characteristics of a strongly correlated electronic system: given a Hamiltonian and an exact eigenstate, what quantity or property can be associated with the regime of strong correlation? Furthermore, what is implied about physical reality when this quantity is large? We require such a quantity to be rigorously defined, physically interpretable, and generally meaningful for both molecules and quantum lattice models (other more technical properties are discussed in the final section). Note that this type of inquiry is distinct from classifying correlation regimes based on which approximate methods are broken. These two approaches to elucidating strong correlation are often conflated.
Another motivating goal of this work is to build a bridge between the chemistry and physics communities, with regard to perspectives and even language surrounding electron correlation. As frequently highlighted by R. Hoffmann,^{24} a paradigmatic example of a concept that is fundamental, chemically-consequential, and beloved by chemists, but whose definition proves more evasive the more one tries to formalise it, is aromaticity and, more interestingly in our view, anti-aromaticity. According to one of the oldest and most widely-used definitions, aromaticity arises in cyclic compounds with 4N + 2 electrons in a conjugated π-orbital system, while antiaromatic compounds have 4N electrons. Baird’s rule further states that a system with an aromatic (antiaromatic) ground-state will have anti-aromatic (aromatic) low-lying vertical excited states.^{25} Admittedly, such definitions and counting rules are not mathematically rigorous, yet empirically these concepts are seen to hold. For example, evidence of aromaticity includes literal aromaticity (smell), unusual stability/inertness to chemical reactivity, and magnetic shielding effects due to ring currents; while antiaromaticity implies an unusually unstable state that tends to rapidly react or distort geometrically.^{26} A beautiful and unexpected result of this work is that antiaromatic molecules share a formal quantum-mechanical footprint with seemingly disparate phenomena ranging from bond-stretching and super exchange physics to antiferromagnetic order in some polynuclear transition metal catalysts and cuprate materials. Indeed, we venture to propose that the footprint is due to the elephant in the room – strong correlation.
Regarding 1: the expansion coefficients of a multiconfigurational wavefunction depend on the choice of single-particle basis; ref. 2 shows a particularly stark example in which delocalized vs. localized orbitals imply qualitatively different assessments of multiconfigurational character. It has also been shown that most mono-nuclear transition metal coordination complexes are qualitatively well described by a single determinant plus an appropriate level of dynamic correlation,^{3} and that exchange-coupled polynuclear transition metal compounds can often be described by a single spin-adapted configuration state function.^{2,27}
Regarding 3: in our view, the characterization of a system as strongly correlated should not depend on the partitioning of a Hilbert space into distinguishable subspaces, as is often required to compute metrics such as entanglement entropy. As in 1, orbital entanglement entropy metrics have been shown to be strongly dependent on the choice of single-particle basis employed.^{12}
Regarding 4: while the overlap of a given wavefunction with FCI, 〈Φ|Ψ_{FCI}〉, is useful for finite systems such as molecules (as it reports on the percentage of the FCI Hilbert space that is relevant to that wavefunction), the focus on the case where |Φ〉 is the mean-field state is not relevant in the thermodynamic limit in light of the well-known “orthogonality catastrophe” initially described by Van Vleck:^{28} given N infinitely separated He atoms, and assuming that 〈Φ_{i}|Ψ_{FCI}〉 = 1 − ε for each of the He atoms, then the overlap for the composite system (1 − ε)^{N} will decay exponentially with N, even for weakly correlated systems.
Regarding 5, Mazziotti^{53} has pointed out that the 1-RDM, and thus its eigenvalues, cannot capture spin-correlations which need not decay with the 1/R interaction (and therefore can be non-zero even when the correlation energy is zero). We note also that the NOON spectrum will be identical in, e.g., H_{2} in the dissociation limit for both singlet and triplet states; this contradicts the qualitative difference in the wavefunctions: and |Ψ_{T}〉 = |σ,σ*〉. If two subspaces are assumed distinguishable, one can write: (L = left, R = right), which exhibits the famous “spooky action at a distance”, also known as spin correlations. Although the UHF wave function recovers the correct energy with a spin-symmetry broken single determinant, it does so for reasons other than having the exact many-particle wave function. Despite the ability of fractional NOONs to signal a multiconfigurational wave function, we are interested in quantities that can also capture spin-correlations/entanglements even in the dissociation limit. In what follows, we also highlight an example where the NOONs fail to distinguish two geometric configurations of H_{4} with different electronic structures.
Regarding 6, it is possible that for a weakly correlated state, the T_{1} norm from a CCSD wave function can be large; this would simply reflect a reference determinant with suboptimal orbitals. While a large T_{2} norm from CCSD can possibly be a symptom of strong correlation, in this work we are interested in how strong correlation is manifested in the true (FCI) wave function.
Regarding 7, the definition of correlation functions with respect to real-space sites is natural for lattice models and specific systems such as well-localized μ-oxo-bridged transition metals. However, such a construction is more challenging to systematically define for, e.g., Be_{2} at short interatomic separation, and (more obviously) atoms like cerium with complicated low-lying states.^{29}
Regarding 8: as mentioned in our response to point 5, in this work we are interested in quantities and properties of the exact wave function or quantities derived from the 2-RDM which can, in principle, be exactly mapped to the exact wave function. Although the size-consistency of GHF and subsequent low-order perturbation theories indeed enables qualitatively correct energies at that level of theory to signal correlation regimes in which energies can be predicted tractably, this metric is out of the scope of this work since we are interested in physical properties of exact eigenstates (which, like spin-entanglements, can persist even when the correlation energy is zero). Additionally, there are many ways to break low-order single-reference perturbation theory that are not (at least directly) physically meaningful, including the idea of “artificial” symmetry breaking in the HF state.^{3,30} In an effort to report on physically-meaningful, i.e. “essential”, symmetry breaking, in this work we will report numerical values of 〈S^{2}〉 from an unrestricted CCSD wavefunction (vide infra). Finally, we note that, GHF aside, redefining the 0th order Hamiltonian can often lead to qualitatively correct energies from low-order perturbation theories, as in CASPT2 or NEVPT2.
(2) |
^{1}Γ^{p}_{q} = 〈Φ|c^{†}_{p}c_{q}|Φ〉 | (3) |
^{2}Γ^{pq}_{rs} = 〈Φ|c^{†}_{p}c^{†}_{q}c_{r}c_{s}|Φ〉 | (4) |
The reduced density matrices can be related to the p-matrices:^{38}
(5) |
(6) |
Beyond the mean-field approximation, the introduction of non-independence between the particles may be evaluated in a statistical sense by a moment-generating function of stochastic variables to evaluate the cumulants of the expansion;^{43} a mapping by way of the antisymmeterized logarithm^{44} yields the cumulants of the ^{p}Γ, ^{p}Δ, a quantity which is generated from the connected portions of the respective ^{p}Γ from the generalized Wick’s theorem. The properties of these terms, especially up through the ^{4}Δ with their role in establishing N-representability criteria for the reconstruction of the 2-RDM,^{31} have been well reviewed in the literature and have been explicitly connected to other quantities such as CC amplitudes.^{44–52}
As in traditional statistical theory, in which the first-order cumulant is identical to the mean, the ^{1}Δ is identical to the ^{1}Γ and yields no additional unique information over the above quantity. As the lowest unique quantity, of particular interest among these is the cumulant of the 2-RDM, which is defined in terms of the 2-RDM and the 1-RDM (^{1}Γ):
^{2}Δ^{ij}_{kl} = ^{2}Γ^{ij}_{kl} − ^{1}Γ^{i}_{k}^{1}Γ^{j}_{l} + ^{1}Γ^{i}_{l}^{1}Γ^{j}_{k} | (7) |
The relation between the spin components of the two systems is shown in the following:
(8) |
(9) |
• For the specific case of H_{2} with a single Slater determinant wave function,^{65}
(10) |
• For finite molecules, spin is a well-defined quantum number, and yet spin-symmetry breaking in a single-determinant wave function can recover more accurate energies in certain cases than the spin-pure counterpart. Let us consider again the case of stretched H_{2} (which is isoelectronic to chemically relevant organic diradicaloids as well as many di-Cu(II) active sites). Here, 〈S^{2}〉_{UHF} goes from zero before the Coulson–Fischer point to one at complete dissociation; in the latter regime the UHF wave function can be expanded as an equal mixture of pure singlet and triplet states. The triplet state in fact has a more accurate electron density distribution than that of the closed-shell spin-pure determinant (although the spin-density distribution is also qualitatively wrong). Thus, the spin-contamination enables a single-determinant to capture (to some extent) an open-shell singlet electronic structure that is required of biradicals, and can be viewed as physically motivated, or “essential” according to Head-Gordon and coworkers.
• Strong spin correlation implies spin-symmetry breaking, though the converse is not true.^{20} In what follows, we will compute 〈S^{2}〉_{UCCSD}, as UCCSD does not contain free parameters and captures to a large extent what is frequently referred to as weak correlation along with some parts of strong correlation. Ref. 23 corroborates our hypothesis that the spin-symmetry breaking in UHF that remains after UCCSD can be attributed to strong spin correlations that are known to break low-order coupled cluster theory.
• Strong spin correlation is one of possibly many components of the as-yet mysterious quantity of interest: strong correlation. Thus, 〈S^{2}〉_{UCCSD} is, by construction, incomplete. In addition, there are a small number of known cases (e.g., two-electron systems including stretched H_{2}) where 〈S^{2}〉_{UCCSD} will always be spin-pure, despite qualitatively different correlation regimes being traversed.
In addition to quantities derived from the 2-RDM and its cumulant, and 〈S^{2}〉_{UCCSD}, we would like to explore an additional metric defined here. The Born–Oppenheimer Hamiltonian can be written as Ĥ = + _{en} + _{ee} + _{nn} = Ĥ_{1e} + _{ee} + E_{nn}, where is the kinetic energy operator, (V_{pqrs} are two-electron integrals), and E_{nn} is a constant energy shift. The first candidate metric is motivated by the large U/t limit of the Hubbard model, which is widely accepted by physicists to exhibit strong correlation, and also the Virial Theorem.^{2} One possible analog for molecules is the ratio:
(11) |
Fig. 1 6-site Hubbard model at half-filling. t = 1, various values of U. ^{2}γ and ^{2}δ are the largest eigenvalues of the specified spin sector of the 2-RDM and cumulant of the 2-RDM. |
We begin our analysis with the well-studied dihydrogen molecule. At equilibrium, a single large-weighted configuration corresponds to the HF solution, with electron fluctuations into the doubly-excited space accounting for the bulk of the correlation energy. Upon stretching, this molecular orbital picture becomes inadequate as two equally-weighted configurations come to dominate the ground-state wave function. Despite the correlation energy tending towards zero at dissociation (in fact, the singlet and triplet states are isoenergetic, and the UHF energy is exact), the entanglement characteristic of the singlet state does not follow such a trend as mentioned above. As shown in Fig. 2, this is well reflected by many of the metrics presently under consideration: ^{2}γ_{αβ} only has its connected portions survive, ^{2}δ_{αβ}, captured in both the trace and squared Frobenius norm of ^{2}Δ. As must follow from the exactness of UCCSD, 〈S^{2}〉_{UCCSD} here does not reflect any spin entanglement. ^{2}Γ_{αα} takes on no values while the elements of ^{1}Γ_{αα} contribute to the increasing squared Frobenius norm of ^{2}Δ as it approaches at dissociation, the previously derived limit for this system.^{54} There is a notable non-monotonicity in the ratio α, which declines as the bond length is stretched beyond 1 Å. This ratio does not appear to be applicable beyond models like the Hubbard model, possibly due to the delicate balance between phenomena such as node-induced confinement, orbital contractions, and polarization effects that can modulate the kinetic energy in molecular systems.^{69}
H_{4} is a challenging molecule that has been the subject of many studies in a variety of geometric configurations.^{70,71} A previous report has shown that the NOONs are identical for square and rhombic geometric configurations, yet notable differences occur in the underlying electronic structures.^{72} Indeed, that report motivated us to look beyond quantities related to ^{1}Γ. We investigate two reaction coordinates – the P4 rectangular D_{2h} dissociation into hydrogen dimers and the S4 square D_{4h} dissociation into hydrogen atoms. In the former, the HF determinant and the lowest energy doubly-excited configuration contribute significantly as the system crosses into the square geometry. There is a region of both spin and spatial frustration as we approach a crossing of the S_{0} and S_{1} surfaces, alongside a near-degenerate T_{1}.^{70} Fig. 3 (left) shows that in the singlet ground-state, the ^{2}Δ quantities correctly flag the electronically challenging region surrounding the square geometry with lattice parameter approaching 1.06 Å (2 Bohr). Spin-symmetry breaking in UCCSD shows similar behavior (though with a slightly wider peak), and we note that at 1.05 Å the UCCSD wave function is in equal parts of singlet and triplet character (averaging out to 〈S^{2}〉 = 1). This is consistent with the open-shell singlet (two electrons in two degenerate orbitals) picture implied by Huckel/MO theory. We again find that the largest eigenvalue of ^{2}Γ decreases in the square regime. In contrast, these metrics corresponding to the quintet state (shown in the right panel of Fig. 3) are all flat. As was the case for H_{2}, the energy ratio α does not seem to contain interpretable information here.
In the case of the S4 dissociation to separate hydrogen atoms (Fig. 4), the ^{2}Δ quantities reveal that, in contrast to the P4 coordinate, S4 does not involve a closed-shell region aptly described by mean-field theory (invoking the statistical sense with which we interpret correlation strength, indeed we can describe such a state as weakly correlated). As was the case in P4, the S4 quintet state (Fig. 4 right panel) is always weakly correlated, with all of our investigated metrics – except α – flat-lining. Interestingly, in the singlet case the squared Frobenius norm and the magnitude of the trace of ^{2}Δ are the only metrics that increase with lattice parameter; 〈S^{2}〉_{UCCSD} decreases (and appears to plateau at a value less than the maximum value of unity from the P4 coordinate), as do the maximum eigenvalues of the cumulant, albeit ever so slightly.
As a brief aside, it has been noted that the S4 model of equivalent side lengths, R_{S4}, has an identical spectrum of NOONs as some trapezoidal configurations in the C_{2v} geometry with one side length t_{2} and the other parallel side consisting of infinitely separated hydrogen atoms.^{73} In the cases of R_{S4} = 2.0 Å and t_{2} = 1.793 Å, although neither can be considered weakly correlated, the S4 example is known to exhibit additional correlation effects. Metrics based on the NOON spectrum of ^{1}Γ are unable to detect any difference (in other words, the ^{1}γ take on identical values). Despite this, differences in the 2-RDM yield differing values derived from the ^{2}Δ in these examples: the trace of the ^{2}Δ evaluates to −0.844 and −0.344 in the S4 and trapezoidal geometries, respectively, while the squared Frobenius norms are 1.37 and 0.52, respectively. Thus, in contrast to the NOONs these ^{2}Δ-based quantities are able to distinguish between two very different regimes of electron correlation.
The six-membered hydrogen ring has been studied extensively, in part because it is isoelectronic to the π system of an aromatic molecule such as benzene (although of course H_{6} involves s-orbitals while the π system of benzene involves out-of-plane p orbitals). While the always-square S4 state of the 4-membered hydrogen ring is generally expected to have a strongly-correlated character across the board, the expected weakly-correlated regime that corresponds to aromaticity in the six-membered ring is well reflected by the ^{2}Δ and 〈S^{2}〉_{UCCSD} metrics in the singlet ground-state before and in the neighborhood of the equilibrium geometry of 1 Å (Fig. 5 left panel). When the lattice parameter is stretched beyond 1 Å the largest eigenvalues, magnitude of the trace, and squared Frobenius norm of ^{2}Δ increase, as does the spin-symmetry breaking from UCCSD. As before, these metrics are largely flat for the highest spin state – in this case a septet (Fig. 5 right panel) – confirming weak correlation. At the end of this section we will return to the low-lying excited states of this model.
The linear symmetric dissociation of HHeH functions as a simple model of superexchange with tunable biradical character, in which the paramagnetic hydrogens are coupled through a diamagnetic helium intermediate. The parallel or anti-parallel coupling of the spins localized on the H atoms results in two prominent low-lying states, ^{3}A_{σu}^{+} and ^{1}X_{σg}^{+}. While even UHF may recapture the singlet–triplet energy splitting (and thus the exchange coupling constant, J) at infinite distance, the wave function itself must include a large weight on the lowest-lying doubly-excited configuration. The moderate to large biradicaloid character after 1.25 Å results in the breakdown of CCSD and CCSD(T).^{74} Fig. 6 shows that the coupling of the hydrogens mediated by the helium center leads to a similar trend as found in stretched H_{2}, though in the singlet HHeH case ^{2}γ_{αα} and ^{2}γ_{ββ} arise from the additional electrons. In addition, the ^{2}γ_{αβ} and ^{2}δ_{αβ} in HHeH follow a similar trend as in stretched H_{2}, with quantitative differences introduced by the intermediate coupling of a third center. Likewise, the squared Frobenius norm of ^{2}Δ approaches a value incrementally higher. While 〈S^{2}〉_{UCCSD} is now able to register a signal, it is slower to reflect the emergence of biradicaloid character in the region of 1.25–1.5 Å, compared to ^{2}Δ quantities.
The beryllium dimer has been the focus of a barrage of experimental and theoretical studies, due in part to the incredibly small measured bond energy of 2.67 kcal mol^{−1}. Another interesting feature is the quasi-degeneracy of the Be 2s and 2p atomic orbitals;^{75} the mixing of these orbitals as the atom centers approach one another gives rise to a notoriously pathological electronic regime. Predictions on Be_{2} were found to be sensitive to choices of active space – RHF and RCAS(4e,4o) yield qualitatively incorrect repulsive curves while RCAS(2e,2o) recovers the correct behavior at the cost of quantitatively incorrect energies. The CCSD curve is also repulsive, and CCSD(T) yields both less consistent and less accurate correlation energies recovering less than 70% of the total.^{76,77} Furthermore, in small basis sets FCI does not bind Be_{2}. In sum, mean-field and even low-order coupled cluster theory are inadequate in describing the beryllium dimer, and the most challenging regime is not at dissociation (though this is still non-trivial due to the near-degeneracy of the 2s and 2p atomic orbitals) but at the equilibrium bond length. As shown in Fig. 7, the pathological behavior at shorter bond lengths is correctly reflected by the ^{2}Δ and 〈S^{2}〉_{UCCSD}; however, the latter decays to zero in the limit of separated Be atoms, which in our view is unphysical: despite being essentially a two-electron problem (assuming large energy separation between the 1s and 2s orbitals), there is non-negligible correlation (“correlation energy” and statistical two-electron correlation) that arises due to the presence of the competing 2p orbitals. ‖^{2}Δ‖^{2} and Tr[^{2}Δ] clearly reflect this near the dissociation limit (as does ^{2}δ_{αβ} to a lesser extent).
Aromaticity in benzene, first studied by Kekulé,^{84} is familiar to every chemist. A less widely known fact is that the first singlet and triplet excited states of a molecule with an aromatic ground state are antiaromatic. Having established a rigorous interpretation of strong correlation based on the cumulant of the 2-RDM, we hypothesize that all antiaromatic systems can be viewed as strongly correlated (though the converse is not true), and that all (unsubstituted) aromatic systems can be viewed as weakly correlated. We will use ^{2}Δ quantities to show evidence that supports this hypothesis (though we realize more cases would be helpful). In passing we note that the ^{2}Δ has been associated with chemical notions involving long-range entanglements; the squared Frobenius norm was used to quantify the r^{−6} van der Waals force.^{85}
We now compute ^{2}Γ and ^{2}Δ for a 6-electron 6 π-orbital active space employing RHF orbitals,^{86} with the knowledge that the inactive orbital space does not contribute to the cumulant.^{44,49} The anti-aromatic character of both the T_{1} and S_{1} states are well documented.^{25,78,87–91} Table 1 shows the usual set of ^{2}Γ and ^{2}Δ-based metrics corresponding to the S_{0} ground-state and the vertically-excited S_{1} and T_{1} states of benzene. The cumulant-based values corresponding to the expected anti-aromatic states are indeed notably larger than those of the ground-state, and the T_{1} state is implied to be slightly more strongly correlated than S_{1}. Table 2 similarly investigates the lowest three eigenstates of the H_{6} ring at 1 Å separation. As was found for benzene, the squared Frobenius norm and trace of the ^{2}Δ show marked increases in value suggesting a transition from a weakly to a strongly correlated regime. It appears to be the case that this simple H_{6} ring model does capture the essential features of aromaticity and antiaromaticity in π-conjugated carbon rings.
State | Max eigenvalues | Trace/squared norm | ||||||
---|---|---|---|---|---|---|---|---|
^{2}γ_{αα} | ^{2}γ_{αβ} | ^{2}δ_{αα} | ^{2}δ_{αβ} | ‖^{1}Γ‖^{2} | ‖^{2}Γ‖^{2} | tr(^{2}Δ) | ‖^{2}Δ‖^{2} | |
S_{0} | 0.451 | 0.474 | 0.041 | 0.088 | 5.354 | 12.529 | −0.323 | 0.374 |
S_{1} | 0.317 | 0.423 | 0.078 | 0.100 | 4.040 | 7.411 | −0.980 | 0.946 |
T_{1} | 0.468 | 0.442 | 0.099 | 0.212 | 4.604 | 9.634 | −0.698 | 1.011 |
State | Max eigenvalues | Trace/squared norm | ||||||
---|---|---|---|---|---|---|---|---|
^{2}γ_{αα} | ^{2}γ_{αβ} | ^{2}δ_{αα} | ^{2}δ_{αβ} | ‖^{1}Γ‖^{2} | ‖^{2}Γ‖^{2} | tr(^{2}Δ) | ‖^{2}Δ‖^{2} | |
S_{0} | 0.482 | 0.491 | 0.031 | 0.044 | 5.754 | 14.032 | −0.123 | 0.127 |
S_{1} | 0.360 | 0.479 | 0.118 | 0.089 | 4.362 | 8.597 | −0.819 | 1.045 |
T_{1} | 0.487 | 0.475 | 0.116 | 0.228 | 4.840 | 10.388 | −0.580 | 0.876 |
As seen in Table 3, the squared Frobenius norm of the ^{2}Γ is constant despite its changing eigenvalues ^{2}γ. The change of the cumulant is dictated by the decrease in the ^{1}γ values, given their inverse relationship. Meanwhile, the triplet, here constrained by the minimal basis into a simple form for all geometries, formally never registers any signal across its spectrum because of the constrained nature of ^{1}γ. Beyond lacking size-extensivity, the ^{2}Γ squared Frobenius norm is incapable of capturing all the correlation reflected in the eigenvalues, while the ^{2}Δ norm and trace incorporate this in a straightforward way. We remark that larger triple- and quadruple-ζ basis sets show similar trends, with only minor numerical differences in the metrics.
Dist. | Singlet | ||||||||
---|---|---|---|---|---|---|---|---|---|
^{1}γ_{αα} | ‖^{1}Γ‖^{2} | ^{2}γ_{αβ} | ‖^{2}Γ‖^{2} | ‖^{2}Δ‖^{2} | |||||
0.50 | 0.995 | 0.005 | 0.990 | 0.497 | 0.036 | 0.036 | 0.003 | 0.250 | 0.010 |
1.00 | 0.969 | 0.031 | 0.940 | 0.484 | 0.087 | 0.087 | 0.016 | 0.250 | 0.066 |
1.50 | 0.873 | 0.127 | 0.778 | 0.437 | 0.166 | 0.166 | 0.063 | 0.250 | 0.295 |
2.00 | 0.713 | 0.287 | 0.590 | 0.356 | 0.226 | 0.226 | 0.144 | 0.250 | 0.661 |
2.50 | 0.595 | 0.405 | 0.518 | 0.298 | 0.245 | 0.245 | 0.202 | 0.250 | 0.830 |
3.00 | 0.539 | 0.461 | 0.503 | 0.270 | 0.249 | 0.249 | 0.230 | 0.250 | 0.867 |
5.00 | 0.501 | 0.499 | 0.500 | 0.250 | 0.250 | 0.250 | 0.250 | 0.250 | 0.875 |
Dist. | Triplet | ||||||||
---|---|---|---|---|---|---|---|---|---|
^{1}γ_{αα} | ‖^{1}Γ‖^{2} | ^{2}γ_{αα} | ‖^{2}Γ‖^{2} | ‖^{2}Δ‖^{2} | |||||
All | 1.000 | 1.000 | 2.000 | 0.500 | 0.500 | 0.500 | 0.500 | 1.000 | 0.000 |
We summarize the properties of the metrics investigated herein in Table 4. A few points are in order to clarify the listed properties: although UCCSD is of course size-extensive, 〈S^{2}〉_{UCCSD} does not scale with particle number, in contrast to the rest of the metrics. This is not necessarily a bad thing, given that comparisons among molecules of different sizes may require something along the lines of division by N for size-extensive metrics. By orbital-invariant, we mean that a fixed wave function will yield a fixed metric no matter which single-particle orbitals are used to represent it. In addition, an ideal metric that tracks the correlation strength will flag strong correlations that arise due to the need to restore symmetries of the system – not just spin-symmetry but spatial and others as well – and should also be meaningful in the thermodynamic limit (〈S^{2}〉 is not a good quantum number in this limit).
Properties | Method | ||||
---|---|---|---|---|---|
〈S^{2}〉_{UCCSD} | NOONs | δ_{σσ′} | Tr(^{2}Δ) | ‖^{2}Δ‖^{2} | |
a When the 1-RDM may be defined uniquely for the system. | |||||
Linear in particle number | × | ✓ | ✓ | ✓ | ✓ |
Spin-entanglement when E_{corr} = 0 | ✓ | × | ✓ | ✓ | ✓ |
Orbital-invariant | × | ✓^{a} | ✓ | ✓^{a} | ✓ |
Reflects all symmetries | × | × | ✓ | ✓ | ✓ |
Valid at thermodynamic limit | × | ✓ | ✓ | ✓ | ✓ |
Pure & ensemble states | × | × | ? | × | ✓ |
A nuanced point about the orbital invariance property of ^{2}Δ quantities: as pointed out recently,^{52} the hydrogen dimer offers a counter-example to when the diagonal elements of the ^{2}Δ alone may serve as a signal of strong correlation. The minimal wave function that incorporates mixing of the configurations, θ, and orbital rotations, ϕ
|Φ〉 = (cosθcos^{2}ϕ + sinθsin^{2}ϕ)|0_{α}0_{β}〉 + (sinθcos^{2}ϕ + cosθsin^{2}ϕ)|1_{α}1_{β}〉 + (cosθ − sinθ)cosϕsinϕ(|0_{α}1_{β}〉 + |1_{α}0_{β}〉) | (12) |
In light of our numerical results and our assessment of the satisfaction of the properties listed above for most chemically-relevant electronic structure problems, both the trace and square norm of the ^{2}Δ are apt metrics that reflect the statistical two-electron interdependence that we are proposing here as a meaningful interpretation of the term “strong correlation”. Given that Tr[^{2}Δ] = (1 − ^{1}Γ)^{1}Γ, i.e. is computable from the 1-RDM alone, this quantity is arguably the most practical for electronic structure calculations of pure states which do not suffer from a non-uniqueness of the natural orbitals at a particular geometry as in the case of stretched H_{2}. In the natural spin orbital basis, , where it is evident that the trace takes on its maximal value when all NOONs are equal (i.e. when the wave function is a superposition of equi-probable many-body basis states). As pointed out previously, the squared Frobenius norm of ^{2}Δ avoids this issue entirely and is generalizable to open/mixed systems while the trace is not. Importantly, both Tr[^{2}Δ] and ‖^{2}Δ‖ remain robust metrics of strong correlation even when the NOON spectrum alone fails to distinguish cases such as square and trapezoidal H_{4}.^{52,59}
In conclusion, after giving a critical overview of different conceptions of strong correlation, we find that the statistical interpretation offered by the trace or square norm of the cumulant of the 2-RDM is rigorously defined, physically meaningful, and computable. We have shown that for a strongly interacting 6-site Hubbard model (half-filled, at large U/t) and for a number of molecular reaction coordinates involving challenging electronic structure regimes, these metrics based on ^{2}Δ indicate what can be justifiably viewed as strong correlation. In addition, a salient result of our work is that we build a bridge between the rather mysterious chemical concept of antiaromaticity and our view of strong correlation.
It is of interest to us whether the physical interpretation of strong correlation, as substantiated by the statistical meaning of the cumulants, can be bridged with chemical notions (in addition to aromaticity and antiaromaticity) or other observables of interest. Another important challenge that we are currently exploring involves obtaining approximate but qualitatively informative 1- and 2-RDMs from scalable models such as quantum Monte Carlo or selected configuration interaction. This will enable reliable assessments of the nature of electron correlations in larger systems of interest to the chemistry and physics communities.
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