Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.


ELECTRON CORRELATION
Electron correlation has often been called the "chemical glue" of nature due to its ubiquitous influence in molecules and solids. 1,2 The problem was studied in the smallest two-electron systems, such as He and H 2 already in the 1920s, and these early efforts are discussed in detail elsewhere. 3−6 The correlation concept usually presupposes an independent particle model, typically Hartree−Fock (HF) mean-field theory, that serves as a reference compared to which the exact solution is correlated. The associated idea of correlation energy goes back to Wigner's work in the 1930s on the uniform electronic gas and metals, 7,8 and it is nowadays commonly defined, following Loẅdin, as the difference between the exact and the HF energy. 3 There is a vast literature on the subject and here we will confine ourselves to some of the most pertinent reviews. Early work on the correlation problem is reviewed by Loẅdin in the 1950s 3 and more recently, 9 as well as by Sinanoglu 10,11 within the context of his many-electron theory. The early perspective of McWeeny is a short but useful discussion of the problem in terms of density matrices. 12 Kutzelnigg, Del Re and Berthier penned an overview on the statistical measures of correlation, 13 and, as P. v. Herigonte, they also reviewed the situation in the seventies. 14 This was later followed up several times by Kutzelnigg. 15, 16 Bartlett and Stanton not only discuss correlation effects in molecules, but also offer a tutorial on post-HF wave function methods. 17 Similarly educational is the chapter written by Knowles, Schuẗz and Werner 18 and the feature articles of Tew, Klopper, Helgaker 19 and Martin. 2 More recent work on the homogeneous electron gas is reviewed by Senatore and March 20 as well as Loos and Gill, 21 the former authors also discuss orbital-based and quantum Monte Carlo methods. For a review taking an entanglement-based approach to the correlation problem, we refer the reader to Chan. 22 There are also several book-length monographs dedicated to the subject by Fulde, 23 March 24 and Wilson, 25 as well as a number of collections. 26,27 In the remainder of this section, we will consider some selected topics about correlation effects before stating the scope of this perspective.
An important aspect of electron correlation is its strength and the various measures proposed to quantify it. For the homogeneous electron gas, where some analytical energy expressions can be obtained at least under some circumstances, the discussion is often concerned with the relative magnitudes of kinetic and potential energies in the limiting cases of high and low electron densities. While Wigner was able to obtain an expression for the correlation energy at the low-density limit, 7,8 in general, perturbation theory diverges already at second order, 28 a problem that can only be remedied partly at the high-density limit following Gell-Mann and Brueckner. 29 In the high-density limit, the kinetic energy dominates over the repulsive potential energy, the electrons are delocalized and behave like a gas, thus, an independent particle model should provide a good description. In the low-density limit, it is the Coulomb interactions responsible for correlation effects that dominate, forcing the electrons to localize on site (form a lattice). Interestingly, the low-and high-density limits have their analogues in molecular systems: 20 in the hydrogen molecule, the high-density case would correspond to short bond lengths and a delocalized molecular orbital 5,30−32 (MO) description, while the low-density limit would correspond to large internuclear separation where the electrons are localized around the nuclei, better described by a local valence bond 5,33−35 (VB) approach. While this is a useful conceptual link, it must be remembered that the homogeneous electron gas is a better description of metallic solids than it is of more inhomogeneous systems like molecules. In the next section, the contrast between solids and molecules with regard to strong correlation will be discussed before we turn our attention fully to molecular systems and examine the various measures used in wave function theory, density functional theory and quantum information theory.
From a wave function point of view, correlation strength is often associated with the nature of the reference function which is usually the HF state, as mentioned above. An interesting point that draws attention to the significance of the chosen reference is the observation that from a statistical point of view HF already contains some correlation introduced by antisymmetrization and the exclusion principle. 13 Following Kutzelnigg, Del Re and Berthier, 13 where the quantities corresponding to the ith and jth particles in an N-electron system can be simply related to the usual expectation values as = · = · N N N r r r r r r , (2) Applying this to position vectors, the implication is that the positions of two electrons are uncorrelated. This is a weaker notion than that of independence, which is often taken to mean lack of correlation. Statistically, independence (or, correlation in the sense of dependence) could be defined 36 by requiring that the two-body cumulant λ 2 = (1, 1 ; 2, 2 ) (1, 1 ; 2, 2 ) (1, 1 ) (2, 2 ) obtained from the one-and two-body reduced density matrices (1-and 2-RDM), γ 1 and γ 2 , should vanish. The fulfilment of this condition would imply a relationship like eq 1 for any observable. This is the sense in which Wigner and Seitz 37,38 understood correlation, and this would correspond to a simple Hartree product of orbitals as a reference function, or, more generally, to a direct product of subsystem wave functions. Antisymmetrization then implies switching from this product reference function to an antisymmetric Slater determinant constructed from spin−orbitals. More generally, it is also possible to use generalized (antisymmetrized/wedge) product functions 39 to obtain the total wave function from antisymmetric group functions of subsystems. Using an antisymmetrized reference, it is customary to talk about Fermi and Coulomb correlation, the former being excluded from the conventional definition of the correlation energy. To make this explicit, the cumulant can be rewritten as = + (1, 1 ; 2, 2 ) (1, 1 ; 2, 2 ) (1, 1 ) (2,2 ) (1, 2 ) (2, 1 ) 2 2 1 1 1 1 (4) Here, the last term accounts for the exchange contribution, and thus, λ 2 only contains the parts of γ 2 that cannot be factorized in terms of γ 1 in any way. In this sense, λ 2 is the most general descriptor of correlation effects. It should also be noted that by using partial trace relations, 36 it can be shown that Tr(λ 2 ) = Tr(γ 1 2 − γ 1 ), and hence correlation effects are also related to the idempotency of the 1-RDM. This notion goes beyond that of Wigner and Seitz and is closer to Loẅdin's idea of correlation. The significance of Fermi correlation in particular was discussed by several authors in the past, 12,13,19 and more recently in a pedagogical manner by Malrieu, Angeli and coworkers. 40 Nevertheless, as pointed out by Kutzelnigg and Mukherjee, 36 eq 4 is not identical to Loẅdin's definition either since it contains no reference to the Hartree−Fock state, apart from that fact that the closed-shell HF determinant, for example, would be enough to make λ 2 vanish and would thus be uncorrelated in this sense. The definition in eq 4 is an intrinsic one in the sense that γ 1 and γ 2 can be calculated for any (reference) state. More generally, any second-quantized operator string can also be evaluated with respect to a general reference (vacuum) state using generalized normal order and the associated general form of Wick's theorem that itself relies on n-body cumulants. 41, 42 The foregoing discussion indicates the role of the reference state in the definition of correlation effects and highlights the fact that reference states should conveniently incorporate certain correlation effects to make subsequent treatment easier. As antisymmetrization dictates using Slater determinants rather than simple orbital products, spin symmetry may be enforced by demanding that the reference be constructed as a fixed linear combination of determinants to account for spin symmetry, 19 and there might be other criteria. 22,43,44 In such cases, one may distinguish between correlation recovered by the reference and residual effects. From such an operational point of view, the question is whether the strongest correlation effects can already be captured at the reference level to ensure a qualitatively correct starting point. In practice, one may choose to build reference states and represent the wave function using the following N-electron basis states: • Determinant (DET): An antisymmetrized product of orbitals. They are eigenfunctions of Ŝz but not necessarily of Ŝ2. • Configuration State Function (CSF): A function that is an eigenstate of both Sẑ and Ŝ2. CSFs can be obtained from linear combinations of determinants, but there are at most as many CSFs as DETs for a given number of unpaired electrons and total spin. • Configuration (CFG): At an abstract level, a configuration is just a string of spatial orbital occupation numbers (0,1,2) for each orbital in a given N-electron wave function. By extension, a configuration is also the set of determinants or CSFs that share the same spatial occupation numbers, but differ in the spin−orbital occupation numbers. We will discuss the role of this choice and offer a classification of wave function expansions and potential reference states based on the properties of the exact solution, which leads us to define what we call the multireference character of the wave function. We will also examine the role of the orbital basis and ask the question how orbital rotations influence this apparent multireference character in a simple model system. On the other hand, we are not concerned here with the practical construction of good reference states, even though the topics discussed undoubtedly have consequences in that regard. At this stage, we simply ask what the difference is between a canonical and a local orbital representation in terms of the apparent multireference character and what the chances are of incorporating the strongest correlation effects into the reference function, be that via enforcing spin symmetry or generating a new many-electron basis by rotating orbitals.

SOLIDS VERSUS MOLECULES
Strong correlation is one of those important but elusive concepts that is often used in discussions of the properties of molecular and solid-state systems. A brief survey of the literature that follows below will reveal that it is often equated with various other notions, such as the multireference character of the wave function, the failure of density functional theory (DFT) and quantum entanglement. It is not immediately clear how these various notions are related to one another or to strong correlation, or whether any one should be preferred over the other. On the other hand, it seems relatively clear that a large part of the discussion on strong correlation focuses on extended solid-state systems, where density functional and dynamical mean-field 45,46 approaches are commonly used, 47 although many-body methods 48 and quantum Monte Carlo techniques 49,50 are also available as well as a number of exactly solvable model problems. 51 The theoretical tools used to describe periodic solids of infinite extension and finite-size molecular systems can be quite different in ways that reflect the underlying physics. Complexity and locality may have different manifestations for these systems in terms of the number of parameters or the type of basis sets required to describe them. In terms of symmetry, the obvious difference is the presence of translational symmetry in solids. Furthermore, symmetry for an infinite number of degrees of freedom also makes spontaneous symmetry breaking possible in ground-state solutions. 52 For all these reasons, it seems reasonable to separate the discussion of electron correlation in solids from that in molecules.
As mentioned before, in the homogeneous electron gas model strong correlation effects are related to the low density limit. Wigner argued that at this limit the electrons are localized at lattice points that minimize the potential energy 7 and vibrate around the lattice points 8 with just the zero point energy demanded by the uncertainty principle. Leaving the vibrational part aside, the total energy of the system at very low densities has the form 8,24,53,54 where α and β are constants and r is the Fermi radius, which in this context is the radius of a sphere available for a single electron. If the density is low, r is large, and that means that the mean kinetic energy ⟨T⟩ is negligible compared to the mean potential energy ⟨V̂⟩, which is sometimes taken to be a criterion for strong correlation. Wigner was able to obtain a correlation energy formula by assuming a vanishingly small kinetic energy at the low-density limit 7 and interpolating between the low-and high-density limits, and considered the zero point kinetic energy, which would add a term propor-tional to r 3/2 in eq 5, only in a later study. 8,21,55 While the result works reasonably at density values actually present in some solids, some criticisms merit attention especially when application to molecules are concerned. While admitting the good agreement with measurements in metallic systems and similar results derived from plasma theory by Bohm and Pines,56,57 Slater criticized 58 the assumption of uniform positive background, which should be an even more serious problem when considering more inhomogeneous systems, such as the dissociating H 2 molecule. 3 We note in passing that it is possible to account for inhomogeneities in the electron gas, 59 but perhaps the conceptual simplicity of the homogeneous electron gas model is more relevant for the current discussion. Concerns were also raised by Loẅdin 3 about the model not obeying the virial theorem 60 in the sense expected in molecular systems. It has since been shown that the uniform electron gas obeys the more general form of the virial theorem 61,62 6) which means that in the low-density limit, the total energy E is not stationary (although the potential energy may be in the sense needed to find an optimal lattice). In molecular systems, on the other hand, stationary points of the total electronic energy are of special interest as in the case of H 2 at equilibrium and infinite separations. In these cases, the virial theorem takes the form 2⟨T⟩ + ⟨V̂⟩ = 0, which does not allow the kinetic energy to be vanishingly small. 3 In this regard, see also the work of Ruedenberg 63 for an analysis of the formation of the chemical bond that does respect the virial theorem. For the electron gas, the main use of the virial theorem is actually the separation of kinetic and potential energy contributions in the correlation energy. 61 Thus, it may be concluded that the criterion of the relative magnitude of the mean kinetic and potential energy is appropriate within the homogeneous electron gas model, which is a better approximation of metallic solids than molecules. Nevertheless, extensions of the same basic idea are possible. Within the Hubbard model, 64 for example, correlation effects are commonly considered as being strong if the on-site electronic repulsion is larger than the bandwidth (resonance energy) or the hopping term, which favors antiferromagnetic coupling between the various sites. 23 Repulsion is a potential energy term, while the hopping integral has to do with the kinetic energy of electrons. We will discuss the appropriate extension to the molecular Hamiltonian in later sections. As in this perspective we are more concerned with molecular systems, it is especially interesting that solids containing d or f electrons are considered strongly correlated, 23 as this would prompt one to look for strong correlation in molecular systems containing transition metals or lanthanides. Before leaving the discussion of solid state behind, it should also be recalled that sometimes the nature of the problem at hand calls for the combination of solid-state and molecular approaches, as in studying strongly correlated (bond-breaking) problems in surface chemistry via embedding approaches, which pose challenges of their own. 65 The foregoing discussion also underlines the problem of finding quantitative measures of strong correlation, or, perhaps more precisely, of the various other properties associated with it. After reviewing the various attempts made along these lines, we will expound our own perspective on strongly correlated systems in chemistry as viewed through the lens of the multireference character of the wave function expansion. We will use a simple metric that can be obtained from the exact solution and examine its properties using the hydrogen molecule as a model system. As mentioned before, this problem has been studied since the early days of quantum chemistry 3−6 and also has some practical relevance in that it can be taken as a model for the antiferromagnetic coupling in transition metal compounds such as a typical Cu(II) dimer. We note in passing that the hydrogen chain is also a common model problem in the study of extended systems. 66,67

WAVE FUNCTION THEORY
Within the framework of wave function-based post-Hartree− Fock approaches, it is customary to distinguish between single and multireference methods to which the concepts of dynamic and static or nondynamic correlation also correspond. 11,17,18,68 Dynamic correlation is also called short-range correlation as it has to do with the electrostatic repulsion of electrons and the difficulty of describing the resulting Coulomb cusp (see this study 69 for a different perspective). When it comes to other correlation effects, some authors use static and nondynamic correlation interchangeably and sometimes refer to it as longrange correlation. 18,70 Others 17 prefer to reserve static correlation for effects induced by enforcing spin symmetry in order to construct a correct zeroth order description, and call those associated with other sources, such as bond breaking, nondynamical. Tew, Klopper and Helgaker follow a similar distinction except that they include spin symmetry in Fermi correlation and identify static correlation as the correlation associated with near degeneracy and thus needed for a correct zeroth order description. 19 Yet others 71 distinguish within static/nondynamical effects between those that can be captured by spin-unrestricted HF and those that cannot. A summary of more formal attempts to define static and dynamic correlations is also available. 72 The important conclusion from this discussion seems to be a point already raised by Wigner and Seitz 38 that the main difference is between correlation effects that arise from symmetry and those arising from interelectronic repulsion, regardless of how we choose to classify these further. Thus, within symmetry effects one might further distinguish between antisymmetry and spin symmetry, within repulsion effects between those associated with spatial (near) degeneracy and the rest.
Ignoring the question of spin symmetry for the moment, nondynamic/static correlation may be identified with strong correlation in the present context, and thus, it is also associated with multireference methods, where "reference" is a generic term for some function that serves as an initial state on which a more sophisticated Ansatz can be built. This also suggests that there are two ways of dealing with static correlation, either improving the reference or improving the wave function Ansatz built on it. In principle, single reference methods will yield the exact solution if all possible excitation classes are included in the Ansatz, and it was furthermore demonstrated that they can also be adapted to strong correlation by removing some of the terms from the working equations. 73 More pragmatically, one might ask at what excitation levels an Ansatz, such as coupled cluster (CC), needs to be truncated to yield acceptable results for some properties, 74,75 but this is costly on the one hand and does not offer a simple picture of strong correlation either. One way to improve the reference is to find the exact solution within at least a complete active space (CAS). 76 Unfortunately, this requires the construction of many-electron basis states the number of which scales exponentially with the size of the active space. However, as observed by Chan and Sharma, 77 this apparent exponential complexity does not always reflect the true complexity of the system since the principle of locality often forces a special structure on the wave function expansion. This entails that the amount of information needed to construct the wave function should be proportional to the system size, which agrees well with the observation that static correlation in molecules can typically be tackled with only a handful of terms in the wave function expansion. 18 As an illustration, the hydrogen molecule around the equilibrium bond length is well-described by a single-determinant built from canonical orbitals and associated with the configuration σ g 2 , while at larger separations another contribution from σ u 2 becomes equally important. 18,77 In the latter limit, the wave function takes the following bideterminant form where i and a are the occupied and unoccupied molecular orbitals in the minimal basis HF solution, and the shorthand notation |ii| is used to denote determinants, with the overbar signifying a β spin. While these canonical orbitals are delocalized, μ and ν denote atomic orbitals, each localized on one of the hydrogens and orthonormal at large distances. As is well-known, 4,6,78 the molecular orbital (MO) and valence bond (VB) approaches offer equivalent descriptions of this wave function, but the locality of the VB picture leads to an interpretation in terms of "left−right" correlation, i.e., the phenomenon that at large internuclear separation an electron preferably stays close to the left nucleus if the other one is close to the right nucleus, and vice versa. This form of static correlation can be accounted for in a spin-unrestricted singledeterminant framework leading to different orbitals for different spins beyond a certain bond distance, but, for molecular systems, approximations preserving the symmetries of the exact wave function are usually preferable. It is all the more interesting that the constrained-pairing mean-field theory of Scuseria and co-workers is designed to capture strong correlation 43 by breaking and restoring various symmetries, 79 while Kutzelnigg took a different route toward obtaining a similarly economic reference function by separating bondbreaking correlation effects via a generalized valence bond (GVB)/antisymmetrized product of strongly orthonormal geminals (APSG) approach. 44 It is also apparent from these considerations that most traditional methods follow the "bottom-up" approach in which a systematic Ansatz is built on a simple reference function and then truncated using some simple criterion (e.g., the excitation level). The typical arsenal of this approach includes many body perturbation theory, CI and CC approaches in both the single and multireference cases. 17,18,80 The most popular of them is CC theory, 81 which was introduced into chemistry by Čizěk, 82 Paldus and Shavitt, 83 was reformulated in terms of a variational principle by Arponen 84 and whose mathematical properties are still an active area of research. 85,86 Even more actively studied are its local variants 87,88 and its explicitly correlated, 89 excited state 90 and multireference 91 extensions. Beyond these methods, there is also room for a "top-down" approach in which one aims directly at the exact solution (full CI, full CC) and applies some strategy to neglect contributions that are less important while trying to save as much of the accuracy as possible. The relative sparsity illustrated above with the H 2 example can be exploited in selected CI methods of the CIPSI (configuration interaction with perturbative selection through iterations) type 92 for much larger systems. The CIPSI method is the first of many selected CI (SCI) approaches that attempt to solve the problem of selecting relevant contributions in a deterministic fashion, but it is also possible to apply stochastic sampling processes, including QMC approaches 93 such as the FCIQMC method. 94,95 FCIQMC can be performed without approximation, in which case the method is exact but suffers from a sign problem in general. 96 Instead, the initiator approximation is typically applied, which involves truncating contributions between certain determinants with low weight. 97,98 These and other methods have been reviewed in more detail recently by Eriksen, 99 including Eriksen and Gauss' own approach based on many body expansions. 100,101 Another approximate FCI method particularly suitable for strongly correlated systems, the density-matrix renormalization group 77,102 (DMRG) approach will be discussed later among entanglement-based approaches. We also note that some of these methods have been analyzed in terms of scaling and the gains due to the reduction in the number of important contributions in the wave function expansion 103−106 (see also some remarks on orbital rotations below).
It has been observed that multireference character, interpreted here as the fact that the wave function expansion contains more than one determinant of significant weight for some system, does not necessarily imply strong correlation, and the additional criteria of the breakdown of single reference perturbation theory and the appearance of degeneracies were suggested. 107 More explicitly, strong correlation could be said to occur whenever the perturbation (the difference between the full Hamiltonian and the Fockian) is larger than the gaps between the Hartree−Fock orbital energies (spectrum of the Fockian). 22 This could be regarded as an extension of the similar definition within the Hubbard model. For nonmetallic states, strong correlation would also be implied if the correlation energy is larger than the excitation energy to the first excited state of the same symmetry as the ground state. 108 Nevertheless, the multireference character can be used to characterize the exact wave function, even if its usefulness for approximate approaches may be questioned. This leads us toward a definition of multireference character via the full configuration interaction expansion = C I I I (8) where Φ I are some basis states, typically determinants. If any one coefficient C I dominates (see more later) the expansion, then the associated Φ I could be taken as the reference in a single reference (determinant) treatment. Thus, multireference character is associated with wave function coefficients C I which for orthonormal basis states are equivalent to the overlap between the exact state Ψ and the basis state Φ I . Unfortunately, this criterion is impractical to decide in advance whether a multireference treatment of a system is necessary, and other diagnostics, 109 such as the one relying on the norm of the singles vector (T 1 ) in a CC calculation 110 truncated at the level of single and double excitations, were also found insufficient 109 or even misleading. 111 Metrics relying on natural orbital populations are more promising as they give information about the number of electrons uncoupled by correlation driven processes. 75,112,113 The deviation of the onebody reduced density matrix from idempotency has also been used as a measure 114,115 and well as measures associated directly with the two-body cumulant. 44,115,116 More pragmatically, the static correlation energy has also been defined as the exact correlation energy obtained from a minimal basis calculation involving valence electrons, the idea being that in such a calculation the most important configurations would be represented, although this approach also needs a hierarchy of approximations to be useful. 117 A few other schemes have also been suggested to decompose the correlation energy into dynamical and nondynamical parts using simple formulas, 70,115 and more recently, using cumulants. 115 For the purposes of this perspective, the FCI-based criterion in eq 8 will be sufficient as it enables us to base our discussion on the properties of the exact wave function. While it is common to use determinants as basis states, this is not the only possibility. Using CSFs or CFGs for example would yield basis states that can be written as sums of determinants or constructed in some other way. 103,118 We will investigate their effects on the multireference character in a later section. Already in eq 7, while both the MO and VB descriptions are multideterminant expressions, multiple configurations only enter into the (canonical) MO picture. We will also examine the role of orbital transformations in defining the multireference character of the wave function. Again, in eq 7, the concept of left−right correlation seems to be connected more to the local VB picture than to the MO one. Our primary goal here is to comment on multireference character, rather than other notions about strong correlation, but before doing that, we briefly examine other approaches to the problem.

DENSITY FUNCTIONAL THEORY
In their recent paper, Perdew et al. remark that strong correlation is sometimes taken to mean "everything that density functional theory gets wrong". 119 Most practical DFT approaches rely on the Kohn−Sham procedure which in most cases corresponds to a single-determinantal implementation. Unlike the Hartree−Fock solution, however, the Kohn−Sham determinant is not an approximation to the exact wave function; rather, it is a tool to obtain the ground-state density or spin densities. There seem to be two opinions in the literature on what this signifies. Perdew et al. argue that as a consequence, the Kohn−Sham determinant need not exhibit the symmetries of the Hamiltonian as those will be reflected in the density in a more indirect fashion. This also implies that despite the appearance of a mean-field approach, Kohn−Sham DFT is in principle able to account for correlation effects. 120 Others argue 121−123 that spin-adapted reference functions should be used at least in some open-shell cases which may be multideterminantal, as it is done in the spin-adapted Hartree− Fock case in wave function theory. In any case, a single determinant may not always suffice and the question remains to what extent DFT can treat strongly correlated systems. In trying to answer this question, Perdew et al. 120 distinguish between strong correlation arising in condensed matter models and static correlation originating in near or exact degeneracies, such as the closing HOMO−LUMO gap in the H 2 bond dissociation process. On problems of the latter type, they note the typical failure of common semilocal approximations.
While multideterminental techniques have been introduced in DFT, 121−123 it is more common to rely on broken symmetry DFT 78,124−126 in which multiplet energies are obtained from a combination of single-determinant energies. This scheme may or may not hold exactly, depending on the specific case. 121,126 Perhaps the simplest example of spin-symmetry breaking is the fact that the spin-unrestricted approach yields only one of the determinants |μν ̅ | or |μ ̅ ν| in eq 7 which can be interpreted as a resonance structure in the valence bond sense. Although such determinants are not pure spin states, they are degenerate. From such broken symmetry solutions and monodeterminantal pure spin states, it is often possible to calculate the energies of multideterminantal spin states: in H 2 , the appropriate combination of the energy of the broken symmetry solution and the triplet energy yields the energy of the excited-state singlet. 78 The use of this simple two-level model in the description of magnetic coupling is analyzed in detail elsewhere. 127 The more general broken symmetry (BS) DFT procedure starts with a mixture of canonical orbitals obtained from a closed-shell calculation and optimizes the mixing angles during the iterative solution to obtain a mixture of multiple configurations. 123 A similar idea underlies the permuted orbital (PO) approach where the broken symmetry states are generated by permuting orbitals, typically the HOMO and the LUMO, 123 and can be used to construct multideterminant spin eigenfunctions. Based on the capabilities of the BS and PO approaches, Cremer proposes a DFT-oriented multireference typology: 123 Type 0 wave functions need a single configuration or determinant and the spin-unrestricted approach can handle them; Type I consists of a single configuration with multiple determinants for which the PO approach is assigned; Type II is the case of multiple configurations each with a single determinant to be treated by BS-DFT; Type III systems with multiple configurations each with multiple determinants are the most difficult and can be handled by explicitly multireference methods, although possibly the PO approach might work. While we find the distinction between determinant and configuration/CSF references commendable, in the cases studied by Cremer there was no need to distinguish between configurations and CSFs, a feature that we will come back to in our own classification. As for interpreting broken symmetry solutions, it should be recalled that while the exact functional for finite systems is not symmetry broken, approximations are. Strong correlation in the exact wave function describes "fluctuations" in which the spins on the different nuclei are interchanged (|μν ̅ | vs |μ ̅ ν|). 128 Symmetry breaking in approximate methods is in fact taken to reveal strong correlations 119 with a mechanism associated with it: at large distances, when H 2 becomes a pair of H atoms, an infinitesimal perturbation can reduce the wave function and the spin densities to a symmetry broken form, in which an electron with a given spin is localized around one H atom. 119,128 All this is in line with Anderson's famed essay 129 on emergent phenomena on different scales involving increasingly larger numbers of particles, though it also implies that symmetry breaking may be more relevant for solids than for molecules. 52,119 It may be worth pointing out that the physical contents of broken symmetry wave functions can be conveniently "read" by means of the corresponding orbital transformation (COT). 130 This transformation orders the spin-up and spindown orbitals into pairs of maximum similarity. This results in a wave function that can be read analogously to a generalized valence bond (GVB) wave function 131,132 consisting of doubly occupied orbitals, singlet coupled electron pairs in two different orbitals and unpaired (spin-up) electrons. Obviously BS-DFT can properly represent the singlet coupled pairs, but the analysis shows that the spatially nonorthogonal pseudosinglet pairs obtained from the COT serve a similar purpose, namely to variationally adjust the ionic-and neutral components of the wave function. 127 This way of reading BS wave functions has found widespread application in the field of metal-radical coupling (e.g., examples 133−135 ). A more thorough review is available. 136 The theorems due to Hohenberg and Kohn 137 form the basis of the variational formulation of DFT, although questions about their precise meaning 138 remain pertinent enough to motivate further research, 139,140 not to mention the fact that these theorems do not generalize easily to excited states. 141 The favorable linear scaling property of DFT approaches is often derived from Kohn's principle of nearsightedness, 142,143 i.e., that under certain conditions the density at some point is mostly determined by the external potential in a local region around that point. Practical approaches to density functional theory often consist of proposing new functional forms 144 in lieu of the unknown exact density functional of the energy. In contrast, the energy is a known functional of the 2-RDM, 114,145,146 although enforcing constraints 147 to actually optimize this functional is not trivial. Fortunately, the energy can also be written as a functional of the 1-RDM, 148 and different variants of density matrix functional theory are usually parametrized in terms of the 1-RDM and the 2-body cumulant. 149−152 Like DFT, this approach also relies on approximate functionals, but it manages to overcome some of the failings of the former. The appropriate constrains for the 1-RDM itself 153 are relatively easy to enforce, although the reconstructed 2-RDM should then still satisfy its own Nrepresentability conditions. 151 One might then construct functionals 154 inspired by wave function approaches, such as geminal theories, or use auxiliary quantities to parametrize the cumulant. 155 In density cumulant functional theory 150,156 (DCFT), the functional is given in terms of the best idempotent estimate to the 1-RDM and the 2-body cumulant, and the critical N-representability conditions affect only the relatively small part due to the latter. Connections with traditional DFT can also be established, e.g., in the DCFT case, an analogue of the DFT exchange-correlation functional can be obtained by neglecting the Coulomb terms in which the nonidempotent part of the 1-RDM occur. 150 The nearsightedness principle applies to the 1-RDM as well, 142 and density matrix functionals have the advantage over DFT that the kinetic, Coulomb and exchange energies are all known functionals of the 1-RDM, only the correlation energy functional remains unknown. 157 Excited state applications of RDM functionals have also been considered. 152,158 A collection of studies on the various aspects of RDM-based methods is available, 159 while the relationship between wave function based methods, DFT and RDM functionals is discussed in Kutzelnigg's review. 157 In the current context, it is relevant to mention that some density matrix functionals have been applied with success to strongly correlated problems. 152,155,160−164 So far we have only considered functional approaches in which the variable is the density or the RDMs, but it is also possible to set up a variational formulation 165−167 for functionals of Green's functions, which themselves can be regarded as generalizations of RDMs with their own cumulant expansion. 168 All three of these quantities form the basis for embedding theories enabling the treatment of large systems. 169 Dynamical mean-field theory 45,46,170 is a much applied method in solid state physics that can be derived from such a variational approach and has also been investigated from the perspective of molecular quantum chemistry. 171,172 Another method that falls into this category is the popular GW approximation of Hedin. 173−175 The properties of these and other similar methods have also been analyzed from the perspective of strong correlation, 170,172,176,177 excited states 174,178 and symmetry breaking. 177,179 Although both have some success in multireference systems, GW approaches are typically applied to weakly correlated systems while dynamical mean-field approaches have succeeded where DFT or other Green's function approaches failed. 170,172,176 It emerges from this discussion that the DFT concept of strong correlation is related on the one hand to its monodeterminant nature and to symmetry breaking on the other. Measures have been proposed along these lines. Thus, Yang and co-workers relate static correlation to fractional spins, and present an extension to DFT covering fractional spins including a measure based on this extension. 180,181 Grimme and Hansen offer a measure as well as a visualization tool for static correlation effects based on fractional occupation numbers obtained from finite temperature DFT calculations. 182 Head-Gordon and co-workers use spin (and sometimes spatial) symmetry breaking as a criterion and suggest a measure based on spin contamination, 107 which may be related to natural orbital occupation numbers, a criterion also used by others. 123 Still other authors use checks based on the idempotency of the one-body density, which is essentially used here as a check on the monodeterminant nature of the wave function under consideration. 114,183

QUANTUM INFORMATION THEORY
Quantum information theory offers a different perspective on the same phenomenon, albeit one that can be related to more traditional quantum chemical concepts. 108 The starting point is a two-level quantum system, a qubit, that serves as an analogue of classical bits. We take spin-up and spin-down states to form the qubit in the following. A spin-up state can be labeled |0⟩ and a spin-down state as |1⟩, and a qubit can be any linear combination of these two basis states, 184 |m⟩ = α|0⟩ + β|1⟩. If the spin of an electron prepared in such a state is measured in a field oriented along an axis parallel to the spin-up and spindown basis states, then the probability of finding the electrons in state |0⟩ or |1⟩ is |α| 2 and |β| 2 , respectively. Along the same lines, one may prepare two electrons. Assuming at first that the two electrons do not interact before the measurement, the possible two-electron states may be described in the basis of the product states |0⟩ ⊗ |0⟩, |0⟩ ⊗ |1⟩, |1⟩ ⊗ |0⟩ and |1⟩ ⊗ |1⟩. They remain distinguishable and they are also independent in the sense that measuring the first qubit yields results that only depend on how the first electron was prepared and is not influenced by the second electron. Imagine now bringing spin-1/2 particles close enough together so that their spins align in an antiparallel fashion corresponding to the ground state and then separating them again far enough so that one is located at site A the other at site B. The wave function that describes this state is In classical computing, the value of the first bit in a two-bit string does not yield information about the value of the second bit, in the same way as the independently prepared quantum systems do not. The two-qubit quantum state in eq 9 is different because measuring the spin only at site A also reveals what the result of the measurement will be at site B. Thus, the two qubits at site A and B are entangled 185 in the sense that the result of a measurement at A determines that at B. It is also often said that they are correlated, although entanglement is only a special kind of quantum correlation. 186 This is different from the definition of strong correlation in terms of perturbation strength, but has the advantage that it is more easily quantifiable. 22 Note that eq 9 describes a system of distinguishable spin-1/2 particles, such as an electron and a positron, or two electrons at a large distance. Although electrons are identical particles, in cases like the dissociating H 2 molecule, each electron can be identified as the one localized around a specific nucleus if the two nuclei are separated enough so that the electronic wavefunctions no longer overlap. 187 Comparing eq 9 to the VB wave function of H 2 at large bond distances in eq 7 and assuming that μ is at site A and ν at site B reveals that since the tensor product here is ordered w.r.t. the particle labels, e.g., μ(2)α(2)ν(1)β(1) → |1⟩ B ⊗ |0⟩ A . Thus, eq 7 is the antisymmetrized version of eq 9. In a sense, this indicates another kind of symmetry breaking. In spin-unrestricted approaches discussed in the DFT context, only one of the determinants |μν ̅ | or |μ ̅ ν| in eq 7 are present at large bond distances, i.e., it is known which spin is at site A (or B) but not which particle carries that spin. On the other hand, eq 9 breaks antisymmetry as a natural consequence of the assumption of distinguishability: in this case it is known which particle is at site A (or B), but not the spin of that particle, which is the question a subsequent measurement can help resolve. As this discussion suggests, the established notion of entanglement 184 assumes distinguishable subsystems, although extensions to indistinguishable particle systems exist. 188 This should perhaps be kept in mind when comparing with quantum chemical notions, where indistinguishability of electrons in molecules is assumed (cf. also the discussion on product functions and generalized product functions 39 earlier). Still, the resonance structures of VB/GVB theories, for example, assume a degree of distinguishability and entanglement may be a useful tool in dealing with them. 22 It is also worth mentioning that when measuring the spin of a single electron as discussed above, it is always possible to rotate the measurement field so that, say, the spin-up state is measured with probability 1. This is not possible for the singlet described in eq 9, since the expectation value of one of the spins along any axis is zero. Due to this rotational invariance, the invariance of entanglement measures with respect to the choice of basis is often emphasized, 185 but this should not be confused with orbital rotations to be discussed below.
It may be said that the VB wave function is created by identifying the entangled local states (orbitals) associated with chemical bonds. In fact, just as traditional wave function approaches start from the independent particle model and build more accurate Ansaẗze by incorporating the most important excitations, one might start a similar hierarchy around independent subsystems as a starting point which explores the Hilbert space by encoding more entanglement. 22 This leads to matrix product states (MPS) in general and the density matrix renormalization group (DMRG) approach in particular. 22 The seniority-based CC theory of Scuseria and coworkers 189 can also be interpreted on a quantum information theory basis. 190,191 This approach is built on orbital pairs rather than single orbitals with seniority being the number of unpaired electrons in a determinant that can be used to select the determinants for inclusion in a wave function expansion. The claim is that strong correlation can be treated by lowseniority contributions in a suitable orbital basis, 189 and it is possible to set up an entanglement-based measure to be used as a cost function. 191 One measure of entanglement is based on the von Neumann entropy (the "quantum Shannon entropy") of a state and the quantum relative entropy 184 which provides a measure of the distinguishability of two states. 186 The relative entropy of entanglement can be given as 22 11) in terms of the singular values σ i corresponding to a Schmidt decomposition: 184 if A and B are distinguishable subsystems of the whole system, then a pure state |Ψ⟩ can be expanded in the basis of functions of the type where |Ψ A ⟩, |Ψ B ⟩ are basis states in A and B, and the singular values σ i are those of the expansion coefficient matrix C AB . If only one singular value is nonzero, the entropy is zero and |Ψ⟩ is separable; otherwise, it is entangled, and it is maximally entangled if all singular values are the same. 22,185 Observe that entanglement has to do with the characterization of subsystems and depends on the partitioning of the system. For example, in eq 9, it is the two subsystems at sites A and B that are entangled because the total state cannot be written in the simple form |m⟩ A ⊗ |n⟩ B . For further considerations on superselection rules and classical and quantum correlation types, see. 186,191 If the local states are chosen to be single orbitals, then we may speak about (single) orbital entropy 190,192 and use the eigenvalues of the orbital reduced density matrix instead of σ i 2 in the definition of the entropy above, with the summation running over unoccupied, singly occupied spinup and spin-down, and doubly occupied states. Orbital entropy has been used to measure multiconfigurational character 193 and as a basis for automatic active space selection. 194 It has also been applied to the H 2 model system 193 and to a number of other bond formation processes. 193,195 See also another entanglement-based study on 2-electron systems by Huang and Kais. 196 The dependence of entanglement-based measures on the orbital basis has been noted in the literature 186,193 and they have also been used to set up a qualitative distinction between nondynamic, static, dynamic and dispersion correlation effects 190,197 in the sense of Bartlett and Stanton. 17 In earlier approaches, natural orbital occupation numbers have also been substituted for σ i 2 to calculate the von Neumann entropy as a measure of correlation strength. 198 Recently, entropy-based measures have even served as a basis of a geometric interpretation and an upper bound to the correlation energy. 199 Finally, we note that the two-orbital entropy can be defined in an analogous manner, and serves as the basis for orbital interaction (mutual information) measures 190,192,200 that can be used to find an orbital ordering that ensures good convergence in DMRG by placing strongly interacting ones near each other. 200 So far we have discussed the application of quantum information theoretical concepts in quantum chemistry. However, these concepts also underlie the theory of quantum computing in which chemistry has also been identified as one of the potential areas of breakthrough, 201 as exemplified by resource estimates in the study of Reiher et al. on FeMoco. 202 The latter compound served as an example of a strongly correlated system that would especially benefit from the fact that quantum computers are quantum systems themselves that need not explicitly generate the FCI coefficients in eq 8. Thus, we will also briefly mention recent developments in this field. In particular, the need that the initial states should have a good overlap with the target state for certain quantum algorithms to succeed is well-known. 203 The fact that the preparation of spin eigenfunctions on quantum computers has been investigated recently 204,205 can be seen as a step in this direction and it also underpins our efforts in the upcoming sections to provide a more nuanced classification of reference functions than the usual single vs multideterminant one. As far as the potential gains in scaling are concerned, a recent study suggests that the exponential nature of quantum advantage in quantum chemistry 206 cannot be taken for granted in most cases, although the authors are careful to point out that quantum computers may still be very useful in quantum chemistry. Thus, the search is on for specific chemical systems where quantum computers could be most beneficially applied. 207−209

SYMMETRY CONSIDERATIONS
We have already distinguished between DETs, CSFs and CFGs as various choices of basis states. It also follows from the previous discussion that symmetry plays a significant role in the classification of correlation strength. To get a better grip on these notions, let us remind ourselves about how an N-electron wave function is built from one electron functions (orbitals) and how various symmetries may be taken into account in the construction. The simplest approach is to approximate the Nelectron wave function as a Hartree product (HPD) of N orbitals and enforce various symmetries. In general, the operator Q̂encodes a symmetry of the Hamiltonian Ĥif [Ĥ, Q̂] = 0, and thus the exact wave function is a simultaneous eigenstate of Ĥand Q̂. If an approximate wave function does not have a particular symmetry, an appropriate projection operator ÔQ can be constructed that would produce an eigenfunction of Q̂with eigenvalue Q. We are not concerned here with all possible symmetries (e.g., the particle number operator, time reversal), but may limit ourselves to the following scheme: where antisymmetrization ( ) produces a Slater determinant (DET), spin projection (ÔS) a configuration state function (CSF), or, more generally, a number of CSFs belonging to the same configuration (CFG, i.e., a sequence of spatial occupation numbers), while spatial symmetry can also be enforced via ÔR

Journal of Chemical Theory and Computation
pubs.acs.org/JCTC Perspective of some symmetry operation R, producing a spatial symmetryadapted CSF (SA-CSF). We will briefly examine these in the following. It only remains to note here that symmetry adaptation in general leads to higher energies in a variational procedure since it introduces constraints in the variational manifold, a phenomenon sometimes referred to as Loẅdin's dilemma. 210 The original purpose of introducing the Slater determinant 211 was to find an N-electron basis that automatically satisfies the exclusion principle, or, rather, its generalization requiring a Fermionic wave function to be antisymmetric. For this purpose, the HPD is constructed from the product of spin orbitals ϕ i (x i ), which itself is the product of a spatial orbital φ i (r i ) and a spin function σ i (s i ), and the antisymmetrization is carried out by permuting the coordinates x i = (r i , s i ) and summing over the permutations multiplied by their parity Since is a projector multiplied by a constant, i.e., = ! N 2 , Φ is an eigenfunction of . The correlation introduced by this formulation is usually illustrated 19,40 using the case of two parallel electrons, where the spatial part φ 1 (r 1 )φ 2 (r 2 ) − φ 2 (r 1 )φ 1 (r 2 ) is obviously zero at r 1 = r 2 . It has been observed that this Fermi hole is in fact nonlocal, because the spatial part also vanishes if φ 1 (r 1 ) = φ 1 (r 2 ) = ± φ 2 (r 1 ) = ± φ 2 (r 2 ). 40 Moreover, antisymmetrization also introduces anticorrelation in the opposite-spin case, which has been used to explain the fact that same-spin double excitations contribute less to the energy in post-Hartree−Fock calculations than opposite-spin ones. 40 Given that the Slater determinant is a much simpler entity than a general group theoretical construction of the wave function, it could be claimed with some plausibility that "Slater had slain the "Gruppenpest"," 212 i.e., the pestilence of groups that threatened the conceptual clarity of quantum mechanics in his view. Were it only about the antisymmetry requirement, Slater's approach would be preferable to general group theoretical presentations given that it is much more transparent, a fact also responsible to a large extent for the establishment of the Slater determinant as the "default choice" as reference state. However, the question of spin adaptation still remains and is in some sense a natural extension of antisymmetrization. A spin-adapted CSF may be written as where we have retained the simplest Hartree product in the spatial part, but assumed a generalized spin part X S which is an eigenfunction of Ŝ2 with total spin S. Since X S can be written as a linear combination of spin function products of the type appearing in eq 14, it also follows that Θ can be written as a linear combination of determinants, although CSFs can be constructed directly without determinants as intermediates. 103,118 The general construction of X S remains an elaborate group theoretical exercise. Matsen's spin-free approach 213 developed in the 1960s relied on the observation that enforcing the SU(2) spin symmetry in the spin part of the wave function is equivalent to representing the N-particle spatial part in terms of the irreducible representations of the symmetric group S N of permutations. 118 On the other hand, the spin-free representation of the Hamiltonian is given in terms of the generators of the unitary group U(n) in the norbital basis. The latter observation is the basis of the unitary group approach 214 (UGA) and its graphical application 215 (GUGA), which was recently discussed in detail by Dobrautz et al. 216 and the advantages of which have also been exploited in a number of publications. 104−106 We will return to some aspects of spin-adaptation and its consequences for the correlation problem in the next section, and similar considerations will also occur in the section on the role of orbitals.
Finally, we shall briefly consider the problem of spatial symmetry since it does appear in discussions on strong correlation. 107 The prototypical example is the He 2 + system 217,218 which dissociates into a He atom and a He + ion. At large distances, two electrons are localized on one of the He nuclei and one electron around the other. In the minimal basis, this means that the Hartree−Fock orbital associated with the two electrons has a larger spatial extent than the other one, and they do not reflect the point group symmetry of He 2 + . Moreover, there are two symmetry broken solutions depending on whether the two electrons are localized around nucleus A or B, corresponding in essence to two resonance structures. It is possible to enforce spatial symmetry by constructing orbitals of the type φ A ± φ B , where φ A and φ B are orbitals localized around A and B. A determinant built from such orbitals could be expanded in terms of the resonance structures mentioned, but would also include high-energy ionic configurations, giving rise to an artificially large energy at long bond distances. Thus, this is the spatial equivalent of the dissociation problem of the H 2 molecule in which breaking spin (and spatial) symmetry leads to a lower energy. Usually, these problems are regarded multiconfigurational, 218 because more than one configuration is needed to give the correct energy without breaking symmetry. Unfortunately, at the level of approximate wave functions (often used as a reference), enforcing spatial symmetry may easily lead to additional complications. Since infinitesimal changes may disrupt the symmetric arrangement of the nuclei, this may lead to discontinuities in the energy or the need to use suboptimal symmetry-adapted orbitals. 219 Thus, using SA-CSFs may introduce more conceptual uncertainty about symmetry-related correlation effects than simply using DETs or even CSFs (CFGs). We will not consider SA-CSFs here any further, but the review of Davidson and Borden 220 discusses the issue of spatial symmetry breaking in all its aspects, be it "real or artifactual", as they put it, and a series of papers by Čízěk and Paldus on the HF stability conditions 221−223 also devotes considerable space to this problem.
It has been noted before that the various categories for correlation are operational, i.e., what counts as static or nondynamical correlation is practically whatever is covered by choice at the level of the reference calculation. Symmetry on the other hand introduces exact degeneracies that are known a priori and reference functions can be constructed to account for them. Thus, in line with the general thrust of many discussions on the subject, it seems worthwhile to distinguish between correlation effects due to various kinds of symmetry and the rest. We may call these symmetry effects static correlation. 12,17 In the next section, we will discuss accounting for spin symmetry in the basis states. The remaining correlation effects may be weak or strong depending on the weight of the symmetry-adapted reference state in the final calculation. In this regard, near degeneracies are of interest since they do not arise from the symmetry effects discussed so far. The case of the Be atom is instructive: 19 due to the near degeneracy of the 2s and 2p orbitals, the 1s 2 2s 2 configuration does not completely dominate the wave function expansion, the 1s 2 2p 2 configurations must also be considered. Such nondynamical effects are not captured by symmetry restrictions but are usually still included in the reference calculation, leaving any residual correlation effects, i.e., dynamic correlation, to be recovered by the wave function Ansatz built on that reference. Handy and co-workers 70 even go so far as to suggest that even this "angular" correlation in Be, so-called because s and p orbitals are of different angular types, might be regarded as dynamical based on a model in which explicit dependence on the interelectronic distance appears. The Be example also indicates that the weight of the reference state is of crucial importance and what is regarded as nondynamical or strong depends on a threshold value set on this weight. The dependence of weight-based measures on the orbital space will also be addressed in a subsequent section using a simple bond breaking process as an example, another situation in which strong correlation effects are typically expected.

REPRESENTING THE WAVE FUNCTION
It has already been mentioned that DETs are only eigenfunctions of Sẑ, while CSFs are also eigenfunctions of the total spin. The total number of DETs 80 associated with a CFG and with a given spin-projection M S and number of unpaired electrons N is  (17) which is obviously less than the number of determinants. In fact, summing over all possible S values yields the number of determinants. 80 To help ground these notions, consider some simple examples collected in Table 1, and discussed in detail elsewhere. 80 Taking the example of two electrons in two orbitals, the electron pair may be placed on the same orbital as in the configurations 20 and 02, or on different orbitals as in the configuration 11. In the former two cases, a single CSF consisting of a single determinant is enough to describe a closed-shell singlet, hence the success of determinants in many situations in chemistry. In the 11 case, a singlet and a triplet CSF can be built from two determinants, though the other two triplet states need only a single determinant. A slightly more complicated example involves distributing three electrons among three orbitals. For the configuration 111, the S = 3/2 cases yield nothing conceptually new, they are described by CSFs consisting of a single or multiple determinants. On the other hand, the S = 1/2, M S = 1/2 subspace can only be spanned by two degenerate CSFs, themselves consisting of multiple determinants. Since the reference function is often obtained at the Hartree−Fock level, it is worth recalling how this discussion affects such calculations. In more commonly occurring cases, the spin-adapted HF wave function can be constructed from either a single determinant, as in the typical closed-shell case, or from a single CSF, as in a low-spin openshell case. 224 In those open-shell cases, such as S = 1/2, M S = 1/2 for 111, where a configuration cannot be uniquely described by a single CSF, 225 Edwards and Zerner recommend in a footnote that a HF calculation using a chosen CSF should be followed by a configuration interaction (CI) calculation. 225 This might minimally include only the CSFs belonging to the CFG in question, e.g., the two degenerate ones in the 111 case, possibly with or without orbital optimization, though the question remains whether such a small CI calculation would accurately reflect the exact solution. Indeed, Edwards and Zerner recommend CI in the active space, i.e., presumably including multiple configurations that might produce the same total and projected spin (e.g., 201 in the three-electron case). Nevertheless, the expansion coefficients of the CSFs of a single CFG will be of interest here for analytic purposes.
It is worth noting that already at the HF level spin-restricted open-shell CSFs are considerably harder to optimize than the unrestricted determinants. This has led to several ways of circumventing the direct use of CSFs. A simplified version 130 of Loẅdin's projection operator 226 may be used to remove the spin-components that lie nearest to the desired state from the spin-unrestricted determinant, but in any case, spin-projection leads to artifacts if applied after optimization and falls short of size-consistency if made part of it. 227 More recent projection approaches constrain the eigenvalues of the spin-density to obtain results that are identical to the more cumbersome conventional spin-restricted approach. 228 Even more pragmatically, quasi-restricted approaches of various kinds have been proposed to obtain orbitals resembling restricted ones from some simple recipe. 229,230 More complicated Ansaẗze are usually obtained from reference states by (ideally spinadapted) excitation (or ionization) operators, 214 and in a similar vein, it is also possible to define spin-flip operators that change the spin state of the reference. 231 In simple cases, such spin-flip operators may even be spin-adapted themselves. 232 Sometimes symmetry-broken solutions may be used creatively, e.g., to obtain some property of spin-states as in PO/BS-DFT |123̅ | a Spatial orbitals are labelled simply as 1, 2, or 3 in DETs and CSFs, while the corresponding occupation numbers (0, 1 or 2) are indicated in CFGs.

Journal of Chemical Theory and Computation
pubs.acs.org/JCTC Perspective discussed above, or in combining unrestricted determinants for quasi-spin-adaptation, 233 sometimes considerations about the CSF expansion are employed innovatively, e.g., to obtain singlet−triplet gaps from bideterminantal M S = 0 states (see Table 1) in cases in which methods with a single determinant bias encounter multideterminant situations. 234 At this stage, there is some ambiguity whether we associate multireference with the requirement that the exact solution is dominated by more than a single determinant, CSF or configuration. In a normalized FCI expansion of the type in eq 8, a set of basis states dominates the expansion if their cumulative weight, ∑ P |C P | 2 is larger than the rest of the weights combined, i.e., 1 − ∑ P |C P | 2 . Assuming knowledge of the exact solution, the following categories are proposed (cf. alone dominates, and hence this is the true multireference case since no single reference function of the kinds discussed before can be found. Note that the single reference (SR) categories above may overlap, since e.g., the closed-shell ground state of a twoelectron system is at the same time SD, SS and SC. An openshell singlet is SS and SC, but not SD. The three-electron doublet that has two CSFs would be SC but not SS or SD, if the cumulative weight of those CSFs dominate the expansion. All these are examples of various SR expansions. The simplest example of an MR expansion would be a combination of the type 02 + 20, as in the case of the dissociating H 2 molecule. In the next section, we shall discuss how the choice of orbital space impacts this classification. Once spin-coupling is taken care of, an important source of correlation effects is addressed and the question remains how strong the remaining correlation effects are. This reflects the distinction between static and nondynamic correlation as used by Bartlett and Stanton,17 with static correlation being eliminated by spin-adaptation. In this regard, this generalization of the excitation schemes can be compared with entanglement 22 and seniority 189 approaches which also aim at finding a good reference based on different criteria.

THE ROLE OF THE ORBITAL SPACE
The idea that the weight of the reference function in an approximate CI calculation could be used as an indication of multireference character is an old one, but it was also observed long ago that due to the bias of canonical orbitals toward the reference (HF) state, it is of limited use. 110 Furthermore, it was also suggested that such CI calculations should be complete within at least an active space for this metric to be of use. 109 In that case, the leading CAS coefficient in the basis of natural orbitals using determinants or CSFs was found to be a sensible metric. 235,236 The role of the orbital space has been discussed in connection to most approaches discussed in this perspective, 109,123,186,189,193 and orbital localization techniques form a part of practical DMRG 237 calculations as well as reducing the complexity of CAS 238 calculations. Within the DMRG context, orbital optimization techniques have been recently suggested as a means of reducing the multireference character of the wave function. 239 A similar result has been observed in SCI methods, where orbital optimization leads to a more rapid convergence to the FCI solution. 240 The recent work of Li Manni and co-workers on GUGA based FCIQMC 104−106 and its application as a stochastic CASCI solver 241 shed even more light on the role of the orbital space. This methodology has the following features relevant for the current discussion: a) using GUGA to calculate spin-adapted basis states efficiently; b) using a unitary orbital transformation to compress the wave function expansion; c) the combined use of unitary and symmetric group approaches means that the ordering of orbitals also affects the complexity of the wave function. The technique has been used to reduce the number of CSFs expected from the exponentially scaling combinatorial formula to a small number of CSFs, sometimes only a single one. 106 Moreover, not only is the wave function more compact, but the Hamiltonian assumes a convenient block-diagonal form that allows for the selective targeting of excited states. 106 Interestingly, it was also found that the best orbital ordering for a spin-adapted CSF is not identical to the best site-ordering in the DMRG sense. 242 More recent work has also attempted to place the orbital ordering process on a more systematic basis exploiting the commutation properties of cumulative spin operators and the Hamiltonian. 243 Here, the role of the orbital space will be illustrated via a simple example that will remain within a more traditional CAS/FCI context.
In this perspective, we are concerned with the a posteriori characterization of states rather than finding an a priori indicator of multireference character, and hence we will assume that an FCI solution is available. We will illustrate the role of orbital spaces by considering the H 2 bond dissociation problem using canonical and localized orbitals. 244 We will define the multireference character M as = | where the summation runs over the DETs or CSFs belonging to the dominant CFG. This measure will converge to 1 for infinitely many coefficients assuming they are all equal, i.e., the worst MR case. In SR cases, M should be close to 0, and in the H 2 case M = 1/3 corresponds to two configurations of equal weight. In practice, one might choose a different threshold on the cumulative weight of the configuration. For example, requiring the dominant cumulative weights to be above 80% leads to M ≤ 1/9. If such a dominant configuration exists (and if it were known in advance), it would be the ideal reference function on which a correlated Ansatz could be built. Figure 1 shows the value of M at different internuclear separations D. In the canonical basis, a single configuration 20 is dominant around the equilibrium distance, while at infinite separation the CFGs 20 and 02 contribute equally. Thus, in the Journal of Chemical Theory and Computation pubs.acs.org/JCTC Perspective latter case, the expansion is multiconfigurational as well as multideterminantal. Evaluating M in the basis of local orbitals yields to the opposite results. The local orbitals are each essentially localized on one atom, and at long bond distance, a single CSF of the 11 configurations can describe the dissociation. Thus, the expansion is SC and SS, but not SD. At shorter bond lengths, the localized expansion is much less appropriate as seen from the rising multireference character. Changing from the canonical to the local orbital basis has a similar effect on entanglement-based metrics: 193 in the canonical basis entanglement changes between its minimal and maximal values as D grows and the other way around in the local basis since in the latter all occupation patterns are equally likely at small D. This ties in also with our earlier remarks on the relationship between VB and MO approaches. It has been argued earlier that an important source of correlation effects may be eliminated by spin adaptation. Yet, because of Loẅdin's dilemma, the constrained reference energy may increase and this could in turn lead to an increase in the correlation energy. In the H 2 example, the spin-restricted closed-shell reference function has been found qualitatively adequate around the equilibrium bond length, but at large distances it leads to a catastrophic increase in the magnitude of the reference and correlation energies. Using canonical orbitals, the remedy involves combining the 20 and 02 configurations. Thus, it seems that enforcing spin symmetry makes the problem more complicated by requiring a multiconfigurational treatment whereas the spin-unrestricted single determinant also converges to the qualitatively correct energy at large bond lengths. This is because it reduces to one of the symmetry broken determinants |μν ̅ | or |μ ̅ ν| in this limit. In general, these broken symmetry determinants produce an energy that lies between the open shell singlet in eq 7 and triplet states that can also be constructed from them, and as such, they are not a good approximation to either. At large bond distances, however, the singlet−triplet gap closes and the energy converges to the correct limit while the wave function is not the exact one. 18,23 While symmetry breaking may be desirable under some circumstances, for finite molecular systems symmetric solutions are in general preferred. In the open-shell singlet case, for example, spin-adaptation is indispensable for a qualitatively accurate description. It is here that orbital rotations may play some role: by localizing the orbitals at large distances, it is possible to convert the multiconfigurational representation of the dissociating H 2 molecule in the canonical basis into a single configurational open-shell singlet in the local basis.
The fact that the multireference character depends on the orbital representation warns against assigning the multiconfigurational nature of the wave function expansion as a physical property of a state of the system. At the very least, one should also demand that there be no unitary transformation originating in the orbital space that removes the multireference character of an expansion completely. Of course, the analysis as presented here is not a practical recipe for computation, since it assumes knowledge of the FCI solution, although it may be used fruitfully in interpreting less costly wave functions of the CASCI or CASSCF type. Furthermore, it does underline the potential of orbital rotations in reducing the multireference character. This important subject will be analyzed in more detail in a forthcoming publication.

OUTLOOK
The distinctions we introduced among single-determinantal, single spin-coupling, single configurational and multiconfigurational expansions should have a relevance in any chemical system, as also pointed out by other authors. 17,19 These categories form a clear hierarchy of SR measures in the sense that they are increasingly less restrictive: SD implies SS and both SD and SS imply SC. Unlike in extended systems, where symmetry broken solutions are often preferred, symmetryadapted wave functions are often needed to reliably predict certain chemical properties of molecules. Again, unlike in periodic solids, strongly interacting regimes are usually easier to localize, and hence the complexity of the wave function can often be reduced. If spin coupling effects can be incorporated into a single reference function, then the remaining correlation effects might be weak, and in cases when they are not, such as in certain regions of bond dissociation curves, they can be typically taken care of by small active space calculations. As observed by Malrieu et al., 245 the fact that one can always build a small complete active space representation by placing two electrons in a pair of localized bonding and antibonding orbitals that accounts for such correlation effects is perhaps the best unbiased confirmation of Lewis' idea of the chemical bond consisting of an electron pair. 246 We have discussed the effect of orbital rotations on the multireference character in a simple bond dissociation process and pointed out that such orbital rotations have the potential of reducing it. If by strong correlation we also mean the large size of the active space necessary to remedy these nondynamical effects, then it seems likely that most molecular systems are not strongly correlated, with the possible exception of multisite antiferromagnetically coupled systems. 107,247−251 Thus, the complexity of most finite chemical systems should be well below exponential and should in most cases involve only a few configurations. Devising an appropriate metric that can help find these a priori is a challenge, but measures based on leading coefficients, density matrices, natural orbitals and orbital entanglement have been used for this purpose in the literature.