Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins “Conjecture”

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

ABSTRACT: Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible.The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals.The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture.Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.T he adjective "complex" describes two distinct character- istics of a (nonrelativistic) electronic wave function Ψ ≡ Ψ(x 1 , ..., x N ) 1 that differ vastly in their importance to the art and science of quantum-chemical calculations.Whereas neither computation nor interpretation of Ψ is significantly affected by its complex-valuedness (which in any case comes into play only in the presence of external magnetic fields), they are both severely impacted by its complexity.The alleviation of this impact, which has been the target of the century-long efforts of chemists, physicists, and mathematicians, can be attained by either replacing Ψ with its reduced-complexity equivalents or constructing it from less complex quantities.
Within the first of these two approaches, the functional for the electronic energy E[Ψ] = ⟨Ψ|H ̂|Ψ⟩, where H ̂is the electronic Hamiltonian, is replaced by its analogue involving a contraction bilinear in Ψ such as, to list the most popular choices, 2  (2) or the one-electron density (3) The obvious advantage of such an approach is the Nindependent complexity of 2 Γ, 1 Γ, and ρ 1 .However, this simplification carries a heavy price of the respective functionals being far more convoluted than E [Ψ].In the case of E[ 2 Γ], this means an explicit functional whose domain is extremely complicated due to the requirement of the N-representability of 2 Γ. 3 Consequently, rigorous electronic structure calculations based upon E[ 2 Γ] turn out in practice to be computationally as expensive as those invoking E[Ψ]. 4 Although the functional E[ 1 Γ] is not explicit, its unknown part is much smaller in magnitude than that of E[ρ 1 ].On the other hand, the Nrepresentability conditions for ρ 1 are trivial, 5−7 whereas those for 1 Γ are less so. 8The relatively low cost of calculations employing E[ 1 Γ] and E[ρ 1 ] comes at the expense of lost rigor as these functionals require empirical parameterization that, especially in the case of E[ρ 1 ], is usually detrimental to their overall reliability (which should not be confused with the smallness of the average error). 9To put it in a nutshell: the complexity of the Ψ equivalent + the complexity of the energy functional that takes it as the argument = the computational effort + the severity of parameterization = const � like everywhere else, there is no free lunch in quantum chemistry! 10,11he other approach to the complexity mitigation involves expressing the N-electron Ψ in terms of a (in general infinite) sum of products of quantities that depend on fewer than N coordinates {x k }.Although many variants of this tensor decomposition are possible 12 (including those based upon geminals 13−17 ), the vast majority of electronic structure calculations employs its most extreme form where = { } j j , ..., N

1
, that carries it all the way down to the orthonormal one-electron eigenfunctions (commonly known as spinorbitals) {ψ j } ≡ {ψ j (x 1 )} of the sẑ(σ 1 ) operator.This representation of Ψ, which is just the FCI (full configuration interaction) expansion written in terms of the Hartree products rather than the Slater determinants, is exact provided that {ψ j } is a complete set. 18In practice, however, only a finite number M of the spinorbitals can be included, resulting in an approximate (square-normalized) N-electron wave function Ψ̅ ≡ Ψ̅ (x 1 , ..., x N ) given by a linear combination of Hartree products.The coefficients { } C of this linear combination are not fully independent as they have to reflect the constraints imposed upon Ψ̅ by its permutational antisymmetry, the expectation values of the spin operators S 2 and S ̂z, and the point group symmetry (if present).The incorporation of other fundamental properties of the exact Nelectron wave function Ψ into its approximate counterpart Ψ̅ is far less trivial.Thus, the nodal structure of Ψ (which has unexpected features even in the deceptively simple case of the helium atom 19,20 ) can be reproduced, depending on the electronic state, quite easily 21 or with considerable difficulty, 22 whereas the electron-electron coalescence cusps 23 cannot be reproduced at all.
From the practical standpoint, the convergence of the approximate expectation value ⟨O̅ ⟩ ≡ ⟨Ψ̅ |O ̂|Ψ̅ ⟩, where O învolves one-and/or two-electron operators, to its exact counterpart ⟨O⟩ ≡ ⟨Ψ|O ̂|Ψ⟩, is of great importance to the electronic structure theory.In the case of variational approaches (such as FCI), in which the coefficients { } C are determined by minimization of ⟨H̅ ⟩ subject to the aforementioned constraints, this convergence is obviously most rapid for the electronic energy.Even so, the decay of the truncation error ΔE(N, M) = ⟨H̅ ⟩ − ⟨H⟩ with M turns out to be quite slow as indicated by its M → ∞ asymptotic lower bound being of the order of . 24 The dependence of both the accuracy of Ψ̅ and the magnitudes of the optimal { } C on the one-electron functions constituting the basis set {ψ j } j=1 M (which is a subset of {ψ j }) brings to the forefront two distinct but intimately related problems of identifying the spinorbitals that, for a given Ψ and M, produce (a) the most accurate Ψ̅ and (b) the largest number of { } C with small magnitudes.The introduction of the natural spinorbitals {ϕ n } ≡ {ϕ n (x 1 )} that, together with their respective occupation numbers {ν n } (ordered nonascendingly), enter the Schmidt decomposition (5) of the 1-matrix, has been prompted by their promise of providing significant insights into the solution of these twin problems.−28 In fact, although Ψ̅ constructed from the first M natural spinorbitals minimizes for a given M the wave function error 29 it does not minimize ΔE(2, M). 24 Even worse, such a choice of M spinorbitals minimizes neither ΔΨ(N, M) 30 nor ΔE(N, M) 31 for any N > 2. In addition, although the employment of the natural spinorbitals drastically reduces (from ∼M 2 to ∼M) the number of the nonzero coefficients among { } C in the truncated FCI expansions of wave functions describing two-electron systems, 29 it is not optimal for N > 2. 31 In light of these observations, it is quite surprising that the original claim 25 remains generally accepted. 32On the other hand, it should be emphasized that the accuracy gains and/or the reductions of computational effort realized upon replacing the natural spinorbitals with their optimal counterparts are marginal in comparison with those observed upon analogous replacement involving, e.g., the Hartree-Fock spinorbitals. 33lthough the initial interest in the natural spinorbitals and their spatial components {φ n } ≡ {φ n (r⃗ 1 )}, known as natural orbitals (NOs), has substantially waned in the wake of the aforementioned studies, the recent years have witnessed a renaissance of research on these one-particle functions spurred by both a growing realization of their usefulness in theoretical nuclear physics 34−36 and the developments concerning the one-electron reduced density matrix functional theory (1-RDMFT). 37Indeed, the substantial progress in the understanding of the properties of the E[ 1 Γ] functional has been greatly facilitated by the adoption of its equivalent E[{ϕ n }, {ν n }] that stems from the decomposition 5.This is so because the ensemble N-representability of 1 Γ is assured upon the satisfaction of the trivial constraints ∀ n 0 ≤ ν n ≤ 1 and ∑ n ν n = N. 38 Although proper minimization of E[ 1 Γ] requires 8 maintaining the pure N-representability of 1 Γ through additional enforcement of the generalized Pauli constraints (GPCs), 39 distinguishing between these two domains of E[ 1 Γ] rapidly becomes irrelevant with increasing N and M. 40 Elucidation of the structure of E[ 1 Γ] is furthered by examination of the constraints imposed upon this functional by certain universal properties of the NOs.Although some of these properties, such as the presence of cusps at the positions of nuclei and the long-range exponential decay, have been uncovered soon after the introduction of the concept of the NOs, 2 most of them have come to light only very recently thanks to the discovery of the fifth-order off-diagonal cusp in 1 Γ 41 that gives rise to the zero-energy Schrodinger equation satisfied asymptotically by both φ n (r⃗ 1 ) and ν n at the limit of n → ∞. 42 The availability of this equation has opened an avenue to facile derivation (which otherwise is quite complicated 43−45 ) of various asymptotic power laws, e.g.

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where is the on-top two-electron density and the NOs corresponding to different spin components are counted separately (Figure 1).Similar power laws and the n → ∞ asymptotics of φ n (r⃗ 1 ) are obtained for non-Coulombic systems such as the contactium 46 and the anyons. 47Generalizations of the NOs, 48 such as the energy natural orbitals 49−51 and the natural transition 52 / binatural orbitals 53 are amenable to an analogous treatment.Among them, those involving the eigenfunctions and eigenvalues of the one-electron kinetic energy density matrices 24 and 42,54 are of particular interest as they yield asymptotic expressions that account for the (rather irregular) 2 expectation value with n (Figure 2).Even more importantly, these expressions lead to the aforementioned estimate of 24 It is this estimate, which is equally applicable to truncated FCI expansions and the functionals E[ 1 Γ] of the 1-matrices constructed from finite numbers of natural (spin)orbitals, that enters the considerations concerning the potential advantages (or the lack thereof) of 1-RDMFT over the wave functionbased methods of quantum chemistry.The point of departure for these considerations is the partitioning where  ( , ; , ) is the two-electron density cumulant (the 2-cumulant), 55,56 that allows splitting E[ 1 Γ] into two terms, namely, the explicit E HF [ 1 Γ] (where HF stands for Hartree-Fock) and the unknown correlation component U[ 1 Γ] of the electronelectron repulsion energy (also known as the intrinsic correlation energy 57 ).The first of these terms, which is system-specific, reads pq pq qp pqrs pr qs ps qr rspq HF where 1 Γ pq = δ pq ν p (which follows from the Schmidt decomposition 5), 1 , and The second term, which is universal, involves the constrained search 58 [ ] = U g min pqrs pqrs rspq over all the 2-cumulant tensors yielding Nrepresentable 2 that conform to the partial trace relations 59,60 = 1 2 ( ) and where 1 Γ ≡ { 1 Γ pq } and the elements of the vector 1 .In addition, for electronic states that are the eigenstates of the S 2 and S ̂z operators with the eigenvalues of + ( 1) and ± , respectively, the elements of these tensors have to satisfy the constraint 59 . 59Both the corresponding [ ] 2 1 , which follows from eq 8, and 1 Γ derive from the same electronic wave function Ψ[ 1 Γ].Therefore, for the 1matrix that minimizes ] is the ground-state electronic wave function (within the manifold of those with a given spin multiplicity) and [ ] is the corresponding ground-state 2-cumulant.The ground-state 2matrix that results from the substitution of this 2-cumulant into eq 7 yields in turn the ground-state on-top two-electron density that enters the condition 6 for the occupation numbers pertaining to the ground-state 1-matrix.Consequently, any genuine 1-RDMFT formalism, whether exact or approximate, produces the 2-matrix that allows calculation of any twoelectron ground-state property.Moreover, such a formalism yields the pairs 1 Γ / [ ] 2 1 that conform to not only the oneindex contractions 11 and 12 but also to the identity [ ] x x x x ( , ; , ) and the condition 6.
It transpires from eqs 10−13 that evaluation of U[ 1 Γ] requires computation of the sum g g ( , , , ) , where the tensor { } g g ( , , , ) ( , , , ) is an unknown function of the vector , the matrices 1 Γ and , and the tensor g ≡ {g rspq } of the electron-electron repulsion integrals.It should be emphasized that, except in the cases of spin-unpolarized systems ( = 0) and those with maximum spin polarization ( = N 1 2 ), both and have to be included in the argument list of 2 .Moreover, neglecting the dependence of 2 on g gives rise to specious paradoxes exhibited by spurious "algebraic functionals" 61−64 unless the system in question is an ensemble of noninteracting one-and two-electron subsystems.
The computational cost of evaluating the electronic energy within a given quantum-chemical formalism involving M spinorbitals grows like M κ (the FCI approach being an exception).It is therefore convenient to introduce the concept of the numerical rank κ as the means for categorizing approximations to E[ 1 Γ] according to their computational efficiency.With a diagonal 1 Γ, only the two-electron integrals {J pq } ≡ {g pqpq } and {K pq } ≡ {g qppq } enter the second sum in the r.h.s. of eq 9, allowing its evaluation from precomputed {g rspq } with ∼M 2 operations.Therefore, the HF-like E HF [ 1 Γ] is a numerical rank-2 functional.The desire to retain this rank upon the addition of U[ 1 Γ] and thus to keep the 1-RDMFT electronic structure calculations affordable for large systems, has been the driving force behind the development of the socalled JK-only functionals, 65 in which the constrained search 10 is circumvented with the approximation where F J (ν) and F K (ν) are some (more or less) empirical functions of the vector ν ≡ {ν p }. 66 Despite their computational attractiveness, the JK-only functionals are fundamentally flawed due to their failure to reproduce the correct behavior of U[ 1 Γ] at the weak-correlation limit 67 and the unphysical nature of the constraints imposed upon U[ 1 Γ] by the approximation 14. 59 Nevertheless, in light of the above discussion, such functionals are compatible with the genuine 1-RDMFT formalisms as long as they produce 1-matrices whose eigenvalues (i.e., the occupation numbers) satisfy the condition 6.These observations prompt one to contemplate the possibility of numerical rank-0 and rank-1 approximations to U[ 1 Γ].At the first glance, development of such approximations appears to make little sense due to the predominance of the computational cost associated with the evaluation of E HF [ 1 Γ] in the resulting E[ 1 Γ].Nevertheless, studies of the functionals belonging to both of these classes have been published.As the numerical rank-0 expressions cannot involve any summation over the indices of spinorbitals, they admit Although not investigated thus far, an analogous linear regression between U[ 1 Γ] and N = Tr 1 Γ is expected to exist.The story behind the numerical rank-1 functionals is considerably more convoluted.It commences with "a conjecture" published by Collins in 1993. 70Supported with neither rigorous theoretical arguments nor a single piece of numerical evidence, and lacking any characterization of its domain of applicability, the claim that E corr is proportional to the Jaynes entropy 71 S J = −∑ p ν p ln ν p computed from the occupation numbers pertaining to the ground-state 1-matrix is obviously a speculative statement rather than a conjecture.Nevertheless, publication of this claim has spurred several numerical studies 72−77 that, despite some assertions to the contrary, have not confirmed its validity.In particular, the linear regressions between the truncation errors in E corr and S J that arise from the employment of diverse basis sets in conjunction with approximate electron-correlation formalisms in calculations on the members of the lithium isoelectronic series 72 and various small molecules 73 simply reflect the regular reduction of these errors upon the application of increasingly higher levels of theory.However, they do not provide any numerical evidence for the proportionality between the E corr and S J of different electronic systems.Indeed, in the case of a homogeneous electron gas, the ratio E S corr J has been found to vary significantly with the gas density. 74Yet another study, involving the members of the helium isoelectronic series and the two-electron harmonium atoms with varying confinement strengths, has uncovered the simultaneous weak-correlation limits of E corr → const and S J → 0 that are irreconcilable with the original claim. 75In the cases of the H 2 76,77 and N 2 77 molecules, rough linear regressions between E corr and S J have been found to hold only for internuclear distances close to their equilibrium values.Turning Collins' claim into a proper conjecture requires (1) replacing E corr with the intrinsic correlation energy U[ 1 Γ], (2) replacing the Jaynes entropy S J with a quantity S nf [ 1 Γ] compatible with the particle-hole equivalence, 78 and (3) defining the necessary conditions for a linear dependence of The first of these requirements is trivial to implement, whereas the second one is readily fulfilled by the adoption of the nonfreeness 79 (a.k.a."the particle-hole symmetric correlation entropy" 80 )  (16)   as the correct measure of "the correlation entropy".The suitability of these two definitions is supported by the plots of U[ 1 Γ] against S J 81 or S nf [ 1 Γ] 82 that produce straight lines for the H 2 , O 2 , and H 2 O molecules at different geometries (note, however, that the nonzero intercepts of these lines imply a linear dependence rather than a direct proportionality).On the basis of these limited numerical results, one is tempted to formulate "the modified Collins conjecture" that, for a given set {Z K } of nuclear charges, the equality where a and b constants depending on {Z K } and N, and 1 Γ is the ground-state 1-matrix, approximately holds for all the sets of the nuclear positions {R ⃗ K }, i.e., for all the points on a given ground-state potential energy hypersurface.However, this conjecture is easily refuted by the following simple Gedankenexperiment.
Consider an atom or a molecule in the ground electronic state whose ionization energy is greater than that of the hydrogen atom.The modified Collins conjecture posits that the equality 17 holds for all the systems composed of (at arbitrary geometry) and an arbitrary number of protons located at arbitrary positions.Consider two cases: (1) all the protons located at infinity, i.e., at infinite distances from both and each other, and (2) some of these protons merged with some of the nuclei of .Since the protons at infinity contribute to neither U[ 1 Γ] nor S nf [ 1 Γ], one concludes that the validity of the modified Collins conjecture on the entire ground-state potential energy hypersurface of some system implies its validity with the same constants a and b on the entire ground-state potential energy hypersurfaces of all the systems isoelectronic with it.
Next, apply the above conclusion to being the helium atom.Per simple arguments of the perturbation theory, 67 the intrinsic correlation energy U[ 1 Γ] of a member of the helium isoelectronic series with the nuclear charge Z tends to a constant at the limit of Z → ∞, whereas the leading terms in the analogous asymptotics of the deviations of the occupation numbers from their uncorrelated values of 0 and 1 are proportional to Z −2 , implying Z −2 ln Z as the leading asymptotics of S nf [ 1 Γ] at Z → ∞.Thus, the modified Collins conjecture fails.
Similarly, consider being the B + cation (note that the ionization energy of the beryllium atom is smaller than that the hydrogen atom).In this case, the leading term of the large-Z asymptotics of U[ 1 Γ] is proportional to Z.As there are certain NOs with occupation numbers tending to constants at Z → ∞ (because of the s/p asymptotic degeneracy), 83 S nf [ 1 Γ] goes to a constant at that limit.The modified Collins conjecture fails again.
These failures bring back the domain-of-applicability question that can be conclusively answered only with rigorous analytic considerations.Due to its definition invoking the constrained search, U[ 1 Γ] is not readily amenable to the analysis of its purported linear dependence on S nf [ 1 Γ].However, such an analysis can be carried out with ease for the responses of these two quantities to one-electron perturbations.Thus, the first-order response of the groundstate electronic energy to the perturbation λ∑ k=1 N χ(r⃗ k ) involving a one-electron operator with the matrix elements 1 Γ pq χ qp by virtue of the Hellmann-Feynman theorem 84−87 (here and in the following, the standard notation , etc., is used).On the other hand, as E HF [ 1 Γ] is not stationary with respect to variations in 1 Γ.

Consequently,
The Journal of Physical Chemistry Letters pubs.acs.org/JPCLPerspective where (see eq 9) is the matrix element of the generalized Fock operator f ̂.Moreover, Thus, for the equality 17 to hold not only at a given point of the ground-state potential energy hypersurface but also in its vicinity (and/or also in the presence of, e.g., electric field), U χ [ 1 Γ] has to be proportional to S nf χ [ 1 Γ] for all the perturbations λ χ.In other words, the condition has to be satisfied for all 1 Γ χ ≡ { 1 Γ pq χ } such that Tr 1 Γ χ = 0. Needless to say, its equivalent where is a constant, is not satisfied by the natural spinorbitals and their occupation numbers.Consequently, eq 17 is not valid even locally.However, eq 22 may hold for a particular 1 Γ χ despite the violation of condition 23.This observation may explain the approximate linear dependence of U[ 1 Γ] on S nf [ 1 Γ] observed for certain nuclear motions. 81,82On the other hand, one-electron properties (e.g., multipole moments) computed as respective derivatives of the energy functional E HF [ 1 Γ] + a S nf [ 1 Γ] + b with respect to the perturbation strength λ are expected to be inaccurate.
In all of the above considerations, the 1-matrix that enters As the computational cost of evaluating S nf [ 1 Γ] is dominated by that of procuring this 1 Γ (or at least an accurate approximation of it), the practical importance of such a formulation of the Collins conjecture is nil.In order to address this issue, the minimization of the functional E ̃[1 Γ] = E HF [ 1 Γ] + a S nf [ 1 Γ] + b has been proposed as the means for computing approximate ground-state electronic energies. 82,88However, such an approach is riddled with inconsistencies.First of all, even if one assumes that the equality U[ 1 Γ] = a S nf [ 1 Γ] + b holds for any ground-state 1-matrix, its validity for arbitrary 1 Γ cannot be ascertained.This observation, with the de facto limited domain of applicability of eq 17 even for the ground-state 1-matrices, makes the derivation of eq 23, which relies on arbitrary variations of 1 Γ, mathematically suspect.Second, the lack of the dependence of a S nf [ 1 Γ] + b on g gives rise to a paradox that concerns the first-order response of the ground-state electronic energy to a twoelectron perturbation.Per the Hellmann-Feynman theo-rem, 84−87 this response involves the 2-matrix derived from the minimizing 2-cumulant [ ] 2 1 .In the case of E ̃[1 Γ], this 2matrix turns out to be given by the partitioning 7 with a vanishing 2-cumulant, which violates the sum rules 11 and 12 unless 1 Γ is idempotent.This paradox can be resolved only by either setting the coefficient a to zero (which yields the Hartree-Fock energy shifted by a constant b) or making a and/or b g-dependent.
The minimizer 1 Γ ̃of E ̃[1 Γ] is obtained by solving the system of equations 23 with ν, h, and g replaced by their counterparts ν, h ̃, and g̃pertaining to 1 Γ ̃rather than to 1 Γ.The p = q component of this system implies a linear relationship between the expectation values {f pp [ 1 Γ ̃]} of the generalized Fock operator and { } ln 1 p p , which rules out 1 Γ ̃being the 1-matrix of a system of correlated electrons (Figure 3).This is so because, as already mentioned in this Letter, such a system is characterized by the large-p asymptotics of ν p ∼ p −8/3 and t p ∼ p 2/3 , the latter carrying over to {f pp [ 1 Γ ̃]} due to the predominance of the kinetic energy term.Although the same objection can be raised about the Hartree-Fock approximation, it should be emphasized that, in contrast to the E ̃[1 Γ] functional, which originates from the fiat approximation 17 and is claimed to include electron correlation effects, E ̃HF [ 1 Γ] derives from a well-defined variational wave function that is Coulombically uncorrelated by default.
In light of these observations, it is justified to view the formalism that invokes the minimization of E ̃[1 Γ] as just a semiempirical one-particle theory employing spinorbitals and their occupation numbers as auxiliary quantities.Regarding it as a variant of 1-RDMT 82 is not warranted. 89ne may wonder whether employing some other numerical rank-1 functional D[ 1 Γ] in place of S nf [ 1 Γ] would provide a better approximation to U[ 1 Γ] that, while not disposing of the aforedescribed inconsistencies, would nevertheless result in an improved conformance of 1 Γ ̃with the large-p asymptotics of ν p and f pp .Since such a replacement would turn eq 23 into ) as a possible candidate for the replacement functional.However, although the overall dependence of (1 ) 1/4 is indeed linear, the correlation between these two quantities remains poor (Figure 4).Even worse, repetition of the aforepresented Gedankenexperiment produces the leading large-Z asymptotics of ∼Z −3/2 and ∼1 for U[ 1 Γ] of the members of the helium and beryllium isoelectronic series, respectively.This recurring failure underscores the necessity of incorporating some dependence on g in the constants a and/or b that, in addition to eliminating the inconsistencies, would provide "the missing powers" of Z in these asymptotics.Some efforts in this direction have been recently reported. 90The attempts at improving the performance of approximate formulations of the density functional Reliability of quantum-chemical calculations based upon the density functional theory (DFT) and its 1-matrix analogue (1-RDMFT) hinges upon limiting to the absolute minimum the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible.Until recently, the constraints pertaining to the 1-matrix functionals have been far less numerous than those employed in the development of their one-electron density counterparts.However, since the former functionals are usually formulated in terms of the natural orbitals, bringing the recently uncovered universal properties of these quantities into play permits reduction of this disparity.This development is an example of novel applications of the natural orbitals that, despite failing to deliver upon the initial promise of maximizing the accuracy of approximate wave functions, remain among the most important concepts of the electronic structure theory.
The benefits of employing a broad instrumentarium of analytical and numerical tools in the validation of the 1-matrix functionals are vividly demonstrated by the insights gained with the critical review of the three incarnations of the socalled Collins conjecture presented in this Letter.The first of these incarnations, published in the form of an unsubstantiated claim 70 unsupported by numerical evidence, 72−77 has been later turned into a proper conjecture with the help of rigorous  definitions of the correlation energy 57 and entropy, 79 and the identification of individual potential energy hypersurfaces as probable domains of its applicability. 81,82However, this second incarnation, called "the modified Collins conjecture" in this Letter, does not withstand theoretical scrutiny, invoking a simple helium/beryllium Gedankenexperiment.In addition, its tentative domain of applicability is ruled out by arguments based upon the first-order response theory.The observation that neither of these incarnations is of any practical importance has given rise to the idea of deploying the modified Collins conjecture in conjunction with a variational formalism.However, the resulting third incarnation, which inherits the flaws with its predecessors, cannot be regarded as a variant of the 1-RDMFT formalism.Attempts at amelioration of these difficulties by replacing the nonfreeness with another correlation strength measure compatible with the universal properties of the natural orbitals produce functionals that again fail the Gedankenexperiment.These failures clearly demonstrate that the numerical rank-1 functionals are merely conduits for incorporation (in a crude manner) of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy. 94 note on computational details: the data displayed in Figures 1−4 have been derived from electronic wave functions described elsewhere. 24

Figure 1 .
Figure 1.An example of the large-n behavior of ν n : n 8/3 ν n vs n for the 1-matrices of the ground states of the H 3 + molecular cation at the equilateral geometries with the internuclear distances (a) 1.65 [au] and (b) 6.0 [au].The blue, magenta, red, cyan, green, and orange dots denote the data pertaining to the NOs with, respectively, the A 1 ′, A 2 ′, E′, A 1 ″, A 2 ″, and E″ symmetries; identical NOs with two different spin components are counted only once.

Figure 2 .
Figure 2.An example of the large-n behavior of t n : n −2/3 t n vs n for the 1-matrices of the ground states of the H 3 + molecular cation at the equilateral geometries with the internuclear distances (a) 1.65 [au] and (b) 6.0 [au].See Figure 1 for the color coding.
functional D[ 1 Γ] should conform to the large-p scaling of

Figure 3 .
Figure 3.An example of the incompatibility of the 1-matrix with the condition 23: f pp vs ln 1 p

Figure 4 .
Figure 4.An example of the improved compatibility of the 1-matrix with the condition 24: f pp vs [ ] (1 2 ) (1 ) p p p

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69e only possible dependence of the approximate U[ 1 Γ] on its argument.Indeed, the function G(N) = c (N − 1), where −0.045 < c < −0.030, has been found68to crudely estimate the correlation energy, calculated according to the definition69