Towards a Formal Definition of Static and Dynamic Electronic Correlations Cite This: Phys

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Towards a formal definition of static and dynamic electronic correlations Cite this: Phys. Chem. Chem. Phys., 2017, 19,12655 Carlos L. Benavides-Riveros, *a Nektarios N. Lathiotakisb and Miguel A. L. Marquesa

Some of the most spectacular failures of density-functional and Hartree–Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the Received 20th February 2017, N-representability problem of the one-body density matrix for pure states, we propose a method to Accepted 6th April 2017 quantify the static contribution to the electronic correlation. By studying several molecular systems we DOI: 10.1039/c7cp01137g show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural rsc.li/pccp spin–orbital in such quantification.

1 Introduction or an antisymmetrized geminal power) within quantum Monte Carlo methods.4,5 The so-called Jastrow antisymmetric geminal The concept of correlation and more precisely the idea of ansatz accounts for inter-pair interactions and multiple resonance correlation energy are central to quantum chemistry. Indeed, structures, maintaining a polynomial scaling cost, comparable to the electron-correlation problem (or how the dynamics of each that of the simpler Jastrow single determinant approach. Highly electron is aﬀected by the others) is perhaps the single largest correlated systems, such as diradical molecules (the orthogonally 1 source of error in quantum-chemical computations. The success twisted ethylene C2H4 and the methylene CH2, for example) and

of Hartree–Fock theory in providing a workable upper bound for bond stretching in H2O, C2 and N2, are well described using such a the ground-state energy is largely due to the fact that a single method.6,7 Slater determinant is usually the simplest wave function having In recent years considerable efforts have been made to the correct symmetry properties for a system of fermions. Since characterize the correlation of a quantum system in terms of more the description of interacting fermionic systems requires multi- meaningful quantities, such as the Slater rank for two-electron Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. determinantal reference wave functions, the correlation energy is systems,8,9 the entanglement classification for the three-fermion commonly defined as the difference between the exact ground- case,10 the squared Frobenius norm of the cumulant part of the state and the Hartree–Fock energy.2,3 Beyond Hartree–Fock two-particle reduced density matrix11 or the comparison with uncor- theory, numerous other methods (such as configuration inter- related states.12 Along with Christian Schilling, we have recently action or coupled-cluster theory) aim at reconstructing the part stressed the importance of the energy gap in the understanding of of the energy missing from a description based on a single- the electronic correlation.13 Notwithstanding, these measures do not determinantal wave function. Indeed, one common indicator of draw a distinction between qualitatively different kinds of electronic the accuracy of a model is, by and large, the percentage of the correlations. In quantum chemistry, for instance, it is customary to correlation energy it is able to recover. distinguish between static (or nondynamic) and dynamic correla- For small molecules, variational methods based on configuration tions. The former corresponds to configurations which are nearly interaction techniques well describe electronic correlations. degenerate with respect to the reference Slater determinant (if any), However, due to their extreme computational cost, configuration- whilst the latter arises from the need for mixing the Hartree–Fock interaction wavefunctions are noteworthily difficult to evaluate for state with higher-order excited states.14,15 Heuristically, one usually larger systems. Among other procedures at hand, the correlation states that in systems with (strong) static correlation the wave- can be treated efficiently by applying a Jastrow correlation term to function differs qualitatively from the reference Slater determinant, an antisymmetrized wave function (e.g. a single Slater determinant while strong dynamic correlation implies a wavefunction including a large number of excited determinants, all with comparable, small occupations. Some of the most spectacular failures of the a Institut fu¨r Physik, Martin-Luther-Universita¨t Halle-Wittenberg, 06120 Halle (Saale), Germany. E-mail: [email protected] Hartree–Fock theory and density functional theory (with standard b Theoretical and Physical Chemistry Institute, National Hellenic Research exchange–correlation functionals) are related to an incorrect 16 Foundation, GR-11635 Athens, Greece description of static correlation.

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It is commonly believed that, to a large extent, both static 2 The generalization of the Pauli and dynamic contributions should be included in the global principle computation of the electronic correlation. Yet there are few systems for which one can distinguish unambiguously between In a groundbreaking work, aimed at solving the quantum marginal these two types of correlations. For instance, the ground state of problem for pure states, Alexander Klyachko generalized the Pauli helium has no excited electronic states nearby, leading there- exclusion principle and provided a set of constraints on the natural fore to the absence of static correlation. In the dissociation occupation numbers, stronger than the Pauli principle.23 Although 17 limit of H2 a state with fractional occupations arises and the rudimentary schemes to construct such constraints are, to some correlation is purely static. According to Hollett and Gill,18 extent, common in the quantum-chemistry literature,27 it was not static correlation comes in two ‘‘flavors’’: one that can be only with the work of Klyachko that this rich structure could be captured by breaking the spin symmetry of the Hartree–Fock decrypted. The main goal of this section is to review the physical

wave function (like in stretched H2) and another that cannot. consequences of such a generalization. N The measures of correlation proposed so far purport to include Given an N-fermion state |Ci A 4 [H1], with H1 being the both static and dynamic correlations, although in an uncontrolled one-particle Hilbert space, the natural occupation numbers are 19 manner. the eigenvalues {ni} and the natural spin–orbitals are the

For pure quantum states, global structural features of the eigenvectors |jii of the one-body reduced density matrix, wave function can be abstracted from local information alone. X r^1 NTrN1½jCihCj ¼ nijiji hjji : (1) Multiparticle entanglement, for instance, can be completely i classified with a more accesible one-particle picture.20 Such a characterization is addressed by a finite set of linear The natural occupation numbers, arranged in decreasing order inequalities satisfied by the eigenvalues of the single-particle ni Z ni+1, fulfill the Pauli condition n1 r 1. The natural spin- 21 states. Furthermore, by using the two-particle density matrix orbitals define an orthonormal basis B1 for H1 and can also be and its deviation from idempotency, it is possible to propose used to generate an orthonormal basis BN for the N-fermion Hilbert N j j a criterion to distinguish static from dynamic correlation, space HN 4 [H1], given by the Slater determinants | i1 iNi = j j which for two-fermion systems only requires the occupancies | i1i44| iNi. For practical purposes, the dimension of the one- 22 of the natural orbitals. Needless to say, grasping global particle Hilbert space H1 is usually finite. Henceforth, HN,d denotes information of a many-body quantum system by tackling an antisymmetric N-particle Hilbert space with an underlying only one-particle information is quite remarkable, mainly d-dimensional one-particle Hilbertspace.Inprinciple,thetotal because in this way a linear number of degrees of freedom d dimension of H is , but symmetries usually lower it. is required. N,d N Recent progress in the N-representability problem of the It is by now known that the antisymmetry of N-fermion pure one-body reduced density matrix for pure states provides an quantum states not only implies the well-known Pauli exclusion extension of the well-known Pauli exclusion principle.23 This principle, which restricts the occupation numbers according to

extension is important because it provides stringent constraints 0 r ni r 1, but also entails a set of so-called generalized Pauli beyond those from the Pauli principle, which can be used, constraints.23,28–30 These take the form of independent linear Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. among others, to improve reduced-density-matrix functional inequalities theories.24–26 Our main aim in this paper is to employ the Xd generalized Pauli exclusion principle to establish a general 0 i Djð~nÞkj þ kjni 0: (2) criterion to distinguish static and dynamic contributions to i¼1 the electronic correlation in fermionic systems. Here the coefficients ki A Z and j =1,2,...,n o N.Accordingly, The paper is organized as follows. For completeness, Section j N,d for pure states the spectrum of a physical fermionic one-body 2 summarizes the key aspects of the so-called generalized Pauli reduced density matrix must satisfy a set of independent linear exclusion principle and its potential relevance for quantum inequalities of the type (2). The total number of independent chemistry. In Section 3, we discuss a Shull–Lo¨wdin-type functional inequalities n depends on the number of fermions and for three-fermion systems, which can be constructed by using the N,d the dimension of the underlying one-particle Hilbert space. pertinent generalized Pauli constraints along with the spin For instance,29 n =4,n =4,n = 31, n = 52, n = 93, symmetries. Since this functional depends only on the occupation 3,6 3,7 3,8 3,9 3,10 n = 14, n = 60, n = 125 and n = 161. numbers, it is possible to distinguish the correlation degree of 4,8 4,9 4,10 5,10 From a geometrical viewpoint, for each fixed pair N and d, the so-called Borland–Dennis setting (with an underlining six- the family of generalized Pauli constrains, together with the dimensional one-particle Hilbert space) by using one-particle normalization and the ordering condition, forms a ‘‘Paulitope’’:† information alone. In Section 4 we discuss a formal way to - a polytope P of the allowed vectors n (n )d . The physical distinguish static from dynamic correlation. Section 5 is devoted N,d i i=1 relevance of this generalized Pauli exclusion principle has been to investigate the static and dynamic electronic correlation in

molecular systems. We compare our results with the well-known † Norbert Mauser coined the term ‘‘Paulitope’’ during the Workshop Generalized von-Neumann entanglement entropy. The paper ends with a Pauli Constraints and Fermion Correlation, celebrated at the Wolfgang Pauli conclusion and an appendix. Institute in Vienna in August 2016.

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24,31–41 already stressed, among others, in quantum chemistry, in n1 + n6 = n2 + n5 = n3 + n4 = 1 (5) open quantum systems42 or in condensed matter.43 and the inequality: The generalized Pauli exclusion principle is particularly

relevant whenever the natural occupation numbers of a given n1 + n2 + n4 r 2. (6) system saturate some of the generalized Pauli constraints.44 This This latter inequality together with the decreasing ordering rule so-called ‘‘pinning’’ eﬀect can potentially simplify the complexity defines a polytope in R6, called here the Borland–Dennis ofthewavefunction.37 In fact, whenever a constraint of the sort - Paulitope. Conditions (5) imply that, in the natural orbital basis, (2) is saturated or pinned (namely, Dj(n)=0),anycompatible - every Slater determinant, built up from three natural spin– N-fermion state |Ci (with occupation numbers n) belongs to the orbitals, showing up in the configuration expansion (4), satisfies null eigenspace of the operator

0 1 d |jijjjki =(nˆ7s + nˆs)|jijjjki, (7) Dˆj = kj + kj nˆ1 +...+ kj nˆd, (3) for s A {1, 2, 3}. Therefore, each natural spin–orbital belongs to where nˆ denotes the number operator of the natural orbital |j i i i one of the three different sets, say j A {j , j }, j A {j , j } of |Ci. This result not only connects the N- and 1-particle i 1 6 j 2 5 and j A {j , j }. Consequently, the dimension of the total descriptions, which is in itself striking, but provides an important k 3 4 Hilbert space is eight. selection rule for the determinants that can appear in the configu- In the symmetry-adapted description of this system, the spin ration interaction expansion of the wave function. Indeed, for a given - of three natural orbitals points down, and the spin of the other wave function |Ci, whenever Dj(n) = 0, the Slater determinants for three points up. The corresponding one-body reduced density which the relation Dˆj|ji ji i =0doesnotholdarenotpermitted 1 N matrix (a 6 6 matrix) is a block-diagonal matrix that can be in the configuration expansion of |Ci.Inthisway,pinnedwave written as the direct sum of two (3 3) matrices (say, r^m and functions undergo an extraordinary structural simplification which r^k), one related to the spin up and the other one related to the suggests a natural extension of the Hartree–Fock ansatz of the form: X spin down. For the doublet configuration, each acceptable C c j ...j : j i¼ i1;...;iN i1 iN (4) Slater determinant contains two spin orbitals pointing up (for fg2i ;...;i I 1 N Dj instance) and one pointing down. It follows that

Here, stands for the family of configurations that may contribute r^m r^k IDj Tr = 2 and Tr = 1. (8) to the wave function in case of pinning to a given generalized Pauli To meet the decreasing ordering of the natural occupations as constraint D.37 Theseremarkableglobalimplicationsofextremal j well as the representability conditions (5), two of the first three local information are stable, i.e. they hold approximately for spectra occupation numbers must belong to the matrix whose trace is close to the boundary of the allowed region.45 equal to two.35 It is straightforward to see that the only These structural simplifications can be used as a variational admisible set of occupation numbers are the ones lying in ansatz, whose computational cost is cheaper than configuration the hyperplane A : interaction or other post-Hartree–Fock variational methods.30,37,45 1

For a given Hamiltonian Hˆ, the expectation value of the energy n1 + n2 + n4 = 2 (9) hC|Hˆ|Ci is minimized with respect to all states |Ci of the form (4),

Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. (equivalently, n + n + n = 1), which saturates the generalized i.e., with natural occupation numbers saturating some specific 3 5 6 Pauli constraint (6), or in the hyperplane A : generalized Pauli constraint. For the lithium atom, a wave function 2

with three Slater determinants chosen in this way accounts for n1 + n2 + n3 = 2 (10) more than 87% of the total correlation energy.37 For harmonium (a (equivalently, n + n + n = 1). Note that the two hyperplanes system of fermions interacting with an external harmonic potential 4 5 6 intersect on the line n = n = 1. In Fig. 1 hyperplanes A and and repelling each other by a Hooke-type force), this method 3 4 2 1 A are shown within the Pauli hypercube n 1. accounts for more than 98% of the correlation energy for 3, 4 2 1 r As stated above, pinning of natural occupation numbers and 5 fermions.46 undergoes a remarkable structural simplification of the wave In a nutshell, the main aim of the strategy is to select the most - functions compatible with n. For the case of the Borland– important configurations popping up in an efficient configuration Dennis setting, eqn (9) implies for the corresponding wave interaction computation. We expect that these are the first function the condition (nˆ + nˆ + nˆ )|Ci =2|Ci, while the configurations to appear in approaches whose attempt is also 1 2 4 constraint (10) implies (nˆ + nˆ + nˆ )|Ci =2|Ci. Consequently, a to choose (deterministically47 or stochastically48,49) the most 1 2 3 wave function (the so-called Borland–Dennis state) compatible important Slater determinants. 35 with the hyperplane A1 can be written in the form pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 3 The Borland–Dennis setting jiCBD ¼ n3jij1j2j3 þ n5jij1j4j5 þ n6jij2j4j6 ; (11) 3.1 A Lo¨wdin–Shull functional for three-fermion systems 1 where n Z n + n and n . A wave function compatible with 3 5 6 3 2 The famous Borland–Dennis setting H3,6, the rank-six approxi- the hyperplane A2 reads mation for the three-electron system, is completely characterized pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 50 by 4 constrains: the equalities ji¼C2 n4jiþj1j2j4 n5jiþj1j3j5 n6jij2j3j6 ; (12)

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The ‘‘Hartree–Fock’’ point (n1, n2, n3) = (1, 1, 1), which

corresponds to the single Slater determinant |j1j2j3i. Note that it does not coincide in general with the Hartree–Fock state, since it is described in the natural-orbital basis set. However, - we call it so because its spectrum is nHF = (1, 1, 1, 0, 0, 0). 2 2 2 The point P ; ; , which corresponds to the a 3 3 3 strongly (static) correlated state: 1 jiCa ¼ pffiffiffiðÞjij j j þ jij j j þ jij j j : (13) 3 1 2 4 1 3 5 2 3 6 1 1 The point P 1; ; . These occupation numbers corre- b 2 2 spond to the state 1 ji¼Cb pffiffiffiðÞjiþj j j jij j j : (14) 2 1 2 3 1 4 5

In quantum information theory, this state is said to be bisepar- Fig. 1 The hyperplanes 2 = n + n + n and 2 = n + n + n , subjected to 1 2 4 1 2 3 able because one of the particles is disentangled from the other the conditions 1 Z n1 Z n2 Z n3 Z 0.5 and 2 Z n1 + n2 + n4. The blue dot ones.55 1 1 3 3 1 3 3 1 is the Hartree–Fock point (1, 1, 1). The red ones are 1; ; , ; ; and The point P ; ; , which corresponds to the (static) 2 2 4 4 2 c 4 4 2 2 2 2 ; ; . correlated state: 3 3 3 1 1 ji¼Cc pffiffiffijiþj j j ðÞjiþj j j jij j j : (15) 2 1 2 3 2 1 4 5 2 4 6 1 where n4 r n5 + n6 and n4 2. Note that, just like in the famous Lo¨wdin–Shull functional for two-fermion systems,51 the wave Points Pb and Pc lie in the intersection of A1 and A2, namely, 1 function is explicitly written in terms of both the natural the degeneracy line n ¼ n ¼ . Since n and n are identical, 3 4 2 3 4 occupation numbers and the natural orbitals. Likewise, any the choice of the highest occupied natural orbital |j3i and the sign dilemma that may occur when writing the amplitudes of lowest unoccupied natural orbital |j4i is not unique anymore the states (11) and (12) can be dodged by absorbing the phase into and the indices 3 and 4 can be swapped in (14) and (15) without the spin–orbitals. Moreover, only doubly excited configurations are changing the spectra. permitted here. For |C i such double excitations are referred to BD These four points are important because they belong to two the Slater determinant whose one-particle density matrix is the different correlation regimes. On the one hand, the states |Cai best idempotent approximation to the true one-particle density and |Cbi exhibit static correlation, as they are equiponderant matrix. The state |C i is orthogonal to the state |j j j i. Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. 2 1 2 3 superpositions of Slater determinants. On the other, the state Interestingly, a non-vanishing overlap of a wave function with |Cci is the superposition of two states (|j1j4j5i and |j2j4j6i) this latter state can only be guaranteed if the sum of the first and the nearly degenerate |j1j4j5i and its correlation is also three natural occupation numbers is greater than two.52 Both static. This is reminiscent of the zero-order description of the |C i and |C i lead to diagonal one-particle reduced density BD 2 beryllium ground state, for which the 2s and 2p orbitals are matrices. nearly degenerate and the state is a equiponderant superposition of For any given Slater determinant, the seniority number is three Slater determinants plus a highly weighted reference state.1 defined as the number of orbitals which are singly occupied. P In Fig. 2 the entanglement entropy S ¼ ni ln ni is plotted Such an important concept is used in nuclear and condensed i as a function of n and n for the hyperplanes A and A . The matter physics to partition the Hilbert space and construct 2 3 1 2 entanglement entropy of the state |j j j i is zero since it is compact configuration-interaction wave functions.53,54 Since 1 2 3 uncorrelated. The entropies of the states |C i and |C i are the wave functions (11) and (12) are eigenfunctions of the spin b c 1.3862 and 1.8178, respectively. As one might expect, the operators, each Slater determinant showing up in these expansions configurations present in A all are strongly correlated. For is also an eigenfunction of such operators. The latter is only possible 2 the highest correlated state |C i, S = 1.9095. The particular if one orbital is doubly occupied and therefore the seniority number a structure of the states |Cai,|Cbi and |Cci prompt us to say that of each Slater determinant is 1. The seniority number of |CBDi the correlation effects of the states lying in the hyperplane A2 and |C2i is also 1. are all due to static effects, while the states in A1 are due to both static and dynamic effects. 3.2 Correlations According to the particle-hole symmetry, when applied to In Fig. 1 we illustrate the discussion of the previous section, a three-electron system, the nonzero eigenvalues and their highlighting four special configurations, namely multiplicities are the same for the one- and the two-body reduced

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active active where |C i A HNr,dr. The first r natural occupation numbers are saturated to 1. The remaining d r occupation

numbers (nr+1,...,nd) satisfy a set of generalized Pauli constraints active and lie therefore inside the polytope PNr,dr.ThespaceHNr,dr is called here the ‘‘active Hilbert space’’. For instance, for the

‘‘Hartree–Fock’’ space HN,N, the corresponding zero dimensional active active Hilbert space is H0,0 . It is possible to characterize a hierarchy of active spaces active by the eﬀective dimension of HNr,dr and the number of Slater determinants appearing in the configuration interaction expansion of |Cactivei.40,45 For the ‘‘active’’ Borland–Dennis active setting H3,6 we can apply the same considerations discussed in the last subsections: if the corresponding constraint (6) is

saturated, the wave function fulfills (nˆr+1 + nˆr+2 + nˆr+4)|Ci = 2|Ci, and the set of possible Slater determinants reduces to just three, taking thus the form: ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi active p p C ¼ nrþ3 jrþ1jrþ2jrþ3 þ nrþ5 jrþ1jrþ4jrþ5 pffiffiffiffiffiffiffiffiffi þ nrþ6 jrþ2jrþ4jrþ6 ; (18)

1 provided that nrþ3 and nr+3 Z nr+5 + nr+6. Fig. 2 Entanglement entropy of the hyperplanes A1 and A2. 2

matrices. Thus, the results in this section based on one-particle 4 Correlations and correlation information alone are also valid at thelevelofthe2-particlepicture. measures

3.3 Borland–Dennis for three active electrons Even if the peculiar role played by electronic correlations in In the configuration interaction picture, the full wavefunction quantum mechanics was noticed from the onset, the problem is to be expressed in a given one-electron basis as a linear of how to measure quantum correlations is still a subject of 58,59 combination of all possible Slater determinants save symme- intense research. The degree of entanglement D(C)ofan tries. On the basis of natural orbitals, it reads arbitrary vector |Ci can be expressed by its projection onto the X nearest normalized unentangled (or uncorrelated) pure state:60,61 jCi¼ ci i jji ...ji i (16) 1 N 1 N 2 1i1 o o iN d DðCÞ¼1 max jhCjFij ; (19) F Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. in a similar fashion to the Hartree–Fock ansatz (4). It is well where the maximum is over all unentangled states, normalized so known that the expansion (16) contains a very large number of that hF|Fi = 1. Although this measure sounds conventional, it configurations that are superfluous or negligible for computing has the merit of being zero whenever |Ci is uncorrelated. molecular electronic properties. In practice, the configurations ~ 2 This measure (and the minimum minF~A JC FJ , where considered effective are sparse if an arbitrary threshold for the F F denotes the set of unnormalized unentangled (or uncorre- value of the amplitudes in (16) is enforced.56 As such, one often lated) pure states60) is also important in the realm of quantum introduces the notion of active space to select the most relevant chemistry. In the Appendix we state and prove that the set of configurations at the level of the one-particle picture. A complete pure quantum systems with predetermined energy is connected: activespaceclassifiestheone-particleHilbertspaceincore(fully given a Hamiltonian Hˆ there are two wavefunctions |c i and |c i occupied), active (partially occupied) and virtual (empty) spin– 1 2 with energies E and E whose distance Jc c J2 is bounded by orbitals. The core spin–orbitals are pinned (completely populated) 1 2 1 2 afunctionof|E E | (see Theorem 2 in the Appendix). Recently, and are not treated as correlated. Adding active-space constraints 1 2 we have shown that when |Ci is the ground state of a given improves the estimate of the ground-state energy in the framework Hamiltonian the measure (19) is closely related to the concept of of reduced-density-matrix theory.57 correlation energy as understood in quantum chemistry.13 Akey The generalized Pauli principle can shed some light on this ingredient in such connection turns out to be the energy gap important concept.40,45 In fact, for the case of the r core (and within the symmetry-adapted Hilbert subspace. consequently d r active orbitals) the Hilbert space HN,d is core active isomorphic to the wedge product Hr,r 4 HNr,dr. Hence, a 4.1 Dynamic correlation wave function |Ci A H can be written in the following way: N,d The N-particle description of a quantum system and its reduced active |Ci =|j1jri 4 |C i, (17) one-fermion picture can be related in meaningful ways. In eﬀect,

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D(C) can be bounded from above and from below by the After this long discussion, it is natural to define all the points 1 l -distance of the natural occupation numbers. In fact, the lying on the hyperplane A2 as statically correlated. The hyper-

distance between a wave function |Ci and any Slater determinant plane A1 contains the uncorrelated Hartree–Fock state and the j ...j 30 correlation of the rest of the states present is due to static as well | i1 iNi satisfies as dynamic effects. dið~nÞ 2 dið~nÞ - 1 j ...j jC ; (20) The ‘‘static’’ states |C (Z)i that lead to the occupancies n (Z) 2minðN; d NÞ i1 iN 2 s s as defined in eqn (22) read rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi where d is the dimension of the underlyingP one-particleP Hilbert 1ffiffiffi 1 1 space, as defined in Section 2, and dið~nÞ ð1 niÞþ ni is jiCsðZÞ p jij1j2j3 þ þ Zjij1j4j5 þ Zjij2j4j6 ; i2i i2=i 2 4 4 - the l1-distance between n (the natural occupation numbers of (24) |Ci) and the natural occupation numbers of the Slater determinant 1 in display (here i {i1,...,iN}). This result is also valid for Hartree– where 0 Z . As stated before, since n and n are identical, 4 3 4 Fock or Brueckner orbitals.13,62 In particular, the l1-distance to the the choice of the highest occupied natural orbital and the Hartree–Fock point is given by X X lowest unoccupied natural orbital is not unique and the indices dHF ð~nÞ¼ ð1 niÞþ ni: (21) 3 and 4 can be swapped without changing the spectra. However, iN i 4 N by doing so the resulting state is orthogonal to |j1j2j3i. 2 - The L -distance (eqn (19)) between the Borland–Dennis state In Fig. 3 we plot dHF(n)/2 for the points in the hyperplane A1.As 2 - (11) and |C (Z)i is given by J (Z) 1 |hC |C (Z)i| . expected, d (n ) = 0. More interestingly, the correlation increases s s BD s HF HF The minimum of this distance depends only on the value of monotonically with n3. All the points on the degeneracy line 1 the natural occupation number corresponding to the highest n3 = n4 are at the same l -distance from the Hartree–Fock point. - occupied natural orbital. To see this note that the minimum of In effect, dHF(ns(Z))/2 = 1, where Js(Z) is attained when dJs(Z)/dZ = 0, which occurs at 3 3 1 1 1 1 n5 n6 ~nsðZÞ þ Z; Z; ; ; þ Z; Z ; (22) Z ¼ : The distance is therefore 4 4 2 2 4 4 41ðÞ n3 ! pffiffiffiffiffi 2 1 n with 0 Z is the set of points lying on the intersection line ffiffiffi3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 n2 pnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 n3 JsðÞ¼Z 1 p þ þ 4 2 21ðÞ n 21ðÞ n 1 3 3 (25) n3 ¼ n4 ¼ . Moreover, dHF(A2)/2 = 1, for all the points lying on 2 1 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 n þ 1 n ¼ n ðÞ1 n ; the hyperplane A2. 2 3 3 2 3 3 4.2 Static correlation which is zero when the correlation of the Borland–Dennis state 1 Roughly speaking, the idea of static correlation is associated is completely static and is 2 when the state is the uncorrelated with the presence of a wave function built up from an equiponderant Hartree–Fock state. 1 superposition of more than one Slater determinant, namely, We can also investigate the l -distance between any state

Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. - - n A and the state n (Z) (22), namely 1 A1 s pffiffiffiffiðÞjF1iþþjFmi ; (23) m dsð~nÞ¼min dist1ðÞ~n;~nsðZÞ : (26) Z For the Borland–Dennis setting H , the hyperplane A contains 3,6 2 Note that the l1-distance reads states with three configurations being almost equiponderant. 3 3 1 dist1ðÞ¼~n;~nsðZÞ 2 n1 þ Z þ n2 Z þ n3 : 4 4 2

- - So, to minimize dist1(n, ns,(Z)) is the same as minimizing 3 3 1 n1 Z þ n2 þ Z. Since the l -sphere with radius z 4 4 - centered at n is the convex hull of the vertices ((n1 z, n2, n3), (n1, n2 z, n3), (n1, n2, n3 z)), the minimum of the distance is 3 1 1 dsð~nÞ¼2 n1 n2 þ þ n3 ¼ 4 n3 ; (27) 2 2 2

which depends on n3 alone. Note that for the Hartree–Fock - - point ds(nHF)/2 = 1 and remember that by our definition ds(n)= - 1 A Fig. 3 l -Distance of the hyperplane A1 with respect to the Hartree–Fock 0ifn A2. Remarkably, Z*, the minimizer of Js(Z), is also a - - point ~nHF. minimizer of dist1(n, ns(Z)). This latter statement can be proved

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PCCP Paper

- by noting that the first occupation number of ns(Z*) satisfies 5 Correlation in molecular systems

To illustrate these concepts we plot in Fig. 4 our measures of 3 n1 n2 2 4n3 þ 2n1 n1 n1ðÞ¼Z þ ¼ n1; static and dynamic correlation for the ground states of the 4 41ðÞ n3 41ðÞ n3 21ðÞ n3 diatomic molecules H2 and Li2 as a function of the interatomic 1 distance. In the same plot we can also see the von-Neumann since n3 . Equivalently, n (Z*) r n . Therefore, 2 2 2 entropy and the value of the highest occupancy of the highest occupied natural spin–orbital. These numbers were obtained 3 3 1 from CAS-SCF calculation using the code Gamess64 and cc-pVTZ dist ðÞ¼~n;~n ðÞZ 2 n Z þ n þ Z þ n 1 s 1 4 2 4 3 2 basis sets with all electrons active and as large as possible active space of orbitals. 3 1 1 ¼ 2 n1 þ n2 þ n3 ¼ 4 n3 ; The molecule H2, and in particular its dissociation limit, is 2 2 2 65 the quintessential example of static correlation. It is well known that the restricted Hartree–Fock approach describes which is the minimum (27). very well the equilibrium chemical bond, but fails dramatically as the molecule is stretched. Around the equilibrium separation, 4.3 Correlation measures Psta is close to zero and Pdyn reaches its maximum. There is a The comparison of the distances (21) and (26) allows us to change of regime of around 1.5 Å because the static correlation distinguish between dynamic and static correlations. The distance begins to grow rapidly. Beyond this point, the restricted Hartree– - to the Hartree–Fock point dHF(n) can be viewed as a measure of the Fock theory is unable to predict a bound system anymore. At the dynamic part of the electronic correlation, as it quantifies how dissociation limit, the correlation is due to static effects only, as much a wave function differs from the uncorrelated Hartree–Fock expected. Both measures allow us to observe the smooth increase - state. The static distance ds(n) can be viewed as a measure of of static effects when the molecule is elongated. A different the static part of the correlation, as it quantifies how much a situation can be observed for the diatomic Li2. For lengths smaller wave function differs from the set of static states. One expects - E for helium dHF(nHe) 0 while for H2 at infinite separation - d s(nH2,N) = 0. For convenience we renormalize these two l1-distances by means of

dHFð~nÞ dsð~nÞ Pstað~nÞ¼ and Pdynð~nÞ¼ : (28) dHFð~nÞþdsð~nÞ dHFð~nÞþdsð~nÞ

- - Since the measures Pdyn(n) and Psta(n) are normalized (while - - dHF(n) and ds(n) are not), they are much more useful to compare different systems. In this way, when the correlation is due to - - Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. static effects Psta(n) = 1 and Pdyn(n) = 0, while the contrary occurs for a completely dynamic state. For instance, one expects for He - E - Psta(nHe) 0 while for H2 at infinite separation Pdyn(nH2,N)=0. These quantities have the merit of being zero or one when the correlation is completely dynamic or completely static and hence they separate the correlation in two contributions. It is worth saying that our considerations do not only apply for the Borland–Dennis setting but also for larger ones. Recall 0 that for the settings HN,d and HN,d0, such that d o d A N, the corresponding polytopes satisfy: PN;d ¼ PN;d0 j . ndþ1¼¼nd0 ¼0 It means that, intersected with the hyperplane given by 30 nd+1 = = nd0 = 0, the polytope PN,d0 coincides with PN,d. Therefore, we have completely characterized the static states up to six dimensional one-particle Hilbert spaces. By choosing

n1 = 1, we freeze one electron, we are effectively dealing with a two active-electron system and our measures can also be used for characterizing the correlation of two-electron systems. Note that for this latter case the natural occupation numbers Fig. 4 Correlation curves for H2 and Li2. Psta and Pdyn as well as the are evenly degenerated, a very well known representability occupancy of the highest occupied natural spin–orbital and the von condition for systems with an even number of electrons and Neumann entropy S are plotted as functions of the interatomic distance 63 time-reversal symmetry. (in Å). The equilibrium bonding length is 0.74 Å for H2 and 2.67 Å for Li2.

Paper PCCP

than the bond length the static correlation decreases as the with full control over the quantum state,66 this model may be

distance increases. The energy, the static correlation and the considered as a simplified tight-binding description of the Hr

von-Neumann entropy reach their minimum around 2.9 Å, very molecule. For the case of H3 the correlation measures are close to the equilibrium bonding length, while the dynamic plotted as a function of the interatomic distance (in Å) and for correlation as well as the occupancy of the highest occupied the Hubbard model as a function of the coupling U/t. In both natural spin–orbital acquired their maximum. Beyond that cases, as the molecule is elongated or the interaction in the value, the static correlation grows and the dynamic correlation Hubbard model is enhanced, the energy gap (the energy diﬀerence decreases slowly. This behaviour changes around 4 Å, where the between the first-excited and the ground states) shortens and the static correlation speeds up. electronic correlation increases, leading to the appearance of static 13 In Fig. 5 we plot the correlation measures for three-electron eﬀects. While H3 exhibits a behaviour essentially similar to H2,

systems: the ground state of the equilateral H3 and the three- the Hubbard model shows two diﬀerent regimes of correlation. site three-fermion Hubbard model, which is very well known as it is For positive values of the relative coupling, the static correlation analytically solvable.13,43 The Hamiltonian (in second quantization) plays a prominent role. In particular, beyond U/t = 3.2147, the

of the one-dimensional r-site Hubbard model reads system lies in the hyperplane A2 of the Borland–Dennis setting X X (10) and the correlation is completely static. For this strongly t y H^ ¼ c c þ h:c: þ 2U n^i n^i ; (29) correlated regime, the ground state can be written as a equi- 2 is ðiþ1Þs " # i;s i ponderant superposition of three Slater determinants. For negative values of the coupling there is always a fraction of i A {1, 2,...,r}, where c† and c are the fermionic creation and is is the correlation due to dynamic effects. In that limit the ground annihilation operators for a particle on the site i with spin s A {m,k} state is written as a superposition of two Slater determinants with and nˆ = c† c . The first term in eqn (29) describes the hopping is is is different amplitudes. between two neighboring sites while the second represents the It is known that static and dynamic electronic correlations on-site interaction. Periodic boundary conditions for the case play a prominent role in the orthogonally twisted ethylene.6 In r = 3 are also assumed. Achieved experimentally very recently fact, the energy gap between the ground state and the first excited state shortens when the torsion angle around the CQC double bond is increased. While the ground state of the planar ethylene is very well described by a single Slater determinant, at ninety degrees at least two Slater determinants are needed, resembling the dihydrogen in the dissociation limit. In Fig. 6 we plot the correlation energy and the energy gap of ethylene as a function of the torsion angle, using the CAS-SCF(12,12) method and a cc-pVDZ basis set. In Fig. 7 we plot the correlation measures for ethylene along the torsional path. For the planar geometry the correlation is almost completely dynamic and the situation remains in this way until the torsional degree reaches Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. 601. From this angle the static correlation shows up. At 801 the static and dynamic correlation are equally important in the total electron correlation of ethylene. When orthogonally twisted, the

Fig. 5 Correlation curves for H3 and the three-fermion three-site Hubbard

model. Psta and Pdyn as well as the occupancy of the highest occupied natural

spin–orbital and the von Neumann entropy S are plotted as functions of the Fig. 6 Correlation energy and energy gap for the twisted ethylene C2H4 interatomic distance (in Å) or the coupling strength. as a function of the torsion angle around the CQC double bond.

PCCP Paper Appendix Lemma 1 Let Hˆ be a Hamiltonian on the Hilbert space H with a unique

ground state |f0i with energy e0 and an energy gap egap = e1 e0,

where e1 is the energy of the first excited state. Then, for any |ci A 13 2 H with energy Ec = hc|Hˆ|ci we have |hf0|ci| Z (e1 Ec)/egap.

Theorem 2 Let Hˆ be a Hamiltonian on H with a unique ground-state.

Let SE be the set of pure states with expected energy E: SE = c A c ˆ c e { H|h |H| i = E}. If |E1 E2| o , then SE2 has an element

close to an element of SE1. Proof. Let us write the spectral decomposition of the Hamil- Fig. 7 Correlation curves for ethylene. Psta and Pdyn as well as the P ^ occupancy of the highest occupied natural spin–orbital and the von tonian in the following way: H ¼ eijifi hjfi , with e0 o e1 r e2 Neumann entropy S are plotted as functions of the torsion angle around i the CQC double bond. .... A wavefunction |c i A S can be written in the eigenbasis r P 1 E1 of Hˆ as jic1 ¼ aijifi : If E2 = E1 there is nothing to prove. i correlation of ethylene is 90% due to static effects and there is Without the loss of generality, let us assume that E2 o E1, take still an important part due to dynamic correlation. Remarkably, 0 r a0 r 1 and choose a state |c2i in SE as a superposition of the rise of static correlation around 601 coincides with the 2 |c1i and the ground state |f0i, namely: |c2i = a|c1i + b|f0i with increase of correlation energy. From this perspective, the gain a and b positive real numbers smaller than 1. Normalization of total correlation is mainly due to static effects. 2 2 dictates that a + b +2aba0 = 1 and the energy constraint reads 2 2 a E1 + b e0 +2aba0e0p= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2. It is easy to see that both conditions 2 2 6 Summary and conclusion translate into: a ¼ ðÞE2 e0 =ðÞE1 e0 and b =[a a0 + 2 1/2 (1 a )] aa0. Using the fact that 0 r a, b, a0 r 1 and Thanks to the generalization of the Pauli exclusion principle, it Lemma 1 one obtains (for E1 r e1): is possible to relate equiponderant superpositions of Slater qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 determinants to certain sets of fermionic occupation numbers a0b ¼ a0 a a0 þ ðÞ1 a aa0 a0 a a0 þ ðÞ1 a aa0 lying inside the Paulitope. In this paper, we have proposed a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi general criterion to distinguish the static and dynamic parts of e1 E1 e1 E2 E2 e0 feðÞ0; e1; E1; E2 : the electronic correlation in fermionic systems, by tackling only e1 e0 e1 e0 E1 e0 one-particle information. By doing so, we provided two kinds of 1 Now we can compare the states |c1i and |c2i: l -distances: (a) to the Hartree–Fock point, which can be viewed rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as a measure of the dynamic part of the electronic correlation, 2 E1 E2 Published on 05 May 2017. Downloaded by Martin-Luther-Universitaet 11/16/2020 1:53:17 PM. kkc1 c2 2 2yðÞe1 E1 geðÞ0; e1; E1; E2 ; and (b) to the static states, which can be viewed as a measure of E1 e0 the static part of the correlation. We gave some examples of y y physical systems and showed that these correlation measures where the Heaviside function reads (x)=0ifx o 0 and (x)=1 Z correlate well with our intuition of static and dynamic correlation. if x 0 and g() = max(0, f ()). Though we focused our attention on two and three ‘‘active’’- fermion systems, the results can in principle be generalized to Acknowledgements larger settings. So far, the complete set of generalized Pauli constraints is only known for small systems with three, four and We thank D. Gross, C. Schilling and M. Springborg for helpful five particles. There is, however, an algorithm which provides in discussions. We acknowledge financial support from the GSRT principle the representability conditions for larger settings.23 of the Hellenic Ministry of Education (ESPA), through the In this paper we have highlighted the paramount importance ‘‘Advanced Materials and Devices’’ program (MIS:5002409) of the occupancy of the highest occupied natural spin–orbital in (N. N. L.) and the DFG through Project No. SFB-762 and No. the understanding of the static correlation. The quantities we MA 6787/1-1 (M. A. L. M.). proposed in this paper can allow us to construct reliable ways to separate dynamic and static correlations and, more importantly, References to better understand the qualitative nature of the correlation present in real physical and chemical electronic systems. They 1 D. P. Tew, W. Klopper and T. Helgaker, J. Comput. Chem., can also be a tool for analysing the failures of quantum many 2007, 28, 1307–1320. body theories (like density functional theory).67 Recent progress 2 E. Wigner, Phys. Rev., 1934, 46, 1002–1011. in fermionic mode entanglement can also shed more light in 3 P.-O. Lo¨wdin, Phys. Rev., 1955, 97, 1509. these directions.68,69 4 M. Casula and S. Sorella, J. Chem. Phys., 2003, 119, 6500–6511.

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