Entanglement Measures for Single- and Multireference CorrelationEffectsKatharina Boguslawski,† Pawel Tecmer,† Ors Legeza,*,‡ and Markus Reiher*,†

†ETH Zurich, Laboratory of Physical Chemistry, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland‡MTA-WRCP Strongly Correlated Systems “Lendulet” Research Group, H-1525, Budapest, Hungary

ABSTRACT: Electron correlation effects are essential for an accurate abinitio description of molecules. A quantitative a priori knowledge of thesingle- or multireference nature of electronic structures as well as of thedominant contributions to the correlation energy can facilitate the decisionregarding the optimum quantum chemical method of choice. We proposeconcepts from quantum information theory as orbital entanglement measuresthat allow us to evaluate the single- and multireference character of anymolecular structure in a given orbital basis set. By studying these measures wecan detect possible artifacts of small active spaces.

SECTION: Molecular Structure, Quantum Chemistry, and General Theory

The correlation energy is defined as the difference betweenthe ground-state energy of a one-determinant wave

function, the Hartree−Fock determinant, and the exact solutionof the Schrodinger equation. Qualitatively speaking, it is causedby electronic interactions1 beyond the mean-field approach.Even though an exact separation of the correlation energy intoindividual contributions is not possible, one usually dividescorrelation effects into three different classes which are denoteddynamic, static, and nondynamic. Although unique definitions ofnondynamic, static, and dynamic electron correlation do notexist, the dynamic part is considered to be responsible forkeeping electrons apart and is attributed to a large number ofconfigurations, that is, Slater determinants or configurationstate functions, with small (absolute) coefficients in the wavefunction expansion, whereas the nondynamic and staticcontributions involve only some determinants with large(absolute) weights, which are necessary for an appropriatetreatment of the quasi-degeneracy of orbitals.2−4 In particular,static electron correlation embraces a suitable combination ofdeterminants to account for proper spin symmetries and theirinteractions, whereas nondynamic correlation is required toallow a molecule to separate correctly into its fragments.3,4

An accurate treatment of dynamic, nondynamic, and staticcorrelation effects is covered by the full configurationinteraction (FCI) solution.5 However, its steep and unfavorablescaling with the size of the molecule limits the applicability ofthe FCI approach to systems containing a small number ofelectrons and small atomic basis sets. To study larger(chemically interesting) molecules, we need to approximatethe FCI wave function, which can be achieved by either single-reference or multireference quantum chemical methods.

Whereas single-reference approaches like, for instance,Møller−Plesset perturbation theory or the coupled cluster(CC) ansatz, are able to capture the largest part of the dynamiccorrelation energy, the missing nondynamic and staticcontributions can be recovered by a multireference treatment.However, the application of multireference methods representsa far more difficult and computationally expensive taskcompared with a single-reference study of electronic structures.Furthermore, employing any wave-function-based electroncorrelation approach requires some a priori knowledge aboutthe interplay of dynamic, nondynamic, and static electroncorrelation effects.Over the past few decades, a number of different diagnostic

tools have been developed to characterize the single- ormultireference nature of molecular systems to validate thequality and performance of single-reference quantum chemicalmethods. For instance, if the absolute or squared weight of thereference configuration (the |C0| coefficient) obtained from aCI calculation is above a certain threshold (|C0| > 0.95 or C0

2 >0.90), then the electronic structure is considered to be of single-reference nature.6 However, the weight of the principalconfiguration can only be considered to be reliable if, forinstance, a complete-active-space self-consistent-field(CASSCF) calculation is feasible and comprises all criticalorbitals in the active space. As an alternative measure, Lee etal.6−8 proposed to analyze the Euclidean norm of the t1amplitudes optimized in a CC calculation, which is usually

Received: August 31, 2012Accepted: October 9, 2012

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denoted as T1 diagnostics. It was shown that single-referenceCC can be considered to be accurate when the T1 diagnostic issmaller than 0.02 for main group elements6,7,9 and 0.05 fortransition metals10 and actinide compounds,11,12 respectively.Because, however, the above-mentioned criteria have not beenrigorously defined and turn out to be system- and method-dependent, additional measures abbreviated by D1 and D2,which are based on single and double excitations, wereintroduced to assess the quality of a single-reference CCcalculation.13−15 Although such diagnostic tools can revealdeeper insights into the electronic structure of molecules, theypresent a posteriori analysis of the performance of quantumchemical calculations and strongly depend on whether a givenquantum chemical method is computationally feasible.Because electron correlation effects are caused by the

interaction of electrons that occupy specific orbitals used toconstruct the Slater determinant basis, an intuitive way to studyelectron correlation is to measure the interaction among anypair of orbitals or the interaction of one orbital with theremaining ones, which are incorporated in an FCI wavefunction. The interaction between orbitals or electrons can becalculated by employing concepts from quantum informationtheory like the von Neumann entropy or the mutualinformation.16 So far, such correlation measures have beenevaluated by employing the one-particle reduced density matrixin terms of natural occupation numbers17 (the eigenvalues ofthe one-particle reduced density matrix), the two-particlereduced density matrix or its cumulant in terms of theirFrobenius norm,18−23 and the weights from excited config-urations of some wave function expansion.24 A differentcorrelation measure also connected to the one-particle reduceddensity matrix is the distribution of effectively unpairedelectrons, which is utilized to estimate the radical character ofa molecule of any spin.25−29

In this work, we seek a different way of quantifying electroncorrelation effects. In contrast with the approaches mentionedabove, our ansatz is based on many-particle reduced densitymatrices whose eigenvalue spectra are employed to classify theentanglement of orbitals comprised in the active space.Furthermore, to exclude any method-dependent error in thediagnostic analysis like the restriction to a predefined excitationhierarchy or to some zeroth-order wave function, theeigenvalues of the many-particle density matrices need to bedetermined under the constraint that no artificial truncation ofthe complete N-particle Hilbert space is performed. An efficientapproach to approximate systematically the FCI solution evenfor large molecules and complicated electronic structures is toapply conceptually different electron correlation methods likethe density matrix renormalization group (DMRG)30−32

algorithm developed by White,33 which allows us to treatlarge active orbital spaces34,35 without a predefined truncationof the complete N-particle Hilbert space. In particular, DMRGcan be considered to be a CASCI method36 whose wavefunction is constructed from many-particle basis states that aredetermined from the eigenvalue spectrum of the many-particlereduced density matrix of the system under study. These many-particle basis states are optimized iteratively in one sweep of theDMRG algorithm (to obtain a converged energy, several ofsuch sweeps are necessary). Thus the eigenvalues of the many-particle reduced density matrix that will enter our diagnosticanalysis can be easily determined from the DMRG wavefunction at the end of one DMRG sweep.

Different entanglement measures were introduced into theDMRG algorithm almost a decade ago that are routinelyapplied to accelerate DMRG convergence toward the globalenergy minimum and paved the way for black-box DMRGcalculations.30,37 In this respect, some of us38 employed the vonNeumann entropy for subsystems containing one single orbitalfrom the complete set of active orbitals to quantify thecorrelation between this orbital and the remaining set oforbitals contained in the active space. The single-orbitalentropy s(1)i can be determined from the eigenvalues of thereduced density matrix wα,i of a given orbital i

∑= −α

α αs w w(1) lni i i, ,(1)

Because this reduced density matrix is defined for one singleorbital, we will use the label one-orbital reduced density matrixinstead. In particular, the one-orbital reduced density matrixcan then be determined from the many-particle reduced densitymatrix by tracing out all other orbital-degrees of freedom exceptthose of orbital i, which leads to a density matrix of dimensionfour (in the case of spatial orbitals) spanned by the basis statesof a one-orbital Fock space. Furthermore, the one-orbitaldensity matrix is determined for a system (the active orbital i),which is embedded in an environment (all other active spaceorbitals), and thus correlation effects are incorporated in itseigenvalue spectrum. As a consequence, s(1)i can besignificantly different for two orbitals i and j even if similarnatural occupation numbers are obtained and the single-orbitalentropy can thus be understood as an appropriate measure ofhow strongly one orbital interacts with all other orbitals. Thetotal quantum information encoded in the wave function39 canbe determined from the set of single orbital entropies and reads

∑=I s(1)i

itot(2)

To allow a balanced treatment of electron interaction, Rissleret al.40 presented a scheme to determine the informationalcontent of any pair of orbitals using the von Neumann entropy.The so-called mutual information Ii,j quantifies the correlationof two orbitals embedded in the environment comprising allother active orbitals; that is, it quantifies how much one activeorbital i knows about any other active orbital j16

= − −I s s s(2) (1) (1)i j i j i j, , (3)

where i = 1...k is the orbital index and runs over all k one-particle states. In the above equation, s(2)i,j is the two-orbitalentropy between a pair i, j of orbitals40

∑= −α

α αs w w(2) lni j i j i j, ; , ; ,(4)

and represents the two-side analogon of eq 1. wα;i,j are theeigenvalues of the two-orbital reduced density matrix that canbe obtained from the many-particle density matrix byintegrating out all other orbital degrees of freedom exceptthose corresponding to orbitals i and j. The two-orbital reduceddensity matrix is of dimension 16 because the pair of orbitals{i,j} can be occupied by zero to four electrons (in the case ofspatial orbitals). The single-orbital entropy and the mutualinformation thus represent convenient measures of entangle-ment, and because of their general definition they can beemployed to quantify different types of correlation present inarbitrary quantum chemical systems.

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To demonstrate our approach, we consider the [Fe(NO)]2+

molecule embedded in a point charge field (see Figure 1a),

which we recently identified as a difficult system for standardelectron correlation approaches because it requires a balancedtreatment of both static, nondynamic and dynamic correlationeffects.35,41 Figure 2 shows the mutual information and single-orbital entropies obtained from a DMRG calculation where 13electrons have been correlated in 29 orbitals. (See also ref 35for further details.) Note that the entanglement diagramspresented here and in ref 35 differ because in this work themutual information and single orbital entropies are displayedfor a different orbital ordering and are determined from aconverged DMRG calculation, whereas in our previous paper35

the mutual information and single orbital entropies weredetermined after three DMRG sweeps. These differences are,however, only of quantitative nature, and similar entropydiagrams can be obtained for a converged DMRG calculationpresented in ref 35 despite the orbital ordering employed in theDMRG calculations. (The orbital ordering strongly effectsDMRG convergence but not the entanglement diagrams.)Those active orbitals that are important for nondynamic andstatic electron correlation are displayed in Figure 2 togetherwith four double-d-shell orbitals. By examining Figure 2, we candistinguish three different interaction strengths of the mutual

information which are indicated by blue, red, and green lines.The blue lines connect each bonding and antibondingcombination of the Fe 3d and NO π* orbitals (there are twoof them, nos. 5−9 and 6−8) and account for an accuratetreatment of nondynamic correlation effects, whereas the redlines connect orbitals that are usually included in standardelectron correlation calculations to account for static correlationeffects. The dynamic correlation energy is eventually capturedby excitations into orbitals, which are connected via green lines.A similar pattern can be observed for the single orbital

entropies where we can distinguish three different orbitalblocks: orbitals (i) with large single orbital entropies (>0.5),(ii) with medium-sized single orbital entropies (0.1 < s(1)i <0.5), and (iii) with small or almost zero single orbital entropies(0 < s(1)i < 0.1). Furthermore, orbitals that are stronglyentangled with at least one other orbital (blue and red lines inthe mutual information diagram of Figure 2) correspond tolarge single orbital entropies. Therefore, we can identify orbitalswith both large Ii,j and large s(1)i to be important fornondynamic correlation effects, whereas intermediate valuesof Ii,j and s(1)i indicate orbitals that are crucial for a correctdescription of static electron correlation. The (dominant) partof the dynamic electron correlation energy is then captured byexcitations into orbitals that are weakly entangled with all otherorbitals (green lines in the mutual information diagram ofFigure 2 and small s(1)i). Note that the double-shell orbitalsindeed recover a large part of the dynamic correlation energy(largest values for s(1)i in the third block). It was shown by usthat the missing dynamic correlation effects induced by theremaining virtual orbitals are of considerable importance for anaccurate description of the electronic structure of the[Fe(NO)]2+ molecule.35 However, our recent work presentedin ref 35 was based on the comparison of spin densitydistributions only, whereas the entanglement-based analysiselaborated here allows us to identify directly the importance ofcorrelation effects through interactions among orbitals. Becausethese virtual orbitals are solely connected by green lines andcorrespond to small s(1)i, their contribution to the electronicenergy is of pure dynamic nature and thus a standard CASSCFmodel of the electronic structure of [Fe(NO)]2+ is not

Figure 1. Structures of the bare and ligated iron nitrosyl complexes,[Fe(NO)]2+ and FeL(NO), respectively. The [Fe(NO)]2+ molecule issurrounded by four point charges of −0.5 e at a distance of dpc = 1.131Å from the iron center. For FeL(NO), a structure optimization wasperformed enforcing Cs symmetry (BP86/TZP).

Figure 2. Mutual information and single orbital entropies s(1) for a DMRG(13,29) calculation employing the DBSS approach with a minimum andmaximum number of renormalized active system states set to 128 and 1024, respectively, and a quantum information loss of 10−5 for the[Fe(NO)]2+ molecule surrounded by four point charges at a distance of dpc = 1.131 Å from the iron center. The orbitals are numbered and sortedaccording to their (CASSCF) natural occupation numbers. Each orbital index in panel b indicates one molecular orbital. The orbital index in panel band the orbital number in panel a correspond to the same natural orbital. The total quantum information is Itot = 4.103.

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sufficient. Note that this could be cured by, for instance,applying second-order perturbation theory upon a CASSCFreference function.42,43

Next, we can investigate the changes in the electronicstructure when the point charge environment is substituted byone or several ligand molecules. We replace the point chargesby a small model molecule of a salen ligand where the aromaticrings have been substituted by CH2 units as displayed in Figure1b. The mutual information and single orbital entropiesobtained from a DMRG calculation correlating 21 electronsin 35 molecular orbitals and imposing Cs symmetry are shownin Figure 3. In particular, we obtain similar entanglementdiagrams for the ligated iron nitrosyl compound as found forthe small [Fe(NO)]2+ molecule, that is, three groups of orbitalsthat can be classified by their (combined) Ii,j and s(1)icontributions. Again, the two bonding and antibondingcombinations of the Fe 3d and NO π* orbitals (nos. 23−24and 6−8, which are shown in Figure 3) are important for acorrect description of nondynamic correlation effects, yet, incontrast with the bare [Fe(NO)]2+ complex, the Fe 3dxy orbitalnow interacts with one ligand σ orbital, and its bonding andantibonding combinations are strongly entangled (nos. 22−25,which are also shown in Figure 3). The single orbital entropyprofile further indicates that these orbitals need to beconsidered for an accurate treatment of static correlation andare thus, together with the remaining highly entangled orbitals(nos. 4 to 9), that is, those that would have been included inany standard (minimum) active space calculation, mandatory tocapture the static correlation energy. However, we now observea great number of orbitals that are weakly entangled and thatcomprise (very) small single orbital entropies. Hence, theinfluence of dynamic correlation increases after ligation and canbe referred to the additional virtual ligand orbitals that areavailable for possible excitations. Simultaneously, the contribu-tion of static electron correlation decreases, which can beexplained by the reduced number of statically entangled orbitals(the red lines in Figure 3) and the smaller single orbitalentropies (compare Figures 3 and 2). Under the process ofligation, the multireference character of the electronic wavefunction depletes, and thus we must be able to account

properly for dynamic correlation effects in a quantum chemicaldescription of the ligated iron nitrosyl compound.The single-orbital entropies and mutual information patterns

can be employed to analyze possible artifacts that emerge fromcalculations with active spaces that are too small. Figure 4summarizes the Ii,j and s(1)i diagrams obtained for differentchoices of the active space of the [Fe(NO)]2+ moleculeembedded in a point-charge field. For the CAS(11,9)calculation, static and nondynamic electron correlation effectsare overestimated compared with the DMRG(13,29) referencecalculation (note the increased number of blue and red lines aswell as the larger values for s(1)i). Simultaneously, thecontributions from dynamic correlation are decreased. Theseartifacts can beat least to some extentresolved if the activespace is enlarged. Including additional Fe 3d orbitals into theactive space in a CAS(11,11) calculation partially corrects thedescription of static correlation (cf. orbital nos. 3 and 9 are lessentangled) but does not account for an accurate description ofdynamic correlation; for example, orbital nos. 6 and 9 stillappear to be important for static electron correlation. In theDMRG(13,29) reference calculation, however, they areconnected solely through dynamic correlation effects, andthus the static correlation energy is overemphasized. If twoadditional double-shell orbitals are included upon the CAS-(11,11) calculation, then a great part of the dynamic correlationenergy can be accounted for in a standard CAS(11,14)SCFcalculation (note that one additional virtual ligand orbital wasrotated in the active space despite the four Fe 3d-double shellorbitals). Nevertheless, the pattern in the mutual informationdoes not improve compared with the CAS(11,11) calculationand the static correlation energy is still overestimated (cf.orbital nos. 6 and 9 remain strongly entangled).Similar conclusions can be drawn when analyzing the

evolution of the single-orbital entropies with respect to thedimension of the active orbital space. For CAS(11,9), the s(1)ivalues are in general too large, which coincides with theoverestimation of static and nondynamic correlation effects,whereas the dynamic contributions are diminished. Theseartifacts can be corrected by extending the dimension of theactive orbital space and thus allowing for a better treatment ofdynamic correlation. However, if the nondynamic correlation

Figure 3. Mutual information and single orbital entropies s(1) for a DMRG(21,35) calculation employing the DBSS approach with a minimum andmaximum number of renormalized active system states of 128 and 1024, respectively, and a quantum information loss of 10−5 for the ligated ironnitrosyl complex as displayed in Figure 1b. The orbitals are numbered and sorted according to their (CASSCF) natural occupation numbers andsorted according to their irreducible representation. Each orbital index in panel b indicates one molecular orbital. The orbital index in panel b and theorbital number in panel a correspond to the same natural orbital.

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energy is overestimated, then the static electron correlationcontributions will be underestimated, which can be observed intoo small single orbital entropies compared with an FCIreference and vice versa, which results in too large values fors(1)i. Although the wrong estimate of static and nondynamicelectron correlation caused by a too small dimension of theactive orbital space can be corrected by including additional

virtual orbitals like Fe 3d double-shell orbitals, the improve-ments are insufficient because the contributions of these virtualorbitals to the dynamic correlation energy remain under-estimated compared with the DMRG(13,29) referencecalculation. Furthermore, extending the active space fromCAS(11,11) over CAS(11,14) to DMRG(13,29) leads to Itotvalues of 3.678, 3.909, and 4.103, respectively. Hence, more

Figure 4. Mutual information and single orbital entropies s(1) for DMRG(11,y) calculations determined for different numbers of active orbitals in[Fe(NO)]2+ surrounded by four point charges at a distance of dpc = 1.131 Å from the iron center. For each DMRG calculation, the number ofrenormalized active-system states was increased until the CASSCF reference energy was obtained. The orbitals are numbered and sorted accordingto their (CASSCF) natural occupation numbers. Each orbital index indicates one molecular orbital. The orbital index in panel b and the orbitalnumber in panel a correspond to the same natural orbital. For CAS(11,9), CAS(11,11), and CAS(11,14), the total quantum information correspondsto 3.761, 3.678, and 3.909, respectively.

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(dynamic) electron correlation is recovered when the activespace is increased. For CAS(11,9), however, dynamiccorrelation is considerably underestimated, which leads tooverrated static correlation effects and results in a too largevalue for the total quantum information of Itot = 3.761 whencompared with the CAS(11,11) calculation.In this work, we have presented a quantitative measure to

assess electron correlation effects that are independent of thereference wave function and do not require an a prioriknowledge about the single- or multirefence character of theelectronic structure. In our analysis, the DMRG algorithm wasemployed, which allows us to approach systematically the FCIsolution. The static, nondynamic, and dynamic contributions tothe correlation energy can be distinguished by examining theentanglement patterns of orbitals. We demonstrated that thesingle- or multirefence nature of electronic structures isencoded in the mutual information and single orbital entropies.These quantities do not significantly depend on the accuracy ofour DMRG calculations and can be already obtained from fastand inexpensive DMRG sweeps. The cost for these DMRGsweeps needed to acquire the entanglement measures is thusnegligible. Expensive in terms of computing time is thecalculation of the two-electron integrals in the molecular orbitalbasis, which, however, is a mandatory step in any correlationtreatment and thus a prerequisite of the correlation treatmentchosen after the evaluation of the entanglement measures. Ofcourse, the DMRG sweeping may also be continued untilconvergence is reached if an alternative like a CC model is notexpected to yield more accurate results or in a shorter time,respectively. The entanglement analysis proposed here can beperformed in any orbital basis without loss of generality and canprovide insight into which quantum chemical method tochoose for an accurate description of the molecule under study.Furthermore, we highlighted the artifacts emerging from smallactive space calculations.

■ COMPUTATIONAL DETAILSAll DMRG calculations as well as the calculation of the mutualinformation and single orbital entropies have been performedwith the Budapest DMRG program.44 As orbital basis, thenatural orbitals obtained from preceding CASSCF calculationsemploying the Molpro program package45 in a cc-pVTZ basisset46,47 are used comprising 11 electrons correlated in 14orbitals (for both the bare and ligated iron nitrosyl complex).For the FeL(NO) molecule, the CASSCF active spacecalculations comprised all Fe 3d orbitals, both NO π* orbitals,two NO σ orbitals, which interact with Fe 3d orbitals, and fourFe 3d double-shell orbitals (excluding dx2−y2), while additionalligand orbitals have been included in the DMRG calculations.For [Fe(NO)]2+, we refer the reader to ref 41. To accelerateDMRG convergence, the dynamic block state selection (DBSS)approach39,48 and the dynamically extended active spaceprocedure38 were employed, whereas the orbital ordering wasoptimized for each active space calculation according to ref 37.The small active space calculations, that is, CAS(11,9),CAS(11,11), and CAS(11,14), are performed in the corre-sponding CASSCF natural orbital basis. All DMRG calculationshave been converged until chemical accuracy (ΔE < 1.5mHartree) has been reached within the active space: ΔE([Fe-(NO)]2+) = 1.3 mHartree, while ΔE(FeL(NO) = 0.4mHartree. In particular, for CAS(11,9), CAS(11,11), andCAS(11,14), the corresponding CAS(11,y)SCF energy hasbeen reproduced.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: olegeza@szfki.hu; markus.reiher@phys.chem.ethz.ch.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe gratefully acknowledge financial support by the Swissnational science foundation SNF (project 200020-132542/1)and from the Hungarian Research Fund (OTKA) under grantno. K73455 and K100908. K.B. thanks the Fonds derChemischen Industrie for a Chemiefonds scholarship. O.L.acknowledges support from the Alexander von Humboldtfoundation and from ETH Zurich during his time as a visitingprofessor.

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The Journal of Physical Chemistry Letters Letter

dx.doi.org/10.1021/jz301319v | J. Phys. Chem. Lett. 2012, 3, 3129−31353135