Entanglement Measures for Single- and ... - American Chemical Society

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Entanglement Measures for Single- and Multireference Correlation Effects Katharina Boguslawski,† Pawel ̷ Tecmer,† Ö rs Legeza,*,‡ and Markus Reiher*,† †

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

‡

ABSTRACT: Electron correlation effects are essential for an accurate ab initio description of molecules. A quantitative a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chemical method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any molecular structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.

SECTION: Molecular Structure, Quantum Chemistry, and General Theory

T

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 dynamic correlation energy, the missing nondynamic and static contributions can be recovered by a multireference treatment. However, the application of multireference methods represents a far more difficult and computationally expensive task compared with a single-reference study of electronic structures. Furthermore, employing any wave-function-based electron correlation approach requires some a priori knowledge about the interplay of dynamic, nondynamic, and static electron correlation effects. Over the past few decades, a number of different diagnostic tools have been developed to characterize the single- or multireference nature of molecular systems to validate the quality and performance of single-reference quantum chemical methods. For instance, if the absolute or squared weight of the reference configuration (the |C0| coefficient) obtained from a CI calculation is above a certain threshold (|C0| > 0.95 or C20 > 0.90), then the electronic structure is considered to be of singlereference nature.6 However, the weight of the principal configuration can only be considered to be reliable if, for instance, a complete-active-space self-consistent-field (CASSCF) calculation is feasible and comprises all critical orbitals in the active space. As an alternative measure, Lee et al.6−8 proposed to analyze the Euclidean norm of the t1 amplitudes optimized in a CC calculation, which is usually

he correlation energy is defined as the difference between the ground-state energy of a one-determinant wave function, the Hartree−Fock determinant, and the exact solution of the Schrödinger equation. Qualitatively speaking, it is caused by electronic interactions1 beyond the mean-field approach. Even though an exact separation of the correlation energy into individual contributions is not possible, one usually divides correlation effects into three different classes which are denoted dynamic, static, and nondynamic. Although unique definitions of nondynamic, static, and dynamic electron correlation do not exist, the dynamic part is considered to be responsible for keeping electrons apart and is attributed to a large number of configurations, that is, Slater determinants or configuration state functions, with small (absolute) coefficients in the wave function expansion, whereas the nondynamic and static contributions involve only some determinants with large (absolute) weights, which are necessary for an appropriate treatment of the quasi-degeneracy of orbitals.2−4 In particular, static electron correlation embraces a suitable combination of determinants to account for proper spin symmetries and their interactions, whereas nondynamic correlation is required to allow a molecule to separate correctly into its fragments.3,4 An accurate treatment of dynamic, nondynamic, and static correlation effects is covered by the full configuration interaction (FCI) solution.5 However, its steep and unfavorable scaling with the size of the molecule limits the applicability of the FCI approach to systems containing a small number of electrons and small atomic basis sets. To study larger (chemically interesting) molecules, we need to approximate the FCI wave function, which can be achieved by either singlereference or multireference quantum chemical methods. © XXXX American Chemical Society

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

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denoted as T1 diagnostics. It was shown that single-reference CC can be considered to be accurate when the T1 diagnostic is smaller than 0.02 for main group elements6,7,9 and 0.05 for transition metals10 and actinide compounds,11,12 respectively. Because, however, the above-mentioned criteria have not been rigorously defined and turn out to be system- and methoddependent, additional measures abbreviated by D1 and D2, which are based on single and double excitations, were introduced to assess the quality of a single-reference CC calculation.13−15 Although such diagnostic tools can reveal deeper insights into the electronic structure of molecules, they present a posteriori analysis of the performance of quantum chemical calculations and strongly depend on whether a given quantum chemical method is computationally feasible. Because electron correlation effects are caused by the interaction of electrons that occupy specific orbitals used to construct the Slater determinant basis, an intuitive way to study electron correlation is to measure the interaction among any pair of orbitals or the interaction of one orbital with the remaining ones, which are incorporated in an FCI wave function. The interaction between orbitals or electrons can be calculated by employing concepts from quantum information theory like the von Neumann entropy or the mutual information.16 So far, such correlation measures have been evaluated by employing the one-particle reduced density matrix in terms of natural occupation numbers17 (the eigenvalues of the one-particle reduced density matrix), the two-particle reduced density matrix or its cumulant in terms of their Frobenius norm,18−23 and the weights from excited configurations of some wave function expansion.24 A different correlation measure also connected to the one-particle reduced density matrix is the distribution of effectively unpaired electrons, which is utilized to estimate the radical character of a molecule of any spin.25−29 In this work, we seek a different way of quantifying electron correlation effects. In contrast with the approaches mentioned above, our ansatz is based on many-particle reduced density matrices whose eigenvalue spectra are employed to classify the entanglement of orbitals comprised in the active space. Furthermore, to exclude any method-dependent error in the diagnostic analysis like the restriction to a predefined excitation hierarchy or to some zeroth-order wave function, the eigenvalues of the many-particle density matrices need to be determined under the constraint that no artificial truncation of the complete N-particle Hilbert space is performed. An efficient approach to approximate systematically the FCI solution even for large molecules and complicated electronic structures is to apply conceptually different electron correlation methods like the density matrix renormalization group (DMRG)30−32 algorithm developed by White,33 which allows us to treat large active orbital spaces34,35 without a predefined truncation of the complete N-particle Hilbert space. In particular, DMRG can be considered to be a CASCI method36 whose wave function is constructed from many-particle basis states that are determined from the eigenvalue spectrum of the many-particle reduced density matrix of the system under study. These manyparticle basis states are optimized iteratively in one sweep of the DMRG algorithm (to obtain a converged energy, several of such sweeps are necessary). Thus the eigenvalues of the manyparticle reduced density matrix that will enter our diagnostic analysis can be easily determined from the DMRG wave function at the end of one DMRG sweep.

Different entanglement measures were introduced into the DMRG algorithm almost a decade ago that are routinely applied to accelerate DMRG convergence toward the global energy minimum and paved the way for black-box DMRG calculations.30,37 In this respect, some of us38 employed the von Neumann entropy for subsystems containing one single orbital from the complete set of active orbitals to quantify the correlation between this orbital and the remaining set of orbitals contained in the active space. The single-orbital entropy s(1)i can be determined from the eigenvalues of the reduced density matrix wα,i of a given orbital i s(1)i = −∑ wα , i ln wα , i α

(1)

Because this reduced density matrix is defined for one single orbital, we will use the label one-orbital reduced density matrix instead. In particular, the one-orbital reduced density matrix can then be determined from the many-particle reduced density matrix by tracing out all other orbital-degrees of freedom except those of orbital i, which leads to a density matrix of dimension four (in the case of spatial orbitals) spanned by the basis states of a one-orbital Fock space. Furthermore, the one-orbital density matrix is determined for a system (the active orbital i), which is embedded in an environment (all other active space orbitals), and thus correlation effects are incorporated in its eigenvalue spectrum. As a consequence, s(1) i can be significantly different for two orbitals i and j even if similar natural occupation numbers are obtained and the single-orbital entropy can thus be understood as an appropriate measure of how strongly one orbital interacts with all other orbitals. The total quantum information encoded in the wave function39 can be determined from the set of single orbital entropies and reads

Itot =

∑ s(1)i i

(2)

To allow a balanced treatment of electron interaction, Rissler et al.40 presented a scheme to determine the informational content of any pair of orbitals using the von Neumann entropy. The so-called mutual information Ii,j quantifies the correlation of two orbitals embedded in the environment comprising all other active orbitals; that is, it quantifies how much one active orbital i knows about any other active orbital j16 Ii , j = s(2)i , j − s(1)i − s(1)j

(3)

where i = 1...k is the orbital index and runs over all k oneparticle states. In the above equation, s(2)i,j is the two-orbital entropy between a pair i, j of orbitals40 s(2)i , j = −∑ wα ; i , j ln wα ; i , j α

(4)

and represents the two-side analogon of eq 1. wα;i,j are the eigenvalues of the two-orbital reduced density matrix that can be obtained from the many-particle density matrix by integrating out all other orbital degrees of freedom except those corresponding to orbitals i and j. The two-orbital reduced density matrix is of dimension 16 because the pair of orbitals {i,j} can be occupied by zero to four electrons (in the case of spatial orbitals). The single-orbital entropy and the mutual information thus represent convenient measures of entanglement, and because of their general definition they can be employed to quantify different types of correlation present in arbitrary quantum chemical systems. 3130



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To demonstrate our approach, we consider the [Fe(NO)]2+ molecule embedded in a point charge field (see Figure 1a),

information which are indicated by blue, red, and green lines. The blue lines connect each bonding and antibonding combination of the Fe 3d and NO π* orbitals (there are two of them, nos. 5−9 and 6−8) and account for an accurate treatment of nondynamic correlation effects, whereas the red lines connect orbitals that are usually included in standard electron correlation calculations to account for static correlation effects. The dynamic correlation energy is eventually captured by 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 orbital blocks: 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 strongly entangled with at least one other orbital (blue and red lines in the mutual information diagram of Figure 2) correspond to large single orbital entropies. Therefore, we can identify orbitals with both large Ii,j and large s(1)i to be important for nondynamic correlation effects, whereas intermediate values of Ii,j and s(1)i indicate orbitals that are crucial for a correct description of static electron correlation. The (dominant) part of the dynamic electron correlation energy is then captured by excitations into orbitals that are weakly entangled with all other orbitals (green lines in the mutual information diagram of Figure 2 and small s(1)i). Note that the double-shell orbitals indeed recover a large part of the dynamic correlation energy (largest values for s(1)i in the third block). It was shown by us that the missing dynamic correlation effects induced by the remaining virtual orbitals are of considerable importance for an accurate description of the electronic structure of the [Fe(NO)]2+ molecule.35 However, our recent work presented in ref 35 was based on the comparison of spin density distributions only, whereas the entanglement-based analysis elaborated here allows us to identify directly the importance of correlation effects through interactions among orbitals. Because these virtual orbitals are solely connected by green lines and correspond to small s(1)i, their contribution to the electronic energy is of pure dynamic nature and thus a standard CASSCF model 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 is surrounded by four point charges of −0.5 e at a distance of dpc = 1.131 Å from the iron center. For FeL(NO), a structure optimization was performed enforcing Cs symmetry (BP86/TZP).

which we recently identified as a difficult system for standard electron correlation approaches because it requires a balanced treatment of both static, nondynamic and dynamic correlation effects.35,41 Figure 2 shows the mutual information and singleorbital entropies obtained from a DMRG calculation where 13 electrons have been correlated in 29 orbitals. (See also ref 35 for further details.) Note that the entanglement diagrams presented here and in ref 35 differ because in this work the mutual information and single orbital entropies are displayed for a different orbital ordering and are determined from a converged DMRG calculation, whereas in our previous paper35 the mutual information and single orbital entropies were determined after three DMRG sweeps. These differences are, however, only of quantitative nature, and similar entropy diagrams can be obtained for a converged DMRG calculation presented in ref 35 despite the orbital ordering employed in the DMRG calculations. (The orbital ordering strongly effects DMRG convergence but not the entanglement diagrams.) Those active orbitals that are important for nondynamic and static electron correlation are displayed in Figure 2 together with four double-d-shell orbitals. By examining Figure 2, we can distinguish three different interaction strengths of the mutual

Figure 2. Mutual information and single orbital entropies s(1) for a DMRG(13,29) calculation employing the DBSS approach with a minimum and maximum 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 sorted according to their (CASSCF) natural occupation numbers. Each orbital index in panel b indicates one molecular orbital. The orbital index in panel b and the orbital number in panel a correspond to the same natural orbital. The total quantum information is Itot = 4.103. 3131



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

properly for dynamic correlation effects in a quantum chemical description of the ligated iron nitrosyl compound. The single-orbital entropies and mutual information patterns can be employed to analyze possible artifacts that emerge from calculations with active spaces that are too small. Figure 4 summarizes the Ii,j and s(1)i diagrams obtained for different choices of the active space of the [Fe(NO)]2+ molecule embedded in a point-charge field. For the CAS(11,9) calculation, static and nondynamic electron correlation effects are overestimated compared with the DMRG(13,29) reference calculation (note the increased number of blue and red lines as well as the larger values for s(1)i). Simultaneously, the contributions from dynamic correlation are decreased. These artifacts can beat least to some extentresolved if the active space is enlarged. Including additional Fe 3d orbitals into the active space in a CAS(11,11) calculation partially corrects the description of static correlation (cf. orbital nos. 3 and 9 are less entangled) but does not account for an accurate description of dynamic correlation; for example, orbital nos. 6 and 9 still appear to be important for static electron correlation. In the DMRG(13,29) reference calculation, however, they are connected solely through dynamic correlation effects, and thus the static correlation energy is overemphasized. If two additional double-shell orbitals are included upon the CAS(11,11) calculation, then a great part of the dynamic correlation energy can be accounted for in a standard CAS(11,14)SCF calculation (note that one additional virtual ligand orbital was rotated in the active space despite the four Fe 3d-double shell orbitals). Nevertheless, the pattern in the mutual information does not improve compared with the CAS(11,11) calculation and 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 the dimension of the active orbital space. For CAS(11,9), the s(1)i values are in general too large, which coincides with the overestimation of static and nondynamic correlation effects, whereas the dynamic contributions are diminished. These artifacts can be corrected by extending the dimension of the active orbital space and thus allowing for a better treatment of dynamic correlation. However, if the nondynamic correlation

sufficient. Note that this could be cured by, for instance, applying second-order perturbation theory upon a CASSCF reference function.42,43 Next, we can investigate the changes in the electronic structure when the point charge environment is substituted by one or several ligand molecules. We replace the point charges by a small model molecule of a salen ligand where the aromatic rings have been substituted by CH2 units as displayed in Figure 1b. The mutual information and single orbital entropies obtained from a DMRG calculation correlating 21 electrons in 35 molecular orbitals and imposing Cs symmetry are shown in Figure 3. In particular, we obtain similar entanglement diagrams for the ligated iron nitrosyl compound as found for the small [Fe(NO)]2+ molecule, that is, three groups of orbitals that can be classified by their (combined) Ii,j and s(1)i contributions. Again, the two bonding and antibonding combinations of the Fe 3d and NO π* orbitals (nos. 23−24 and 6−8, which are shown in Figure 3) are important for a correct description of nondynamic correlation effects, yet, in contrast with the bare [Fe(NO)]2+ complex, the Fe 3dxy orbital now interacts with one ligand σ orbital, and its bonding and antibonding combinations are strongly entangled (nos. 22−25, which are also shown in Figure 3). The single orbital entropy profile further indicates that these orbitals need to be considered for an accurate treatment of static correlation and are thus, together with the remaining highly entangled orbitals (nos. 4 to 9), that is, those that would have been included in any standard (minimum) active space calculation, mandatory to capture the static correlation energy. However, we now observe a great number of orbitals that are weakly entangled and that comprise (very) small single orbital entropies. Hence, the influence of dynamic correlation increases after ligation and can be referred to the additional virtual ligand orbitals that are available for possible excitations. Simultaneously, the contribution of static electron correlation decreases, which can be explained by the reduced number of statically entangled orbitals (the red lines in Figure 3) and the smaller single orbital entropies (compare Figures 3 and 2). Under the process of ligation, the multireference character of the electronic wave function depletes, and thus we must be able to account 3132



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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 of renormalized active-system states was increased until the CASSCF reference energy was obtained. The orbitals are numbered and sorted according to their (CASSCF) natural occupation numbers. Each orbital index indicates one molecular orbital. The orbital index in panel b and the orbital number in panel a correspond to the same natural orbital. For CAS(11,9), CAS(11,11), and CAS(11,14), the total quantum information corresponds to 3.761, 3.678, and 3.909, respectively.

virtual orbitals like Fe 3d double-shell orbitals, the improvements are insufficient because the contributions of these virtual orbitals to the dynamic correlation energy remain underestimated compared with the DMRG(13,29) reference calculation. Furthermore, extending the active space from CAS(11,11) over CAS(11,14) to DMRG(13,29) leads to Itot values of 3.678, 3.909, and 4.103, respectively. Hence, more

energy is overestimated, then the static electron correlation contributions will be underestimated, which can be observed in too small single orbital entropies compared with an FCI reference and vice versa, which results in too large values for s(1)i. Although the wrong estimate of static and nondynamic electron correlation caused by a too small dimension of the active orbital space can be corrected by including additional 3133



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(dynamic) electron correlation is recovered when the active space is increased. For CAS(11,9), however, dynamic correlation is considerably underestimated, which leads to overrated static correlation effects and results in a too large value for the total quantum information of Itot = 3.761 when compared with the CAS(11,11) calculation. In this work, we have presented a quantitative measure to assess electron correlation effects that are independent of the reference wave function and do not require an a priori knowledge about the single- or multirefence character of the electronic structure. In our analysis, the DMRG algorithm was employed, which allows us to approach systematically the FCI solution. The static, nondynamic, and dynamic contributions to the correlation energy can be distinguished by examining the entanglement patterns of orbitals. We demonstrated that the single- or multirefence nature of electronic structures is encoded in the mutual information and single orbital entropies. These quantities do not significantly depend on the accuracy of our DMRG calculations and can be already obtained from fast and inexpensive DMRG sweeps. The cost for these DMRG sweeps needed to acquire the entanglement measures is thus negligible. Expensive in terms of computing time is the calculation of the two-electron integrals in the molecular orbital basis, which, however, is a mandatory step in any correlation treatment and thus a prerequisite of the correlation treatment chosen after the evaluation of the entanglement measures. Of course, the DMRG sweeping may also be continued until convergence is reached if an alternative like a CC model is not expected to yield more accurate results or in a shorter time, respectively. The entanglement analysis proposed here can be performed in any orbital basis without loss of generality and can provide insight into which quantum chemical method to choose for an accurate description of the molecule under study. Furthermore, we highlighted the artifacts emerging from small active space calculations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support by the Swiss national science foundation SNF (project 200020-132542/1) and from the Hungarian Research Fund (OTKA) under grant no. K73455 and K100908. K.B. thanks the Fonds der Chemischen Industrie for a Chemiefonds scholarship. Ö .L. acknowledges support from the Alexander von Humboldt foundation and from ETH Zurich during his time as a visiting professor.

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REFERENCES

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COMPUTATIONAL DETAILS All DMRG calculations as well as the calculation of the mutual information and single orbital entropies have been performed with the Budapest DMRG program.44 As orbital basis, the natural orbitals obtained from preceding CASSCF calculations employing the Molpro program package45 in a cc-pVTZ basis set46,47 are used comprising 11 electrons correlated in 14 orbitals (for both the bare and ligated iron nitrosyl complex). For the FeL(NO) molecule, the CASSCF active space calculations comprised all Fe 3d orbitals, both NO π* orbitals, two NO σ orbitals, which interact with Fe 3d orbitals, and four Fe 3d double-shell orbitals (excluding dx2−y2), while additional ligand orbitals have been included in the DMRG calculations. For [Fe(NO)]2+, we refer the reader to ref 41. To accelerate DMRG convergence, the dynamic block state selection (DBSS) approach39,48 and the dynamically extended active space procedure38 were employed, whereas the orbital ordering was optimized 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 corresponding CASSCF natural orbital basis. All DMRG calculations have been converged until chemical accuracy (ΔE < 1.5 mHartree) has been reached within the active space: ΔE([Fe(NO)]2+) = 1.3 mHartree, while ΔE(FeL(NO) = 0.4 mHartree. In particular, for CAS(11,9), CAS(11,11), and CAS(11,14), the corresponding CAS(11,y)SCF energy has been reproduced. 3134



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