Characterizing Bonding Patterns in Diradicals and Triradicals by Density-Based Wave Function Analysis: a Uniform Approach Natalie Orms,† Dirk R

Article

Cite This: J. Chem. Theory Comput. 2018, 14, 638−648 pubs.acs.org/JCTC

Characterizing Bonding Patterns in Diradicals and Triradicals by Density-Based Wave Function Analysis: A Uniform Approach Natalie Orms,† Dirk R. Rehn,§,‡ Andreas Dreuw,‡ and Anna I. Krylov*,†

† Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, United States ‡ Interdisciplinary Center for Scientiﬁc Computing, Ruprecht-Karls University Heidelberg, Im Neuenheimer Feld 205A, 69120 Heidelberg, Germany

*S Supporting Information

ABSTRACT: Density-based wave function analysis enables unambiguous comparisons of the electronic structure computed by different methods and removes ambiguity of orbital choices. We use this tool to investigate the performance of different spin- flip methods for several prototypical diradicals and triradicals. In contrast to previous calibration studies that focused on energy gaps between high- and low spin-states, we focus on the properties of the underlying wave functions, such as the number of effectively unpaired electrons. Comparison of different density functional and wave function theory results provides insight into the performance of the different methods when applied to strongly correlated systems such as polyradicals. We show that canonical molecular orbitals for species like large copper-containing diradicals fail to correctly represent the underlying electronic structure due to highly non-Koopmans character, while density-based analysis of the same wave function delivers a clear picture of the bonding pattern.

1. INTRODUCTION represented by a single Slater determinant; the wave function in Chemists define diradicals and triradicals as species with two this case is a mixture of covalent and charge-resonance − fi and three unpaired electrons.1 4 This bonding pattern arises con gurations, and the molecule can be described as a when two (or three) electrons are distributed in two (three) closed-shell species. In between there is a continuum of intermediate bonding patterns, which are sometimes described nearly degenerate molecular orbitals. Figure 1 shows all 2 fi as diradicaloids. Detailed analysis of different types of diradical resulting con gurations with positive spin-projections. For 1,2 same-spin electrons, there are only two possible arrangements: electronic structure can be found in classic papers; for a ± fi ± review of triradicals, one can consult ref 3. Ms 1 (con guration A(i) in Figure 1 has Ms = 1) and Ms fi The wave functions composed by Slater determinants in 3/2 (con guration B(i) in Figure 1 has Ms = 3/2). For the states with lower spin-projections, more configurations can be which two electrons occupy the same MO, e.g., (iv) and (v) in A and (v)-(x) in B, are commonly referred to as of “closed- generated, as illustrated in Figure 1. ” The energy gaps, relative state ordering, and relative weights shell type (although they can correspond to completely ffi λ unpaired electrons, as in the case of A (v) with λ = 1), in of the Slater determinants (i.e., coe cients in Figure 1) “ ” depend on the nature and energy separation of the frontier contrast to open-shell wave functions (i)-(iii) in A and (i)- molecular orbitals (MOs). The character of the MOs also (iv) in B (which can correspond to a purely ionic and, therefore, closed-shell pattern). Here we follow this accepted determines the character of the wave function, e.g., whether it 4 has predominantly covalent (i.e., two electrons residing on terminology for consistency with other studies. An important ff difference between the two types of open-shell wave functions di erent parts of a molecule) or ionic character. ffi λ To illustrate this point, consider a simple diradical, such as a is that in A(v) the coe cients depend on the orbital degeneracy, whereas in A(ii)-(iii) the relative weights of the molecule with a broken bond (i.e., H2 at dissociation limit or twisted ethylene). In this case, the two MOs are a bonding and contributing Slater determinants are determined by spin σ σ* π symmetry alone.1 antibonding pair (such as and orbitals in stretched H2 or and π* in twisted ethylene). When the two MOs are exactly In model pedagogical examples of two-electrons-in-two- degenerate, λ = 1 in wave functions (iv) and (v) in Figure 1A. orbitals or three-electrons-in-three-orbitals, the number of Wave function (iii) corresponds to the covalent diradical singlet unpaired electrons can be unambiguously determined from state (H·H·), which is exactly degenerate with the triplet state, the wave functions. For example, in diradicals, (i) and (ii) whereas (v) represents a purely ionic (charge-resonance) state correspond to the two unpaired electrons, whereas the relative (i.e., H−H+ ± H+H−). This situation is often described as a perfect diradical. If the gap between bonding and antibonding Received: October 1, 2017 orbitals is large, λ becomes small, and the ground state can be Published: December 21, 2017

Figure 1. Wave functions of di- (A) and triradicals (B) that are eigenfunctions of Ŝ2. In both panels, wave function (i) corresponds to the reference- state wave function: the high-spin Ms = 1 triplet state of a diradical or the Ms = 3/2 quartet state of a triradical. Wave functions (ii)-(v) in A and (ii)- fi (x) in B correspond to the low-spin states: Ms = 0 singlets and triplets, or Ms = 1/2 doublets and quartets. For simplicity, only con gurations with positive spin-projection are shown. weight of purely covalent configuration in (v) is determined by occupations are two (for occupied orbitals) and zero (for λ, i.e., λ = 1 corresponds to the 2 unpaired electrons. However, virtuals). For a two-determinantal wave function (v) with λ =1, this simple picture does not apply for realistic many-body wave natural occupations are 1. For multiconfigurational wave functions. Even in the simplest case of two electrons, the functions, the partial occupations of natural orbitals can be dynamic correlation and the arbitrariness in orbital choice result used to derive an effective number of the unpaired electrons. in multiconfigurational wave functions whose character cannot For example, for the perfectly diradical wave functions, such as fi λ be easily assigned on the basis of just 2 leading con gurations. A(i)-(iii) and (iv) with =1inFigure 1, n1 = n2 = 1 and the In many-electron systems the contributions of electrons number of unpaired electrons is 2, whereas for the closed-shell λ occupying lower MOs give rise to through-bond interactions case, A(iv) with =0,n1 =2,n2 = 0 and the number of between the radical centers, further complicating the wave unpaired electrons is 0. function analysis. Although the number of effectively unpaired Several ways to compute an effective number of unpaired electrons is not an observable physical property, it is related to electrons from the one-particle density matrices have been 8−10 the bonding pattern, which, in turn, manifests itself in concrete proposed. In this work, we make use of the two indices, nu 3−5 9 structural, spectroscopic, and thermochemical quantities. and nu,nl, proposed by Head-Gordon as an extension of work Thus, the ability to assign an effective number of unpaired by Yamaguchi et al.:8 electrons to a particular electronic wave function is valuable for n = ∑ min(nn , 2− ) chemical insight, akin to other methods of wave function u ii̅ ̅ (5) analysis.6,7 i 8−10 Several solutions toward this goal have been proposed. 22 6,7,11−14 n = ∑ nn(2− ) As with many other wave function analysis tools, they unl, ii̅ ̅ (6) 15 ψ i exploit the concept of natural orbitals. Natural orbitals ( i) are eigenstates of the one-particle reduced density matrix In both equations, the sum runs over all natural orbitals and the contributions of the doubly occupied and unoccupied ψ()xcx= ∑ ϕ () i ji j (1) orbitals are exactly zero. As one can easily verify, both formulas j yield correct answers for the model examples from Figure 1, i.e., 2 for A(i)-(iii) and (iv) with λ = 1, 0 for A(iv) with λ = 0, and γccn= (2) so on. The two expressions differ by how they account for partially occupied orbitals. Numerical experimentation has γ ≡⟨Ψ| †̂ |Ψ̂ ⟩ pq pq (3) shown that for many-electron wave functions, eq 6 consistently † gives physically meaningful values, whereas n often produces where p̂and q̂are the creation and annihilation operators u ϕ ϕ the number of unpaired electrons which is too high (as corresponding to p and q MOs. The respective eigenvalues compared to chemical intuition). This happens because the nu (ni) are non-negative, add up to the total number of electrons, index does not suppress dynamic correlation contributions and can be interpreted as orbital occupations. Natural orbitals ff (which come from a large number of small natural occupation a ord the most compact representation of the electron density numbers) to the total number of unpaired electrons, whereas 2 nu,nl emphasizes radical character at ni values near one and ρ()xxxnx==∑∑γϕpq p () ϕ q ()i ψi () (4) closed-shell character for ni values close to zero and two. Below pq i ’ we refer to nu,nl as Head-Gordon s index. and reflect multiconfigurational character of wave functions. For the two-electrons-in-two-orbitals case, the coefficient λ 5 For example, for a single Slater determinant, natural and nu,nl are related by the following expression:

639 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

32λ4 analysis of the underlying wave functions has been performed, 35,39 nunl, = 24 with an exception of several recent studies. One interesting (1+ λ ) (7) question is how the level of correlation (or the functional, in SF-DFT methods) affects the effective number of unpaired As illustrated by the numerical examples in ref 5, nu,nl provides a much more robust and reliable measure of the electrons. This question is addressed here, by means of the diradical character than λ. Furthermore, unlike λ’s, n does not analysis of the density matrices of states obtained by the u,nl ff − depend on orbitals used in the correlated calculation di erent post-Hartree Fock and TDDFT methods. (unrestricted or restricted open-shell Hartree−Fock, Kohn− The structure of the paper is as follows. In the next section, Sham orbitals, etc.). we describe theoretical methods and outline computational In addition to enabling calculations of the number of protocols. We then present our results, beginning with a − effectively unpaired electrons, natural orbitals afford visual- comparison of the equilibrium geometries computed by Kohn ization of the frontier orbitals that are associated with correlated Sham and SF-TDDFT. We use meta-benzyne to illustrate the − many-electron wave functions, thus departing from canonical failure of standard Kohn Sham DFT to describe singlet-state Hartree−Fock MOs. Comparing natural occupations of the structures of strongly correlated systems. We proceed to discuss   frontier orbitals with the computed number of effectively wave function properties in particular, nu and nu,nl using unpaired electrons informs us of how well the frontier orbital the example of H2 along the dissociation coordinate. In Section picture (e.g., two-electrons-in-two-orbitals representation of 3.3, we analyze the character of EOM-SF-CCSD wave functions diradicalas) represents the reality. In the idealized diradical case, of methylene, the di- and tridehydrobenzynes, and the triradical only two frontier natural orbitals contribute to eq 6. xylylenes. We then compare these results with the analysis of Because the one-particle density matrix is a reduced quantity, SF-TDDFT and SF-ADC(2) wave functions. Included in the which can be computed for any wave function (or for Kohn− SF-TDDFT and SF-ADC(2) results section are the CUA- Sham DFT and TD-DFT states), the analyses based on density QAC02 and CITLAT bicopper complexes. We conclude by ff matrices afford comparisons of bonding patterns computed by summarizing relative performance of the di erent approaches. different methods and are also orbital-invariant. The utility of state or transition density matrices as well as difference density 2. THEORETICAL METHODS AND COMPUTATIONAL matrices are well recognized in quantum chemis- DETAILS − − try.6,7,11 14,16 21 Here we apply state-density based analysis As illustrated in Figure 1, diradical character results in tools to characterize the electronic structure of several multiconfigurational wave functions. This multiconfigurational prototypical diradicals and triradicals. character arises due to electronic near-degeneracies of the Polyradicals play important roles in fields as diverse as frontier molecular orbitals. Standard electronic structure photochemistry,22 atmospheric chemistry,23 and molecular methods,40 which are based on the hierarchical improvements magnetism.24 Depending on the type of frontier MOs, they of a single-determinantal Hartree−Fock wave function, fail in can be described as all-σ or all-π, σπ, or spatially separated situations where more than one Slater determinant has a large atom-centric. The degree of interaction and (nominal) bonding contribution. The Kohn−Sham DFT also breaks down in this between unpaired electrons differs for each type and each case. Such strongly correlated systems are sometimes treated by molecular species. We consider the following diradicals: multireference methods based on a multiconfigurational SCF methylene (CH2, same-center diradical); ortho-, meta-, and ansatz and separate description of static and dynamic para-benzyne (σσ); 1-(2-dehydroisopropyl)-4-dehydrobenzyne correlation. Here we employ an alternative approach, the SF (σπ), wherein radical electrons are localized on the tertiary method,25,41 which allows one to describe multiconfigurational carbon of the propane substituent and the para position of the wave functions of the diradical and triradical types in a simple benzene ring; binuclear copper complexes CUAQAC02 and single-reference framework. CITLAT (spatially separated atom-centric). Model triradicals The SF approach is based on the observation that high-spin σ include 1,3,5- and 1,2,4-tridehydrobenzyne (all- ) and 5- states, such as Ms = 1 triplet or Ms = 3/2 quartet, can be well dehydro- and 2-dehydro-meta-xylylene (σπ). Together, these described by a single determinant and that the low-spin states molecules constitute a diverse benchmark set, representative of (singlets and doublets) are formally single excitations from the various bonding patterns one encounters in open-shell species. respective high-spin reference states. Thus, in the SF approach, In this paper we employ several variants of spin-flip (SF) − a high-spin state is used as the reference state, and the target methods.25 29 The SF approach affords a robust and accurate manifold of states is generated by applying a linear spin-flipping description of diradicals and triradicals within a black-box excitation operator to the reference state: single-reference formalism. The performance of different Ψst, = R̂ Ψ t variants of the SF method has been extensively benchmarked, M0sss==−=M1M1 (8) focusing on the energy gaps between the electronic states and − sometimes their structures.27 37 The most insightful compar- dq, ̂ q ΨM1/2= = R M1=− ΨM3/2= (9) isons are based on photoelectron spectra, which provide s s s fi information about the electronic energies and the struc- For diradicals, a triplet reference (Ms = 1 con guration (i) in 34,35 tures. These studies illustrated that the SF-CCSD method panel A of Figure 1) is used, and a quartet reference (Ms = 3/2 provides reliable energy gaps, with errors close to 1 kcal/mol, as configuration (i) in panel B of Figure 1) is used for triradicals. well as reliable photoelectron spectra.34,35 Subchemical Using different approaches to describe correlation in the accuracy can be achieved by including the effect of higher reference state gives rise to different SF methods. For example, excitations perturbatively.38 Within SF-DFT, the best perform- applying this strategy to a Kohn−Sham determinant leads to ance is delivered by B5050LYP (in collinear formulation,27 SF-TDDFT27,28 (in this case, the operator R̂includes only where a large fraction of exact exchange is needed) and by single excitations that flip the spin of an electron). In wave PBE50 (within noncollinear formulation).28 However, no function methods, one can use an uncorrelated reference

640 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

Figure 2. Structures of the four benzyne diradicals (A−D), methylene (E), and four triradicals (F−I) included in this study. Numbers in italics − − correspond to the optimized lowest high-spin state (Ms = 1 triplets for A E and Ms = 3/2 quartets for F I), while underlined numbers correspond to the optimized lowest energy singlet (diradicals) or doublet (triradicals) state computed by SF-TDDFT. Additional structural data for compounds D and H is presented in the SI.

(Hartree−Fock), which gives rise to SF-CIS,25,42 or a correlated 2. In calculations of the CUAQAC02 and CITLAT binuclear one, such as MP2 or CCSD. The accuracy of SF calculations copper diradicals, we used X-ray crystal structures from the can be systematically improved (up to the exact result) by Cambridge Crystal Structure Database.47,48 The perchlorate increasing the level of correlation. In this paper, we consider counterion from the crystal structure of CITLAT was omitted. two wave function approaches: one based on coupled-cluster These structures are shown in the SI. theory (EOM-SF-CCSD or SF-CCSD)26,30 and one based on To assess the effect of the method on the structure, for the algebraic-diagrammatic construction (ADC) scheme for the selected systems, we performed additional geometry optimiza- polarization propagator43 (SF-ADC).29,36 In EOM-SF-CCSD, tions of the singlet and triplet (diradicals) or doublet and the operator R̂includes single and double excitations that flip quartet (triradicals) Kohn−Sham references. Relevant Carte- the spin of an electron. In the SF-ADC(2) method,29 both sian coordinates are provided in the Supporting Information single and double excitations are included, but the doubly (SI). excited determinants are treated only in zeroth order of For all benchmark systems, we present vertical energy gaps, ff perturbation theory. The mathematical structure of the second- computed as energy di erences of the target SF states at the order correction is similar to the one of the SF-CIS(D∞) ground-state equilibrium geometry (optimized by SF-TDDFT). 44 Energy gaps are defined as follows method; however, in CIS(D∞) it is evaluated perturbatively only once using CIS wave functions, whereas in ADC(2) it is ΔEE=− E included already within the iterative solution for the ADC low‐‐ spin high spin (10) vectors, similar to the CC2 method.45 For selected examples, i.e., negative ΔE corresponds to the low-spin ground states we also present results for SF-TDDFT.27,28 (singlets or doublets); note that E ‑ is the energy of the M As one can see, all configurations that appear in electronic high spin s = 0 triplet or Ms = 1/2 quartet state and not the energy of the states shown in Figure 1 are treated on an equal footing within reference state. Natural orbitals and Head-Gordon’s indices of the SF formalism. In diradicals, the target-state manifold different states are also computed at the respective equilibrium comprises the singlets and the Ms = 0 component of the triplet geometries. We used the cc-pVTZ basis in all calculations. state. Likewise, in triradicals the target-state manifold comprises For selected examples, we compare the following levels of the doublets and the low-spin (Ms = 1/2) component of the theory: EOM-SF-CCSD, SF-ADC(2)-s, and SF-TDDFT. In quartet state. Because of the balanced description of all target SF-TDDFT we employed the following functionals: PBE50 states, the energy gaps between the target SF states are more (50% PBE49 and 50% Hartree−Fock exchange with 100% PBE accurate than energy gaps between the reference and the target correlation), B5050LYP (50% Hartree−Fock +8% Slater +42% states. SF methods can be employed to optimize the geometries Becke50 for exchange, with 19% VWN + 81% LYP51 for 26−29 of the target states using analytic gradients. SF wave correlation),27 B97,52 and LDA (Slater exchange with VWN functions can be used to construct one-particle density matrices correlation) functionals. We used the collinear kernel with that can then be analyzed using natural orbitals and wave B5050LYP27 and noncollinear kernel28,53,54 with all other function analysis tools. functionals. 2.1. Computational Details. We performed geometry We used the libwfa module12,13 of Q-Chem to compute and optimizations of the high-spin triplet and quartet states by visualize natural orbitals and Head-Gordon’s indices. We report DFT/B5050LYP/cc-pVTZ. To obtain the structures of the nu,nl, eq 6, and natural occupations of the frontier orbitals. lowest closed-shell singlet or doublet states, we used SF- Because the present implementation of the SF methods is not TDDFT/B5050LYP/cc-pVTZ. For methylene, we used the spin-adapted, the SF states show some (usually small) spin- FCI/TZ2P structures.46 These structures are shown in Figure contamination even if ROHF references are employed.

641 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

Consequently, α and β frontier orbitals (and the respective occupations) are slightly diﬀerent. Below we report spin- average natural occupation, ni̅:

nnn̅ =|αβ + | (11) We also report the diﬀerence between the α and β natural occupations (Δ = |nα − nβ|), which provides an additional measure of spin-contamination. In all ﬁgures, we show only orbitals of the triplet states of diradicals, because the shapes of frontier natural orbitals for singlet and triplet states are indistinguishable for all systems considered in this study. We only show α-orbitals, as the shapes of paired α and β natural Figure 3. Optimized geometries of the lowest singlet state computed α by regular Kohn−Sham DFT (left) and SF-TDDFT (center) and orbitals are the same. For triradicals, we show orbitals for the high-spin triplet state of meta-benzyne (B5050LYP/cc-pVTZ). doublet and quartet states. Distance and angle of separation between radical sites are shown. The Q-Chem electronic structure package was used for all Head-Gordon’s indices for the 3 structures are computed using SF- 55,56 calculations, and orbitals were rendered using Jmol. CCSD wave functions for the lowest singlet state.

3. RESULTS AND DISCUSSION electronic structure method is used for the wave function 3.1. Equilibrium Geometries. The extent of open-shell analysis. We illustrate this point using meta-benzene. character has concrete structural implications,3,4,57 e.g., in Figure 3 shows optimized structures for meta-benzyne. The molecules with large diradical character the structures of triplet rightmost structure is computed by B5050LYP for the high-spin and singlet states are rather similar, in contrast to closed-shell triplet state in which the two electrons are completely unpaired. systems. Consequently, the choice of the electronic structure Because the distance between the two radical centers is large, method is important for obtaining accurate structures. Figure 2 the singlet-state wave function shows considerable diradical shows the optimized geometries for the two lowest spin-states character as indicated by the relatively large value of nu,nl (0.9). of each of the di- and triradicals included in this study (see The central structure is computed by SF-DFT/B5050LYP for Section 2.1 for details). For D and H, we also show the the lowest singlet state. As one can see, the distance between structures computed by the standard Kohn−Sham DFT for the the two radical centers is shorter by about 0.3 Å, as compared low-spin states (Figure S2 in the SI). to the triplet-state structure, due to a partial bond formed by For species with well-separated radical centers, like para- the two electrons. At this geometry, the singlet-state wave benzyne and 1-isopropyl-4-benzyne, the SF optimized geom- function has moderate diradical character (nu,nl = 0.3). The etry of the singlet or doublet state closely resembles the leftmost structure is computed using regular restricted Kohn− optimized geometry of the respective high-spin state, indicative Sham DFT/B5050LYP. Because this approach is not capable of of weakly interacting unpaired electrons. For species with describing diradical character, the optimized structure is modest di- or triradical character, the singlet or doublet bicyclic, with a short distance between the two radical centers structures are markedly different from the high-spin geometries, (this structure is similar to the CCSD structure reported by due to the bonding interactions between the unpaired Crawford and co-workers58). The singlet-state SF-CCSD wave electrons.57 function computed at this geometry shows nearly perfect The structures of triplet or quartet states can be accurately closed-shell character (nu,nl = 0.03). This example illustrates the computed by using standard Kohn−Sham DFT for high-spin importance of performing calculations at the nuclear geometry states (these structures are very close to the structures that corresponds to the correct electronic configuration of a computed for the corresponding SF-DFT states). In cases of molecule (i.e., in the case of meta-benzyne, of a singlet with very weak diradical/triradical character, the structures of the moderate diradical character). − ’ low-spin states can also be computed by regular Kohn Sham 3.2. Head-Gordon s Indices along the H2 Dissociation DFT; however, in cases of relatively strong di/triradical Curve. Before proceeding to wave function analysis in character, one needs to employ the SF-DFT approach, in diradicals and triradicals, let us consider a model example, for order to correctly capture multiconfigurational character of the which CCSD is an exact, dissociation curve of the dihydrogen underlying wave functions. The most notable example is meta- molecule. | − | ff benzyne shown in Figure 3, where the angle of separation Figure 4 shows nu, nu,nl, nu nu,nl , and the di erence in total between the radical carbons in the SF optimized singlet B5050LYP and CCSD state energies as a function of geometry is roughly 20 degrees less than in the structure internuclear distances R for the lowest singlet state of H2. optimized by regular closed-shell Kohn−Sham DFT. Tabulated raw data is provided in the SI. Using a method that fails to correctly describe open-shell As expected, around equilibrium the number of unpaired character can lead to catastrophically wrong structures.4 One electrons is small. As the internuclear distance R increases, well studied example is meta-benzyne.58 In this system, single- diradical character increases, reaching 2 at the dissociation limit. reference methods such as regular Kohn−Sham DFT and At the equilibrium and at the dissociation limit, SF-TDDFT CCSD underestimate the diradical character and yield agrees well with SF-CCSD (exact result). However, at (incorrect) bicyclic structures, whereas methods that do not intermediate distances, SF-TDDFT underestimates the number include dynamic correlation (valence bond, CASSCF) exag- of unpaired electrons. Compare, for example, the two blue gerate the distance between the radical centers. The structure curves, which show the respective nu,nls. At 2 Å, the exact wave used in wave function analysis has a strong effect on the function has 1 unpaired electron, whereas the SF-TDDFT wave bonding pattern. That is, using a wrong structure will produce function has only 0.25. Only 0.5 Å further SF-DFT develops an incorrect number of unpaired electrons, even if an accurate open-shell character yielding nu,nl = 1. This lag is observed for nu

642 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

3.3. EOM-SF-CCSD Energies and Wave Function Character in Diradicals and Triradicals. Table 1 shows ⟨ 2̂⟩ energy separations, nu,nl, and S for the lowest states of di- and | − | triradical benzynes and methylene. nu and nα nβ for each state are provided in the SI. All states considered suﬀer very little from spin-contamination. For high-spin states, nu,nl is very close to the ideal values of 2 unpaired electrons for triplet states and 3 unpaired electrons for quartet states. In contrast, nu,nl for singlets and doublets depends very much on the nature of the di- or triradical ground state. For species with a singlet ground state, nu,nl of the ground state ranges from closed-shell values close to zero (for singlets with modest radical character) to values close to two (for open-shell singlets and strong diradical character). The same is true for triradicals with doublet ground states; namely, “closed-shell” doublets like those observed in F and G have nu,nl values close to 1, while open-shell doublets such as the ground state of I have nu,nl values close to 3. We Δ observe that radical character (nu,nl) increases and E becomes more positive as the distance between the radical carbons increases from A−D.

Table 1. EOM-SF-CCSD Energy Gaps (eV) and Wave a Function Properties of the Lowest Two States | − | − Δ ⟨ 2̂⟩ Figure 4. nu, nu,nl, and nu nu,nl (top) and EB5050LYP ECCSD ( E) nu,nl S | B5050LYP − CCSD| Δ and nu,nl nu,nl ( n, bottom) computed for the lowest singlet singlet/ triplet/ singlet/ triplet/ state of H2 along the bond-stretching coordinate using EOM-SF- molecule ΔE doublet quartet doublet quartet CCSD and SF-TDDFT/B5050LYP with the cc-pVTZ basis set. CH2 0.94 0.25 2.00 0.00 2.00 A −2.46 0.16 2.00 0.00 2.00 B −1.90 0.26 2.00 0.01 2.01 C −0.24 1.45 2.00 0.01 2.01 and nu,nl. Strict SF-ADC(2) does not give exact excited states due to the zeroth-order treatment of the doubly excited D 0.24 2.00 2.01 0.01 2.04 − determinants, but improving the description of the doubly F 2.21 1.28 3.06 0.88 3.99 − excited configurations to first order, as in SF-ADC(2)-x and SF- G 2.00 1.16 3.02 0.82 3.83 ADC(3), yields exact states for H , as SF-CCSD (only the H 0.41 3.01 3.02 0.79 3.83 2 − energy differences, such as singlet−triplet gaps, are exact but I 0.14 3.01 3.01 0.78 3.80 a not the total state energies). Singlets and Ms = 0 triplets for diradicals, and Ms = 1/2 doublets and Regardless of the level of theory, at the dissociation limit the quartets for triradicals. ff di erence between nu and nu,nl is small. However, at small and intermediate distances we observe noticeable discrepancy between nu and nu,nl. Compared to nu,nl, around the equilibrium Natural orbitals of the lowest singlet/doublet and triplet/ nu overestimates diradical character, and at longer distances it quartet states of methylene and the di- and triradical benzynes underestimates it. As expected, maximum deviation occurs are shown in Figure 5. With the exception of methylene, D, H, around the natural orbital occupation numbers of 0.25 and 1.75 and I, all di- and triradical frontier orbitals are of the σ type, electrons, where the quadratic nature of the expression for nu,nl consistent with the molecular orbital patterns reported by suppresses and enhances radical character, respectively.9 The Krylov and Cristian.57 The natural orbitals in D, H, and I are of σπ point along the dissociation curve where the nu and nu,nl curves type. ≈ intersect at nu = nu,nl 1 depends on the level of theory. This The values of n̅show spin-averaged occupancy of each spatial large discrepancy between the two quantities is only observed orbital. The trends in orbital occupancy are consistent with nu,nl:  for the singlet state nu and nu,nl equal exactly 2 for the high- increased radical character is ascribed to ground states with spin reference and low-spin triplet states at every point along natural orbitals n̅-values close to 1. Values in parentheses (Δn) the dissociation curve from 0.74 to 5.00 Å (raw data is provided indicate spin imbalance in orbital occupancy arising due to in the SI). spin-incompleteness of the underlying wave function. For The lower panel of Figure 4 shows the difference between diradicals, Δn-values are close to the ideal value of zero. the SF-CCSD and SF-DFT total energies and the respective Doublet tridehydrobenzynes also exhibit ideal Δn-values, with nu,nl. At large internuclear separation (>3 Å), the two methods only one predominantly singly occupied natural orbital hosting yield identical nu,nl, and the respective potential energy curves the odd electron that gives rise to the doublet. Quartet states of are nearly parallel. However, at shorter distances (less than 3 the tridehydrobenzynes, and open-shell doublet and quartet Å), where we observe large discrepancies between the number states of the xylylene triradicals, feature natural orbitals that are of unpaired electrons computed by SF-CCSD and SF-DFT, we predominantly singly occupied (n̅close to 1) but suffer from also observe large nonparallelity errors in the SF-DFT potential some spin-imbalance. This imbalance appears to be minor and energy curve. This example illustrates that the errors in SF-DFT does not strongly impact other properties of these states, like energies originate from errors in underlying densities. the ⟨Ŝ2⟩ values or the sign and the magnitude of ΔE.

643 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

Figure 5. EOMSF-SF-CCSD/cc-pVTZ natural orbitals of lowest singlet/doublet and triplet/quartet (low-spin) states of A−I. n̅= |nα + nβ|, with Δn | − | = nα nβ provided in parentheses. n̅s and n̅t correspond to n̅values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. ff For all systems, we observe that the di erence between nu,nl representation of the wave function in the basis of natural and nu is much smaller for the low-spin triplet states than for orbitals provides a more facile interpretation of the underlying the singlet states. This is partially because in the limiting case of electronic structure than the canonical MOs. Figure 6 shows 2 unpaired electrons the difference between the two indices is singly occupied natural and canonical MOs for the triplet states expected to vanish. For example, the two indices give identical of the two binuclear copper diradicals alongside their spin- results for the singlet state of dihydrogen at the dissociation difference densities. limit and identical results for the triplet states at all distances, The highest canonical MOs in CUAQAC02 and CITLAT yet for many-electron electron examples, even for triplet states, (which have 202 and 278 electrons, respectively) appear to be nu assumes values that are slightly larger than 2. This can be delocalized. Furthermore, despite low spin-contamination of − α β attributed to the fact that nu,nl suppresses contributions from the the Kohn Sham triplet reference, and MOs cannot be dynamic correlation. Thus, the second reason for much smaller easily matched. Thus, by considering canonical MOs only, it is discrepancy between the two indices for the triplet states difficult to ascribe overall orbital character or localization to the relative to the singlets can be attributed to the smaller dynamic unpaired electrons that give rise to the diradical states. In correlation, which is characteristic of the triplet wave contrast, the frontier natural orbitals obtained from the triplet 25 functions. Interestingly, the high-spin reference states show density matrix have expected shapes that can be described as dxy ff a larger di erence between nu,nl and nu than the respective low- or dyz orbitals localized on the two copper centers. The spin- spin components of the same multiplicity. density differences (shown in Figure 6) are consistent with the 3.4. Molecular Magnets: Natural Orbitals versus shapes of frontier natural orbitals. Molecular Orbitals. Two binuclear copper diradicals, 3.5. Comparison between EOM-SF-CCSD, SF-TDDFT, CUAQAC02 and CITLAT (Figure S1), are included in the and SF-ADC(2)-s wave functions. Tables 2, 3, 4, and 5 show SF-TDDFT wave function analysis benchmark study. Their DFT and ADC(2) energy gaps and state properties for the low- experimental exchange-coupling constants (equal to ΔE within lying spin-states of di- and triradical benzynes, methylene, − | − | the Heisenberg Dirac-van-Vleck model, which assumes weak CUAQAC02, and CITLAT. nu and nα nβ for triradicals are interactions between the unpaired electrons) are −286 and 113 provided in the SI. Figures S4−S7 in the SI show natural cm−1, respectively.47,48 This example illustrates that a compact orbitals of the relevant spin-states at each level of theory.

644 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

Table 3. SF-TDDFT/cc-pVTZ Energy Splittings (eV) and Wave Function Properties of Lowest Ms = 1/2 Doublet and a Quartet States of Triradicals

⟨ ̂2⟩ nu,nl S molecule method ΔE doublet quartet doublet quartet F PBE50 −2.18 1.17 3.01 0.87 3.96 B5050LYP −2.44 1.09 3.00 0.84 3.88 LDA −2.80 1.01 3.00 0.78 3.80 B97 −2.83 1.02 3.00 0.80 3.81

G PBE50 −2.64 1.09 3.00 0.79 3.80 B5050LYP −2.82 1.06 2.01 0.79 3.79 LDA −3.16 1.01 3.00 0.76 3.75 B97 −3.16 1.01 3.00 0.78 3.77

H PBE50 0.39 3.06 3.05 1.42 3.81 B5050LYP 0.23 3.04 3.04 0.99 4.10 LDA 0.28 3.00 3.00 0.79 3.80 B97 0.46 3.01 3.01 1.12 3.64 Figure 6. Spin-diﬀerence densities, unrestricted singly occupied (SO) molecular and natural frontier orbitals of high-spin triplet states of − CUAQAC02 (left) and CITLAT (right) computed by B5050LYP/cc- I PBE50 0.38 3.01 3.04 1.79 3.21 pVTZ. B5050LYP −0.14 3.01 3.03 1.77 3.16 LDA 0.03 3.00 3.00 1.25 3.29 Table 2. SF-TDDFT and ADC(2)-s Energy Splittings (eV) B97 0.19 3.00 3.00 1.74 2.91 a and Wave Function Properties of Lowest Singlet and Ms =0 Results for four density functionals are shown. Triplet States of Diradicals −1 ⟨ 2⟩ Table 4. SF-TDDFT Energy Splittings (cm ) and Wave nu,nl S Function Properties of Lowest Singlet and Ms = 0 Triplet molecule method ΔE singlet triplet singlet triplet States of Binuclear Copper Diradicals CH PBE50 1.03 0.30 2.00 0.01 2.02 2 n ⟨Ŝ2⟩ B5050LYP 0.30 0.08 2.00 0.01 2.01 u,nl LDA 0.93 0.33 2.00 0.01 2.01 molecule method ΔE singlet triplet singlet triplet B97 0.23 0.04 2.00 0.01 2.00 CUAQAC02 PBE50 −123 1.97 2.00 0.01 2.01 ADC(2) 1.02 0.33 2.01 B5050LYP −148 1.96 2.00 0.01 2.01 LDA −1420 0.47 2.00 0.01 2.01 A PBE50 −2.38 0.09 2.00 0.03 2.02 B97 71 2.59 2.69 1.07 1.77 B5050LYP −2.66 0.04 2.00 0.02 2.02 CITLAT PBE50 100 2.00 2.00 0.01 2.02 LDA −2.86 0.01 2.00 0.01 2.02 B5050LYP 77 2.00 2.00 0.01 2.01 B97 −2.84 0.01 2.00 0.02 2.01 LDA 411 1.96 2.00 0.01 2.01 ADC(2) −2.46 0.32 2.17 B97 228 1.98 2.00 0.01 2.01

B PBE50 −1.85 0.19 2.00 0.07 2.11 methods and reasonable agreement with EOM-SF-CCSD for B5050LYP −2.04 0.12 2.00 0.05 2.07 organic radicals. LDA and B97 fail for binuclear copper radicals LDA −2.23 0.01 2.00 0.02 2.03 with respect to energies, overestimating the degree of closed- B97 −2.22 0.03 2.00 0.03 2.05 shell character of the low-spin states relative to PBE50 and ADC(2) −1.97 0.41 2.17 B5050LYP. The two nonhybrid functionals appear sufficient for describing the underlying wave function character of di- and C PBE50 −0.19 1.47 2.00 0.03 2.03 triradicals with respect to the shapes of natural orbital, with the B5050LYP −0.23 1.29 2.00 0.02 2.02 notable exception of CUAQAC02 and the B97 functional, LDA −0.52 0.30 2.00 0.01 2.02 where the dyz orbitals predicted by EOM-CCSD, PBE50, B97 −0.40 0.63 2.00 0.02 2.02 B5050LYP, and LDA are not the frontier natural orbitals (B97 ADC(2) −0.24 1.58 2.18 frontier natural orbitals of CUAQAC02 and CITLAT shown in Figure S4 in the SI). LDA and B97 appear to systematically D PBE50 0.24 2.02 2.02 0.33 2.10 underestimate diradical character of singlet states as reflected by B5050LYP 0.16 2.01 2.02 0.23 2.10 lower nu,nl and ns̅values, relative to PBE50 and B5050LYP. LDA 0.20 2.00 2.00 0.02 2.03 SF-ADC(2) has been used to compute the energy splittings B97 0.44 2.00 2.00 0.66 1.47 ’ − and Head-Gordon s indices of CH2 and A D (Tables 2 and 5). ADC(2) 0.20 2.24 2.24 For these molecules, the singlet−triplet energy splittings (ΔE) computed with SF-ADC(2) agree very favorably with the ones Diradicals with states affected by spin-contamination typically obtained at the EOM-SF-CCSD level of theory. The largest | − | ’ have large nα nβ values. There is agreement in natural orbital deviation is 0.08 eV found for CH2. Inspecting Head-Gordon s   character and sometimes energetics among DFT index nu,nl for these molecules (Table 2), we note a good overall

645 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article n n a Table 5. u and u,nl at the SF-TDDFT and ADC(2)-s/cc-pVTZ Levels of Theory triplet singlet | − | | − | molecule method nu nu,nl nu nu,nl nu nu,nl nu nu,nl

CH2 PBE50 2.014 2.000 0.014 0.447 0.303 0.144 B5050LYP 2.010 2.000 0.010 0.217 0.079 0.138 LDA 2.008 2.000 0.008 0.469 0.332 0.137 B97 2.013 2.000 0.013 0.166 0.045 0.121 ADC(2) 2.114 2.009 0.105 0.578 0.326 0.252

A PBE50 2.016 2.000 0.016 0.246 0.091 0.155 B5050LYP 2.012 2.000 0.012 0.171 0.044 0.127 LDA 2.010 2.000 0.010 0.070 0.007 0.063 B97 2.014 2.000 0.014 0.095 0.013 0.082 ADC(2) 3.133 2.171 0.962 1.458 0.323 1.135

B PBE50 2.070 2.003 0.067 0.399 0.192 0.207 B5050LYP 2.045 2.001 0.044 0.302 0.116 0.186 LDA 2.020 2.000 0.020 0.092 0.011 0.081 B97 2.033 2.001 0.032 0.154 0.030 0.124 ADC(2) 3.146 2.174 0.972 1.546 0.406 1.140

C PBE50 2.019 2.000 0.019 1.272 1.470 0.198 B5050LYP 2.014 2.000 0.014 1.129 1.286 0.157 LDA 2.010 2.000 0.010 0.450 0.304 0.146 B97 2.014 2.000 0.014 0.693 0.627 0.066 ADC(2) 3.159 2.179 0.980 2.377 1.577 0.800

D PBE50 2.161 2.019 0.142 2.164 2.019 0.145 B5050LYP 2.142 2.015 0.127 2.118 2.010 0.108 LDA 2.019 2.000 0.019 2.018 2.000 0.018 B97 2.055 2.002 0.053 2.047 2.002 0.045 ADC(2) 3.796 2.244 1.552 3.785 2.237 1.548

CUAQAC02 PBE50 2.010 2.000 0.010 1.845 1.973 0.128 B5050LYP 2.009 2.000 0.009 1.814 1.961 0.147 LDA 2.004 2.000 0.004 0.567 0.466 0.101 B97 2.815 2.685 0.130 2.754 2.589 0.165

CITLAT PBE50 2.011 2.000 0.011 1.964 1.998 0.034 B5050LYP 2.010 2.000 0.010 1.951 1.996 0.045 LDA 2.006 2.000 0.006 1.804 1.959 0.155 B97 2.009 2.000 0.009 1.861 1.978 0.117 a Values are provided for the lowest singlet and Ms = 0 triplet states of each diradical benchmark system. Results for four density functionals are compared. agreement with the EOM-SF-CCSD values. The SF-ADC(2) 3.6. Adiabatic Singlet−Triplet and Doublet-Quartet values are consistently slightly larger by about 0.15. Further Gaps. All energy diﬀerences between low-lying spin states detailed investigations of the quality of the wave functions reported here are vertical energy gaps, computed at the obtained with SF-ADC approaches, including higher orders of equilibrium geometries of the ground state by optimizing the perturbation theory, will be the topic of future work. lowest energy SF-TDDFT/cc-pVTZ state. In order to assess the absolute accuracy of the methods by comparison against Natural orbitals of the binuclear copper complexes are of dxy experimental values, one needs to consider adiabatic gaps, as and dyz type and are well-localized on copper centers. The 28,30,31,60 unpaired electrons of CITLAT (the copper diradical with a was done in previous benchmark studies. As a guidance for readers, here we provide a compilation of adiabatic energy positive ΔE) exhibit some through-space antibonding inter- gaps for the benchmark compounds considered here (with the action with p-orbitals of neighboring oxygen atoms in the exception of D, CUAQAC02,andCITLAT)reported bonding plane. This interaction is absent in CUAQAC02 (the elsewhere28,30,31,60 and are summarized in Tables S5 and S6 Δ copper diradical with a negative E), in which singly occupied in the SI. Note that in the reported experimental values zero- natural orbitals can be described as the dyz orbitals that give rise point energies are subtracted. As one can see, EOM-SF- to δδ* type interactions and bond orders greater than three CCSD(dT) is within 1 kcal/mol from the experiment. EOM- between transition metal nuclei in more strongly interacting SF-CCSD delivers consistent performance and is very close to bimetallic complexes.59 EOM-SF-CCSD(dT). This comparison justiﬁes using EOM-

646 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648 Journal of Chemical Theory and Computation Article

SF-CCSD vertical gaps as the benchmark for SF-DFT and Present Address § ADC(2). Division of Theoretical Chemistry and Biology, School of Biotechnology, Royal Institute of Technology, SE-106 91 4. CONCLUSION Stockholm, Sweden. Analysis of the natural orbitals derived from the one-particle Funding density matrices provides insight into the extent and type of This work is supported by the Department of Energy through radical character for a variety of di- and triradical species. This the DE-FG02-05ER15685 grant (A.I.K.). analysis affords quantitative comparisons of electronic properties beyond energy differences computed by different methods. Notes In particular, using these tools one can compare the The authors declare the following competing financial performance of wave function and DFT methods. In agreement interest(s): A.I.K. is a member of the Board of Directors and with earlier benchmark studies, we observe good agreement a part-owner of Q-Chem, Inc. between EOM-SF-CCSD and SF-DFT (when using recom- mended functionals). The comparison of natural orbitals and ■ REFERENCES their occupations computed for the lowest singlet and triplet (1) Salem, L.; Rowland, C. The electronic properties of diradicals. and doublet and quartet states indicates good agreement Angew. Chem., Int. Ed. Engl. 1972, 11,92−111. between EOM-SF-CCSD and TDDFT in most cases. We (2) Bonacič-Koutecký ,́ V.; Koutecky,́ J.; Michl, J. Neutral and charged observe very good agreement among DFT functionals with biradicals, zwitterions, funnels in S1, and proton translocation: Their regard to the character of frontier natural orbitals; however, the role in photochemistry, photophysics, and vision. Angew. Chem., Int. respective occupations (and, consequently, the effective Ed. 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648 DOI: 10.1021/acs.jctc.7b01012 J. Chem. Theory Comput. 2018, 14, 638−648