J. Chem. Phys. 152, 094108 (2020); https://doi.org/10.1063/1.5143318 152, 094108

© 2020 Author(s).

Effective Hamiltonians derived fromequation-of-motion coupled-cluster wavefunctions: Theory and application to theHubbard and Heisenberg HamiltoniansCite as: J. Chem. Phys. 152, 094108 (2020); https://doi.org/10.1063/1.5143318Submitted: 21 December 2019 . Accepted: 16 February 2020 . Published Online: 03 March 2020

Pavel Pokhilko , and Anna I. Krylov

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Effective Hamiltonians derivedfrom equation-of-motion coupled-cluster wavefunctions: Theory and application to the Hubbardand Heisenberg Hamiltonians

Cite as: J. Chem. Phys. 152, 094108 (2020); doi: 10.1063/1.5143318Submitted: 21 December 2019 • Accepted: 16 February 2020 •Published Online: 3 March 2020

Pavel Pokhilko and Anna I. Krylova)

AFFILIATIONSDepartment of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA

a)Author to whom correspondence should be addressed: krylov@usc.edu

ABSTRACTEffective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exactmany-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not trivial. In this contribution,we apply Bloch’s formalism to equation-of-motion coupled-cluster wave functions to rigorously derive effective Hamiltonians in Bloch’s anddes Cloizeaux’s forms. We report the key equations and illustrate the theory by application to systems with two or three unpaired electrons,which give rise to electronic states of covalent and ionic characters. We show that Hubbard’s and Heisenberg’s Hamiltonians can be extracteddirectly from the so-obtained effective Hamiltonians. By establishing a quantitative connection between many-body states and simple mod-els, the approach facilitates the analysis of the correlated wave functions. We propose a simple diagnostic for assessing the validity of themodel space choice based on the overlaps between the target- and model-space states. Artifacts affecting the quality of electronic structurecalculations such as spin contamination are also discussed.

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I. INTRODUCTION

Coarse graining is commonly used in computational chem-istry and physics. It can be deployed in the real space, when agroup of atoms is replaced by an effective particle, or in the Hilbertspace, when a group of many-body electronic states is representedby a set of states expressed in a compact basis of model states.1

Coarse-graining is exploited in a number of classic models serv-ing as a foundation of modern solid-state physics: tight bind-ing,2,3 Drude–Sommerfeld’s model,4–6 and Hubbard’s7 and Heisen-berg’s8–10 Hamiltonians. These models explain macroscopic proper-ties of materials through effective interactions whose strengths aretreated as model parameters. The values of these parameters aredetermined either from more sophisticated theoretical models or byfitting to experimental observables.

For example, in Hubbard’s model, the model space comprisesconfigurations obtained by distributing N electrons over M sites and

each site is represented by just one localized orbital, as illustratedin Fig. 1 for the three-electrons-in-three-centers case. Hubbard’sHamiltonian includes electron-hopping terms (one-electron transi-tions facilitated by the couplings between neutral and ionic configu-rations) described by the parameter t and the site energies (energiesof the ionic configurations) described by the parameter U,

HHub =M

∑i≠j∑σtija†

iσajσ + UM

∑iniαniβ, (1)

where indices i, j denote localized orbitals,11 σ denotes spin part αor β, operators a†

iσ and aiσ are the creation and annihilation oper-ators of an electron on a localized orbital i with spin σ, and niσ isa particle number operator, niσ = a†

iσaiσ . While the configurationalspace in Hubbard’s model is the same as in the complete-active spacewave function (using the same orbital space), the two models arenot the same: Eq. (1) does not admit two-electron interactions, such

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FIG. 1. Configurations entering three-electron-in-three-centers model Hamiltoni-ans. Each center (or site) is represented by one localized orbital. The configu-rations are built from different distributions of three electrons on three localizedorbitals (numbered 1, 2, and 3 and colored with yellow, green, and violet circles,respectively). Each box represents one configuration. Heisenberg’s model spaceincludes only open-shell configurations in which the orbitals are singly occupied.A half-filled (i.e., three-electrons-in-three orbitals) Hubbard’s model space alsoincludes ionic configurations in which localized orbitals can host two electrons.

as direct exchange terms that couple first three configurations fromFig. 1. Although Hubbard’s model parameterizes only one-particleinteractions between the configurations, it includes the Coulombrepulsion U between the conducting (active) electrons, making itsufficiently flexible to describe non-trivial physical phenomena, suchas Mott–Hubbard’s phase transition.12

Heisenberg’s model8–10 describes the interaction betweenopen-shell configurations in terms of local spin (an effective quan-tity that we discuss below). This model is most commonly usedto describe magnetic properties of solids.12 It can be writtenas

HHeis = −∑A<B

JABSASB, (2)

where A, B enumerate radical (or magnetic) centers, JAB is an effec-tive exchange constant, and SI is an effective localized spin operatorassociated with center I. Heisenberg’s Hamiltonian can be derivedfrom Hubbard’s Hamiltonian through degenerate13 and canonicalperturbation theories;14 thus, it can be considered an effective the-ory with respect to Hubbard’s model. Although Heisenberg’s modelspace does not admit ionic configurations, their effect is folded intothe effective exchange parameters.

Effective Hamiltonian theory provides a powerful frame-work for a rigorous construction of effective Hamiltonians frommulticonfigurational many-electron wave functions computed abinitio. Pioneering works of Kato,15 Ôkubo,16 Bloch,17 and desCloizeaux18 have established the foundations of the operatoreffective Hamiltonian theory. Further development came from

Löwdin’s partitioning19 technique, Feshbach’s formalism,20 andgeneralizations.21–24 The history and recent developments of effec-tive Hamiltonian theories are summarized in comprehensivereviews.25,26

Effective theories fulfill a dual role. On one hand, they can beused to develop new electronic structure methods with the system-atically improved description of effective Hamiltonians. For exam-ple, perturbative construction of the wave operator and effectiveHamiltonians has been exploited in the development of various mul-tireference perturbative27–29 and coupled-cluster30–32 methods. Onthe other hand, effective theories can be used as an interpretationtool by providing an essential description of the complex electronicstructure.

The distinction between using effective Hamiltonians for anal-ysis of the electronic structure and method development is notalways binary. For example, introduction of Hubbard’s repulsionterm U into a density functional expression allowed Anisimov andco-workers to develop a widely used DFT+U method.33–35 Mayhalland Head-Gordon noted that effective exchange couplings J from asingle spin-flip (SF) calculation agree with the couplings extractedfrom more computationally expensive n-SF calculations,36 whichled to the development of an effective computational scheme forstrongly correlated systems exploiting a coarse-graining idea.37 InMayhall’s protocol, a single spin-flip calculation is performed first.Then, under the assumption of Heisenberg’s physics, the exchangecouplings are extracted, and an effective Hamiltonian is constructedto compute the spin states that are not accessible in the singlespin-flip calculation.

The parameters of effective Hamiltonians also can be obtainedindirectly. Experimentally, exchange couplings can be extractedfrom the temperature dependence of magnetic susceptibility,electron paramagnetic resonance, and neutron scattering experi-ments.38–44 Theoretically, if there are known relations between thestates of interest, such as the Landé interval rule,45 advanced elec-tronic structure methods can be easily applied.46 Even more indi-rectly, exchange couplings can be extracted from contaminatedsolutions47 obtained in broken-symmetry density functional theory(BS-DFT) or broken-symmetry coupled-cluster48 calculations.

Numerous studies49–52 have used Bloch’s and des Cloizeaux’sformalisms to build Heisenberg’s and Hubbard’s effective Hamil-tonians from configuration interaction (CI) wave functions. In thiscontribution, we apply Bloch’s formalism to the equation-of-motioncoupled-cluster (EOM-CC) wave functions to rigorously deriveeffective Hamiltonians in Bloch’s and des Cloizeaux’s forms. Wereport the key equations and illustrate the theory by examples ofsystems with several unpaired electrons, giving rise to electronicstates of covalent and ionic characters. Our goal is to establish arigorous mapping between the EOM-CC solutions and effectiveHamiltonians and to provide a theoretical basis for the extraction ofeffective parameters from the EOM-CC calculations. These param-eters can be directly compared with the parameters extracted fromexperimental measurements, facilitating unambiguous comparisonbetween the theory and experiment. In addition, such a mappingprovides physical insights into the complex electronic structure ofstrongly correlated systems. Finally, the theoretical developmentspresented here serve as a stepping stone toward the developmentof coarse-grained models of electron correlation in large systems,following Mayhall’s and Head-Gordon’s ideas.36,37

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The structure of this paper is as follows: In Sec. II, we pro-vide an overview of the EOM-CC theory and Bloch’s formalism. Wethen apply Bloch’s formalism to the EOM-CC wave functions andderive the working expressions. Next, we consider several moleculesfor which we discuss Hubbard’s and Heisenberg’s Hamiltonians rig-orously constructed from the EOM-SF-CCSD solutions. Wheneverpossible, we compare the results from extraction through the Landérule and from the effective Hamiltonians.

II. THEORYA. Equation-of-motion coupled-cluster theory

Similar to CI, EOM-CC methods53–56 utilize linear parameteri-zation of the wave function,

∣ΨRI ⟩ = RI ∣Φ0⟩, (3)

⟨ΨLI ∣ = ⟨Φ0∣L†

I , (4)

where ΨI is a EOM target state, R and L are general excitationoperators, and Φ0 is the reference determinant, which defines theseparation between occupied and virtual orbital spaces. The equa-tions for the amplitudes of the EOM operators are derived vari-ationally, leading to a CI-like eigenproblem.57 In contrast to CI,the EOM theory employs a (non-Hermitian) similarity-transformedHamiltonian,

H = e−THeT , (5)

where T is an excitation operator. BecauseH has the same spectrumas the bare Hamiltonian regardless of the choice of T, solving theEOM eigenproblem in the full configurational space, i.e., when Rincludes all possible excitations, recovers the exact (full CI or FCI)limit. In practical calculations, these operators are truncated, mostoften to single and double excitations, giving rise to EOM-CCSDansatz, and the choice of the operator T becomes important. Forexample, choosing T = 1 leads to plain CISD (CI with singles anddoubles), but taking T from the coupled-cluster equations for thereference state amounts to including correlation effects and ensuressize-intensivity. Most often, T is truncated at the same level as R; forexample, in EOM-CCSD, one uses

T =∑iatai a

†i +14∑ijab

tabij a†b†ji + . . . , (6)

where i, j, . . . and a, b, . . . denote occupied and virtual (withrespect to Φ0) orbitals. When T satisfies the CC equations, then thereference determinant Φ0 is also an eigen-state of H.

Because H is non-Hermitian, the left and right EOM eigenstatesare not Hermitian conjugates of each other,

HRI ∣Φ0⟩ = EIRI ∣Φ0⟩, (7)

⟨Φ0∣L†I H = ⟨Φ0∣L†

I EI , (8)

but are often chosen to form a biorthogonal set,

⟨Φ0∣L†I RJ ∣Φ0⟩ = δIJ . (9)

Because the excitation operators T and R commute, one canalso write the EOM states as

∣ΨRI ⟩ = RI ∣ΨCC⟩, (10)

where

∣ΨCC⟩ = eT ∣Φ0⟩. (11)This form shows clearly that the EOM-CC states include higher exci-tations than the respective CI states by virtue of the wave operatoreT and that the EOM states are excited states with respect to thereference-state CC wave function. Note that in the derivations belowwe use EOM states as defined by Eqs. (3) and (4) rather than (10)and (11).

Different choices of R allow access to different manifolds oftarget states, giving rise to a variety of EOM-CC methods.53–56 Com-mon choices include excitation,58 spin-flip,59–61 ionization,62,63 orelectron-attachment64 operators. For example, spin-flip operatorsare

RSF=∑

iarai a

†i +14∑ijab

rabij a†b†ji + . . . , (12)

LSF =∑ialai a

†i +14∑ijab

labij a†b†ji + . . . , (13)

where the number of creation operators corresponding to α orbitalsis not equal to the number of annihilation operators correspondingto α orbitals, giving rise to a non-zero MS (spin-projection) of R (incontrast, the EOM-EE operators are of the MS = 0 type).

B. Effective Hamiltonians1. General formalism

Effective theories operate with target (large) and model (small)spaces, as illustrated in Fig. 2. Both spaces describe the same setof eigenstates, but using different sets of configurations. The targetspace contains long multiconfigurational expansions, i.e., it can be afull Hilbert space of a many-body system. The model space containsexpansions over a small set of configurations.

FIG. 2. Target and model spaces are connected through a wave operator Ω. Thetarget space can be the full configurational (i.e., Hilbert) space of the system, whilethe model space is a subspace of the full configurational space.

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Bloch’s theory usually starts from target states, which are eigen-vectors of the Hermitian Hamiltonian,

H∣Ψμ⟩ = Eμ∣Ψμ⟩, (14)

P =n

∑μ∣Ψμ⟩N−2

μ ⟨Ψμ∣. (15)

Here, the eigenstates Ψμ are not necessarily normalized to one. Pdenotes a projector onto the target space, Nμ is the normalizationfactor of the eigenstate Ψμ, Eμ is the energy of Ψμ, and n is the dimen-sion of the target space. The selection of the eigenstates in the targetspace is determined by the problem at hand—we include only thoseeigenstates that are relevant to a specific physics. The number ofstates included in the model space (i.e., the dimension of the modelspace) is the same as that in the corresponding target space. EffectiveHamiltonians are defined in the model space such that, by construc-tion, their eigenvalues reproduce the eigenvalues of the target-spaceHamiltonian,

Heff∣ΨR

μ⟩ = Eμ∣ΨRμ⟩, (16)

P0 =n

∑μ∣ΨR

μ⟩⟨ΨLμ ∣, (17)

where ∣ΨRμ⟩ and ⟨ΨL

μ ∣ are right and left eigenstates of the effectiveHamiltonian, and P0 is the projector onto the model space. In otherwords, the idea behind the effective Hamiltonian entails the trans-formation of the interaction between the model vectors to reproducethe energies in the target space. The model space is connected to thetarget space through a wave operator Ω,

∣Ψμ⟩ = Ω∣Ψμ⟩. (18)

The relations (14)–(18) lead to generalized Bloch’s equation, estab-lishing a connection between the wave operator and the effectiveHamiltonian,

HΩP0 = ΩP0Heff P0. (19)

Effective Hamiltonians in Eq. (16) are defined up to a similar-ity transformation. Therefore, a constraint is needed for a uniquedefinition. Bloch’s constraint (which we denote through a subscript“B”), also known as intermediate normalization, is

P0ΩBP0 = P0. (20)

This constraint leads to the following expressions for the waveoperator and effective Hamiltonian:21

ΩB = P(P0PP0)−1 (only in subspaces), (21)

HeffB = P0H

effB P0 = P0HΩBP0. (22)

Equations (21) and (22) mean that if the exact eigen-states andeigen-energies are known, one can select a model space, computethe projectors (and, consequently, the wave operator), and constructHeff

B , which, by construction, will yield the same eigen-energies as H.As an interesting example of an effective theory, one can

consider coupled-cluster theory. The exponential operator eT with

untruncated T has the meaning of the wave operator connecting themodel space (a single Slater determinant) with the exact (FCI) space.Coupled-cluster equations require that the reference determinantΦ0

is the eigenfunction of the effective HamiltonianH,

∣Φ0⟩ΩÐ→ ∣ΨCC⟩ = Φ0. (23)

Thus, eT maps the exact many-body ground state into a single Slaterdeterminant.

Bloch’s effective Hamiltonian is not Hermitian, which may notbe convenient for the analysis. Des Cloizeaux gave a recipe for pro-ducing a Hermitian effective Hamiltonian18 by means of symmet-ric orthogonalization of Bloch’s eigenvectors. An arbitrary Hermi-tian effective Hamiltonian can be obtained from Bloch’s effectiveHamiltonian through a similarity transformation A. For any Bloch’seigenprojector, this can be written as

A−1∣ΨR

B,μ⟩Eμ⟨ΨLB,μ∣A = ∣ΨH,μ⟩Eμ⟨ΨH,μ∣, (24)

A−1∣ΨR

B,μ⟩ = ∣ΨH,μ⟩, (25)

⟨ΨLB,μ∣A = ⟨ΨH,μ∣, (26)

∣ΨRB,μ⟩ = AA

†∣ΨL

B,μ⟩. (27)

Since Bloch’s eigenvectors are biorthogonal,

AA†=∑

μ∣ΨR

B,μ⟩⟨ΨRB,μ∣, (28)

(AA†)−1=∑

μ∣ΨL

B,μ⟩⟨ΨLB,μ∣, (29)

A is defined up to a unitary transformation. Polar decomposition ofa matrix gives unique positive-semidefinite Hermitian matrix P anda unitary part U,

A = PU,

P = (AA†)

1/2. (30)

To make the Hermitian effective Hamiltonian physically close toBloch’s effective Hamiltonian, one has to minimize the degree ofunitary rotations, i.e., consider only the square root (AA†

)1/2. It is

easy to show (see the Appendix) that, similar to Löwdin’s symmetricorthogonalization, this transformation changes the eigenvectors to aminimal possible extent.

2. Effective Hamiltonians from EOM-CCHere, we treat the EOM-CC space as the target space:

∣ΨRμ⟩ = Rμ∣Φ0⟩ and ⟨ΨL

μ ∣ = ⟨Φ0∣L†μ. Our model space depends on

a problem and will be defined for each specific case; it will con-tain a subset of single EOM-CC amplitudes. The interaction betweenthe configurations in this target space is described throughH. Aftersubstitution into Eq. (22) and cancellation of the normalizationprefactors, the final expressions are

Iμ′ ,μ′′ =∑μ⟨Φμ′ ∣ΨR

μ⟩⟨ΨLμ ∣Φμ′′⟩, (31)

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HeffB = ∑

ν,μ,μ′ ,μ′′∣Φν⟩⟨Φν∣ΨR

μ⟩Eμ⟨ΨLμ ∣Φμ′⟩(I−1

)μ′ ,μ′′⟨Φμ′′ ∣. (32)

Here, we assumed that the model functions Φi are orthonormal.Eigenfunctions of Bloch’s Hamiltonian are

∣ΨRB,μ⟩ =∑

ν∣Φν⟩⟨Φν∣ΨR

μ⟩, (33)

⟨ΨLB,μ∣ = ∑

μ′ ,μ′′⟨ΨL

μ ∣Φμ′⟩(I−1)μ′ ,μ′′⟨Φμ′′ ∣. (34)

Assuming that ⟨Φν∣ΨRμ⟩ and ⟨ΨL

μ ∣Φμ′⟩ are invertible, one can getbiorthogonality of Bloch’s eigenstates,

LB†RB = LEOM†I−1REOM = 1, (35)

where columns of RB and LB are Bloch’s eigenvectors in the basisof model functions and columns of REOM and LEOM are overlaps ofthe EOM vectors with model functions ⟨Φν∣ΨR

μ⟩ and ⟨Φν∣ΨLμ⟩. This

relation reveals the physical meaning of I: it is a metric tensor for theEOM vectors projected onto the model space; thus, it quantifies howwell the model states can represent the target states. This observationsuggests using the overlaps (specifically, the lowest absolute value ofthe eigenvalues of I) as a diagnostic of whether a particular modelspace is capable of accurate representation of the given target space.

Des Cloizeaux’s transformation is obtained in a similar way,

AA†= REOMREOM

†, (36)

AC = (AA†)

1/2, (37)

where A denotes similarity transformation A in a basis of orthonor-mal model functions.

III. NUMERICAL EXAMPLES AND DISCUSSIONA. Computational details

We implemented the tools for constructing the effective Hamil-tonians from the EOM-CC wave functions within the Q-Chempackage.65,66 The illustrative examples in this work use EOM-SF-CCSD,57,61 but the same workflow can be executed with otherEOM-CC methods. Briefly, the calculation proceeds as follows:

1. Perform an SCF calculation for a high-spin reference state.2. Determine single occupied orbital space by computing corre-

sponding α and β orbitals by means of singular value decom-position (SVD) of the αβ block of the overlap matrix.

3. Localize orbitals within singly occupied space.4. (Optional) Apply the open-shell frozen natural orbital approx-

imation (OSFNO)67 in the virtual space.5. Construct the model space using an appropriate subset of

EOM-CC amplitudes (see Secs. III D and III E for specificexamples).

6. Build effective Hamiltonians in Bloch’s and des Cloizeaux’sforms from the EOM-CC energies and relevant amplitudes, asdescribed above.The validity of the selected model space can be assessed by

considering the overlaps between the target and model spaces.Once the effective Hamiltonian is computed, one can extractparameters of Hubbard’s or Heisenberg’s Hamiltonians and assess

the validity of these models. Here, we considered non-relativisticEOM-CC calculations. The algorithm can be generalized toinclude perturbative treatment of spin–orbit couplings68 to enablethe extraction of other magnetic properties, such as magneticanisotropies. Additional details of the algorithm are given below; anexample of an input is provided in the supplementary material.

To illustrate the theory, we constructed effective Hamiltoniansfor four representative molecules (structures are shown in Fig. 3)with two and three strongly correlated electrons (i.e., diradicals69

and triradicals70):Hubbard’smodel: propane-1,3-diyl (molecule 1) and pentane-

1,3,5-triyl (molecule 2). These molecules are examples of organicdi- and triradicals. Molecule 1 (di-methilene-methane) was used toillustrate the construction of effective Hamiltonians within the CIframework.51 We also consider Heisenberg’s model for these sys-tems. The geometries were optimized with CCSD/cc-pVTZ and aregiven in the supplementary material.

Heisenberg’s Model: PATFIA (magnet 1, Ref. 71) and tris-(3-acetylamino-1,2,4-triazolate) motif of HUKDOG (magnet 2,Ref. 72). These systems are examples of single molecule magnets(SMMs)—they can also be described as di- and tri-radicals, but withmetallic radical centers. The geometry of magnet 1 optimized withωB97X-D/cc-pVTZ for the triplet state was taken from Ref. 46. Asin Ref. 46, the ferrocene group was not included in the calculations.We also excluded water molecules, sulfate groups, and perchlorateanions in the structure of magnet 2. The geometry of magnet 2was optimized with ωB97X-D/cc-pVTZ and is given in the supple-mentary material. We optimized the geometry of magnet 2 withfinite thresholds: 10−4 tolerance on the maximal gradient compo-nent (a.u.) and 10−6 a.u. on the energy change; these thresholds aretighter than those defined in Q-Chem’s default criteria for geometryoptimization. The resulting geometry is slightly asymmetric withinthese criteria (for example, Cu–Cu distances are 3.416, 3.420, and3.418 Å). Tighter thresholds lead to a symmetric geometry. TheC3 symmetry point group has degenerate irreducible representa-tions, which leads to problems with degenerate EOM solutions, asdescribed in Ref. 68. To circumvent these problems, we used the

FIG. 3. Structures of the considered systems. Color code: copper (bronze), nitro-gen (blue), oxygen (red), carbon (gray), and hydrogen (white). In molecule 1and molecule 2, representing organic di/triradicals, the unpaired electrons arelocalized on the odd carbon atoms. In single-molecule magnets (magnet 1 andmagnet 2), the unpaired electrons are localized on the copper centers.

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lower symmetry geometry, which we also provide in the supplemen-tary material. In this non-symmetric geometry, the doublet states aresplit by only 2.5 cm−1.

In the SMM calculations, we used EOM-SF-CCSD withOSFNO truncation of the virtual space67 with the total popula-tion threshold of 99%. We used the single precision73 capabilityof the libxm tensor contraction library74 in calculations of copperSMMs. Unrestricted Hartree–Fock references were used in all corre-lated calculations. Core electrons were frozen. All calculations wereperformed with the Q-Chem software.65,66

B. Model space selectionStarting from a high-spin reference determinant, we define the

open-shell orbitals through SVD of the overlap matrix of α occupiedand β virtual spin-orbitals, as in the OSFNO scheme67 and in Ref. 37.The singular vector pairs corresponding to open-shell α and β spin-orbitals have near unity singular values. All other singular values arezero (ROHF) or small (UHF), enabling the separation of the open-shell subspace. This is followed by the localization of open-shellorbitals. Here, we used Foster–Boys’ localization criterion75 and agradient-based algorithm76–78 with the BFGS optimizer. The modelspaces were generated by one-electron spin-flipping excitations act-ing on the reference determinant and constrained to open-shell sub-spaces. For Heisenberg’s model space, we considered only open-shelldeterminants, while for Hubbard’s model space, we also includedionic determinants (shown in Fig. 1).

C. Extraction of parameters from effectiveHamiltonians

To construct Heisenberg’s Hamiltonian, one should first com-pute the effective Hamiltonian using the model space with open-shell configurations only (see Fig. 1). Then, the parameters forHeisenberg’s model (JAB) are given by the sum of matrix elementpairs between the respective configurations.

For Hubbard’s Hamiltonian, the model space should includeboth open-shell and ionic configurations. Once the effective Hamil-tonian is constructed, the extraction of the parameters is also ratherstraightforward. A simple relation between the matrix elements andeffective parameters is established by the substitution of Hubbard’sHamiltonian definition and second quantization algebra. Let A andB subscripts denote radical centers A and B, CS and OS denoteclosed-shell and open-shell determinants, and REF denotes a high-spin reference state (in this example, it is a high-spin triplet state).The expressions for the diagonal and off-diagonal matrix elementsare

⟨CSA∣HHub∣CSA⟩ = ⟨CSA∣U∑iniαniβ∣CSA⟩

= U⟨CSA∣(1 + 0)∣CSA⟩ = U, (38)

⟨CSA∣HHub∣OSA⟩ = ⟨CSA∣∑i≠j∑σtija†

iσajσ ∣OSA⟩

= ⟨REF∣(a†A,βaB,α)

†⎛

⎝∑i≠j∑σtija†

iσajσ⎞

⎠a†A,βaA,α∣REF⟩

= ⟨REF∣(a†A,βaB,α)

†tBA(a†AαaBα)a

†A,βaA,α∣REF⟩ = −tBA.

(39)

D. Hubbard’s effective HamiltonianLocalized orbitals (shown in Fig. 4 and Fig. S1) enable unequiv-

ocal identification of covalent or ionic character of the electronicconfigurations constructed from these orbitals. Bloch’s and desCloizeaux’s effective Hamiltonians for molecules 1 and 2 built forcovalent and ionic configurations from EOM-SF-CCSD/cc-pVTZare shown in Fig. 4 and Eqs. (S1)–(S3) in the supplementary mate-rial. We note that des Cloizeaux’s transformation either does notchange the matrix elements of the effective Hamiltonian on the diag-onal (molecule 1) or alter them in a minimal way (molecule 2).Des Cloizeaux’s transformation averages the off-diagonal matrixelements yielding a Hermitian effective Hamiltonian. The valid-ity of the selected model spaces is reflected by the weights ofthe leading configurations shown in Table S1 in the supplemen-tary material and by the smallest eigenvalue of metric tensor I.For molecule 1, it is 0.76, confirming that the selected modelspace adequately represents the chosen set of the target states. Formolecule 2, we obtain a smaller value (0.45), suggesting that someof the selected states are represented not as well as the states inmolecule 1.

The obtained matrix elements of effective Hamiltoniansdirectly correspond to the coupling parameters in Hubbard’s Hamil-tonian, as explained in Sec. III C. In particular, the elementscorresponding to hopping coupling t and energies of ionic con-figurations U [see Eq. (39)] are shown in blue in Fig. 4. How-ever, Hubbard’s model does not include two-electron couplings,such as an effective direct exchange between the open-shell deter-minants and effective interaction between the ionic determinants.The singlet–triplet gap (EOM-SF-CCSD/cc-pVTZ) between the twolowest states in molecule 1 is 0.077 eV; by construction, it is repro-duced exactly by Bloch’s and des Cloizeaux’s effective Hamiltonians.The contribution of the ionic configurations and the validity of Hub-bard’s model are revealed by the magnitudes of the matrix elementsof effective Hamiltonians. The truncated (inexact) effective Hamil-tonian, including only open-shell determinants (Fig. 4), gives thesinglet–triplet gap of 0.084 eV. This observation can also be ratio-nalized from perturbative arguments: the effective direct exchangesplits the open-shell states in the first order of degenerate pertur-bation theory. The indirect exchange from the ionic configurationscomes only from the second-order perturbation, having a magni-tude of 0.12/4 = 2.5 ⋅ 10−3 eV, which is by one order of magnitude

FIG. 4. Localized orbitals of molecule 1 and Bloch’s effective Hamiltonian (eV)constructed with EOM-SF-CCSD/cc-pVTZ. All energies are shifted to achieve zerotrace of the open-shell submatrix. CS and OS denote closed-shell ionic determi-nants and open-shell determinants, respectively. Values in blue mark the numberscorresponding to Hubbard’s Hamiltonian parameters; values in red (antidiagonal)mark the numbers that do not enter standard Hubbard’s Hamiltonian.

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smaller than the direct exchange. This analysis reveals the nature ofinteraction of open-shell configurations in this system: the singlet–triplet gap primarily comes from the effective direct exchange, ratherfrom the interaction with the ionic configurations. The standardHubbard Hamiltonian (blue numbers in Fig. 4) does not includethese interactions; consequently, its diagonalization yields a poorvalue of the singlet–triplet gap (−0.008 eV—wrong magnitude andsign). To accommodate this physics, Hubbard’s Hamiltonian can beextended to include the direct exchange contribution, as was done inRef. 52.

It is instructive to compare the open-shell sub-block ofthis effective Hamiltonian with Heisenberg’s Hamiltonian con-structed considering only open-shell configurations [shown in thesupplementary material, Eqs. (S4) and (S5)]. By construction, thisHamiltonian yields exactly the same singlet–triplet gap as the oneconstructed using Hubbard’s model space. The coupling matrixelements give the values of effective exchange constants J, whichinclude all many-body effects present in the EOM-SF-CCSD wavefunctions. These effects include the contributions of the ionic config-urations, which are treated explicitly in the case of Hubbard’s modelspace. The difference between the OS–OS couplings from Heisen-berg’s Hamiltonian (0.038 eV) and the matrix elements between theOS configurations from the Hamiltonian constructed using Hub-bard’s space (0.042 eV) quantify the contribution of the ionic con-figurations into the effective J values. Finally, the lowest absoluteeigenvalue of I is 0.96, meaning that Heisenberg’s model captureslow-energy physics in molecule 1 very well.

The considered triradical—molecule 2—exhibits similar trendsand similar effective interaction constants shown in Eqs. (S2) and(S3) and Fig. S1. Although all the energies in Bloch’s and desCloizeaux’s effective Hamiltonians have been shifted to producea zero trace in the open-shell submatrix, the individual ener-gies of the open-shell determinants are not zero. This is likelydue to some degree of spin contamination of the EOM-SF-CCSDstates used to build the effective Hamiltonians. Spin contamina-tion is especially large in the highest ionic states: the corresponding⟨S2⟩ values are 1.06 and 1.17. This degree of spin contamina-

tion can deteriorate the quality of energy, resulting in somewhathigher energies of the corresponding ionic configurations: 6.5 eVand 6.3 eV. These configurations are not well represented by themodel space, resulting in a relatively small lowest eigenvalue of Iof (0.45), which is smaller than the corresponding value (0.76) ofmolecule 1. We note that there are several spin-contaminated elec-tronic states with lower energies that are not represented well bythe model space; these states were not included in the effectiveHamiltonian. Interestingly, Heisenberg’s model space for this sys-tem [see Eqs. (S6) and (S7) in the supplementary material] showsa much better overlap, with the lowest eigenvalue of I of 0.96,indicating that open-shell states are described well by this effectiveHamiltonian.

E. Heisenberg’s effective HamiltonianFigures 5 and 6 show localized open-shell orbitals and the effec-

tive Hamiltonians constructed from the open-shell determinantsexpressed using these orbitals. The complex character of the wavefunctions has been illustrated before,46 yet the SVD-transformedopen-shell orbitals yield a relatively simple expansion of wave

FIG. 5. Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’s(right, bottom) effective Hamiltonians for magnet 1. All matrix elements are incm−1.

functions over this small set of configurations. Open-shell configu-rations in these orbitals are the dominant configurations contribut-ing more than 90% of the total amplitude sum (Table S2), whichconfirms the validity of Heisenberg’s model in representing thesesystems. Our proposed diagnostic values, the lowest absolute eigen-values of I, equal 0.91 and 0.93, which means that the model spacerepresents the target wave functions very well.

We note that by no means, the quality of this compact modelspace means that the effect of other electronic configurations in thefull Hilbert space (e.g., double EOM-SF-CCSD amplitudes) can beneglected. To illustrate this point, we computed the eigenvalues ofH in the space of single excitations only, as in EOM-SF-CCSD-S;79

the results are shown in Table S3 in the supplementary material. Theenergy gaps between the target states are reproduced reasonably well(albeit not exactly); however, the resulting states are severely spin-contaminated.

The obtained effective Hamiltonians shown in Figs. 5 and 6were shifted to achieve a zero trace, as in the case of Hub-bard’s Hamiltonian. Although the diagonal elements are small, theydeviate from zero by 0.9–2.6 cm−1. This artifact is a violation oftime-reversal symmetry, which can be explained by a small spincontamination of EOM-SF-CCSD states entering the effective

FIG. 6. Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’s(right, bottom) effective Hamiltonians of magnet 2. All matrix elements arein cm−1.

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TABLE I. Exchange coupling constants, J (cm−1), extracted from the EOM-SF-CCSD/cc-pVDZ calculations in different ways.

Magnet 1 Magnet 2

Bloch −70.40 −96.73, −93.61, −97.51Des Cloizeaux −70.40 −96.72, −93.64, −97.50Landé rule −70.47 −95.96a

aHere, we used a three-center generalization of the Landé rule:80 E(Q) − E(D) = − 32 J.

Energies of the two doublet states were averaged.

Hamiltonian construction. The effective exchange coupling con-stants J (shown in Table I) were extracted from the off-diagonal ele-ments of the effective Hamiltonians and through the Landé intervalrule.45 Bloch’s and des Cloizeaux’s forms of effective Hamiltonianspredict nearly identical values of J, slightly different from the Landérule. This can be attributed to small non-zero diagonal terms con-tributing to eigenvalues and entering the energy gaps used in theLandé rule.

IV. CONCLUSIONWe generalized Bloch’s formalism to a non-Hermitian Hamil-

tonian (H) and presented the application of the resulting the-ory to the EOM-CC wave functions. We compared Bloch’s andCloizeaux’s versions of the effective Hamiltonian approach anddemonstrated that they perform similarly. The constructed Bloch’sand des Cloizeaux’s effective Hamiltonians provide a direct way tointerpret the EOM-CC wave functions in terms of effective theo-ries. The theory is formulated using model spaces built from local-ized open-shell orbitals computed via SVD of the overlap betweenthe α and β HF orbitals (such as in OSFNO). The localizationof open-shell orbitals allows one to quantify the ionic and cova-lent characters of the wave functions. The effective Hamiltonians,which are constructed in a rigorous manner, supply the interac-tion constants for the model Hamiltonians, such as the Heisenbergand Hubbard Hamiltonians. The physical quality of the selectedmodels can be assessed by considering the eigenvalues of I, thematrix that quantifies the overlap between the model and targetspaces. We propose to use the lowest absolute eigenvalue of I asa diagnostic in the construction of the effective Hamiltonians. Weobserve that Hubbard’s model space gives smaller values of thediagnostic than Heisenberg’s model space. This can be attributedto the effect of ionic configurations: the states with ionic charac-ter are known to have large dynamic correlation (due to Coulombrepulsion), which leads to more multiconfigurational wave func-tions and, therefore, reduces the overlap with model spaces. Thequality of electronic structure calculations, such as the degree ofspin contamination, enters the constructed effective Hamiltoniansand affects the respective parameters. In particular, spin contami-nation leads to a small difference in magnetic exchange couplingsextracted in different ways. This work provides a foundation for thedirect comparison of the results of many-body calculations with theexperimentally derived effective parameters for magnetic systemsand for further development of coarse-grained approaches to strongcorrelation.

SUPPLEMENTARY MATERIAL

See the supplementary material for Hubbard’s Hamiltonian,localized open-shell orbitals of molecule 2, Heisenberg’s Hamilto-nians for molecule 1 and molecule 2, leading amplitudes in open-shell orbital subspace and overlaps, ligand impact on J values ofmagnet 2, relevant Cartesian coordinates, and an example of aninput.

ACKNOWLEDGMENTSThis work was supported by the Department of Energy through

the Grant No. DE-SC0018910. The authors thank Dr. MaristellaAlessio for carefully reading the manuscript and providing a valu-able feedback.

The authors declare the following competing financial inter-est(s): A.I.K. is the President and a part-owner of Q-Chem, Inc.

APPENDIX: PROOF OF OPTIMALITY OF DESCLOIZEAUX’s TRANSFORMATION

Let us find a similarity transformation A of Bloch’s effectiveHamiltonian such that the eigenvectors of the resulting Hermitianeffective Hamiltonian differ from Bloch’s eigenvectors to a minimalextent. For this purpose, we formulate the problem as the overlapmaximization with right and left Bloch’s eigenvectors,

∑μ(⟨ΨL

B,μ∣ΨH,μ⟩ + ⟨ΨH,μ∣ΨRB,μ⟩)

=∑μ(⟨ΨH,μ∣A−1

∣ΨH,μ⟩ + ⟨ΨH,μ∣A∣ΨH,μ⟩)→ max. (A1)

This problem can be viewed as a trace maximization: Find sucha unitary matrix U for a given A that maximizes Tr(AU) +Tr(U−1A−1). Based on singular value decomposition A = WΣV† andCauchy–Schwartz inequality, one can prove81 that the maximum ofboth Tr(AU) and Tr(U−1A−1) is achieved at Uopt = VW†. Therefore,AUopt = WΣV†VW† = WΣW†, which is the polar part of A.

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69L. Salem and C. Rowland, “The electronic properties of diradicals,” Angew.Chem., Int. Ed. 11, 92 (1972).70A. I. Krylov, “Triradicals,” J. Phys. Chem. A 109, 10638 (2005).71C. López, R. Costa, F. Illas, C. de Graaf, M. M. Turnbull, C. P. Landee,E. Espinosa, I. Mata, and E. Molins, “Magneto-structural correlations in binuclearcopper(II) compounds bridged by a ferrocenecarboxylato(−1) and an hydroxo- ormethoxo-ligands,” Dalton Trans. 13, 2322 (2005).72S. Ferrer, F. Lloret, I. Bertomeu, G. Alzuet, J. Borrás, S. García-Granda,M. Liu-González, and J. G. Haasnoot, “Cyclic trinuclear and chain of cyclictrinuclear copper(II) complexes containing a pyramidal Cu3O(H) core. Crystalstructures and magnetic properties of [Cu3(μ3-OH)(aaat)3(H2O)3](NO3)2 ⋅ H2O[aaat = 3-acetylamino-5-amino-1,2,4-triazolate] and [Cu3(μ3-OH)(aat)3(μ3-SO4)] ⋅ 6H2On [aat = 3-acetylamino-1,2,4-triazolate]: New cases of spin-frustrated systems,” Inorg. Chem. 41, 5821 (2002).73P. Pokhilko, E. Epifanovskii, and A. I. Krylov, “Double precision is not neededfor many-body calculations: Emergent conventional wisdom,” J. Chem. TheoryComput. 14, 4088 (2018).74I. Kaliman and A. I. Krylov, “New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculationswith over 1000 basis functions on a single compute node,” J. Comput. Chem. 38,842 (2017).75S. F. Boys, “Construction of some molecular orbitals to be approximatelyinvariant for changes from one molecule to another,” Rev. Mod. Phys. 32, 296(1960).76T. E. Abrudan, J. Eriksson, and V. Koivunen, “Steepest descent algorithms foroptimization under unitary matrix constraint,” IEEE Trans. Signal Process. 56,1134 (2008).77T. Abrudan, J. Eriksson, and V. Koivunen, “Conjugate gradient algorithmfor optimization under unitary matrix constraint,” Signal Process. 89, 1704(2009).78S. Lehtola and H. Jónsson, “Unitary optimization of localized molecularorbitals,” J. Chem. Theory Comput. 9, 5365 (2013).79A. Sadybekov and A. I. Krylov, “Coupled-cluster based approach for core-ionized and core-excited states in condensed phase: Theory and application todifferent protonated forms of aqueous glycine,” J. Chem. Phys. 147, 014107(2017).80B. S. Tsukerblat, B. Ya. Kuyavskaya, M. I. Belinskii, A. V. Ablov, V. M.Novotortsev, and V. T. Kalinnikov, “Antisymmetric exchange in the trinuclearclusters of copper (II),” Theor. Chim. Acta 38, 131 (1975).81See https://web.archive.org/web/20160321110525/https://math.stackexchange.com/questions/754301/maximizing-the-trace for sketch of the proof.

J. Chem. Phys. 152, 094108 (2020); doi: 10.1063/1.5143318 152, 094108-10

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