Effective Hamiltonians Derived from Equation-Of-Motion Coupled-Cluster

Home , Coupled cluster, Tight binding

Eﬀective Hamiltonians derived from equation-of-motion coupled-cluster wave-functions: Theory and application to the Hubbard and Heisenberg Hamiltonians

Pavel Pokhilkoa and Anna I. Krylova a Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482

Effective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exact many-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 (EOM-CC) wave functions to rigorously derive effective Hamiltonians in the Bloch’s and des Cloizeaux’s forms. We report the key equations and illustrate the theory by examples of systems with electronic states of covalent and ionic characters. We show that the Hubbard and Heisenberg Hamiltonians are extracted directly from the so-obtained effective Hamiltonians. By making quantitative connections between many-body states and simple models, the approach also facilitates the analysis of the correlated wave functions. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed.

I. INTRODUCTION

Coarse graining is commonly used in computational chemistry and physics. It is exploited in a number of classic models serving as a foundation of modern solid-state physics: tight binding[1, 2], Drude–Sommerfeld’s model[3–5], Hubbard’s [6] and Heisenberg’s[7–9] Hamiltonians. These models explain macroscopic properties of materials through effective interactions whose strengths are treated as model parameters. The values of these parameters are determined either from more sophisticated theoretical models or by fitting to experimental observables. For example, Hubbard’s model describes electron hopping (one-electron transitions driven by the couplings) by parameter t and the energies of the ionic configurations by the parameter 2

1 2 3

Heisenberg

Hubbard

FIG. 1: Configurations entering different three-electron-in-three-centers model Hamiltonians. The configurations are built from different distributions of three electrons on three localized orbitals (numbered 1, 2, and 3 and colored with yellow, green, and violet circles, respectively). Each line represents one configuration. Heisenberg’s model space includes only open-shell configurations in which the orbitals are singly occupied. A half-filled (i.e., three-electrons-in-three orbitals) Hub- bard’s model space also includes ionic configurations in which localized orbitals can host 2 electrons.

X X † X HHub = tijaiσajσ + U niαniβ, (1) i6=j σ i

† where indices i, j denote localized orbitals and σ denotes spin part α or β, operators aiσ and aiσ are the creation and annihilation operators of an electron on a localized orbital i with spin † σ, and niσ is a particle number operator: niσ = aiσaiσ. Although Hubbard’s model param- eterizes only one-particle interactions between the configurations, it includes the Coulomb repulsion U between the conducting (active) electrons, making it sufficiently flexible to describe non-trivial physical phenomena, such as Mott–Hubbard phase transition[10]. Heisenberg’s model[7–9] describes the interaction between open-shell configurations in 3

terms of local spin (an eﬀective quantity which we will discuss below). This model is most commonly used to describe magnetic properties of solids[10]. It can be written as

X HHeis = − JABSASB, (2) A

where A, B enumerate radical centers, JAB is an effective exchange constant, SI is an effective localized spin operator associated with center I. Heisenberg’s Hamiltonian can be derived from Hubbard’s Hamiltonian through degenerate[11] and canonical perturbation theories[12]; thus, it can be considered an effective theory with respect to Hubbard’s model. Effective Hamiltonian theory provides a powerful framework for a rigorous construction of effective Hamiltonians from multiconfigurational many-electron wave functions com- puted ab initio. Pioneering works of Kato [13], Okuboˆ [14], Bloch [15], des Cloizeaux [16] have established the foundations of the operator effective Hamiltonian theory. Further development came from Löwdin’spartitioning[17] technique, Feshbach’s formalism[18], and generalizations[19–22]. The history and recent developments of effective Hamiltonian theories are summarized in comprehensive reviews[23, 24]. Effective theories fulfill a dual role. On one hand, they can be used to develop new electronic structure methods by systematically improvable description of effective Hamiltonians. For example, perturbative construction of the wave operator and effective Hamiltonians have been exploited in the development of various multireference perturbative[25–27] and and coupled-cluster[28–30] methods. On another hand, effective theories can be used as an interpretation tool by providing an essential description of complex electronic structure. The distinction between using effective Hamiltonians for analysis of electronic structure and method development is not always binary. For example, introduction of Hubbard’s repulsion term U into a density functional expression allowed Anisimov and co-workers to develop a widely used DFT+U method[31–33]. Mayhall and Head-Gordon noticed that the effective exchange couplings J from single spin-flip (SF) calculations agree with the couplings extracted from more computationally expensive n-SF methods[34], which lead to the development of an effective computational scheme for strongly correlated systems exploiting a coarse-graining idea[35]. In Mayhall’s protocol, a single spin-flip calculation is performed first. Then, under the assumption of Heisenberg’s physics, the exchange couplings are extracted, and an effective Hamiltonian is constructed to compute the spin states that 4 are not reachable by a single spin-flip calculation. The parameters of effective Hamiltonians also can be obtained indirectly. Experimen- tally, exchange couplings can be extracted from temperature dependence of magnetic sus- ceptibility, electron paramagnetic resonance, and neutron scattering experiments[36–42]. Theoretically, if there are known relations between the states of interest, such as Landéin- terval rule[43], advanced electronic structure methods can be easily applied[44]. Even more indirectly, broken-symmetry density functional theory (BS-DFT) can be used to extract exchange coupling from contaminated solutions[45]. Numerous studies[46–49] have used Bloch’s and des Cloizeaux’s formalisms to build Heisenberg’s and Hubbard’s effective Hamiltonians from configuration interaction (CI) wave functions. In this contribution, we apply Bloch’s formalism to the equation-of-motion coupled-cluster (EOM-CC) wave functions to rigorously derive effective Hamiltonians in the Bloch’s and des Cloizeaux’s forms. We report the key equations and illustrate the theory by examples of systems with electronic states of covalent and ionic characters. Our goal is to establish theoretical connection between the EOM-CC methods and effective Hamilto- nians and to provide a theoretical basis for extraction of effective parameters from EOM-CC calculations. The structure of the paper is as follows. In the next section, we provide an overview of the EOM-CC theory and Bloch’s formalism. We then apply Bloch’s formalism to the EOM- CC wave functions and derive the working expressions. Next, we consider several molecules for which we discuss Hubbard’s and Heisenberg’s Hamiltonians rigorously constructed from the EOM-SF-CCSD solutions. Whenever possible, we compare the results from extraction through the Landérule and from the effective Hamiltonians. 5

II. THEORY

A. Equation-of-motion coupled-cluster theory

Similarly to CI, EOM-CC methods[50–53] utilize linear parameterization of the wave function:

R ˆ |ΨI i = RI |Φ0i , (3) L ˆ † hΨI | = hΦ0| LI , (4)

where ΨI is a EOM target state, Rˆ and Lˆ are general excitation operators, and Φ0 is the reference determinant, which deﬁnes the separation between occupied and virtual orbital spaces. The equations for the amplitudes of the EOM operators are derived variationally, leading to a CI-like eigenproblem[54]. In contrast to CI, the EOM theory employs a (non- Hermitian) similarity-transformed Hamiltonian:

ˆ ˆ H¯ˆ = e−T Heˆ T , (5)

where Tˆ is an excitation operator. Because H¯ˆ has the same spectrum as the bare Hamilto- nian regardless of the choice of Tˆ, solving the EOM eigen problem in the full conﬁgurational space, i.e., when Rˆ includes all possible excitations, recovers the exact (full CI, FCI) limit. In practical calculations, these operators are truncated, most often to single and double excitations giving rise to EOM-CCSD ansatz, and the choice of the operator Tˆ becomes important. For example, choosing Tˆ = 1ˆ leads to plain CISD, but taking Tˆ from the coupled-cluster equations for the reference state results in including correlation eﬀects and ensures size- intensivity. Most often, T is truncated at the same level as R, e.g., in EOM-CCSD one would use

X 1 X Tˆ = taa†i + taba†b†ji + ..., (6) i 4 ij ia ijab

where i, j, . . . and a, b, . . . denote occupied and virtual (with respect to Φ0) orbitals. When

Tˆ satisﬁes CC equations, then the reference determinant Φ0 is also an eigen-state of H¯ . Because H¯ˆ is non-Hermitian, the left and right EOM eigenstates are not Hermitian 6 conjugates of each other:

ˆ H¯ RÎ |Φ0i = EI RÎ |Φ0i , (7) † ˆ † hΦ0| LÎ H¯ = hΦ0| LÎ EI , (8) but are often chosen to form a biorthogonal set:

ˆ† ˆ hΦ0|LI RJ |Φ0i = δIJ . (9)

Because the excitation operators Tˆ and Rˆ commute, one can also write the EOM states as

R ˆ |ΨI i = RI |ΦCC i , (10) where

Tˆ |ΨCC i = e |Φ0i . (11)

In this form, one can see that: (i) the EOM-CC states include higher excitations than the respective CI states by virtue of the wave operator eTˆ and (ii) the EOM states are excited states with respect to the reference-state CC wave function. Different choices of Rˆ allow access to different manifolds of target states, giving rise to a variety of EOM-CC methods[50–53]. Common choices include excitation[55], spin-flip[56– 58], ionization[59, 60] or electron-attachment[61] operators. For example, spin-flip operators are

X 1 X RˆSF = raa†i + raba†b†ji + ..., (12) i 4 ij ia ijab X 1 X LˆSF = laa†i + laba†b†ji + ..., (13) i 4 ij ia ijab where the number of creation operators corresponding to α orbitals is not equal to the number of annihilation operators corresponding to α orbitals, giving rise to a non-zero MS of Rˆ (in contrast, the EOM-EE operators are of the Ms=0 type). 7

B. Eﬀective Hamiltonians

1. General formalism

Target space, P

+ +c1 +c2

1 Ω−

Model space, P0

FIG. 2: Target and model spaces are connected through a wave operator Ω. The target space can be the full conﬁgurational space of the system, while model space is a subspace of the full conﬁgurational space.

Eﬀective theories operate with target (large) and model (small) spaces. Bloch’s theory usually starts from target states, which are eigenvectors of the Hermitian Hamiltonian:

H |Ψµi = Eµ |Ψµi , (14) n X −2 P = |Ψµi Nµ hΨµ| . (15) µ

Here the eigenstates Ψµ are not necessarily normalized to one. P denotes a projector onto the target space. Eﬀective Hamiltonians are deﬁned in the model space such that its eigenvalues reproduce the eigenvalues of the target-space Hamiltonian:

eﬀ ˜ R ˜ R H |Ψµ i = Eµ |Ψµ i , (16) n X ˜ R ˜ L P0 = |Ψµ i hΨµ | , (17) µ

˜ R ˜ L where |Ψµ i and hΨµ | are right and left eigenstates of the eﬀective Hamiltonian, P0 is the 8 projector onto the model space. In other words, the idea behind the eﬀective Hamiltonian entails the transformation of the interaction between the model vectors to reproduce the energies in the target space. The model space is connected to the target space through the wave operator Ω:

|Ψµi = Ω |Ψ˜ µi . (18)

The relations (14–18) lead to generalized Bloch’s equation establishing a connection between the wave operator and the eﬀective Hamiltonian:

eﬀ HΩP0 = ΩP0H P0 (19)

Effective Hamiltonians in Eq. 16 are defined up to a similarity transformation. Therefore, a constraint is needed for a unique definition. Bloch’s constraint, also known as intermediate normalization, is written as

P0ΩBP0 = P0 (20)

This constraint leads to the following expressions for the wave operator and eﬀective Hamiltonian[19]:

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

eﬀ eﬀ HB = P0HB P0 = P0HΩBP0. (22)

As an interesting example of an effective theory, one can consider EOM-CC. The expo- nential operator eTˆ with untruncated Tˆ has the meaning of the wave operator connecting the model space with the exact (FCI) space. The reference determinant Φ0 and RÎ Φ0 are the eigenfunctions of the effective Hamiltonian H¯ˆ :

Ω |Φ0i −→|ΨCC i = Cˆ0Φ0, (23) Ω RÎ |Φ0i −→ RÎ |ΨCC i = CÎ Φ0, (24)

Here Cˆ0Φ0 and CˆI Φ0 are the FCI solutions. 9

Bloch’s effective Hamiltonian is not Hermitian, which may not be convenient for the analysis. Des Cloizeaux gave a recipe for producing a Hermitian effective Hamiltonian[16] by means of symmetric orthogonalization of Bloch’s eigenvectors. An arbitrary Hermitian effective Hamiltonian can be obtained through Bloch’s effective Hamiltonian through a similarity transformation A. For any Bloch’s eigenprojector, this can be written as

Since Bloch’s eigenvectors are biorthogonal,

† X ˜ R ˜ R AA = |ΨB,µi hΨB,µ| , (29) µ †−1 X ˜ L ˜ L AA = |ΨB,µi hΨB,µ| (30) µ

A is deﬁned up to a unitary transformation. Polar decomposition of a matrix gives unique positive-semideﬁnite Hermitian matrix P and a unitary part U:

A = P U, P = (AA†)1/2. (31)

To make the Hermitian effective Hamiltonian physically close to Bloch’s effective Hamilto- nian, one has to minimize the degree of unitary rotations, i.e., consider only the square root (AA†)1/2. One can show (see Appendix) that, similarly to Löwdin’ssymmetric orthogonalization, this transformation changes the eigenvectors to a minimal possible extent.

2. Eﬀective Hamiltonians from EOM-CC

R ˆ It is convenient to start from the EOM-CC space as the target space: |Ψµ i = Rµ |Φ0i and L ˆ† hΨµ | = hΦ0| Lµ. The interaction between the conﬁgurations in this target space is described through H¯ˆ . After substitution into Eq. 22 and cancellation of the normalization prefactors, 10 the ﬁnal expressions are:

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

˜ R X R |ΨB,µi = |Φνi hΦν|Ψµ i , (34) ν L X L −1 ˜ 0 00 hΨB,µ| = hΨµ |Φµ i I µ0,µ00 hΦµ | . (35) µ0,µ00

R L Assuming that hΦν|Ψµ i and hΨµ |Φµ0 i are invertible, one can get biorthogonality of Bloch’s eigenstates:

† † −1 LB RB = LEOM I REOM = 1, (36)

where columns of RB and LB are Bloch’s eigenvectors in the basis of model functions, and R columns of REOM and LEOM are overlaps of the EOM vectors with model functions hΦν|Ψµ i L and hΦν|Ψµ i. This relation provides a physical meaning to I: it is a metric tensor for the EOM vectors projected onto the model space. Des Cloizeaux’s transformation is obtained in a similar way:

† † AA = REOMREOM , (37)

† 1/2 AC = (AA ) , (38) where A denotes similarity transformation A in a basis of orthonormal model functions. 11

III. NUMERICAL EXAMPLES AND DISCUSSION

A. Computational details

We implemented the tools for constructing the eﬀective Hamiltonians from the EOM- CC wavefunctions within the Q-Chem package [67, 68]. 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 corresponding α and β orbitals by means of singular value decomposition (SVD) of the αβ block of the overlap matrix.

3. Localize orbitals within singly occupied space.

4. (Optional) Apply open-shell Frozen Natural Orbital approximation (OSFNO)[64] in the virtual space

5. Construct the model space using an appropriate EOM-CC model (here, EOM-SF- CCSD).

6. Build eﬀective Hamiltonians in Bloch’s and des Cloizeaux’s forms.

Here we considered non-relativistic EOM-CC calculations. The algorithm can be generalized by including perturbative account of spin–orbit interactions[73] to enable the extraction of other magnetic properties, such as magnetic anisotropies. Additional details of the algorithm are given below; an example of input is provided in the SI. We characterized the following systems by means of the eﬀective Hamiltonians:

1. Hubbard model: propane-1,3-diyl (Molecule 1) and pentane-1,3,5-triyl (Molecule 2). The geometries of both systems were optimized with CCSD/cc-pVTZ and are given in the SI. Molecule 1 (di-methilene-methane) has also been used as a test for the construction of eﬀective Hamiltonians within CI frameworks[48].

2. Heisenberg–Dirac–Van Vleck Hamiltonian: PATFIA (Magnet 1, Ref. [62]) and tris- (3-acetylamino-1,2,4-triazolate) motif of HUKDOG (Magnet 2, Ref. [63]), copper single molecule magnets (SMMs). Geometry of Magnet 1 optimized with ωB97X- D/cc-pVTZ for the triplet state was taken from Ref. [44]. As in Ref. [44], the ferrocene 12

group was not included in the structure. We also did not consider water molecules, sulfate groups, and perclorate anions in the structure of Magnet 2. The geometry of Magnet 2 was optimized with ωB97X-D/cc-pVTZ and is given in the SI.

We used EOM-SF-CCSD with open-shell frozen natural orbital (OSFNO) truncation of the virtual space[64] with the total population threshold of 99% for the SMM calculations. We used single precision[65] capability of libxm tensor contraction library [66] in calculations of copper SMMs. UHF reference determinant was used in all correlated calculations. Core electrons were frozen. All the calculations were performed with the Q-Chem software [67, 68].

Molecule 1 Magnet 1

Molecule 2 Magnet 2

FIG. 3: Structures of the considered systems. Color code: Copper (bronze), nitrogen (blue), oxygen (red), carbon (gray), hydrogen (white).

B. Model space selection

Starting from a high-spin reference determinant, we deﬁne the open-shell orbitals through SVD of the overlap matrix of α occupied and β virtual spin-orbitals, as in the OSFNO scheme[64] and in Ref. [35]. The singular vector pairs corresponding to open-shell α and β spin-orbitals have near unity singular values. All other singular values are zero (ROHF) or small (UHF), enabling the separation of the open-shell subspace. This is followed by the localization of open-shell orbitals. Here we used Foster–Boys’ localization criterion[69] and a gradient-based algorithm[70–72] with the BFGS optimizer. The model spaces considered in this work were generated by one-electron spin-ﬂipping excitations acting on the reference determinant and constrained to open-shell subspaces. For Heisenberg’s model space we 13 considered only open-shell determinants, while for Hubbard’s model space we also included ionic determinants (shown in Figure 1).

C. Hubbard’s eﬀective Hamiltonian

CS OS OS CS CS 4.424 0.088 -0.089 -0.130 OS 0.056 0.000 -0.042 -0.144 OS -0.144 -0.042 0.000 0.056 CS -0.130 -0.089 0.088 4.424

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 zero trace of open-shell submatrix. CS label denotes closed-shell ionic determinants; OS label denotes open-shell determinants. Blue color marks the numbers corresponding to Hubbard’s Hamiltonian parameters; red color on antidiagonal marks the numbers that do not enter the conventional Hubbard Hamiltonian.

Localized orbitals (shown in Figures 4 and S1) enable unequivocal identification of covalent or ionic character of the electronic configurations constructed from these orbitals. Bloch’s and des Cloizeaux’s effective Hamiltonians for Molecules 1 and 2 built for covalent and ionic configurations from EOM-SF-CCSD/cc-pVTZ are shown in Figure 4 and Eqs. S1, S2, S3 in the SI. We note that des Cloizeaux’s transformation either does not change the matrix elements of effective Hamiltonian on the diagonal (Molecule 1) or alter them in a minimal way (Molecule 2). Des Cloizeaux’s transformation averages the off-diagonal matrix elements yielding a Hermitian effective Hamiltonian. The obtained matrix elements of effective Hamiltonians directly correspond to the coupling parameters in Hubbard’s Hamiltonian. In particular, the elements corresponding to hopping coupling t and energies of ionic configurations U are colored in blue in Figure 4. However, it does not include two-electron couplings, such as an effective direct exchange between the open-shell determinants and effective interaction between the ionic determinants. Singlet–triplet gap (EOM-SF-CCSD/cc-pVTZ) between the lowest states in Molecule 1 is 0.077 eV. This gap is reproduced exactly by Bloch’s and des Cloizeaux’s effective Hamilto- nians. The contribution of the ionic configurations and the validity of Hubbard’s model are 14 revealed by the magnitudes of the matrix elements of effective Hamiltonians. The truncated (inexact) effective Hamiltonian to open-shell determinants (Figure 4) gives the singlet–triplet gap of 0.084 eV. This observation can be also rationalized from perturbative arguments: the effective direct exchange splits the open-shell states in the first order of degenerate perturbation theory. The indirect exchange from the ionic configurations comes only from the second-order perturbation, having a magnitude of 0.12/4 = 2.5 · 10−3 eV, which is by one order of magnitude smaller that the direct exchange. These observations reveal the nature of interaction of open-shell configurations in this system: the singlet–triplet gap primarily comes from the effective direct exchange rather from the interaction with the ionic configurations. To accommodate this physics, Hubbard’s Hamiltonian can be extended to include direct exchange contribution, as was done, for example, in Ref. [49]. The considered triradical—Molecule 2—shows similar trends and similar effective interaction constants shown in Eq. S2, S3, and Figure S1. Although all the energies in Bloch’s and des Cloizeaux’s effective Hamiltonians has been shifted to produce a zero trace in the open-shell submatrix, the individual energies of the open-shell determinants are not zero. This is likely due to some degree of spin contamination of the EOM-SF-CCSD states used to build effective Hamiltonians. Spin contamination is especially significant for the highest ionic states: the corresponding S2 values are 1.06 and 1.17. This degree of spin contamination can deteriorate the quality of energy, resulting in somewhat higher energies of the corresponding ionic configurations: 6.5 and 6.3 eV. We note that there are several spin-contaminated electronic states with lower energies that are not represented well by the model space; these states were not included in the effective Hamiltonian.

D. Heisenberg’s eﬀective Hamiltonian

We optimized the geometry of Magnet 2 with ﬁnite thresholds: 10−4 tolerance on the maximal gradient component (a.u.) and 10−6 a.u. on the energy change; these thresholds are tighter that Q-Chem’s default criteria for geometry optimization. The resulting geometry is slightly asymmetric within these criteria (for example, Cu–Cu distances are 3.416, 3.420, and

3.418 A).˚ Tighter thresholds lead to a symmetric geometry. C3 symmetry point group has degenerate irreducible representations, which leads to issues with degenerate EOM solutions, as described in Ref. [73]. To circumvent these problems, we used the lower symmetry 15

OS OS OS -1.48 35.32 B OS 35.08 1.48 OS OS OS -1.48 35.20 C OS 35.20 1.48

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 in cm−1.

OS OS OS OS 0.91 48.21 46.95 B OS 48.51 -2.59 48.57 OS 46.66 48.94 1.68 OS OS OS OS 0.90 48.36 46.82 C OS 48.36 -2.58 48.75 OS 46.82 48.75 1.67

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 are in cm−1.

geometry, which we also provide in the SI. In this non-symmetric geometry, the doublet states are split by only 2.5 cm−1. Figures 5 and 6 show localized open-shell orbitals and the effective Hamiltonians constructed from the open-shell determinants expressed using these orbitals. Although com- plicated character of the wave functions has been shown before[44], the SVD-transformed open-shell orbitals yield a relatively simple expansion of wave functions over configurations. Open-shell configurations in these orbitals are the dominant configurations contributing more than 90% of the total amplitude sum (Table S1). The obtained effective Hamiltonians shown in Figures 5 and 6, were shifted to achieve 16

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, -97.51 Des Cloizeaux -70.40 -96.72, -93.64, -97.50 Landérule -70.47 -95.96a a 3 Here we used a triangular generalization of the Landérule[74]: E(Q) − E(D) = − 2 J. Energies of the two doublet states were averaged. a zero trace, as in the case of Hubbard’s Hamiltonian. Although the diagonal elements are small, they deviate from zero by 0.9–2.6 cm−1. This artifact is a violation of time-reversal symmetry, which can be explained by a small spin contamination of EOM-SF-CCSD states entering the effective Hamiltonian construction. The effective exchange coupling constants J (shown in Table I) were extracted from the off-diagonal elements of the effective Hamiltonians and through the Landéinterval rule[43]. Bloch’s and des Cloizeaux’s forms of effective Hamiltonians predict nearly the same values of J, which is slightly different from the Landé rule. This can be attributed to small non-zero diagonal terms contributing to eigenvalues and entering the energy gaps used in Landérule.

IV. CONCLUSION

We generalized Bloch’s formalism to a non-Hermitian Hamiltonian (H¯ ) and presented the application of the resulting theory to the EOM-CC wave functions. We compared Bloch’s and Cloizeax’s versions of the effective Hamiltonian approach and demonstrated that they perform similarly. The constructed Bloch’s and des Cloizeaux’s effective Hamiltonians provide a direct way to interpret the EOM-CC wave functions in terms of effective theories. The theory is formulated using model spaces built from corresponding SVD open-shell HF orbitals (such as in OSFNO). The localization of open-shell orbitals allows one to quan- tify the ionic and covalent characters of the wave functions. The effective Hamiltonians, which are constructed in a rigorous manner, supply the interaction constants for the model Hamiltonians, such as the Heisenberg–Dirac–Van Vleck and Hubbard Hamiltonians. The quality of electronic structure calculation, such as the degree of spin contamination, enters the constructed effective Hamiltonians and affects the respective parameters. In particular, spin contamination leads to a small difference in magnetic exchange couplings extracted 17 in different ways. This work provides a foundation for direct comparison of the results of many-body calculations with the experimentally derived effective parameters for magnetic systems and for further development coarse-grained approaches to strong correlation.

Acknowledgments

This work is supported by the Department of Energy through the DE-SC0018910 grant.

The authors declare the following competing ﬁnancial interest(s): A.I.K. is a member of the Board of Directors and a part-owner of Q-Chem, Inc.

Supplementary Material

Hubbard Hamiltonians, localized open-shell orbitals of Molecule 2, leading amplitudes in open-shell orbital subspace, ligand impact on J values of Magnet 2, relevant Cartesian coordinates, example of an input.

Appendix: Proof of optimality of des Cloizeaux’s transformation

Let us find a similarity transformation A of Bloch’s effective Hamiltonian such that the eigenvectors of the resulting Hermitian effective Hamiltonian differ from Bloch’s eigenvectors to a minimal extent. For this purpose, we formulate the problem as the overlap maximization with right and left Bloch’s eigenvectors:

X ˜ L ˜ ˜ ˜ R hΨB,µ|ΨH,µi + hΨH,µ|ΨB,µi = µ

X −1 hΨ˜ H,µ|A |Ψ˜ H,µi + hΨ˜ H,µ|A|Ψ˜ H,µi → max . (39) µ

This problem can be viewed as a trace maximization: Find such a unitary matrix U for a given A that maximizes Tr(AU) + Tr(U −1A−1). Based on singular value decomposition A = W ΣV † and Cauchy–Schwartz inequality, one can prove[75] that the maximum of both −1 −1 † † † Tr(AU) and Tr(U A ) is achieved at Uopt = VW . Therefore, AUopt = W ΣV VW = W ΣW †, which is the polar part of A. 18

[1] A. W. Maue, Die Oberflächenwellen in der Elektronentheorie der Metalle, Z. Phys. 94, 717 (1935). [2] E. T. Goodwin, Electronic states at the surfaces of crystals: II. The approximation of tight binding: finite linear chain of atoms, Math. Proc. Camb. Philos. Soc. 35, 221 (1939). [3] P. Drude, Zur Elektronentheorie der Metalle, Ann. Phys. 306, 566 (1900). [4] P. Drude, Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomag- netische Effecte, Ann. Phys. 308, 369 (1900). [5] A. Sommerfeld, Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik, Z. Phys. 47, 1 (1928). [6] J. Hubbard and B. H. Flowers, Electron correlations in narrow energy bands, Proc. R. Soc. London A 276, 238 (1963). [7] W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. Phys. 49, 619 (1928). [8] P. A. M. Dirac, On the theory of quantum mechanics, Proc. R. Soc. London A 112, 661 (1926). [9] J. H. van Vleck, The Theory of Electric and Magnetic Susceptibilities. Clarendon Press, Oxford, 1932. [10] S. H. Simon, The Oxford solid state basics. Oxford University Press, 2013. [11] C. L. Cleveland and R. Medina A., Obtaining a Heisenberg Hamiltonian from the Hubbard model, Am. J. Phys. 44, 44 (1976). [12] K. A. Chao, J. Spalek, and A. M. Oles, Kinetic exchange interaction in a narrow S-band, J. Phys. C: Solid State Phys. 10, L271 (1977). [13] T. Kato, On the convergence of the perturbation method. I, Prog. Theor. Phys. 4, 514 (1949). [14]S. Okubo,ˆ Diagonalization of hamiltonian and tamm-dancoff equation, Prog. Theor. Phys. 12, 603 (1954). [15] C. Bloch, Sur la thèoriedes perturbations des étatsliés, Nucl. Phys. 6, 329 (1958). [16] J. des Cloizeaux, Extension d’une formule de lagrange àdes problèmesde valeurs propres, Nucl. Phys. 20, 321 (1960). [17] P.-O. Löwdin,Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism, J. Math. Phys. 3, 969 (1962). 19

[18] H. Feshbach, A unified theory of nuclear reactions. 2, Ann. Phys. (N.Y.) 19, 287 (1962). [19] P. Durand, Direct determination of effective hamiltonians by wave-operator methods. I. Gen- eral formalism, Phys. Rev. A 28, 3184 (1983). [20] C. E. Soliverez, On degenerate time-independent perturbation theory, Am. J. Phys. 35, 624 (1967). [21] C. E. Soliverez, An effective hamiltonian and time-independent perturbation theory, J. Phys. C: Solid State Phys. 2, 2161 (1969). [22] C. E. Soliverez, General theory of effective Hamiltonians, Phys. Rev. A 24, 4 (1981). [23] J. P. Killingbeck and G. Jolicard, The Bloch wave operator: generalizations and applications: Part I. The time-independent case, J. Phys. A: Math. Gen. 36, R105 (2003). [24] G. Jolicard and J. P. Killingbeck, The Bloch wave operator: generalizations and applications: II. the time-dependent case, Journal of Physics A: Mathematical and General 36, R411 (2003). [25] J. Finley, P.-A.˚ Malmqvist, B. O. Roos, and L. Serrano-Andrés, The multi-state CASPT2 method, Chem. Phys. Lett. 288, 299 (1998). [26] A. Zaitsevskii and J.-P. Malrieu, Multi-partitioning quasidegenerate perturbation theory. a new approach to multireference Møller–Plesset perturbation theory, Chem. Phys. Lett. 233, 597 (1995). [27] A.A. Granovsky, Extended multi-configuration quasi-degenerate perturbation theory: The new approach to multi-state multi-reference perturbation theory, J. Chem. Phys. 134, 214113 (2011). [28] B. Jeziorski and J. Paldus, Spin-adapted multireference coupled-cluster approach: Linear approximation for two closed-shell-type reference configurations, J. Chem. Phys. 88, 5673 (1988). [29] N. Oliphant and L. Adamowicz, Multireference coupled-cluster method using a single-reference formalism, J. Chem. Phys. 94, 1229 (1991). [30] C. M. L. Rittby and R. J. Bartlett, Multireference coupled cluster theory in Fock space, Theor. Chim. Acta 80, 469 (1991). [31] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Band theory and Mott insulators: Hubbard U instead of Stoner I, Phys. Rev. B 44, 943 (1991). [32] V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czy˙zyk,and G. A. Sawatzky, Density- functional theory and NiO photoemission spectra, Phys. Rev. B 48, 16929 (1993). 20

[33] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+U method, J. Phys.: Condens. Matter 9, 767 (1997). [34] N. J. Mayhall and M. Head-Gordon, Computational quantum chemistry for single Heisenberg spin couplings made simple: Just one spin flip required, J. Chem. Phys. 141, 134111 (2014). [35] N. J. Mayhall and M. Head-Gordon, Computational quantum chemistry for multiple-site Heisenberg spin couplings made simple: Still only one spin-flip required, J. Phys. Chem. Lett. 6, 1982 (2015). [36] B. Bleaney and K. D. Bowers, Anomalous paramagnetism of copper acetate, Proc. R. Soc. London A 214, 451 (1952). [37] S. A. Al’tshuler; B. M. Kozyrev, Electron Paramagnetic Resonance. Academic Press, New York, 1964. [38] B. Sieklucka and D. Pinkowicz, Molecular Magnetic Materials: Concepts and Applications. Wiley, 2017. [39] I. Mirabeau; M. Hennion; H. Casalta; H. Andres; H. U. Güdel;A. V. Irodova; A. Caneschi,

Low-energy magnetic excitations of the Mn12-acetate spin cluster observed by neutron scattering, Phys. Rev. Lett. 83, 628 (1999). [40] A. Michels, Magnetic small-angle neutron scattering of bulk ferromagnets, J. Phys.: Condens. Matter 26, 383201 (2014). [41] S. Mühlbauer,D. Honecker, E. A. Périgo,F. Bergner, S. Disch, A. Heinemann, S. Erokhin, D. Berkov, C. Leighton, M. R. Eskildsen, and A. Michels, Magnetic small-angle neutron scattering, Rev. Mod. Phys. 91, 015004 (2019). [42] G. Amoretti, R. Caciuffo, J. Combet, A. Murani, and A. Caneschi, Inelastic neutron scattering

below 85 eV and zero-field splitting parameters in the Fe8 magnetic cluster, Phys. Rev. B 62, 3022 (2000). [43] A. Landé,Termstruktur und Zeemaneffekt der Multipletts, Z. Phys. 15, 189 (1923). [44] N. Orms and A. I. Krylov, Singlet-triplet energy gaps and the degree of diradical character in binuclear copper molecular magnets characterized by spin-flip density functional theory, Phys. Chem. Chem. Phys. 20, 13127 (2018). [45] L. Noodleman, Valence bond description of antiferromagnetic coupling in transition metal dimers, J. Chem. Phys. 74, 5737 (1981). 21

[46] C. J. Calzado, J. Cabrero, J. P. Malrieu, and R. Caballol, Analysis of the magnetic coupling in binuclear complexes. II. derivation of valence effective hamiltonians from ab initio CI and DFT calculations, J. Chem. Phys. 116, 3985 (2002). [47] R. Maurice, R. Bastardis, C. de Graaf, N. Suaud, T. Mallah, and N. Guihèry, Universal theoretical approach to extract anisotropic spin hamiltonians, J. Chem. Theory Comput. 5, 2977 (2009). [48] A. Monari, D. Maynau, and J.-P. Malrieu, Determination of spin Hamiltonians from projected single reference configuration interaction calculations. I. Spin 1/2 systems, J. Chem. Phys. 133, 044106 (2010). [49] J.P. Malrieu, R. Caballol, C.J. Calzado, C. de Graaf, and N. Guihéry, Magnetic interactions in molecules and highly correlated materials: Physical content, analytical derivation, and rigorous extraction of magnetic Hamiltonians, Chem. Rev. 114, 429 (2013). [50] A. I. Krylov, Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The hitchhiker’s guide to Fock space, Annu. Rev. Phys. Chem. 59, 433 (2008). [51] R. J. Bartlett, The coupled-cluster revolution, Mol. Phys. 108, 2905 (2010). [52] K. Sneskov and O. Christiansen, Excited state coupled cluster methods, WIREs: Comput. Mol. Sci. 2, 566 (2012). [53] R. J. Bartlett, Coupled-cluster theory and its equation-of-motion extensions, WIREs: Com- put. Mol. Sci. 2, 126 (2012). [54] S. V. Levchenko and A. I. Krylov, Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobutadiene, J. Chem. Phys. 120, 175 (2004). [55] J. F. Stanton and R. J. Bartlett, The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties, J. Chem. Phys. 98, 7029 (1993). [56] A. I. Krylov, Size-consistent wave functions for bond-breaking: The equation-of-motion spin- flip model, Chem. Phys. Lett. 338, 375 (2001). [57] D. Casanova, L. V. Slipchenko, A. I. Krylov, and M. Head-Gordon, Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples, J. Chem. Phys. 130, 044103 (2009). [58] A. I. Krylov, The spin-flip equation-of-motion coupled-cluster electronic structure method for 22

a description of excited states, bond-breaking, diradicals, and triradicals, Acc. Chem. Res. 39, 83 (2006). [59] J. F. Stanton and J. Gauss, Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method, J. Chem. Phys. 101, 8938 (1994). [60] P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, Charge localization and Jahn-Teller distortions in the benzene dimer cation, J. Chem. Phys. 129, 074104 (2008). [61] M. Nooijen and R. J. Bartlett, Equation of motion coupled cluster method for electron attachment, J. Chem. Phys. 102, 3629 (1995). [62] C. 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 binuclear copper(II) compounds bridged by a ferrocenecarboxylato(-1) and an hydroxo- or methoxo-ligands, Dalton Trans. 13, 2322 (2005). [63] S. Ferrer, F. Lloret, I. Bertomeu, G. Alzuet, J. Borrás,S. Garc´ıa-Granda,M. Liu-González, and J. G. Haasnoot, Cyclic trinuclear and chain of cyclic trinuclear copper(II) com-

plexes containing a pyramidal Cu3O(H) core. crystal structures 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). [64] P. Pokhilko, D. Izmodenov, and A. I. Krylov, Extension of frozen natural orbital approximation to open-shell references: Theory, implementation, and application to single-molecule magnets, J. Chem. Phys. , submitted; preprint: http://dx.doi.org/10.26434/chemrxiv.10308053.v1. [65] P. Pokhilko, E. Epifanovskii, and A. I. Krylov, Double precision is not needed for many-body calculations: Emergent conventional wisdom, J. Chem. Theory Comput. 14, 4088 (2018). [66] I. Kaliman and A. I. Krylov, New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions on a single compute node, J. Comput. Chem. 38, 842 (2017). [67] Y. Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y.M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zimmerman, D. Zuev, B. 23

Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y.A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V. Lawler, S.V. Levchenko, C.Y. Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V. Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y.-C. Su, A.J.W. Thom, T. Tsuchimochi, L. Vogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V. Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y. Zhang, X. Zhang, Y. Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y. Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V.A. Rassolov, L.V. Slipchenko, J.E. Subotnik, T. Van Voorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon, Advances in molecular quantum chemistry contained in the Q-Chem 4 program package, Mol. Phys. 113, 184 (2015). [68] A. I. Krylov and P. M. W. Gill, Q-Chem: An engine for innovation, WIREs: Comput. Mol. Sci. 3, 317 (2013). [69] S.F. Boys, Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another, Rev. Mod. Phys. 32, 296 (1960). [70] T. E. Abrudan, J. Eriksson, and V. Koivunen, Steepest descent algorithms for optimization under unitary matrix constraint, IEEE Transactions on Signal Processing 56, 1134 (2008). [71] T. Abrudan, J. Eriksson, and V. Koivunen, Conjugate gradient algorithm for optimization under unitary matrix constraint, Signal Processing 89, 1704 (2009). [72] S. Lehtola and H. J´onsson, Unitary optimization of localized molecular orbitals, J. Chem. Theory Comput. 9, 5365 (2013). 24

[73] P. Pokhilko, E. Epifanovsky, and A. I. Krylov, General framework for calculating spin–orbit couplings using spinless one-particle density matrices: theory and application to the equation- of-motion coupled-cluster wave functions, J. Chem. Phys. 151, 034106 (2019). [74] B. S. Tsukerblat, B. Ya. Kuyavskaya, M. I. Belinskii, A. V. Ablov, V. M. Novotortsev, and V. T. Kalinnikov, Antisymmetric exchange in the trinuclear clusters of copper (II), Theor. Chim. Acta 38, 131 (1975).

[75] See https://web.archive.org/web/20160321110525/https://math.stackexchange.com/ questions/754301/maximizing-the-trace.