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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 observ- ables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not triv- ial. 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 ex- ploited 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 pa- rameters 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. U: X X y X HHub = tijaiσajσ + U niαniβ; (1) i6=j σ i y 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 y 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 de- scribe 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<B where A; B enumerate radical centers, JAB is an effective exchange constant, SI is an ef- fective 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 construc- tion of effective Hamiltonians from multiconfigurational many-electron wave functions com- puted ab initio. Pioneering works of Kato [13], Okubo^ [14], Bloch [15], des Cloizeaux [16] have established the foundations of the operator effective Hamiltonian theory. Further de- velopment came from L¨owdin'spartitioning[17] technique, Feshbach's formalism[18], and generalizations[19{22]. The history and recent developments of effective Hamiltonian theo- ries are summarized in comprehensive reviews[23, 24]. Effective theories fulfill a dual role. On one hand, they can be used to develop new elec- tronic 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 Land´ein- 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 the- ory 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 Land´erule 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 ^ jΨI i = RI jΦ0i ; (3) L ^ y hΨI j = hΦ0j LI ; (4) where ΨI is a EOM target state, R^ and L^ 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: ^ ^ H¯^ = e−T He^ T ; (5) where T^ is an excitation operator. Because H¯^ has the same spectrum as the bare Hamilto- nian regardless of the choice of T^, solving the EOM eigen problem in the full configurational space, i.e., when R^ includes all possible excitations, recovers the exact (full CI, FCI) limit. In practical calculations, these operators are truncated, most often to single and double excita- tions giving rise to EOM-CCSD ansatz, and the choice of the operator T^ becomes important. For example, choosing T^ = 1^ leads to plain CISD, but taking T^ 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 T^ = taayi + tabaybyji + :::; (6) i 4 ij ia ijab where i; j; : : : and a; b; : : : denote occupied and virtual (with respect to Φ0) orbitals. When T^ satisfies CC equations, then the reference determinant Φ0 is also an eigen-state of H¯ . Because H¯^ is non-Hermitian, the left and right EOM eigenstates are not Hermitian 6 conjugates of each other: ^ H¯ R^I jΦ0i = EI R^I jΦ0i ; (7) y ^ y hΦ0j L^I H¯ = hΦ0j L^I EI ; (8) but are often chosen to form a biorthogonal set: ^y ^ hΦ0jLI RJ jΦ0i = δIJ : (9) Because the excitation operators T^ and R^ commute, one can also write the EOM states as R ^ jΨI i = RI jΦCC i ; (10) where T^ jΨCC i = e jΦ0i : (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 eT^ and (ii) the EOM states are excited states with respect to the reference-state CC wave function. Different choices of R^ allow access to different manifolds of target states, giving rise to a variety of EOM-CC methods[50{53].
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