13.6.1 Overview

Effective Hamiltonians provide a powerful framework for coarse-grained representation of complex electronic structure and for making direct comparisons with the experimentally derived parameters. It is commonly used in magnetic systems.⁷²⁸ Following general theory of effective Hamiltonians,⁷²⁸ Ref. 908 describes how effective Hamiltonians can be extracted from the EOM-CC wave functions by using Bloch’s formalism. This approach provides an exact map of the many-body electronic structure onto a selected model space, yielding rigorous effective Hamiltonians in this model space. Currently, only EOM-SF-CCSD wave functions and Heisenberg’s and Hubbard’s model spaces are supported. The implemented procedure is $N$-to-$N$ map, meaning that the number of model states is the same as the number of electronic states used to build the effective Hamiltonian. In the case of the Heisenberg’s Hamiltonian, the number of states $N$ should be the same as the number of open-shell electrons, since this is the number of open-shell configurations that a single spin-flipping excitation can provide. In the case of the Hubbard’s Hamiltonian, $N$ should equal to the total number of configurations that a single spin-flipping excitation can generate in the open-shell subspace (squared number of open-shell orbitals). Once CC_MAKE_EFF_HAM $rem variable is set to 1, the electronic states are read from the list specified in the $eff_ham section. Right and left amplitudes of these EOM states are used for the effective Hamiltonian construction. The Hamiltonians are constructed in two forms: the Bloch’s form (non-Hermitian) and des Cloizeaux’s form (Hermitian). If CC_OSFNO is invoked (recommended), the correspondence between the open-shell orbitals is established through the SVD procedure,⁹⁰⁶ followed by Boys’ localization of these orbitals. If these orbitals are desired, the orbital canonicalization should be disabled with CC_CANONIZE = FALSE. If GUI is set to 2, the orbitals used in CCMAN2 calculations are dropped to the .fchk file and can be visualized by IQmol.

Note:  Beware that localization procedure may fail when there are several orbitals residing on the same radical centers, since in this case Boys localization becomes ill-conditioned.

The syntax of the $eff_ham section is similar to the syntax of the $trans_prop section. First, state_list starts the list of electronic states that are used for the construction of the effective Hamiltonian. The EOM states are abbreviated by the type of the method (currently, only SF_STATES is supported), its irreducible representation (use 1 for OSFNO and localized orbitals), and the number in the particular irreducible representation. The list of states is terminated by end_list. The type of the effective Hamiltonian is specified by the eff_ham keyword, which can be either Heisenberg or Hubbard.

The example below is based on Molecule 1 from Ref. 908. It shows how the Heisenberg Hamiltonian can be constructed from the EOM-SF-CCSD wave-functions. It can be modified to construct the Hubbard Hamiltonian by ($i$) requesting four SF states (that would include two covalent and two ionic states) and ($ii$) setting eff_ham to Hubbard.