New Strategies in Modeling Electronic Structures and Properties …
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excitation energies are best determined using so-called hybrid exchange–correlation
functionals, like PBE0 [1, 112, 113], B3LYP [10, 89], or CAM-B3LYP [169] that
provide good agreement with experimental data or high-level wave-function-based
methods [51, 79, 117, 146, 160].
3.2.8 Targeting Excited States with Wave-Function-Based Approaches
Since electronic spectra are used to identify the oxidation states and ligand effects
in actinide species, reliable theoretical predictions of excitation energies of actinide
compounds are highly important. In CI-type methods, such as MCSCF or DMRG,
the electronic excitation energies are usually obtained by calculating higher roots
of the eigenvalue problem. However, in order to compute excited states in coupled
cluster theory, we have to define a new ansatz. The most popular approaches applied
to actinides are the equation-of-motion (EOM) and linear response (LR) coupled
cluster formulations [11, 12, 122, 148]. In this chapter, we will focus on a different
approach that allows us to include strong correlation effects in excited states: the
Fock-space coupled cluster (FSCC) approach.
3.2.9 Strongly-Correlated Excited States with Fock-Space Coupled
Cluster Theory
The FSCC method belongs to the group of state-universal multi-reference coupled
cluster theories and operates in the Fock space. The key idea behind the FSCC
approach is to find an effective Hamiltonian in a low-dimensional model P space,
with eigenvalues that reliably approximate the desired eigenvalues of the real (physical) Hamiltonian. In the FSCC method, the P space (also called the model space)
contains all active valence orbitals directly involved in the electronic excitations,
while the Q space (also called auxiliary or complementary space) includes all remaining orbitals. Thus, only a few eigenvalues out of the whole spectrum are calculated,
reducing the expensive step of diagonalizing the Hamiltonian matrix. In many practical applications, however, the P and Q spaces are not well separated from each other,
which might result in intruder state problems. They usually manifest as convergence
difficulties for large P spaces, which are particularly desired for modeling electronic
structure of actinides. Such divergencies might occur for a specific molecule, a given
molecular geometry, or basis set. To remedy this problem, the intermediate Hamiltonian (IH) formulation of the FSCC method has been introduced, which imposes
a buffer space between the desired and undesired states. Thus, the P space is further divided into a main P m space and an intermediate P i space. The intermediate
space serves as a buffer between the P m and Q spaces, for which various numerical
procedures have been developed [101].
A characteristic feature of the FSCC approach is the partitioning into sectors (k,
l) depending on the number of electrons removed from or attached to the reference
state. Within the hole-particle formalism, the cluster operator T n is expressed as
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excitation energies are best determined using so-called hybrid exchange–correlation
functionals, like PBE0 [1, 112, 113], B3LYP [10, 89], or CAM-B3LYP [169] that
provide good agreement with experimental data or high-level wave-function-based
methods [51, 79, 117, 146, 160].
3.2.8 Targeting Excited States with Wave-Function-Based Approaches
Since electronic spectra are used to identify the oxidation states and ligand effects
in actinide species, reliable theoretical predictions of excitation energies of actinide
compounds are highly important. In CI-type methods, such as MCSCF or DMRG,
the electronic excitation energies are usually obtained by calculating higher roots
of the eigenvalue problem. However, in order to compute excited states in coupled
cluster theory, we have to define a new ansatz. The most popular approaches applied
to actinides are the equation-of-motion (EOM) and linear response (LR) coupled
cluster formulations [11, 12, 122, 148]. In this chapter, we will focus on a different
approach that allows us to include strong correlation effects in excited states: the
Fock-space coupled cluster (FSCC) approach.
3.2.9 Strongly-Correlated Excited States with Fock-Space Coupled
Cluster Theory
The FSCC method belongs to the group of state-universal multi-reference coupled
cluster theories and operates in the Fock space. The key idea behind the FSCC
approach is to find an effective Hamiltonian in a low-dimensional model P space,
with eigenvalues that reliably approximate the desired eigenvalues of the real (physical) Hamiltonian. In the FSCC method, the P space (also called the model space)
contains all active valence orbitals directly involved in the electronic excitations,
while the Q space (also called auxiliary or complementary space) includes all remaining orbitals. Thus, only a few eigenvalues out of the whole spectrum are calculated,
reducing the expensive step of diagonalizing the Hamiltonian matrix. In many practical applications, however, the P and Q spaces are not well separated from each other,
which might result in intruder state problems. They usually manifest as convergence
difficulties for large P spaces, which are particularly desired for modeling electronic
structure of actinides. Such divergencies might occur for a specific molecule, a given
molecular geometry, or basis set. To remedy this problem, the intermediate Hamiltonian (IH) formulation of the FSCC method has been introduced, which imposes
a buffer space between the desired and undesired states. Thus, the P space is further divided into a main P m space and an intermediate P i space. The intermediate
space serves as a buffer between the P m and Q spaces, for which various numerical
procedures have been developed [101].
A characteristic feature of the FSCC approach is the partitioning into sectors (k,
l) depending on the number of electrons removed from or attached to the reference
state. Within the hole-particle formalism, the cluster operator T n is expressed as