Abstract
The present work is concerned with the development and application of two new multistate multireference perturbation theory methods. In contrast to state-specific perturbation theory methods, multistate methods can simultaneously provide several states that are allowed to mix under the influence of dynamic correlation. The first new method is the 2nd order dynamic correlation dressed complete active space method (DCD-CAS(2)), which is formulated in terms of an effective Hamiltonian that is based on the theory of intermediate effective Hamiltonians. It simultaneously provides the ground state and a few low-lying excited states. The method is orbitally invariant and preserves orbital degeneracies of the underlying complete active space self-consistent field solutions. In cases where model space components become nearly degenerate after the inclusion of dynamic correlation, DCD-CAS(2) is shown to be superior to state-specific 2nd order dynamic correlation methods like the N-electron valence state perturbation theory (NEVPT2).
It was found that DCD-CAS(2) fails in simultaneously describing ligand field and charge transfer states in a balanced way because of the state-averaged 0th order Hamiltonian used in its construction. The multi-partitioning idea allows the use of statespecific 0th order Hamiltonians in a multistate framework and could therefore alleviate the mentioned problem. However, the effective Hamiltonian is non-Hermitian in the traditional formulation of multi-partitioning, which can lead to unphysical behavior especially for nearly degenerate states. In order to achieve a more balanced treatment of states with a different physical character and at the same time have a Hermitian effective Hamiltonian, we combine for the first time multi-partitioning with canonical Van Vleck perturbation theory. At the 2nd order, the result is a Hermitian variant of multi-partitioning quasi-degenerate NEVPT2 (QD-NEVPT2). It is given the acronym HQD-NEVPT2. The effect of model space non-invariance of the method is discussed and the benefit of a Hermitian formulation is highlighted with numerical examples.
Both DCD-CAS(2) and HQD-NEVPT2 are also extended to incorporate spin-dependent relativistic effects into the Hamiltonian. This results in effective Hamiltonians that simultaneously contain the effects of static correlation, dynamic correlation and relativity. All important contributions necessary for the description of magnetic phenomena and electron paramagnetic resonance spectroscopy, namely spin-orbit coupling, magnetic hyperfine coupling, Zeeman interaction, and direct electronic spin-spin coupling, are incorporated. We also suggest a novel analysis of Kramers doublet g-matrices and A-matrices based on the singular value decomposition. It provides not only the magnitude but also the sign of the principal components and allows for a transparent decomposition into different physical contributions.
Tests are performed for excitation energies of first-row transition metal ions, as well as D-tensors and g-shifts of first-row transition metal complexes using minimal active spaces. It is observed that state-mixing effects are usually small in these cases and that the results are comparable to nondegenerate NEVPT2 in conjunction with quasidegenerate perturbation theory (QDPT). Results on EPR parameters of pseudo-squareplanar copper(II) complexes show that state mixing with a ligand-to-metal charge transfer configuration greatly improves the results compared with NEVPT2/QDPT. HQD-NEVPT2 turns out to be more reliable than DCD-CAS(2) for this kind of problem because of its better 0th order description of the involved states. HQD-NEVPT2 is also shown to give good results for the calculation of electronic transitions of the tetrachlorocuprate(II) complex, which is an example where the balance between ligand-field and charge transfer configurations is of utmost importance.
The last part of this work is concerned with the connection of the newly developed multistate methods with the ab initio ligand field theory (AILFT), which has evolved into an important tool for the extraction of ligand field models from ab initio calculations over the last few years. The incorporation of dynamic correlation was previously realized at the level of NEVPT2. The two new versions of AILFT are tested for a diverse set of transition metal complexes. It is found that the multistate methods have, compared to NEVPT2, an AILFT fit with smaller root-mean-square deviations (RMSDs) between ab initio and AILFT energies. Comparison of AILFT excitation energies with the experiment shows that for some systems the agreement gets better at the multistate level because of the smaller RMSDs. However, for some systems the agreement gets worse, which can be attributed to a cancellation of errors at the NEVPT2 level that is partly removed at the multistate level. An investigation of trends in the extracted ligand field parameters shows that at the multistate level the ligand field splitting ∆ gets larger, while the Racah parameters B and C get smaller and larger, respectively. An investigation of the reasons for the observed improvement for octahedral chromium(III) halide complexes shows that the possibility of state mixing relaxes constraints that are present at the NEVPT2 level and that keep ∆ and B from following their individual preferences.
