Potential energy curves for Mo₂: multi-component symmetry-projected Hartree–Fock and beyond

Abstract

The molybdenum dimer is an example of a transition metal system with a formal sextuple bond that constitutes a challenging case for ab initio quantum chemistry methods. In particular, the complex binding pattern in the Mo 2 molecule requires a high-quality description of non-dynamic and dynamic electron correlation in order to yield the correct shape of the potential energy curve. The present study examines the performance of a recently implemented multi-component symmetry projected Hartree-Fock (HF) approach. In this work, the spin and spatial symmetries of a trial wavefunction written in terms of non-orthogonal Slater determinants are deliberately broken and then restored in a variation-after-projection framework. The resulting symmetry-projected HF wavefunctions, which possess well-defined quantum numbers, can account for static and some dynamic correlations. A single symmetry-projected configuration in a D ∞h S-UHF or a D ∞h KS-UHF framework offers a reasonable description of the potential energy curve of Mo 2 , though the binding energy is too small for the former. Our multi-component strategy offers a way to improve on the single configuration result in a systematic way towards the exact wavefunction: in the def2-TZVP basis set considered in this study, a 7-determinant multi-component D ∞h S-UHF approach yields a bond length of 2.01 Å , in good agreement with experimental results, while the predicted binding energy is 39.2 mhartree. The results of this exploratory study suggest that a multi-component symmetry-projected HF stategy is a promising alternative in a high-accuracy description of the electronic structure of challenging systems. We also present and discuss some benchmark calculations based on the CEEIS-FCI (correlation energy extrapolation by intrinsic scaling -full configuration interaction) method for selected geometries.

Half-Projected σ Self-Consistent Field For Electronic Excited States

“…PQT and its variants 39 are found to be effective at recovering the static part of electron correlation that arises from symmetry breaking. [46][47][48][49] Extensions of PQT to DFT 50,51 and Quantum Monte-Carlo 52 (QMC) have also been reported to include dynamic correlation.…”

Variational description of the ground state of the repulsive two-dimensional Hubbard model in terms of nonorthogonal symmetry-projected Slater determinants

“…The FED methodology has already been quite successful in microscopic nuclear structure theory, 56 but it is still in its first steps in both quantum chemistry 49,50 and condensed matter physics. 46,47 We believe that it is a good candidate for further multidisciplinary bridges between these research fields.…”

“…55 Such a parametrization has already been shown to be a useful tool in nuclear structure 56,58,59,96 and condensed matter [43][44][45][46][47] physics but also in quantum chemistry. [48][49][50] All the results discussed in this paper have been obtained with an in-house code where the optimization is handled with a limited-memory quasi-Newton method. 97 In a previous study, 46 we have discussed the computational performance of the FED scheme in the case of 1D systems.…”

“…However, an intuitive physical picture of the basic units of quantum fluctuations in the considered 1D systems has remained an open issue within several approximations. In recent years, both single reference (SR) and multireference (MR) symmetry-projected approximations, [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] routinely used in nuclear structure physics, [55][56][57][58][59][60][61] have been applied to describe correlated electronic systems. It has been shown that MR schemes like the Resonating Hartree-Fock [39][40][41][42][62][63][64][65] (ResHF) and the Few Determinant [46][47][48][49]56,57 (FED) ones, provide a reasonable description of the ground state energies of half-filled and doped 1D Hubbard lattices but also account for the main physical trends in correlation functions, momentum distributions, spectral functions, and density of states.…”

Compact wavefunctions from compressed imaginary time evolution

“…Namely the quality of description and rate of convergence in N c are invariant to invertible local transformations of the state (i.e. atomic orbitals vs. natural orbitals) [19], and the mathematical machinery related to the use and extension of such a wavefunction is already well developed [34][35][36][37] While the method we use for determinant selection is unique, non-orthogonal Slater determinants have been used successfully in valence bond theory [4,5] as well as more recent symmetry breaking and projection methods [38,39]. Unconstrained non-orthogonal Slater determinants have been utilized before, but in a purely variational context [16,20].…”