Low-Lying Electronic States of the Nickel Dimer

Abstract

The generalized Van Vleck second order multireference perturbation theory (GVVPT2) method was used to investigate the low-lying electronic states of Ni2. Because the nickel atom has an excitation energy of only 0.025 eV to its first excited state (the least in the first row of transition elements), Ni2 has a particularly large number of low-lying states. Full potential energy curves (PECs) of more than a dozen low-lying electronic states of Ni2, resulting from the atomic combinations 3F4 + 3F4 and 3D3 + 3D3, were computed. In agreement with previous theoretical studies, we found the lowest lying states of Ni2 to correlate with the 3D3 + 3D3 dissociation limit, and the holes in the d-subshells were in the subspace of delta orbitals (i.e., the so-dubbed δδ-states). In particular, the ground state was determined as X 1Γg and had spectroscopic constants: bond length (Re) = 2.26 Å, harmonic frequency (ωe) = 276.0 cm−1, and binding energy (De) = 1.75 eV; whereas the 1 1Σg+ excited state (with spectroscopic constants: Re = 2.26 Å, ωe = 276.8 cm−1, and De = 1.75) of the 3D3 + 3D3 dissociation channel lay at only 16.4 cm−1 (0.002 eV) above the ground state at the equilibrium geometry. Inclusion of scalar relativistic effects through the spin-free exact two component (sf-X2C) method reduced the bond lengths of both of these two states to 2.20 Å, and increased their binding energies to 1.95 eV and harmonic frequencies to 296.0 cm−1 for X 1Γg and 297.0 cm−1 for 1 1Σg+. These values are in good agreement with experimental values of Re = 2.1545 ± 0.0004 Å, ωe = 280 ± 20 cm−1, and D0 = 2.042 ± 0.002 eV for the ground state. All states considered within the 3F4 + 3F4 dissociation channel proved to be energetically high-lying and van der Waals-like in nature. In contrast to most previous theoretical studies of Ni2, full PECs of all considered electronic states of the molecule were produced.

Potential-Energy Curves for the Ground and Several Electronic States of NdO and NdS

“…To perform bond order analysis of the NdO and NdS systems, the effective bond order (EBO), η, was calculated for the ground state by eq : η = false ∑ j χ j c j 2 false ∑ j c j 2 where χ j is the EBO for the j th CSF and c j 2 is its corresponding weight (Note, c j > 0.05 were included). The EBO for each CSF is defined by eq , χ j = 1 2 false( n b − n ab false) where n b and n ab are the number of bonding and antibonding electrons, respectively. ,, To determine the vibrational frequencies at the optimized MCSCF/GVVPT2 geometries of NdO and NdS, 5-point finite differencing near the minimum of the ground electronic state was performed.…”

“…The EBO for each CSF is defined by eq 2 , where n b and n ab are the number of bonding and antibonding electrons, respectively. 34 , 44 , 45 To determine the vibrational frequencies at the optimized MCSCF/GVVPT2 geometries of NdO and NdS, 5-point finite differencing near the minimum of the ground electronic state was performed.…”

“…The primary-external interactions are perturbatively applied to correct the states within the primary subspace, where the secondary subspace has a buffering role to separate the primary and external subspaces energetically. 34 Many multireference perturbation theory methods are plagued with the intruder state problems and the GVVPT2 scheme overcomes the issue by defining the energy-buffering secondary subspace to produce smooth potential energy surfaces 35 like the ones obtained in this study despite their close proximity with one another. In addition, the macro-configuration approach implemented in the GVVPT2 method divides the molecular orbitals into groups with assigned occupancies 36 while also taking into account physical considerations of the molecular system in constructing the configuration spaces.…”

“…Using the Van Vleck formalism and a unitary wavelike operator to satisfy the generalized Bloch equation allows that the primary–secondary interactions can be solved variationally; therefore, the strongly quasi-degenerate states of the primary and secondary subspaces can be addressed. The primary-external interactions are perturbatively applied to correct the states within the primary subspace, where the secondary subspace has a buffering role to separate the primary and external subspaces energetically . Many multireference perturbation theory methods are plagued with the intruder state problems and the GVVPT2 scheme overcomes the issue by defining the energy-buffering secondary subspace to produce smooth potential energy surfaces like the ones obtained in this study despite their close proximity with one another.…”