Spin−Orbit Effects in the Ground States of Singly Positive and Neutral V₂, VNb, and Nb₂:  INDO/S and Empirical Model Calculations

Abstract

Ground-state spin−orbit splittings in V2 +, VNb+, and Nb2 + were investigated with spin−orbit configuration interaction (SOCI) calculations using the intermediate neglect of differential overlap method for optical spectroscopy (INDO/S). Splittings found for the X4Σ- ground states of V2 + and VNb+ using a CI treatment isolating essential correlation of the ground and isoconfigurational a2Σ+ and b2Σ- state wave functions agreed well with experiment. The size of the splitting for Nb2 + was predicted. In all three cases the a2Σ+ and b2Σ- states were found to be strongly coupled by the spin−orbit operator. Several perturbative models for predicting energies of interacting excited states based on measured splittings were analyzed by comparing to SOCI results for V2, VNb, and Nb2. Second-order perturbation theory, treating only the isoconfigurational states, and neglecting overlap and CI in computing couplings was the best choice of approximations due to cancellation of errors. An empirical model treating the isoconfigurational states of the cations was developed within these guidelines and agrees well with SOCI calculations for V2 +. The empirical wave functions were divergent for Nb2 + due to similar energies of the doublet states. Analysis of correlation effects in the INDO/S model indicates that a charge-iterative method could be effective for systematically correlated semiempirical calculations on metal cluster ions.

Resonant Multiphoton Fragmentation Spectrum of Niobium Dimer Cation

“…This value gives an averaged value for the molecular spin−orbit constant ( A SO ) for B 4 Π Ωu and D 4 Π Ωu excited states: A SO (B 4 Π Ωu (ν′ = 1)) ≈ 183 cm −1 and A SO (D 4 Π Ωu (ν′ = 0)) ≅ 200 cm −1 . From the placement of these subbands, we determined the second-order spin−orbit coupling (Δ) in the ground-state to be 145 cm −1 , in excellent agreement with values of 142 and 145 cm −1 estimated from eqs and as well as the DFT , and INDO/S-SOCI calculations . From the fitting of several relatively weak transitions to excited states from vibrationaly excited ground state, we determined ground-state vibrational frequency (ω′′) of Nb 2 + to be 420 ± 3 cm −1 , which is compatible with the theoretical predictions: ω′′ = 436 cm −1 at the BLYP/LanL2DZ level and 430 cm −1 at the FOCI level .…”

“…To the best of our knowledge, no experimental value for the second-order spin−orbit splitting (Δ) has been reported, except for a few theoretical calculations. The first reported Δ value for Nb 2 + calculated by Simard et al is 142 cm −1 , which is in good agreement with the calculations at the different levels of the DFT (109, 119, and 166 cm −1 ) by Aydin et al and 145 cm −1 from INDO/S-SOCI calculations by O’Brien . However, Balasubramanian and Zhu have performed the FOCI at the CASSCF/MRCI level and they found 18 low-lying electronic states below 16 900 cm −1 .…”

“…If the distance between the X 4 Σ g − (...2σ g 1 δ g 2 ) and 2 Σ g + (...2σ g 1 δ g 2 ) states is higher than 16 900 cm −1 , then, the value of Δ must be less than 40 cm −1 . However, this value seems too small given the trends in measured values of Δ for the V 2 + (Δ = 20 and 21 cm −1 ) and VNb + (Δ = 82 cm −1 ), which have the same ground-state symmetry, X 4 Σ g − , resulting from the similar valence electronic configuration, π u 4 1σ g 2 2σ g 1 δ g 2 (for V 2 + ) and π 4 1σ 2 2σ 1 δ 2 (for VNb + ). ,, In their neutral species, Δ = 75 cm −1 for V 2 , 230 cm −1 for VNb, and 410 cm −1 for Nb 2 , , and these have X 3 Σ g − /X 3 Σ − ground state symmetry that results from the π u 4 1σ g 2 2σ g 2 1δ g 2 (π 4 1σ 2 2σ 2 1δ 2 ) valence electronic configuration.…”

“…As mentioned earlier, an experimental value for the second-order spin−orbit splitting (Δ) has not yet been reported, except a few theoretical calculations: 142 cm −1 by Simard et al; 109, 119, and 166 cm −1 by Aydin et al; and 145 cm −1 from the INDO/S-SOCI calculations by O’Brien . However, Balasubramanian and Zhu have performed the FOCI at the CASSCF/MRCI level and they found 18 low-lying electronic states below 16 900 cm −1 and but no 2 Σ Ωg + state found in this energy region.…”

“…In 1994, an ESR study on Nb 2 + in argon matrices by Weltner et al suggested that the ground-state of Nb 2 + was 2 Σ g + , arising from the electronic configuration 1π u 4 1σ g 2 2σ g 1 δ g 2 . However, all theoretical calculations such as the density functional theory (DFT), ,, complete active space self-consistent field (CASSCF), complete active space self-consistent field calculations followed by multireference configuration interaction (CASSCF/MRCI), and a more recent spin−orbit configuration interaction (SOCI) calculations using the intermediate neglect of differential overlap method for optical spectroscopy (INDO/S) predicted 4 Σ g − as a ground-state of Nb 2 + , arising from the same electronic configuration 1π u 4 1σ g 2 2σ g 1 δ g 2 , consistent with Hund’s rule.…”