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Two independent approaches are applied to study the intra-molecular dynamics in B$4$ electronic ground state. First, a double minimum six-dimensional potential energy surface (PES) is determined in symmetry coordinates for the most stable rhombic (D) B$4$ isomer in its $1$A electronic ground state by fitting to energies calculated by the CCSD(T) approach. The PES exhibits a barrier to the D square structure of 255 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The vibrational levels (J = 0) are calculated variationally using the MULTIMODE} \underline{\textbf{107}}, 10458 (1997).} code, which involves the Watson kinetic energy operator expressed in normal coordinates. The pattern of about 65 vibrational levels up to 1600 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ for all stable isotopomers is analyzed. Analogous to the inversion in ammonia-like molecules, the rhombus rearrangements lead to splittings of the vibrational levels. In B$4$ it is the b ($D4h$) mode which distorts the square molecule to its planar rhombic form. It is shown that the rhombic ground states of B$\mathrm{4+}$ and B$\mathrm{4-}$ consist of two vibronically coupled electronic states. The PES for B$4$ is used also in time-dependent molecular dynamics calculations using the multi-configurational time-dependent Hartree} \underline{\textbf{165}}, 73 (1990).} (MCTDH) method. The 6-D wavepackets are analyzed and the results are compared with time-independent results.