Chemistry

B3LYP/6-311++G(d,p)//B3LYP/6-31G(d) level of theory has been used to investigate the ring size effect on
the Bellus-Claisen rearrangement. The results show the reaction has a high asynchronous character and
there is no direct relationship between ring size and Ea; for example, the Ea is almost the same amount for
three- and five-member rings. Four-member rings however, have an unexpectedly higher Ea. Reactants
which have nitrogen as a heteroatom in their structures have lower Ea in comparison to oxygen containing
rings. The effect of solvent on the Ea is considerable. In all of the cases, by increasing the polarity of
solvents, the amount of Ea decreases. The results show that the effect of solvent on Ea for N as heteroatom
is more than that of O. There is an inverse relation between solvent polarity and Ea. This can be interpreted
as higher stabilization effect of the polar solvents on TSs’ structures comparing to less polar solvents
which leads to lower Ea.

An effective set of the Hartree-Fock (HF) equations are derived for electrons of the muonic systems, i.e., molecules containing a positively charged muon, conceiving the muon as a quantum oscillator, which are completely equivalent to the usual two-component HF equations used to derive stationary states of the muonic molecules.  In these effective equations, a non-Coulombic potential is added to the orthodox coulomb and exchange potential energy terms, which describes the interaction of the muon and the electrons effectively and is optimized during the self-consistent field cycles. While in the two-component HF equations muon is treated as a quantum wave, in the effective HF equations it is absorbed into the effective potential and practically transformed into an effective potential field experienced by electrons.  The explicit form of the effective potential depends on the nature of muon’s vibrations and is derivable from the basis set used to expand the muonic spatial orbital. The resulting effective Hartree-Fock equations are implemented computationally and used successfully, as a proof of concept, in a series of the muonic molecules containing all atoms from the second and third rows of the Periodic Table.  To solve the algebraic version of the equations muon-specific gaussian basis sets are designed for both muon and surrounding electrons and it is demonstrated that the optimized exponents are quite distinct from those derived for the hydrogen isotopes.  The developed effective HF theory is quite general and in principle can be used for any muonic system while it is the starting point for a general effective electronic structure theory that incorporates various types of quantum correlations into the muonic systems beyond the HF equations.

A set of effective electronic-only Kohn-Sham (EKS) equations are derived for the muonic molecules (containing a positively charged muon), which are completely equivalent to the coupled electronic-muonic Kohn-Sham equations derived previously within the framework of the Nuclear-Electronic Orbital density functional theory (NEO-DFT).  The EKS equations contain effective non-coulombic external potentials depending on parameters describing muon’s vibration, which are optimized during the solution of the EKS equations making muon’s KS orbital reproducible.  It is demonstrated that the EKS equations are derivable from a certain class of effective electronic Hamiltonians through applying the usual Hohenberg-Kohn theorems revealing a “duality” between the NEO-DFT and the effective electronic-only DFT methodologies.  The EKS equations are computationally applied to a small set of muoniated organic radicals and it is demonstrated that a mean effective potential maybe derived for this class of muonic species while an electronic basis set is also designed for the muon.  These computational ingredients are then applied to muoniated ferrocenyl radicals, which had been previously detected experimentally through adding muonium atom to ferrocene.  In line with previous computational studies, from the six possible species the staggered conformer, where the muon is attached to the exo position of the cyclopentadienyl ring, is deduced to be the most stable ferrocenyl radical.