Toward a muon-specific electronic structure theory: effective electronic Hartree–Fock equations for muonic molecules

Abstract

An effective set of Hartree-Fock (HF) equations are derived for electrons of 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 a muon is treated as a quantum particle, 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 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.

How intrinsic nuclear nonadiabaticity affects molecular structure, electronic density, and conformational stability: Insights from the multicomponent DFT calculations of Mu/H isotopologues

“…In addition, the exponents of muonic basis functions were carefully arranged to have enough flexibility in the description of muon vibrational anharmonicity. The competence and efficiency of the combined set of electronic and muonic basis functions of the muonium atom in characterizing the muon's anharmonic and anisotropic vibrations have been recently shown within the framework of the effective electronic-only formalism at Hartree-Fock and Kohn-Sham levels of theory [112,113] . The overall levels of theory are denoted as UB3LYP/[6-311++g(d,p)/4s1p:5s5p] and UB3LYP/6-311++g(d,p) for the muoniated and hydrogenated adducts hereafter, respectively.…”

Developing effective electronic-only coupled-cluster and Møller–Plesset perturbation theories for the muonic molecules

“…originate from muon's kinetic energy integral, the muon-electron potential energy integral, and the muon-clamped nucleus potential energy integral, respectively. 56 One may re-interpret equation (2) as the variational integral for the ground state energy of the following effective Schrödinger equation: 56 is adopted for the approximation of muon's distribution that leads to the following effective Hamiltonian:…”

“…More complicated effective Hamiltonians are derivable if more complex oscillator models are employed for muon's vibrations. 56 In order to proceed further let us assume that e  , for a typical closed-shell system, is a Slater determinant formed from N electronic spin-orbitals constructed from 2 N spatial orbitals,   , 1,..., 2 j j N   . The conventional functional variation of equation ( 2): 43 after some mathematical manipulations, yields the following set of the HF differential equations:…”

“…46 Instead of applying the NEO-HF theory directly to the muonic species, we have recently proposed an alternative formalism, which is a reformulation of the NEO-HF theory, and is called the effective HF (EHF) theory. 55,56 Within the context of the EHF theory, the muon disappears from the molecular Hamiltonian with the aid of integration over the muonic coordinates, and a as result, a non-Coulombic potential complements the effective electronic Fock operator. This additional term would be optimized during the self-consistent field solution of the EHF equations, 56 and is directly derivable from the basis set used to expand the muonic orbital manifesting the "back-action" of muon's vibrations on the electronic distribution; after solving the EHF equations, the muonic orbital is completely reproducible from the optimized effective potential.…”

“…55,56 Within the context of the EHF theory, the muon disappears from the molecular Hamiltonian with the aid of integration over the muonic coordinates, and a as result, a non-Coulombic potential complements the effective electronic Fock operator. This additional term would be optimized during the self-consistent field solution of the EHF equations, 56 and is directly derivable from the basis set used to expand the muonic orbital manifesting the "back-action" of muon's vibrations on the electronic distribution; after solving the EHF equations, the muonic orbital is completely reproducible from the optimized effective potential. It is timely to stress that we have recently extended the idea of the effective theory to the NEO density functional theory (NEO-DFT) and derived an effective set of electronic-only Kohn-Sham (EKS) equations for the muonic molecules.…”