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This article is a response to a comment made by Professor Liu regarding a previously published paper. The response addresses questions raised by Professor Liu and focuses on three key aspects of the validity of the paper. It explains the use of different contractions in the construction of the QED Hamiltonians and clarifies the commutation relations used in the calculations. The article also highlights that the formulation of the molecular orbital method described in the paper is independent of the ordering of the operators and can derive expressions for various perturbation theories. The response concludes by stating that alternative criteria for the QED Hamiltonians are not ruled out and could be explored in future research. [Extracted from the article]

Theoretical discussions are given on issues in relativistic molecular orbital theory to which the quantum electrodynamics (QED) Hamiltonian is applied. First, several QED Hamiltonians previously proposed are sifted by the orbital rotation invariance, the charge conjugation and time reversal invariance, and the nonrelativistic limit. The discussion on orbital rotation invariance shows that orbitals giving a stationary point of total energy should be adopted for QED Hamiltonians that are not orbital rotation invariant. A new total energy expression is then proposed, in which a counter term corresponding to the energy of the polarized vacuum is subtracted from the total energy. This expression prevents the possibility of total energy divergence due to electron correlations, stemming from the fact that the QED Hamiltonian does not conserve the number of particles. Finally, based on the Hamiltonian and energy expression, the Dirac–Hartree–Fock (DHF) and electron correlation methods are reintroduced. The QED-based DHF equation is shown to give information on positrons from negative-energy orbitals while having the same form as the conventional DHF equation. Three electron correlation methods are derived: the QED-based configuration interactions and single- and multireference perturbation methods. Numerical calculations show that the total energy of the QED Hamiltonian indeed diverged and that the counter term is effective in avoiding the divergence. The relativistic molecular orbital theory presented in this article also provides a methodology for dealing with systems containing positrons based on the QED Hamiltonian. [ABSTRACT FROM AUTHOR]

The generalized Foldy–Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two‐component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two‐component methods. New two‐component methods were also systematically derived based on the GFW transformation. These various two‐component methods were applied to hydrogen‐like and helium‐like ions. Numerical errors in energy were evaluated and classified into four types: the one‐electron Hamiltonian approximation, the two‐electron operator approximation, the newly defined "picture difference error (PDE)," and the error in determining the transformation, and errors in multi‐electron systems were discussed based on this classification. [ABSTRACT FROM AUTHOR]

A combined method of the Dirac–Hartree–Fock (DHF) method and the reference interaction‐site model (RISM) theory is reported; this is the initial implementation of the coupling of the four‐component relativistic electronic structure theory and an integral equation theory of molecular liquids. In the method, the DHF and RISM equations are solved self‐consistently, and therefore the electronic structure of the solute, including relativistic effects, and the solvation structure are determined simultaneously. The formulation is constructed based on the variational principle with respect to the Helmholtz energy, and analytic free energy gradients are also derived using the variational property. The method is applied to the iodine ion (I−), methyl iodide (CH3I), and hydrogen chalcogenide (H2X, where X = O–Po) in aqueous solutions, and the electronic structures of the solutes, as well as the solvation free energies and their component analysis, solvent distributions, and solute–solvent interactions, are discussed. [ABSTRACT FROM AUTHOR]

The synthesis, structure and reactions of (acylimino)triaryl-lambda(super 5)-bismuthanes and a comparative study of the structure and reactivity of a series of (acylimino)pnictoranes is given. The unique reactivities of (acylimino)triaryl-lambda(super 5)-bismuthanes demonstrate their potential utility in organic synthesis.

The protonation/deprotonation reaction is one of the most fundamental processes in solutions and biological systems. Compounds with dissociative functional groups change their charge states by protonation/deprotonation. This change not only significantly alters the physical properties of a compound itself, but also has a profound effect on the surrounding molecules. In this paper, we review our recent developments of the methods for predicting the K a , the equilibrium constant for protonation reactions or acid dissociation reactions. The p K a , which is a logarithm of K a , is proportional to the reaction Gibbs energy of the protonation reaction, and the reaction free energy can be determined by electronic structure calculations with solvation models. The charge of the compound changes before and after protonation; therefore, the solvent effect plays an important role in determining the reaction Gibbs energy. Here, we review two solvation models: the continuum model, and the integral equation theory of molecular liquids. Furthermore, the reaction Gibbs energy calculations for the protonation reactions require special attention to the handling of dissociated protons. An efficient method for handling the free energy of dissociated protons will also be reviewed. [ABSTRACT FROM AUTHOR]

Initial cleavage of a ring CC bond to form a trans .CHsub2SiHsub2Chsub2Chsub2. diradical is the most likely route from silacyclobutane to ethylene + silene, according to multiconfigurational self-consistent field wave functions and second order perturbation theory. The trans diradical is predicted to be a minimum on the MCSCF ground state potential energy surface. The addition of dynamic correlation leads to the disappearance of the transition state.