An improved ab initio relativistic zeroth-order regular approximation correct to order $1/c2$

The equations of the original ab initio scalar-relativistic zeroth-order regular approximation (ZORA) and the infinite-order regular approximation (IORA) are expanded in orders of $1/c2.$ It is shown that previous ZORA/IORA implementations in ab initio quantum chemistry programs were not correct to order $1/c2,$ but contained imperfections leading to fictitious self-interactions. These errors can be avoided by adding exchange-type terms (coupling the large and small components) to the relativistic ZORA correction to the Hamiltonian, yielding improved ab initio relativistic zeroth- and infinite-order regular approximations that are correct to order $1/c2.$ The new methods have been tested numerically by computing the total energies, orbital energies, and static electric dipole polarizabilities of the rare gas atoms He through Xe.