@article {26622,
	title = {   Relativistic explicit correlation: Coalescence conditions and practical suggestions},
	journal = {Journal of Chemical Physics},
	volume = {136},
	year = {2012},
	pages = {144117},
	abstract = {To set up the general framework for relativistic explicitly\&nbsp;correlated wave function methods, the electron-electron coalescence\&nbsp;conditions are derived for the wave functions of the Dirac-Coulomb\&nbsp;(DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified\&nbsp;Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA)\&nbsp;Hamiltonians. The manipulations make full use of the internal\&nbsp;symmetries of the reduced two-electron Hamiltonians such that the\&nbsp;asymptotic behaviors of the wave functions emerge naturally. The\&nbsp;results show that, at the coalescence point of two electrons, the\&nbsp;wave functions of the DCG Hamiltonian are regular, while those of\&nbsp;the DC and DCB Hamiltonians have weak singularities of the type\&nbsp;$r_{12}^{\nu}$ with $\nu$ being negative and of $\mathcal{O}(\alpha^2)$. The\&nbsp;behaviors of the MDC wave functions are related to the original ones\&nbsp;in a simple manner, while the spin-free counterparts are somewhat\&nbsp;different due to the complicated electron-electron interaction. The\&nbsp;behaviors of the ZORA wave functions depend on the chosen potential\&nbsp;in the kinetic energy operator. In the case of the nuclear\&nbsp;attraction, the behaviors of the ZORA wave functions are very\&nbsp;similar to those of the nonrelativistic ones, just with an\&nbsp;additional correction of $\mathcal{O}(\alpha^2)$ to the\&nbsp;nonrelativistic cusp condition. However, if the Coulomb interaction\&nbsp;is also included, the ZORA wave functions become close to the\&nbsp;large-large components of the DC wave functions. Note that such\&nbsp;asymptotic expansions of the relativistic wave functions are only\&nbsp;valid within an extremely small convergence radius $R_c$ of\&nbsp;$\mathcal{O}(\alpha^2)$. Beyond this radius, the behaviors of the\&nbsp;relativistic wave functions are still dominated by the\&nbsp;nonrelativistic limit, as can be seen in terms of direct\&nbsp;perturbation theory (DPT) of relativity. However, as the two limits\&nbsp;$\alpha\rightarrow0$ and $r_{12}\rightarrow0$ do not commute, DPT is\&nbsp;doomed to fail due to incorrect descriptions of the small-small\&nbsp;component $\Psi^{SS}$ of the DC wave function for $r_{12}\&lt;R_c$.\&nbsp;Another deduction from the possible divergence of \&nbsp;$\Psi^{SS}$ at\&nbsp;$r_{12}=R_c$ is that the DC Hamiltonian has no bound electronic\&nbsp;states, although the last word cannot be said. These findings enrich\&nbsp;our understandings of relativistic wave functions. On the practical\&nbsp;side, it is shown that, under the no-pair approximation,\&nbsp;relativistic explicitly correlated wave function methods can be made\&nbsp;completely parallel to the nonrelativistic counterparts, as\&nbsp;demonstrated explicitly for MP2-F12. Yet, this can only be achieved\&nbsp;by using an extended no-pair projector.},
	url = {http://dx.doi.org/10.1063/1.3702631},
	author = {Zhendong Li and Sihong Shao and Wenjian Liu}
}
