<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Zhendong Li</style></author><author><style face="normal" font="default" size="100%">Sihong Shao</style></author><author><style face="normal" font="default" size="100%">Wenjian Liu</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">   Relativistic explicit correlation: Coalescence conditions and practical suggestions</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Chemical Physics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2012</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.1063/1.3702631</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">136</style></volume><pages><style face="normal" font="default" size="100%">144117</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">To set up the general framework for relativistic explicitly&amp;nbsp;correlated wave function methods, the electron-electron coalescence&amp;nbsp;conditions are derived for the wave functions of the Dirac-Coulomb&amp;nbsp;(DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified&amp;nbsp;Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA)&amp;nbsp;Hamiltonians. The manipulations make full use of the internal&amp;nbsp;symmetries of the reduced two-electron Hamiltonians such that the&amp;nbsp;asymptotic behaviors of the wave functions emerge naturally. The&amp;nbsp;results show that, at the coalescence point of two electrons, the&amp;nbsp;wave functions of the DCG Hamiltonian are regular, while those of&amp;nbsp;the DC and DCB Hamiltonians have weak singularities of the type&amp;nbsp;$r_{12}^{\nu}$ with $\nu$ being negative and of $\mathcal{O}(\alpha^2)$. The&amp;nbsp;behaviors of the MDC wave functions are related to the original ones&amp;nbsp;in a simple manner, while the spin-free counterparts are somewhat&amp;nbsp;different due to the complicated electron-electron interaction. The&amp;nbsp;behaviors of the ZORA wave functions depend on the chosen potential&amp;nbsp;in the kinetic energy operator. In the case of the nuclear&amp;nbsp;attraction, the behaviors of the ZORA wave functions are very&amp;nbsp;similar to those of the nonrelativistic ones, just with an&amp;nbsp;additional correction of $\mathcal{O}(\alpha^2)$ to the&amp;nbsp;nonrelativistic cusp condition. However, if the Coulomb interaction&amp;nbsp;is also included, the ZORA wave functions become close to the&amp;nbsp;large-large components of the DC wave functions. Note that such&amp;nbsp;asymptotic expansions of the relativistic wave functions are only&amp;nbsp;valid within an extremely small convergence radius $R_c$ of&amp;nbsp;$\mathcal{O}(\alpha^2)$. Beyond this radius, the behaviors of the&amp;nbsp;relativistic wave functions are still dominated by the&amp;nbsp;nonrelativistic limit, as can be seen in terms of direct&amp;nbsp;perturbation theory (DPT) of relativity. However, as the two limits&amp;nbsp;$\alpha\rightarrow0$ and $r_{12}\rightarrow0$ do not commute, DPT is&amp;nbsp;doomed to fail due to incorrect descriptions of the small-small&amp;nbsp;component $\Psi^{SS}$ of the DC wave function for $r_{12}&amp;lt;R_c$.&amp;nbsp;Another deduction from the possible divergence of &amp;nbsp;$\Psi^{SS}$ at&amp;nbsp;$r_{12}=R_c$ is that the DC Hamiltonian has no bound electronic&amp;nbsp;states, although the last word cannot be said. These findings enrich&amp;nbsp;our understandings of relativistic wave functions. On the practical&amp;nbsp;side, it is shown that, under the no-pair approximation,&amp;nbsp;relativistic explicitly correlated wave function methods can be made&amp;nbsp;completely parallel to the nonrelativistic counterparts, as&amp;nbsp;demonstrated explicitly for MP2-F12. Yet, this can only be achieved&amp;nbsp;by using an extended no-pair projector.</style></abstract></record></records></xml>