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I apologize for a perhaps newbie question.

I would like to compute the potential energy surface for several small molecular ions, like $\ce{H2+}$ and $\ce{H3+}$, using quantum chemistry software. What are the proper method and basis set to do that and why? I am thinking about ROHF/STO-3G or UHF.

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$\ce{H2+}$ or $\ce{H3+}$ have unpaired electrons because they are ions. Due to this, you should be using an unrestricted shell theoretical model which doesn't restrict electrons to paired (closed shell) orbitals. You should start with unrestricted Hartree-Fock (UHF) and can increase the level of theory to include electron correlation with something like 2nd order Møller Plesset (MP2) or 4th order (MP4).

The electron correlation models acknowledge the fact that electron locations are not totally random, repulsion between electrons is a real effect and can give more realistic results.

With regard to basis set, STO-3G would be considered a starting point. If time and resources allow, you could build up to something like a 6-31G(d,p) which adds p-orbitals to the hydrogen atoms allowing more optimization of molecular orbitals. Increasing the basis set quality approximates the radial components of the atomic orbitals with more Gaussian curves, thereby increasing the possible accuracy of the calculation.

Exploring Chemistry with Electronic Structure Methods - Foresman and Frisch

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    $\begingroup$ For the people who are not familiar with the terminology, could you add references? $\endgroup$ – rcollyer Aug 21 '13 at 20:16
  • $\begingroup$ Thanks for the suggestion, I've added some additional detail and a reference. From the OP's terminology it appears that Gaussian might be the software used so I stuck with their nomenclature. $\endgroup$ – scs217 Aug 21 '13 at 20:32
  • $\begingroup$ $\ce{H3+}$ does not have unpaired electrons. $\endgroup$ – Martin - マーチン Dec 29 '15 at 15:51
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if u accurately want to make ur computations a bigger basis set and high level are necessery (CI(SDQ) or CAS), however so bigger basis set so less human readable are results. If the only aim is to see nice orbitals, minimal basis and DFT or even semiemprical methods should be enough. As stated before unrestricted methods are necessary unless u want to play with half-occupations.

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