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Amongst many models, including the valence bond model (VB) or the molecular orbital (MO) model, which are the ones with best predictive power? (e.g. the MO is thought to predict spectroscopic properties better than the VB)

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The so called gold standard of quantum chemical calculations of ground state energies and properties is usually considered to be Coupled Cluster Singles Doubles (with perturbative) Triples as a method. This method is based on ab initio molecular orbital theory and it is size consistent, but not variational. A basis set of at least a triple zeta quality with additional valence and polarisation functions should be employed, good examples for this are Dunning's (augmented) correlation consistent basis sets, e.g. (aug-)cc-pV{T,Q,5,6}Z, or Weigend's def2-basis set family, def2-TZVPP, def2-QZVP, but there are various others. (If you like to have some citations, drop a note in the comments.) This level of theory is usually denoted as CCSD(T)/TZVP, where TZVP is a placeholder for one of the various basis sets.

In principle CC theory is exact at the complete basis set limit, but in practise, this is not solvable. Another theory that is in principle exact, because they become equivalent, is Full Configuration Interaction, F-CI. The truncated variants CISD, CISD(T), CISDT, ... are variational, but not size consistent. For chemical systems it is much more important, that the calculations are size consistent, therefore these methods lost their importance. An attempt to solve this was to introduce Quadratic Configuration Interaction, QCI, but this is almost the same as coupled cluster theory, so it was not successful either.

The importance of Many Body Perturbation Theory, MBPT, of high orders, like Møller Plesset Perturbation Theory of fourth order, MP4, has very much decreased because of it inefficiency.

There are some cases, which cannot be accurately described by these models. Those are cases, where the ground state cannot be well defined by one determinante, the initial guess wave function obtained from Hartree-Fock, HF. Here you have to use a different description. A Multi Configuration, MC, or Multi Reference, MR, ansatz is necessary. The qualitative equivalent to HF is Complete Active Space Self Consistent Field, CASSCF. Based on this you can use Perturbation Theory to include more correlation effects. The best possible description so far (judging accuracy versus computational cost) is second order perturbation theory on a CASSCF wave function, for short CASPT2. It is also very reliable for spectroscopic calculations. There are also various developments on Multi Reference Configuration Interaction as another theory. (I have limited expertise on that field.)

Another way of treating chemical systems is Valence Bond theory, or with the application to modern computational aspects. Here you propose the mixing of various Lewis like structures, to obtain the ground state energy and properties. The formalism can get quite complicated, maybe this is why its performance for excited states properties is not very accurate at the moment. It is still a very active field of research, by a few scientific groups. (I am not one of them.)

However, affordable spectroscopic calculations have still huge errors and sometimes the numbers can vary quite a bit. Currently most used is a variety of methods derived from Time Dependent Density Functional Theory, TDDFT. An ab initio method of great importance in this field is Configuration Interaction Singles. Also fairly well implemented are approximate coupled cluster techniques, e.g. CC2. The struggle for finding an accurate, robust and computationally cheap method are one of the main goals of modern quantum chemistry. (I can apply these methods fairly well, but I have no deeper insight how they work.)

For extended systems like surfaces and solids there are various approaches. Since they get very demanding very soon, the most commonly used practical approaches are based on plain or hybrid density functionals. They often also employ plane wave basis sets with periodic boundary condition. (Unfortunately I am not an expert on that field either.)

As you can see, there is no definite answer to your question and the development of new and efficient methods is still a very active field. Depending on what type of description you want or need, there are many possibilities. The most advanced models are obviously for ground state energies and properties, since they have been researched for about a century now.

I hope this helps you, I apologise for the chatty answer. If something remains unclear, drop me a note in the comments.

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    $\begingroup$ Very good summary. I would like to add three things: * Not all of these methods works nicely when you have open-shell systems (spin). * Many, if not most, of these methods can be computationally prohibitive in practical studies, especially if you are not in a computational group with good access to computational power. *All these methods are about isolated entities in vacuum, however solvent / environmental effects can often add a whole new layer to the problem. $\endgroup$ – Greg Jul 28 '14 at 17:47

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