Given the fact that H-bonds are dipole-dipole interactions with some covalent character, IMHO, a purely electrostatic description isn't reliable. As you pointed out - and very correctly - you need to take into account dispersion and polarisation effects. 

Normally, MM methodologies account for H-bonds by treating it 'specially' (this, of course, depends upon the FF used). As for ab initio, it really depends. If use DFT, it can give you a decent value for the H-bond, but without any physical meaning - it's just dumb luck, based on error cancellation; whereas if you use variational methodologies, such as MP2, it describes it pretty decently, but at a hefty computational cost. Nowadays, DFT functionals such as the h1 and h2 by Wang et al. finally have a good description of the XC hole of the Hartree-Fock model, and as such, they are able to predict decently H-bonding and even weaker complexes.

So, a straightforward answer would be: the more complete the model, the better the results. But this implies greater computational power. And as such, there are some clever ways to 'compensate' such phenomena. So, at this point, it all resumes to a question of adequating your computational facilities - and time - to the theory, not the reverse. There's always a tradeoff...

But if you think a bit about it, you've already answered your question. Just think about tautomeric effects and the answer reveils itself - is it just electrostatic? No it isn't. Will classical descriptions take that into account? To some extent, but not fully describe it, by obvious reasons. Will ab initio provide better results? Depending on how you define your exchange and correlation parts of your HF hamiltonian. Is DFT better than variational in this field? Again, depends on how it is defined. Lots of variables, but fortunately, in the literature, this is a widely discussed issue. You should be able to find more and better answers in books such as Cramer's Essentials of Computational Chemistry, among others. But it isn't a trivial subject.

I'll get back to you if I come across something that might be of help to you.

typically from computational experience H-bonds are about 60% electrostatic and 40% induction and dispersion. Kindly look at papers published by Steve scheiner if that can help you.

Dear Emanuel,

Your question is a controversial question, depends on the nature of hydrogen donor and hydrogen acceptor,nature can be from electrostatic (i.e, interaction between unperturbed densities) to  dispersion dominant even where  monomers have permanent dipole moment what we have recently shown:

http://www.ncbi.nlm.nih.gov/pubmed/25973745

if you look at the IUPAC description of hydrogen bonding you realize it encompasses large amount of systems. I think Pedas' work would be interesting where Pendas has applied different decomposition scheme + SAPT

http://scitation.aip.org/content/aip/journal/jcp/125/18/10.1063/1.2378807

Best wishes,

Sirous

These two papers evaluate different DFT functionals for the particular case of describing hydrogen bonds. You may find that useful:

http://pubs.acs.org/doi/abs/10.1021/jp0377073

http://pubs.acs.org/doi/abs/10.1021/ct100469b

This paper also describes the nature of hydrogen bond, which as has been said, has been extended to include more subtle dispersion interactions:

http://onlinelibrary.wiley.com/doi/10.1002/anie.201002960/abstract

About the nature of the hydrogen bond see:

1. Pendas M.A., Blanco M.A., Francisco E. The nature of the hydrogen bond: A synthesis from the interacting quantum atoms picture// J. Chem. Phys.- 2006.- Vol. 125, N 18.- P. 184112-184120.

2. Weinhold F., Klein R.A. What is a hydrogen bond? Mutually consistent theoretical and experimental criteria for characterizing H-bonding interactions// Mol. Phys.- 2012.- Vol. 110, N 9-10.- P. 565-579.

Thanks to all of you for your answers. I conclude that a classical electrostatic picture of the hydrogen-bond might be more or less partially wrong. However, at present there is no better description of hydrogen-bonding systems, e.g. aqueous systems, for long time-scales, longer than the picosecond timescale and system-sizes larger than hundreds of atoms. Errors might be cancelled out by bonded interactions and lennard-jones interactions, which are by themselves questionable, since an exponential repulsive dispersion interaction is more realistic, than 1e-12. I mention that a lot of the properties of liquid water could still be described using classical models. Maybe, at some point there will be massive improvements, so that classical pictures of molecular systems will not be necessary any longer. Thanks all of you.   

A much wider perspective can be found at "The theory of Intermolecular Forces" by Anthony John Stone.

http://www.amazon.com/The-Theory-Intermolecular-Forces-Edition/dp/0199672393