tag:blogger.com,1999:blog-5439168179960787195.post4239238749727302304..comments2018-04-18T20:24:40.433+10:00Comments on Condensed concepts: Computational density functional theory (DFT) in a nutshellRoss H. McKenziehttp://www.blogger.com/profile/09950455939572097456noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-5439168179960787195.post-27156448211253102832017-05-10T08:19:41.625+10:002017-05-10T08:19:41.625+10:00BO: I have also arrived to gripe about your use of...BO: I have also arrived to gripe about your use of 'Born-Oppenheimer approximation'. The Wikipedia article quite correctly states that the BO approx is that the nuclear and electronic wavefunctions are separable, that there are no correlation terms. So you can calculate the electronic and nuclear wavefunctions separately, and then combine them. B+O show this by considering the mismatch in mass of electrons and the nucleus, lowering the coupling (cross) terms, and meaning you can calculate your electronic wavefunction in a static snapshot of the nucleus, and the nucleus move in a potential generated by the electrons (the BO energy surface).<br /><br />But what people take the BO approximation to mean is the further approximation is made that the nuclear charges are motionless points, and never bother even considering nuclear motion (zero point or thermal). Some of the path-integral literature explicitly calls this the 'coarse Born-Oppenheimer approximation', which I think is pretty accurate!<br /><br />LDA: The LDA functional is simply that Ex[n(r)] = C * n(r)^1/3 . From a classical electrons in phase space argument (the Slater Exchange energy), C=-(3/4)(3/pi)^1/3 = -0.738... <br />However, as far as I'm aware, all 1980s onwards codes use the Ceperley (1980) MC result, as parametrised by Perdew and Zunger (1981). Quite literally this is a set of 7 parameters for each of the polarised + unpolarised case, used as a set of n, sqrt(n), ln(n). <br />So are LDA calculations ab-initio? I would argue not, though they are fitted to reproduce the HEG results.<br /><br />Incidentally I definitely recommend Thijssen's Computational Physics (2nd edition) for learning about the nuts and bolts of electronic structure.<br /><br />Not only is DFT difficult to systematically improve, but it is not variational. So with HF + post-HF quantum-chemistry methods, anything you do to try and decrease the energy of the system takes it towards the ground state. This includes basis-set expansion. Whereas with DFT, you can be above or below the true answer, and oscillating wrt. basis size.Jarvisthttps://www.blogger.com/profile/04899572225072861017noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-48805074715645502522017-05-04T01:48:41.237+10:002017-05-04T01:48:41.237+10:00The Born-Oppenheimer approximation, in general, do...The Born-Oppenheimer approximation, in general, does NOT treat the nuclei classically. In the case of electronic structure theory, it treats them as infinitely heavy (i.e. they don't have dynamics at all, neither classical nor quantum.) Its quite normal (for molecules) to then treat the nuclear rotations and vibrations quantum mechanically on the resulting potential surface, and then, less commonly, to use perturbation theory to correct for the Born-Oppenheimer approximation.Unknownhttps://www.blogger.com/profile/07594633209626059331noreply@blogger.com