## tisdag 15 november 2016

### realQM vs Hartree-Fock and DFT

I have put up an updated version of realQM (real Quantum Mechanics) to be compared with stdQM (standard QM).

stdQM is based on a linear Schrödinger equation in a $3N$ dimensional wave function with global support for an atom with $N$ electrons, which is made computable in Hartree-Fock and Density Functional Theory DFT approximations reducing the dimensionality to basically 3d.

realQM is based on a system of non-linear Schrödinger equations in $N$ 3d electron wave functions with local disjoint supports, which is computable without approximation. Evidence that realQM describes real physics is given.

#### 11 kommentarer:

1. what do you mean by "DFT approximation"?

2. DFT is to be viewed as an approximation of full multi-d Schrödinger since the exchange- correlation term is unknown and is approximated by HF.

3. To be correct, DFT is not an approximation, the ground state is a functional of the density. The functional is unknown and there are some approximations. The most famous and used one is LDA, then there are few versions of GGA and then there have been many others. HF is not an approximation to DFT. HF gives the exact exchange but there is no correlation term (which is actually defined as the difference between the unknown functional and HF).

4. Multi-d Schrödinger cannot be solved exactly and so can only be solved approximatively and this is the purpose of HF and DFT. To argue DFT is not an approximation makes no sense.

5. There's an error with a quote, the first one in chapter 8. It was not Dirac who said 'Shut up and calculate'. It was David Mermin.

I have actually never seen anyone pin this one on Dirac (It's so unlike Dirac's character to say something in this fashion). Sometimes it is miss pinned on Feynman (It's more in Feynmans spirit).

6. In the text. You have several wavefunctions that have discontinuities.

How do you treat the kinetic energy at the discontinuities?

7. Is it correct to assume that you model an electrons to be the unit charge distributed over some region of space? (Eq. 2.1)

8. Wave functions for different electrons are supposed to meet under continuity and vanishing normal derivative.

9. Thank you for replying.

What then, is the physical mechanism, that makes the charge in one region, ignore the charge from the same electron in another region?
Coulomb's law should still be valid, right? Charge in one point can't distinguish where the charge "over there" originates from. Do you agree?

10. I agree, it is not clear if self-interaction to some degree is present or not, but refined computations with the model may tell. In any case, for Hydrogen with one electron the energy has no contribution from self-interaction, if the model is to agree with observation.