First Molecule HeH+ by RealQM and DFT

Crosscut 3d showing one electron (red) moved from He++ kernel to H+ kernel and remaining electron (yellow) around He++ kernel. Note free boundary developed between electrons starting from initial vertical cut through He++ kernel. Run code below to follow dynamics.

This is a follow up on previous post on the first molecule formed after a Big Bang when one of the two electrons of a Helium atom He joins with an approaching proton H+ to form a helium hydrid ion molecule HeH+ (or rather He+H) built by a cation He+ and a Hydrogen atom H. The energy count in Hartree is as follows:

• Energy of He atom = -2.903
• Energy of He+ and H separated = -2.000 - 0.500 = -2.500
• Energy E of HeH+ molecule = -2.592 
• Dissociation Energy of HeH+ into He+ and H = 0.092  observed
• Energy for formation FE of HeH+ from He and H+ = 0.311 

Let us compare RealQM and DFT as concerns prediction of the observed FE = 0.311. Notice the difference between He plus H+ and He+ plus H. Check by noticing that -2.903 = - 2.500 - 0.311 - 0.092. Notice that the bulk of FE is supplied by exterior forcing to make H+ approach the He++ kernel. 

RealQM code gives FE = 0.301 from E = -2.602. You can follow the transfer of one electron from He to H by running the code starting from two electron half-lobes around the He kernel with supports displayed on red and yellow. You can test a different location of H+ vs He electron split by running this code. In both cases see how one electron dynamically shifts from He to H+ forming a molecule of He+ and H starting from He and H+. 

DFT gives according to chatGPT:

DFT Functional	Predicted Dissociation Energy (Hartree)	Error Trend
LDA (Local Density Approximation)	~ -0.35	Overbinds HeH⁺ (too stable)
GGA (PBE, BLYP)	~ -0.33 to -0.34	Still overestimates bond strength
Hybrid (B3LYP, PBE0)	~ -0.30 to -0.32	Closest to exact (-0.311)
High-Accuracy (CCSD(T), FCI)	-0.311	Exact value

• LDA and GGA functionals overestimate binding, leading to a more negative dissociation energy (~ -0.34 to -0.35 Hartree).
• Hybrid functionals (B3LYP, PBE0) improve accuracy, but they still may predict a slightly too strong bond.
• Post-HF methods (CCSD(T), FCI) match experimental values (-0.311 Hartree).

We see that the precision with standard DFT is not better than RealQM, rather the opposite. It is not clear that DFT can model the dynamics of the shift of one electron from He to H+.

Notice that we are here dealing with the simplest possible problem in quantum mechanics, a molecule with only two electrons as the first molecule formed in the early universe, with H2 coming only later after dissociation of HeH+ into He+ and H (and then formation of H2 under release of energy). Would you expect that DFT after 50 years of massive investment would deliver a very convincing result? Did we get that?