H2 and Helium vs Virial Theorem

The Virial Theorem states that for certain systems $PE =-2*KE$, where $PE$ is Potential Energy and $KE$ Kinetic Energy with $TE=PE+KE=-KE$ Total Energy. 

The ground state of the Hydrogen atom with only one electron shows to be such a system. In atomic units  $TE=PE+KE=-1+\frac{1}{2}=-\frac{1}{2}$.

There seems to be a belief that the Virial Theorem should be satisfied also by atoms/molecules with more than one electron.  

Let us check what Real Quantum Mechanics RealQM gives for H2 and Helium both with two electrons in ground state of minimal total energy.

We have (run this code):

• KE for H2 = 0.9047
• PE for H2 = -2.0752
• TE for H2 =--1.1705       

We have (run this code (fine) or this code (coarse)):

• KE for Helium = 1.6884
• PE for Helium = -4.5887
• TE for Helium  =--2.9003    

We see that in both cases $KE$ is substantially smaller than $-\frac{1}{2}*PE$ and so indicates that the Virial Theorem is not at all valid for the ground states of H2 and Helium. It is the same for atoms and molecules with more than two electrons. RealQM does not produce energies in accordance with the Virial Theorem.

We can see this fact to be the reason why the H2 molecule is formed as two H atoms approach each other  with reduction of $PE$ from -2 to -2.07 by electron densities coming closer to proton kernels, while the kinetic energy is reduced from 1 to 0.9047 as a result of the way electron densities meet in RealQM. This is a key aspect of RealQM in violation of the Virial Theorem.

This fact is also the reason the ground state energy of Helium is as small as -2.903 in agreement with observation, while the $1S^2$ configuration assigned by Standard Quantum Mechanics StdQM only gives -2.75. 

Summary: There is no reason to insist that the Virial Theorem should be valid for atoms and molecules, since that is too simplistic. We see that RealQM captures the essence of H2 and Helium under violation of the Virial Theorem. 

To get perspective check out: Quantum Electrodynamics is rotten at the core.

PS1 In some computations based on StdQM a Virial Theorem is built in, which does not appear to be a good idea since it can introduce too much kinetic energy.

PS2 ChatGPT educated by StdQM insists that the Virial Theorem is valid for H2 and Helium. More than that: ChatGPT claims that solutions to Schrödinger's equation for any atom or molecule satisfies the Virial Theorem and gives a very short proof. Is it credible that the Virial Theorem has this very wide range of applicability? RealQM says no. Does it mean that RealQM is wrong? How is it possible that there is a very simple relation between kinetic and potential energy which add up to the total energy in a very subtle and complicated balance?

PS3 Recall that StdQM is trial-and-error by choice of some trial functions for energy minimisation until desired result, while RealQM is ab initio parameter-free with only choice of mesh resolution.