Covalent Bond: The Quantum Chemistry Enigma

The first challenge for the new modern physics emerging at the turn to the 20th century was to give a theoretical explanation of the observed spectrum of the Hydrogen atom H with one electron around a proton kernel. This was given a breathtakingly convincing answer in terms of the eigenvalues of a mathematical model formulated by the 38 year old Austrian physicist Erwin Schrödinger in 1926, named Schrödinger's Equation SE coming with direct formal extension to many electrons. So was a whole new form of physics as Quantum Mechanics formed, based on SE as a (parameter-free) model of atoms/molecules in the form of a linear partial differential equation with solutions named wave functions $\Psi (x_1,...,x_N)$ as eigenfunctions depending on $N$ 3d spatial coordinates $x_1,...,x_N$ for a system with $N$ electrons. 

SE presents the ground state of an atom as the eigenfunction $\Psi$ with smallest eigenvalue as total energy $E=PE_{ek}+PE_{ee}+KE_e$ where

• PE_{ek} = Potential Energy: electron-kernel: negative 
• PE_{ee} =  Potential Energy: electron-electron: positive 
• KE_e = Kinetic Energy: electron: positive.     

The next challenge was to explain the observed formation of the molecule H2 as a system of two H atoms with a total energy of $E= -1.17$ at kernel distance of 1.4 (atomic units) to be compared with $E=-1$ when widely separated, thus with a binding energy of $0.17$ required to pull the molecule apart. 

In 1927 Heitler and London produced a wave function with a binding energy of about 50% of the observed, claimed to expresses the physics of a covalent bond as being established by the two electrons of the two H atoms somehow "sharing" the region between the kernels. The nature of the Heitler-London wave function still today serves as the main theoretical explanation of covalent chemical bonding as a fundamental theme of theoretical chemistry. It is formed as a superposition of products of wave functions for electron 1 and 2 with association around kernels A and B:   

• $\Psi = \Psi_A(1)\Psi_B(2)+\Psi_B(1)\Psi_A(2)$.

The rationale for binding is presented as follows based on a specific HL wave function of this form:

1. $\Psi$ expresses joint presence of electron 1 and 2 between the A and B in both terms of the superposition, which causes a decrease of $PE_{ek}$. Binding.
2. The joint presence does not increase $PE_{ee}$ because in fact both terms express alternating presence: when 1 is close to A then 2 is close to B and vice versa. This is the key argument.
3. It is claimed that $KE_{e}$ increases very little despite electron concentration between kernels. 

The HL wave function thus serves to indicate qualitative bonding but to get quantity right requires very complex superpositions, with physics difficult to visualise. The explanation builds on a specific effect from superposition combining "shared presence" with "alternating presence" as being contradictory  within classical physics/logic.  

 

RealQM gives a fundamentally different explanation of the physics of the covalent bond of H2 with full quantitive agreement. See also the RealQM book p 187. The essence is easy to understand: The two electrons represent non-overlapping charge densities meeting at a plane orthogonal to the axis between the kernels with continuity of non-zero charge density. This allows charge concentration between kernels without increase of kinetic energy creating binding. The essence is that electron charge densities do not overlap which can be seen being maintained by Coulomb repulsion. RealQM explains binding as a dynamic process driven by Coulomb forces towards energy minimum.  

Here is a comment to the post by chatGPT.