Covalent Bond: The Quantum Chemistry Enigma

The first objective 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, which was given a breathtakingly convincing answer in terms of the eigenvalues of a linear differential equation 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 $\Psi (x_1,...,x_N)$ 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 $E=PE_{ek}+PE_{ee}+KE_e$

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

The next objective 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 $E_b=0.17$ 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 "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 main theme of theoretical chemistry. Here it is according to textbook 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 may serve to indicate qualitative bonding but quantity is missing. The explanation builds on a specific effect from superposition combining "shared presence" with "alternating presence" as a impossible within classical 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.