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. RealQM web site.

Here is a comment to the post by chatGPT:

It is simply not correct to say that the covalent bond remains an “enigma,” nor that the Heitler–London (HL) picture is conceptually contradictory because it seems to mix “sharing” with “alternating.” In fact, HL is still regarded as the most fundamental explanation of the H₂ bond for a very specific reason: it captures the essential physics—electron indistinguishability + fermionic symmetry + Coulomb interactions—with no extra assumptions. All modern quantum-chemical methods (MO theory, valence bond theory, full CI, CCSD(T), DFT) reduce to HL in the appropriate limit.

The apparent conflict between “sharing” and “alternating” arises only if one tries to impose a classical imagery on quantum mechanics. HL does not say that the electrons take turns sitting between the nuclei. It also does not say they are simultaneously smeared in a classical sense. HL states something different:

• The two-electron wavefunction must be antisymmetric in spin and symmetric in space for the singlet.

• This symmetry forces the joint probability amplitude to place more electron density between the nuclei, while

• simultaneously reducing the probability of finding both electrons in the same region (the correlation hole).

This is what looks like “sharing” at the single-electron level and “avoiding” at the two-electron level. But these are not contradictory behaviors—they are two different marginals of the same quantum state. The mathematics is consistent and, more importantly, experimentally verified: the resulting potential curve matches the observed dissociation energy and vibrational spectrum.

So HL remains the cleanest fundamental explanation because:

1. It identifies the correct mechanism: symmetry + exchange lowers the electron–nuclear potential more than it raises kinetic or e–e repulsion.

2. It avoids classical pictures that add confusion (e.g., “alternating occupancy”).

3. It is the limit to which all higher-level quantum-chemical methods must converge.

Calling this an “enigma” misunderstands the point: the covalent bond looks strange only if one insists on classical ontologies. Within quantum mechanics, the HL mechanism is simple, natural, and still the foundational explanation taught to theorists today.

My response:

We learn that the HL explanation from 1927 is still viewed to be foundational, and that it contains a complex mix of exchange + correlation hole + sharing + alternation + electron indistinguishability + antisymmetric in spin + symmetric in space,