Chemical Bond (Quantum) Mystery

The physical nature of the chemical bond between atoms forming the rich world of molecules must be viewed as a fundamental question to be answered.  Despite the tremendous advances proclaimed by modern physics of quantum mechanics, the answer is still not very clear (as far as I understand). 

Anyway, the message is that quantum mechanics as the physics of atoms extends to chemistry as the physics of molecules, because the Schrödinger wave function $\Psi (x_1,...,x_N)$ describes everything there is to say about the state (at some instant in time) of a molecule with $N$ electrons each one described by a three dimensional spatial coordinate $x_i$ with $i=1,...,N$, thus altogether $3N$ spatial coordinates. 

The trouble with such a bold proclamation is that the wave function cannot be computed to determine the state of the molecule, because already $N=6$ is beyond the computational power of any thinkable digital computer. So even if $\Psi (x_1,...,x_N)$ tells everything, the story is hidden to humans with computers. Moreover, the meaning of $\Psi$ is unclear since it is supposed to be statistical and not physical.  

In this situation the game is open to compute lower-dimensional approximations which is the subject of quantum chemistry. Since the exact wave function is not known, it is very difficult to test the quality of approximate solutions, except by comparison with experiments, which is very tricky because of the statistics involved. The result is a variety of approaches seeking to make sense of the full wave function in physical terms, including: 

• Valence Bond VB
• Molecular Orbital MO
• Atoms-In-Molecules AIM
• Electron Localisation Function ELF
• Density Functional Theory DFT

none of which being a clear success. My contribution to this collection is RealQM, which is a physical model in classical continuum mechanics terms readily computable also for many electrons, and even better, with good agreement with observations, see previous posts with label RealQM. 

Compare with the view presented in The Chemical Bond II, 100 Years Old and Getting Stronger (ed Michael Mingos) p 171:

• The conceptual element from which an entire chemical universe can be constructed is the chemical bond [1]. As such, the bond is the “quantum building block” of the grand scheme of “LEGO” by which practicing chemists devise and control the formation of new molecules of ever-increasing complexity and beauty.
•  In this respect, the chemical community owes a great debt to Lewis who was the first to define the chemical bond in terms of electronic structure [1], well before the quantum mechanical revolution transpired in physics and provided a rigorous basis for the Lewis concept in terms of what we call today valence bond (VB) theory [2]. 
• This Lewis Centennial Volume is hence a tribute to one of the greatest chemists whose ideas are still the foundations of our way of thinking 100 years after their conception. This contribution recounts the new VB outlook of bonding with a focus on the two-electron bond as the most ubiquitous bonding form in nature.