Unified theory of chemical bonds
The short answer: no. We do not know how to solve the Schrodinger equation analytically, even for systems as simple as the He atom.

The much longer answer:

The Schrodinger equation is the basis for almost all theoretical chemistry: chemistry that attempts to predict properties of compounds without an actual experiment. The problem is that the Schrodinger equation is unsolvable for any compound with more than two electrons, and so we must make some approximations to solve it. This leads to methods such as Hartree-Fock and Density Functional Theory.

While these approximations are usually pretty good, they are still only approximations, and sometimes stop working correctly. There is an added problem that the Schrodinger equation is only an approximation, and the proper theory to use to describe light-matter interaction is Quantum Electrodynamics, which is incredibly difficult to solve for a single photon reflecting off of a mirror.

Long story short: there may or may not be a "unified theory" for describing exactly how all bonds behave in every situation, but there are currently methods that are good enough to predict a large portion of the behavior that we observe.