Is there any theory more advanced than MOT?
MO theory is more of a descriptive guide than a rigorous mathematical construct. The main idea of molecular orbital theory is that the electrons in atoms can be described by linear combinations of atomic orbitals (LCAOs), which themselves are the exact solutions to the Schrodinger equation for single electron atomic systems.
This idea can be applied in a more mathematically quantitative way by use of the Hartree-Fock (HF) method. HF is essentially MO theory put into practice. HF takes as input the geometry of a system (i.e. the spatial coordinates of all atoms in your molecule) and a set of &quot;basis functions&quot; which are usually sets of 3D gaussian functions centered on the atomic nuclei of the system. These basis functions are your &quot;atomic orbitals&quot; in MO theory. The HF method takes linear combinations of the atomic orbitals to generate a set of molecular orbitals which are usually more delocalized. These are the molecular orbitals of MO theory, and we can calculate them using the HF method running on a computer.
There are several approximations involved in HF theory, which the descriptive MO theory inherits.

The Schrodinger equation (SE) itself is an approximation. The SE is not exact as it fails to account for so-called relativistic effects. These are effects originating from Einstein's theories of special and general relativity. Some examples of relativistic effects are spin-orbit coupling and relativistic contraction of orbitals, especially in heavy atoms. The straightforward HF method doesn't take relativistic effects into account. Starting instead from the Klien-Gordon or Dirac equations will recover (some of) the missing relativistic effects.

The HF method assumes a complete separation between the movement of electrons and nuclei, known as the Born-Oppemheimer approximation. Basically, electrons are assumed to equilibrate instantaneously to the positions of atomic nuclei. This usually isn't a problem because electrons are much lighter than atomic nuclei so they do move much faster. Not a bad approximation in most cases.

Electron correlation. Even if we ignore the previous two approximations, the HF answer (and therefore the MO theory prescription) misses out on a very chemically important effect known as electron correlation. Electron correlation arises because a single set of molecular orbitals combined into a Slater determinant (in order to satisfy the Pauli exclusion principle) is not enough to fully describe the true exact solution to the Schrodinger equation for a molecular system. One must take linear combinations of many Slater determinants to improve on the HF answer. Electron correlation is responsible for effects such as the Van der Waals dispersion force. Another way of thinking about Electron correlation is that it is the energy that arises due to the dynamic Coulomb repulsion between moving electrons (although this picture is incomplete).


In summary, an exact molecular theory would need to account for Einstein's theory of relativity, combine the movement of atomic nuclei with electrons, and account for the dynamic correlations between different electrons in a system, in addition to being a proper quantum mechanical theory.
There are some quantum chemistry theories out there that incorporate relativistic effects into them. For example, the Douglas-Kroll-Hess approach is similar to HF theory, but begins from the Dirac equation and so incorporates relativistic effects explicitly. I believe there are also approaches that don't use the Born-Oppemheimer approximation, though I'm unfamiliar with them.
There is a massive amount of research and development into the so-called post Hartree-Fock methods. All of them attempt to account for electron correlation in various different ways. The main issue with the post HF methods is that the results (energies and wavefunctions) are often incredibly computationally demanding to actually compute, except on very small molecules. The most exact of the post-HF methods is the Full Configuration Interaction (FCI). With a sufficiently large input basis set, FCI produces an exact answer to the Schrodinger equation under the Born Oppenheimer approximation. Other post-HF methods include: Møller-Plesset perturbation theory, Coupled Cluster theory, and Multi-Configurational Self Consistent Field theory. All of these post HF methods are more accurate than the MO theory picture, but they are also way more difficult to visualize and understand for new students.