The problem for atoms with more than one electron is that the equation describes many body wavefunctions. That's the main difficulty. These days it can be solved using pseudopectral methods, though, of course, bound states require special care.

Dear Sadeem, in my humble opinion there are deep problems with the use of Schrodinger equation, not only philosophically (with the meaning of psi function), but also mathematically. Many texbooks just ignore to discuss these problems, although it is known for decades that you should use DFT to solve manybody problems (DFT: density functional theory). 

For more discussions on how to solve wave equation for other atoms in order to arrive at a periodic table close to Mendeleyev's periodic table, you can check may papers written by George Shpenkov in this researchgate, or visit http://shpenkov.janmax.com

Nonetheless, i think there is still formal correspondence between quantum mechanics and classical physics, therefore in some situations qm can still be used successfully. For introduction on correspondence between qm and cm, we can mention that there exists corespondence between Poisson bracket and commutator bracket in qm. There are other examples of such correspondence. Best wishes

To my knowledge only the 2-body problem with central forces has been solved "exactly" either using classical mechanics or quantum mechanics. Systems  with 3 or more particles are solved using approximate methods.

Many body system consists of particles A, B, C, D,... If let the system evolve, then Bohr's theory tells, that in given point can be found all of these particles (they tell "a particle is simultaneously everywhere"). Because particles can not be simultaneously in one place, then the many body system has not a solution in Bohr's philosophy. What is done? The wave function of proton is kept collapsed: proton is in certain place with certain impulse: zero.

And the electron's wave function is free. But because of uncertainty principle, you can not have the proton at your disposal. Therefore, please, abandon the Bohr's philosophy-solipsism, and direct your attention to the David Bohm's theory.

Heavy atoms and many body problem ( of atoms and molecules) could be solved approximately via the DFT techniques which is based on Schrodinger equation and some functionals of energy density etc. One example is  Kohn-Sham equation that is a modified Schrodinger equation. Although all heavy atomic structure and energy levels may not be calculated in this sense , however these DFT methods which basically derived from Schrodinger equation, in association with a  computerized simulation facilitate prediction of many body  properties in a vast area of research.

Thanks to all;

So I conclude from the majority of opinions, which refer to DFT and others that ignoring the the electron-electron interaction and the correlation are the main error sources. But DFT itself has also their problems like intermolecular interactions specially van der Waals forces (dispersion), charge transfer excitations and transition states ....etc

Dear Sadeem, to include the electron-electron interaction, one needs to know: 1) where the electrons are. But due to uncertainty principle the electrons are "everywhere and nowhere."

Dear Sadeem, this problem is discussed in detail in my article "Femtotechnology. First step - a hydrogen atom"  http://vixra.org/pdf/1306.0014v1.pdf ,  http://www.slideshare.net/alexanderilyanok/femtotechnologies-step-i-atom-hydrogen-alexander-ilyanok

Schrodinger's equation cannot be solved exactly for atoms with more than one electron because of the repulsion potential between electrons. You can find more about that in any quantum chemistry textbook. I recommend the book 'Modern quantum chemistry' by Szabo & Ostlund, or the more general book 'Quantum Chemistry' by Levine.

Apart from the fact that the multi-electron Schroedinger equation is very difficult to solve, as pointed out by others here, it is also ultimately not the most accurate description of nature. You would have to use a Dirac equation to include relativistic effects, and full quantum electrodynamics (QED) to account for the quantum nature of the electromagnetic interaction (not only the quantum nature of the electrons). QED effects can only be calculated by approximations (even for the simple hydrogen atom), but this theory still gives the most precise predictions currently available.

Schroedinger is correct equation as far as relativistic effects are ignored. Equation for multi electron atoms is hard to solve, practically impossible, hence approximations a'la Thomas-Fermi, Hartree-Fock, trial wave functions, etc.

Have no idea what is the self-consistent way of describing bound states in the framework of relativity. Bethe-Salpeter is a good start.

Lamb shift, anomalous magnetic moment and other QED effects are for electrons in background field, in and out approximation, not bound states.

Luckily, most of approximations to a non relativistic Schroedinger lead to (satisfactorily) accurate results.

The good example of exact solution of multi-body Schroedinger is Calogero-Sutherland model - N particles all interacting with each other with potential inverse distance squared -shows how drastically exact solution may differ from the approximation.

I agree. 

If you consider electrons in atoms like Mg, Al, Si, P, S and Cl, you find binding energies too big to use the Schrodinger equation. Then you need approximations to Dirac equation to get accuracy.

Even for the state 1S in N, the energy is very big.

It is true that Schroedinger's equation cannot be solved *exactly* for many-electron systems. However, for small atoms, there are incredibly accurate numerical methods, essentially based on the variational principle. Thus for Helium the results are comparable to experimental precision. There are difficulties, however, as pointed out by the other commentators: even in the case of Helium, if we want to attain the desired accuracy, we must include several relativistic corrections to the ordinary Schroedinger equation. These effects become ever more significant as atomic number increases. As to DFT, as far as I can see, it is an approximation to the non-relativistic Schroedinger equation. For atoms, even rather large ones, I believe there are more accurate methods, in particular methods where one has better control on the error. These are so-called configurational integral methods.

There is, of course, no difficulty of principle in evaluating the electronic repulsion. Further, to be accurate, one certainly needs to treat the nucleus as having finite mass and therefore moving. 

I think the problem does not lie with the schrodinger equation, but rather with our inadequate knowledge about the wavefunction pertaining to two-body problems. Many intricacies come into play, such as pauli exclusion principle, electron electron interactions, electron proton interactions and many more. The scenario becomes very much company licated, and even for an exact wavefunction for the same, the equation simply cannot be solved using conventional means. So physiscts resort to numerical methods and other approximate methods such as perturbation theory, variational method etc. Furthermore a complete knowledge about the interactions involved is not known, and therefore the complete wavefunction is not known. Many approximations exist as to the wavefunction such as the Hartree-Fock model and Thomas Fermi model. 

There is a relativistic version of Schrodinger equation which includes spin as a part of the Laplacian operator and yields Dirac energy levels for hydrogen atom and provides good results for heavy quarkonium. Researchers working on atoms other than hydrogen should try to find out how this new wave equation behave for heavier atoms.

Article Quarkonium and hydrogen spectra with spin-dependent relativi...

There is the illusion that it must be possible to find an a priori mathematical description for all deterministic systems. This is just not so. We contantly reach the limits of human hubris. As soon as we put in arbitrary constants that have to be set by hand we are not doing maths but painting a picture.

Andy, I am afraid your last sentence is true.

 Absolutely, an extremely simplified, black and white picture of nature.