Provide a mathematical explanation of why the non-relativistic Schrödinger equation for a Molecular system, cannot be solved analytically?

The **non-relativistic Schrödinger Equation** is:


$\widehat{H}|\psi\rangle=E|\psi\rangle$


Where $\widehat{H}$ is the **Molecular Hamiltonian** in Atomic Units and has the following terms:


$$ \widehat{H}
 =
 -\sum\_{A=1}^M\frac{1}{2M\_A}\nabla^2\_A
 -\sum\_{i=1}^N\frac{1}{2}\nabla^2\_i
 +\sum\_{A=1}^M\sum\_{B>A}^M\frac{Z\_AZ\_B}{R\_{AB}}
 +\sum\_{i=1}^N\sum\_{j>i}^N\frac{1}{r\_{ij}}
 -\sum\_{i=1}^N\sum\_{A=1}^M\frac{Z\_A}{r\_{iA}} $$


Where labels A,B,.. denote nuclei and labels i,j,.. denote electrons. $\nabla^2\_A$ and $\nabla^2\_i$ are the corresponding Laplacians. $M\_A$ is the mass of nuclei. $R\_{AB}$ and $r\_{ij}$ are the distances between corresponding Nuclear pairs and Electron pairs, respectively. $r\_{iA}$ is the inter-nuclear-electron distance. $Z\_A$ is the corresponding Nuclear Charge.


After applying the **Born-Oppenheimer Approximation** and treating the nuclei as stationary, and treating the **Electronic Hamiltonian** separately, i.e.


$$ \widehat{H}\_{ele}
 =
 -\sum\_{i=1}^N\frac{1}{2}\nabla^2\_i
 -\sum\_{i=1}^N\sum\_{A=1}^M\frac{Z\_A}{r\_{iA}}
 +\sum\_{i=1}^N\sum\_{j>i}^N\frac{1}{r\_{ij}} $$


I would like to know a **mathematical explaination** of why we cannot analytically solve the non-relativistic Schrödinger equation using the above electronic hamiltonian? Is it specifically the 3rd term that causes the problem?


In general many-body problems cannot be solved analytically, that is true. I would like to know the explanation for this specific case I mentioned above- for a Molecular System. I haven't found any resources to explain this point in more mathematical detail.


I have been reading Szabo and Ostlund's book. I would like to read more material regarding this specific point. Whether someone has worked this out for simple cases like say H2+ or H2?




I will show specifically the case for a Helium atom, but it would explain in general the difficulty of solving multi-electronic systems. The Hamiltonian for Helium can be represented as given below:


$$\hat{H} = -\frac{1}{2}\nabla^2\_1 - \frac{1}{2}\nabla^2\_2 - \frac{Z}{r\_1} - \frac{Z}{r\_2} + \frac{1}{r\_{12}}$$


This can be re-written as


$$\hat{H} = \hat{H\_1} + \hat{H\_2} + \frac{1}{r\_{12}}$$


where $\hat{H\_i} = -\frac{1}{2}\nabla^2\_i - \frac{Z}{r\_i}; i = 1,2$


If it were not for the repulsion term in the hamiltonian, it would be completely separable. Now, you also have to consider the fact that the wavefunction, $\Psi(r\_1,r\_2)$, essentially is a function of two electrons. If we ignored the electron-electron repulsion term, we could separate the wavefunction into the product of two functions for each electron: $\Psi(r\_1,r\_2) = \psi(r\_1)\cdot\psi(r\_2)$ and solve the differential equation operating $\hat{H\_i}$ on the corresponding $\psi(r\_1)$ or $\psi(r\_2)$. However, this would be like solving for two separate hydrogen atoms but with Z = 2.


The electron repulsion term makes this separation impossible as the term depends on $r\_1$ and $r\_2$. This is why there is a slight burden in solving the non-relativistic Schrödinger equation for a multi-electron system, I believe.


 2

The fact that the Schrödinger equation for a multi-body case cannot be solved, i.e. has no closed form solution, has nothing to do with quantum mechanics itself but is a more general issue with dynamical systems governed by invert square interaction (Coulomb, gravitational force, etc.). This is usually refer to as the [three-body problem](https://en.wikipedia.org/wiki/Three-body_problem) (or more generally the $n$-body problem).


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