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

Question

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?

Answer 1

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.

Answer 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 (or more generally the $n$-body problem).