Molecules

Abstract

Molecules as composite structures of more than one nuclei and electrons are considered and the Born-Oppenheimer method for the formulation of their eigenvalue problem is analyzed. As an example of a simple molecule the Hydrogen Ion is considered and its ground state energy is estimated.

Problems and Exercises

16.1

Calculate the expectation value of the energy of the Hydrogen ion $\epsilon _{\pm }$ for the trial wave function

$$\psi _{\pm }(\mathbf{{r}})\,=\,\frac{N_{\pm }}{\sqrt{\pi a_0^3}}\left( \,e^{-|\mathbf{{ r}}-\mathbf{{ R}}/2|/a_0}\,\pm e^{-|\mathbf{{ r}}+\mathbf{{ R}}/2|/a_0}\,\right) \,.$$

16.2

Write down the electronic Hamiltonian for the Hydrogen molecule. Assume that an acceptable approximation for the ground state of the molecule is a properly symmetrized wave function of two mutually noninteracting electrons, each in a Hydrogen ground state. Find the correction to the energy in the presence of an external homogeneous magnetic field.

16.3

A one-dimensional molecule. Consider a particle moving in one dimension and subject to a double square well potential

$$V(x)\,=\,\left\{ \begin{array}{cc} 0\,&{}\,|x|>L+a\\ \,&{}\,\\ -V_0\,&{}\,a<|x|<L+a\\ \,&{}\,\\ 0\,&{}\,|x|<a \end{array}\right. $$

Assume that the distance between the two wells is much greater than the width of each well, i.e., $a{\gg }L$. If $\psi _1$ is the wave function of the lowest energy bound state localized in the left well and $\psi _2$ is the wave function of the lowest energy bound state localized in the right well, with corresponding energies $E_1=E_2=E_0$, calculate the expectation value of the energy for each of the trial wave functions $\psi \,=\,\frac{1}{\sqrt{2}}\left( \psi _1\pm \psi _2\right) \,$, in terms of the integral $A=\int \,dx\,\psi _1\hat{H}\psi _2$. You may assume that the overlap of $\psi _1$ and $\psi _2$ is very small, so that $A{\ll }E_0$. Which of the two choices corresponds to the ground state of the system?