G25.2666: Quantum Chemistry and Dynamics

Problem set #2
Due: Feb. 15, 2005

1.

Show explicitly that $L_z$ commutes with the Hamiltonian for the H$_2^+$ molecule ion.

2.

In the H$_2^+$ molecule ion, let $R$ be the distance between the protons. Note that the LCAO approximation developed in class is correct when $R$ is large and the electron is on one or the other proton. However, when $R\rightarrow 0$, the ground state eigenfunction should correspond to that of a He$^+$ ion, which is not well reproduced in the LCAO approximation. Thus, consider taking a variational wavefunction of the form:

$$\psi({\bf r}) = C_1\psi_1({\bf r};Z) + C_2\psi_2({\bf r};Z)$$

where

$$\psi_{1(2)}({\bf r};Z) = \left({Z^3 \over \pi a_0^3}\right)^{1/2}e^{-Z\vert{\bf r}+(-)R\hat{z}/2\vert\vert}$$

Let $C_1$, $C_2$ and $Z$ be variational parameters. We expect $Z$ to lie between 1 and 2. What does this trial wavefunction predict for the binding energy and bond length of the H$_2^+$ molecule ion?

3.

Suppose we wish to calculate the energy of the H$_2^+$ molecule ion using simple Gaussian wavefunctions of the form

$$\psi_i({\bf r}) = e^{-\alpha ({\bf r}-{\bf R}_i)^2}$$

where $i=1,2$ indexes the two protons and ${\bf R}_i$ is the position of the $i$th proton. Treating the coefficients in an analogous ``LCAO'' expansion using the Gaussian trial wavefunctions and the width $\alpha$ as variational parameters, calculate the approximate binding energy of H$_2^+$ and its approximate equilibrium bond length. How do these values compare to the exact values? You may make any approximations you think are reasonable related to the distance between the protons. It might help you to know that an integral of the form

$${2 \over \sqrt{\pi}}\int_0^x e^{-t^2} dt = {\rm erf}(x)$$

where ${\rm erf}(x)$ is known as the error function. Note that $d{\rm erf}(x)/dx = (2/\sqrt{\pi})e^{-x^2}$, that $\lim_{x\rightarrow\infty} {\rm erf}(x)=1$, and that ${\rm erf}(x)$ converges very rapidly to 1, e.g. ${\rm erf}(2)=0.99532$. For more information on error functions, see Abromowitz and Stegun Handbook of Mathematical Functions.