The Hydrogen Atom

Whenever introducing a new approximation it is often useful to examine its performance on the simplest model systems. Here the effect of CASE is investigated on the H atom, the attenuated Schrödinger equation for which is
 $$\begin{align} \frac{1}{2} \frac{d^{2}\Psi}{d r^{2}} + \frac{1}{r} \frac{d \Psi}{d r} + \frac{\erfc( \omega r)}{r}\Psi + E \Psi = 0. \end{align}$$

With $\omega = 0$ CASE becomes the traditional Schrödinger equation giving E₀ = -1/2 hartrees and $\Psi_{0}(r) = {\pi}^{-1/2} \exp(-r)$. As $\omega$ is increased the attractive nuclear-electron contribution to the energy rapidly decreases and the total energy rises. The effect on the wavefunction, however, is far less intuitive. A quantitative investigation of $\Psi$ is possible through first-order perturbation theory.

The CASE energy and wavefunction can be expanded as
$$\begin{align}E &= E_{0} + E_{1} + E_{2} + \cdots \\ \Psi &= \Psi_{0} + \Psi^{1}... ...dots\\ \Psi^{1} &= c_{2} \Psi_{0}^{2} + c_{3} \Psi_{0}^{3} + \cdots \end{align}$$
where $\Psi_{0}^{2}$, $\Psi_{0}^{3}$, $\cdots$ are the $\omega = 0$ excited-state wavefunctions. This allows the first-order correction to be expressed as the expectation value of the long-range operator $\hat{L}$ (the `background')
$$\begin{align}E_{1} &= \int \Psi_{0} \hat{L} \Psi_{0} \\ &= \int_{0}^{\infty} 4r... ...mega \sqrt{\pi}}+(1-2\omega^{-2})\exp(\omega^{-2})\erfc(\omega^{-1}) \end{align}$$
and the coefficients of the first-order correction to the wavefunction for any $\omega$ are given by the $\omega = 0$ components
$$\begin{align}c_{k} = \frac{\int \Psi_{0} \hat{L} \Psi_{0}^{k}}{E_{0} - E_{0}^{k}}. \end{align}$$

  

Table 6.1: CASE First-order perturbation theory on the H atom

$\omega$	E	E0 + E1	c2	c3
0	-0.500000	-0.500000	0	0
0.001	-0.498872	-0.498872	3x10-9	1x10-9
0.01	-0.488717	-0.488717	3x10-6	1x10-6
0.1	-0.388270	-0.388266	3x10-3	1x10-3
0.2	-0.282838	-0.282631	2x10-2	6x10-3
0.5	-0.051071	-0.031011	1x10-1	5x10-2

The first-order energy E₀ + E₁, along with the exact energy, which has been obtained using the mathematics package Mathematica [193] by solving equation (6.5) numerically, are listed in Table 6.1 for a selection of $\omega$ values. The table shows that the energy does rise with increasing $\omega$. What is more interesting, however, is the surprisingly close agreement between the exact and first-order energies. This suggests that, at least for small $\omega$ values, $\Psi_{0}$ is an excellent approximation to $\Psi$, or conversely, that attenuation has only a small effect on the wavefunction $\Psi$. This is supported by the first two coefficients c₂ and c₃, presented in Table 6.1, being exceedingly small, indicating that the first-order correction to the wavefunction is negligible.

This simple example above does show that CASE chemistry is not the disaster that might have been expected. A good wavefunction is produced, thanks to the smoothness of the background. The large magnitude of the background does, however, produce large effects on the total energy under attenuation. This is in contrast to variational calculations which usually produce a good energy from a poor wavefunction. CASE is clearly not that useful if total energies are the goal of a calculation, yet if relative energies are the goal, it is conceivable that there may be a systematic cancellation of error and, by virtue of an accurate wavefunction, relative energies could be largely unaffected by the neglect of the background. This is investigated in the next section.