Loss of Symmetry

The exact density matrix is symmetric under ${\bf R}\leftrightarrow{\bf R'}$. Since we have singled out ${\bf R'}$ as the initial point for the imaginary time dynamics, it is not clear that the approximation given in Eq. 3.27 automatically satisfies this condition. For the free particle limit and the harmonic oscillator, where the Gaussian is the exact solution, it obviously does but in general it does not.

As a specific example, we consider again the ground state limit of the hydrogen atom in the Gaussian approximation. Using the ground state values for the variational parameters, ${\bf m}=0$ and $w=9 \pi/\beta$, Eq. 3.27 becomes,

$$\lim_{\beta\to\infty}\;\rho({\bf r},{\bf r}';\beta) = e^{D(r';\beta)} \; (8/9\pi^{2})^{3/2} e^{-8r^{2}/9\pi}\;.$$ (165)

For this to be symmetric under ${\bf r}\leftrightarrow {\bf r}'$, we must have

$$\lim_{\beta\to\infty} D(r';\beta)=-8 r'^{\,2}/9\pi + c(\beta)$$ (166)

and from the result for $\dot{D}$, $\lim_{\beta\to\infty} c(\beta) = 4\beta/3\pi +c_{1}$.

Figure 3.2 compares the $D(r,\beta )$ from the Gaussian VDM with Eq. 3.48 using $c(\beta)=4\beta/3\pi+3/2\ln 2$.

Figure 3.2: $D(r,\beta )$ from the Gaussian approximation in the ground state limit (solid line) of the hydrogen atom. Deviations of this function from linearity indicate a breakdown of symmetry in the Gaussian approximation for $\rho({\bf r},{\bf r}';\beta)$. The dashed line is $-8r^{2}/9\pi+4\beta/3\pi+3/2\ln 2$ expected from the Rayleigh-Ritz ground state Eq. 3.37.

There are several consequences of this small violation of ${\bf R}\leftrightarrow{\bf R'}$ symmetry. As shown generally in the section above, in the $\beta\to \infty$ limit $-\dot{D}$ is the Rayleigh-Ritz variational ground state energy for a Gaussian wave function, which for the hydrogen atom is $E_0=-4/3\pi = -0.4244$. Because of the loss of symmetry this is not the same as the energy given by the estimator

$$\left<E\right> = \left<{\mathcal{H}}\right> \equiv \frac{\mbox{Tr}[{\mathcal{H}}\rho]}{\mbox{Tr}[\rho]}$$ (167)

in the $\beta\to \infty$ limit, which for the hydrogen atom gives the more accurate result $\left<E\right>=-0.4709$. This will be seen again below for the hydrogen molecule where Eq. 3.49 also gives more accurate ground state energies. Other consequences are less pleasant. Although the energy is more accurate the virial theorem, $\left<K\right>=-\left<U\right>/\,2$, between the kinetic and potential energy is violated by about $3\%$ (while both are more accurate than the usual ground state variational Gaussian result). This has consequences for calculating the equation of state particularly at low density. Slightly more complicated, explicitly symmetric forms for the VDM could be used but in this paper we will continue to explore the basic Gaussian approximation.