Let $R_{\mathcal{A}}$ be the Cartan classical domains of type Ⅲ and Ⅳ, and ${\mathrm{\Delta }}_g$ is assumed to be the Laplace-Beltrami operator associated to the Bergman metric $g$ on $R_{\mathcal{A}}$. In this paper, we derive an estimate for ${\lambda }_1({\mathrm{\Delta }}_g)$, which is the bottom of the spectrum of ${\mathrm{\Delta }}_g$ on $R_{\mathcal{A}}$.