1. The problem involves calculating the change in energy $\Delta E$ given by the expression $\Delta E = -2 \cdot bdt \cdot 18 - 101 \cdot bdt \cdot 20\alpha$.\n\n2. We start by writing the expression clearly: $$\Delta E = -2 \times bdt \times 18 - 101 \times bdt \times 20\alpha$$\n\n3. Factor out the common term $bdt$: $$\Delta E = bdt \times (-2 \times 18 - 101 \times 20\alpha)$$\n\n4. Calculate the constants inside the parentheses: $$-2 \times 18 = -36$$ and $$-101 \times 20 = -2020$$, so the expression becomes: $$\Delta E = bdt \times (-36 - 2020\alpha)$$\n\n5. The final simplified expression for the change in energy is: $$\boxed{\Delta E = bdt \times (-36 - 2020\alpha)}$$\n\nThis formula shows how $\Delta E$ depends linearly on $bdt$ and $\alpha$.\n\nNote: The trigonal bipyramidal geometry of $ClO_2^-$ is mentioned but does not affect this algebraic calculation.