In a 3D Cartesian coordinate system, there is a point A (x=1.0 , y = 1.0 , z=0.0 ) that has a 222 point group

In a 3D Cartesian coordinate system, there is a point A (x=1.0 Å, y = 1.0 Å, z=0.0 Å) that has a 222 point group symmetry. It means that there are three 2-fold rotation axes (green-color lines in the following figure), each of which is parallel to the x-axis, y-axis, and z-axis, respectively; these three rotation axes also intercept together at the point A (x=1.0 Å, y = 1.0 Å, z=0.0 Å). For a point B with its Cartesian coordinate (x = 2.0 Å, y = 3.0 Å, z = 4.0 Å), write down the Cartesian coordinates for all its symmetry-equivalent points (including point B itself) due to this 222 point group symmetry in this Cartesian coordinate system; write down the key calculation steps.

 

O (0, 0, 0) 1N 2-fold axis || Z Y -2-fold axis || Y 1 X 2-fold axis || X A (x=1.0 , y = 1.0 , z=0.0 )