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From the Hamiltionian on the previous slide we know that if eta is zero, there is axial symmetry. Since eta is defined as V_xx-V_yy / V_zz and V_xx = V_yy = -1, eta is zero.
From the picture of the toy model we see that the axis system is defined by taking the z-axis along the two electrons. Hence we have total freedom to define the x- and y-axis, as long as they are all perpendicular to each other.
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