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Use the ordered basis $(p_x,p_y,p_z)$ at the origin. Choose $z$ as the principal improper-rotation axis, $x,y$ along the two $C_2^{\prime }$ axes, and the diagonal mirror planes as $x=\pm y$. Use active operations with $\hat{U}(g)f(\mathbf{r})=f(g^{-1}\mathbf{r})$, so the matrix columns give the transformed orbitals. For these real p orbitals the matrices are the ordinary vector transformation matrices.

One representative from each D2d molecular point group class isFor example $S_4^{+}$ sends $p_x\mapsto p_y$, $p_y\mapsto -p_x$ and $p_z\mapsto -p_z$. Choosing the opposite sense of $S_4$ transposes its xy block but leaves every character and the final decomposition unchanged.

Fix the positive lobes of the real atomic orbitals along the printed positive axes, and write ${\varphi }_1=p_x(1)$, ${\varphi }_4=p_x(4)$, ${\varphi }_5=p_y(5)$ and ${\varphi }_8=p_y(8)$. A coordinate model consistent with the figure is $r_1=(0,a,-b)$, $r_4=(0,-a,-b)$, $r_5=(a,0,b)$ and $r_8=(-a,0,b)$, with $a,b>0$. For the D2d molecular point group, take the two transverse twofold axes along $x+y$ and $x-y$, with vertical mirrors $xz$ and $yz$. The $A_2$ character table row on $(E,2S_4,C_2,2C_2^{\prime },2{\sigma }_d)$ is $(1,1,1,-1,-1)$. The character projection operator isThe eight signed images needed for this projection areso $P_{A_2}{\varphi }_1=({\varphi }_1-{\varphi }_4-{\varphi }_5+{\varphi }_8)/4$. Under the standard orthonormal atomic orbital approximation its squared norm is $4/16=1/4$. Therefore the normalized symmetry-adapted linear combination isAn overall minus sign describes the same quantum state; changing one local atomic orbital phase requires changing its coefficient too. If intersite orbital overlap were retained, normalization would instead use the overlap matrix.