Using the character table for the C2v point group, specify the Mulliken label of each of the five d orbitals.

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The d orbitals are found in the transition metals in the center of the periodic table of elements.  For a given energy level there are five associated d orbitals (since the azimuthal quantum number is 2, that means there are five orbitals corresponding to -2, -1, 0, 1, and 2).  The d orbitals have a general octahedral symmetry across the three dimensional Cartesian coordinate axes x, y, and z.  Assuming only electrostatic interactions, the 5 orbitals can be divided into two different energy levels.  The d(z^2) and d(x^2-y^2) orbitals are approximately equal in energy and have the Mulliken designation of eg (the g is a subscript).  The d(xz), d(yz), and d(xy) orbitals are all approximately the same energy level below the other two and have the Mulliken designation of t2g (the 2 and g are subscripts).  In reality, the exact shape of the chemical alters the energy levels of the d orbitals, but this is the generalized form for the Mulliken designation of the d orbitals


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