- Aug 2022
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chem.libretexts.org chem.libretexts.org
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then there must be at least one symmetry operation that can interconvert the two orbitals
It seems that the (x,y) "symmetry degeneracy" implies that the two are both mathematically dependent and they (??) interconvertable via a symmetry operation.
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Both the E and T symbols can carry subscripts and primes that have specific meanings
Oh my lord this is where the t2g and eg crystal field theory separations of the five d-orbitals arise from !
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Figure 2.3.36 Character table for the point group C2v
Just another note- you can find the symmetry of square functions by literally multiplying characters of the irreducible representations that correspond to each linear function column-wise.
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Consider a rotational vector around z that indicates clockwise rotation (Fig. 2.3.37)
SUPER elucidating explanation of how the different rotations can be classified under one of the irreducible representations.
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This means that any linear function of the z-coordinate has the symmetry type A1
Like the wavefunction describing the pz orbital
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For C2v the dimension of the point group is 1+1+1+1=4. The order (h) of the point group is just the sum of all symmetry operations in the point group. For C2v the order is 4.
A clearer definition of the order of a point group, and how to determine it using a character table.
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whenever the characters for the operations in their irreducible representations are the same, they belong to the same class
Another explanation for how symmetry operations are grouped into classes.
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Similarly, because the 3dz2 orbital is a function of z2 and the 3dz2 has the symmetry type A1 any function of z2 has this symmetry type. Analogously, any function which is the product of the x and the z coordinate belongs to the symmetry type B1, any function which is a function of x2-y2 belongs to the symmetry type A1, and any function which is a product of y and z belongs to the symmetry type B2
Very intuitive explanation of the predicted symmetry that arises from multiplying coordinates
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more mathematical perspective
Relating the orbital understanding of symmetry to the symmetry of the mathematical wavefunctions that describe the corresponding orbitals.
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Therefore we can say that the 2s orbital of the O-atom in the water molecule has the symmetry type A1
it seems that this would apply to s-orbitals generally, for all (?) compounds.
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Anti-symmetric means that the algebraic sign of the coordinate changes as we rotate.
And this seems to be the answer to my first question. They just state that the two are seemingly equivalent; anti-symmetric (in this case) seems to mean the sign of coordinate changes during rotation wrt to the principle axis. They say C2 here- is that the same as saying the z-axis?
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The specific characters in an irreducible representation determine the symmetry type of the irreducible representation
This is how they connect the irreducible representations represented by Mulliken symbols to the irreducible representations derived in the previous section of this chapter
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If it is a -1 the algebraic sign of the coordinate changes, if it is +1, it does not
How does this connect to symmetric vs anti-symmetric interpretation from the other article?
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