Give an appropriate set of LCAOs for the indicated geometry. |
1. |
sp linear geometry |
[Solution] |
2. |
sp2 trigonal planar geometry |
[Solution] |
3. |
sp3 tetrahedral geometry (no unique axis) |
[Solution] |
4. |
sp3 tetrahedral geometry (one bond along an axis) |
[Solution] |
5. |
sp3 tetrahedral geometry (pairs of bonds in a plane) |
[Solution] |
6. |
sp3d trigonal bipyramidal geometry |
[Solution] |
7. |
sp3d square pyramidal geometry (axial-base angle = 90°) |
[Solution] |
8. |
sp3d square pyramidal geometry (axial-base angle > 90°) |
[Solution] |
9. |
sp3d2 octahedral geometry |
[Solution] |
10. |
sp2d square planar geometry |
[Solution] |
| | |
1. |
sp linear geometry:
|
ψ1 |
= |
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|
1 |
|
φs |
+ |
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|
1 |
|
φpz |
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|
2 |
|
2 |
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|
ψ2 |
= |
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|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
|
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|
2 |
|
2 |
|
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[Back to Questions] |
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2. |
sp2 trigonal planar geometry:
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
2 |
|
φpx |
|
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|
3 |
|
3 |
|
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|
ψ2 |
= |
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|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φpy |
|
|
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|
|
3 |
|
6 |
|
2 |
|
|
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|
ψ3 |
= |
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|
1 |
|
φs |
− |
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|
1 |
|
φpx |
− |
|
|
1 |
|
φpy |
|
|
|
|
|
3 |
|
6 |
|
2 |
|
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[Back to Questions] |
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3. |
sp3 tetrahedral geometry (no unique axis):
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φpy |
|
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|
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|
4 |
|
4 |
|
4 |
|
4 |
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|
ψ2 |
= |
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|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φpy |
|
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|
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|
4 |
|
4 |
|
4 |
|
4 |
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|
ψ3 |
= |
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|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φpy |
|
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|
4 |
|
4 |
|
4 |
|
4 |
|
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|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φpy |
|
|
|
|
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|
4 |
|
4 |
|
4 |
|
4 |
|
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|
|
[Back to Questions] |
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4. |
sp3 tetrahedral geometry (one bond along an axis):
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
3 |
|
φpz |
|
|
|
|
4 |
|
4 |
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
2 |
|
φpx |
|
|
|
|
|
4 |
|
12 |
|
3 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φpy |
|
|
|
|
|
|
4 |
|
12 |
|
6 |
|
2 |
|
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|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φpy |
|
|
|
|
|
|
4 |
|
12 |
|
6 |
|
2 |
|
|
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|
|
|
|
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|
|
|
|
[Back to Questions] |
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| | |
| | |
5. |
sp3 tetrahedral geometry (pairs of bonds in a plane):
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpx |
|
|
|
|
|
4 |
|
4 |
|
2 |
|
|
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
|
|
|
|
|
4 |
|
4 |
|
2 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpy |
|
|
|
|
|
4 |
|
4 |
|
2 |
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpy |
|
|
|
|
|
4 |
|
4 |
|
2 |
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
| | |
| | |
| | |
6. |
sp3d trigonal bipyramidal geometry:
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
+ |
|
|
3 |
|
φdz2 |
|
|
|
|
|
5 |
|
2 |
|
10 |
|
|
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
3 |
|
φdz2 |
|
|
|
|
|
5 |
|
2 |
|
10 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
+ |
|
|
2 |
|
φpx |
− |
|
|
2 |
|
φdz2 |
|
|
|
|
|
5 |
|
3 |
|
15 |
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φpy |
− |
|
|
2 |
|
φdz2 |
|
|
|
|
|
|
5 |
|
6 |
|
2 |
|
15 |
|
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|
|
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|
ψ5 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φpy |
− |
|
|
2 |
|
φdz2 |
|
|
|
|
|
|
5 |
|
6 |
|
2 |
|
15 |
|
|
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
| | |
| | |
| | |
7. |
sp3d square pyramidal geometry (axial-base angle = 90°):
|
ψ1 |
= |
|
|
1 |
|
φpz |
|
|
|
1 |
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ5 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
| | |
| | |
| | |
8. |
sp3d square pyramidal geometry (axial-base angle > 90°):
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
4 |
|
φpz |
|
|
|
|
5 |
|
5 |
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
20 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
20 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
20 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ5 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
− |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
20 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
| | |
| | |
| | |
9. |
sp3d2 octahedral geometry:
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φdz2 |
|
|
|
|
|
6 |
|
2 |
|
3 |
|
|
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpz |
+ |
|
|
1 |
|
φdz2 |
|
|
|
|
|
6 |
|
2 |
|
3 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φdz2 |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
2 |
|
12 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
− |
|
|
1 |
|
φdz2 |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
2 |
|
12 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ5 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdz2 |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
2 |
|
12 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
ψ6 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdz2 |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
|
5 |
|
2 |
|
12 |
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
| | |
| | |
| | |
10. |
sp2d square planar geometry:
|
ψ1 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ2 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpx |
+ |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ3 |
= |
|
|
1 |
|
φs |
+ |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
ψ4 |
= |
|
|
1 |
|
φs |
− |
|
|
1 |
|
φpy |
− |
|
|
1 |
|
φdx2−y2 |
|
|
|
|
|
4 |
|
2 |
|
4 |
|
|
|
|
|
|
|
|
|
|
[Back to Questions] |
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