Supplementary Problems

Linear Combinations of Atomic Orbitals (LCAOs)

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]

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Solutions

1.

sp linear geometry:

	ψ₁	=			1		φ_s	+			1		φ_{p_z}

					2						2


	ψ₂	=			1		φ_s	−			1		φ_{p_z}

					2						2

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2.

sp² trigonal planar geometry:

	ψ₁	=			1		φ_s	+			2		φ_{p_x}

					3						3


	ψ₂	=			1		φ_s	−			1		φ_{p_x}	+			1		φ_{p_y}

					3						6						2


	ψ₃	=			1		φ_s	−			1		φ_{p_x}	−			1		φ_{p_y}

					3						6						2

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3.

sp³ tetrahedral geometry (no unique axis):

	ψ₁	=			1		φ_s	+			1		φ_{p_z}	+			1		φ_{p_x}	+			1		φ_{p_y}

					4						4						4						4


	ψ₂	=			1		φ_s	+			1		φ_{p_z}	−			1		φ_{p_x}	−			1		φ_{p_y}

					4						4						4						4


	ψ₃	=			1		φ_s	−			1		φ_{p_z}	+			1		φ_{p_x}	−			1		φ_{p_y}

					4						4						4						4


	ψ₄	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_x}	+			1		φ_{p_y}

					4						4						4						4

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4.

sp³ tetrahedral geometry (one bond along an axis):

	ψ₁	=			1		φ_s	+			3		φ_{p_z}

					4						4


	ψ₂	=			1		φ_s	−			1		φ_{p_z}	+			2		φ_{p_x}

					4						12						3


	ψ₃	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_x}	+			1		φ_{p_y}

					4						12						6						2


	ψ₄	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_x}	−			1		φ_{p_y}

					4						12						6						2

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5.

sp³ tetrahedral geometry (pairs of bonds in a plane):

	ψ₁	=			1		φ_s	+			1		φ_{p_z}	+			1		φ_{p_x}

					4						4						2


	ψ₂	=			1		φ_s	+			1		φ_{p_z}	−			1		φ_{p_x}

					4						4						2


	ψ₃	=			1		φ_s	−			1		φ_{p_z}	+			1		φ_{p_y}

					4						4						2


	ψ₄	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_y}

					4						4						2

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6.

sp³d trigonal bipyramidal geometry:

	ψ₁	=			1		φ_s	+			1		φ_{p_z}	+			3		φ_{d_z2}

					5						2						10


	ψ₂	=			1		φ_s	−			1		φ_{p_z}	+			3		φ_{d_z2}

					5						2						10


	ψ₃	=			1		φ_s	+			2		φ_{p_x}	−			2		φ_{d_z2}

					5						3						15


	ψ₄	=			1		φ_s	−			1		φ_{p_x}	+			1		φ_{p_y}	−			2		φ_{d_z2}

					5						6						2						15


	ψ₅	=			1		φ_s	−			1		φ_{p_x}	−			1		φ_{p_y}	−			2		φ_{d_z2}

					5						6						2						15

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7.

sp³d square pyramidal geometry (axial-base angle = 90°):

	ψ₁	=			1		φ_{p_z}

					1


	ψ₂	=			1		φ_s	+			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					4						2						4


	ψ₃	=			1		φ_s	−			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					4						2						4


	ψ₄	=			1		φ_s	+			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					4						2						4


	ψ₅	=			1		φ_s	−			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					4						2						4

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8.

sp³d square pyramidal geometry (axial-base angle > 90°):

	ψ₁	=			1		φ_s	+			4		φ_{p_z}

					5						5


	ψ₂	=			1		φ_s	−			1		φ_{p_z}	+			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					5						20						2						4


	ψ₃	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					5						20						2						4


	ψ₄	=			1		φ_s	−			1		φ_{p_z}	+			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					5						20						2						4


	ψ₅	=			1		φ_s	−			1		φ_{p_z}	−			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					5						20						2						4

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9.

sp³d² octahedral geometry:

	ψ₁	=			1		φ_s	+			1		φ_{p_z}	+			1		φ_{d_z2}

					6						2						3


	ψ₂	=			1		φ_s	−			1		φ_{p_z}	+			1		φ_{d_z2}

					6						2						3


	ψ₃	=			1		φ_s	+			1		φ_{p_x}	−			1		φ_{d_z2}	+			1		φ_{d_x2_−y2}

					6						2						12						4


	ψ₄	=			1		φ_s	−			1		φ_{p_x}	−			1		φ_{d_z2}	+			1		φ_{d_x2_−y2}

					6						2						12						4


	ψ₅	=			1		φ_s	+			1		φ_{p_y}	−			1		φ_{d_z2}	−			1		φ_{d_x2_−y2}

					6						2						12						4


	ψ₆	=			1		φ_s	−			1		φ_{p_y}	−			1		φ_{d_z2}	−			1		φ_{d_x2_−y2}

					6						2						12						4

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10.

sp²d square planar geometry:

	ψ₁	=			1		φ_s	+			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					4						2						4


	ψ₂	=			1		φ_s	−			1		φ_{p_x}	+			1		φ_{d_x2_−y2}

					4						2						4


	ψ₃	=			1		φ_s	+			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					4						2						4


	ψ₄	=			1		φ_s	−			1		φ_{p_y}	−			1		φ_{d_x2_−y2}

					4						2						4

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