Summary of 3d atomic orbitals

	Symmetry	Orbitals
Free atom		E1 m=2, Ψ3,2,2 = (1 / 81√(2π)) (Z/a)7/2 r2 e-Zr/3a sin2θ ei2φ m=1, Ψ3,2,1 = (√2 / 81√π) (Z/a)7/2 r2 e-Zr/3a sinθ cosθ eiφ m=0, Ψ3,2,0 = (1 / 81√(6π)) (Z/a)7/2 r2 e-Zr/3a (3cos2θ -1) m=-1, Ψ3,2,-1 = (√2 / 81√π) (Z/a)7/2 r2 e-Zr/3a sinθ cosθ e-iφ m=-2, Ψ3,2,-2 = (1 / 81√(2π)) (Z/a)7/2r2 e-Zr/3a sin2θ e-i2φ
Octahedral / Tetrahedral	Oh / Td	E1 (eg) x2-y2=(3,2,2)+(3,2,-2) sin2θ cos2φ z2=(3,2,0) (3cos2θ-1) E2 (t2g) xy=(3,2,2)-(3,2,-2) sin2θ sin2φ yz=(3,2,1)-(3,2,-1) sinθ cosθ sinφ xz=(3,2,1)+(3,2,-1) sinθ cosθ cosφ
Triangular bipyramid symmetry		E1 x2-y2, xy E2 xz, yz E3 z2

Angular momentum operator:
L_z=(ħ/i)∂/∂φ

BandGapof3dTMOxides

Band gap of 3d transition metal oxides

	Oxides	Xystal color	band gap (eV)	Oxides	Xystal color	band gap (eV)	Oxides	Xystal color	band gap (eV)
		2+			3+			4+
3d0				Sc2O3	white	6	TiO2	white	3.2
3d1				Ti2O3	violet	4.08	VO2	blue-black	0.7
3d2	TiO	dark	0	V2O3	black	0.2	CrO2	black	0
3d3	VO	black, grey, or green	0	Cr2O3	green	3	MnO2	gray to black	3.4
3d4
3d5	MnO	green	3.6-4.3	Fe2O3	red-brown	Xystal 2.0/film 2.7
3d6	FeO	black
3d7	CoO	grey or olive green	2.4
3d8	NiO	green	3.0-4.3

 

As we can see that the color does not correlate with band gap, except for the case that the material is always black when the gap is lower than 1.8 eV. This is because the optical gap we normally refer is actually charge transfer gap which bare huge absorption strength which color properties can be determined by the excitation that has much less absorption strength, e.g. on-site forbidden excitation. See this post.

Polaron reference

Holstein T, Annals of physics, V8,p 325 (1959)
Li ZZ, Theory of Solid State Physics, p 294, (1985)
Yuri M. Galperin, Introduction to Modern Solid State Physics,

Patterson J.,Bailey B., Solid-State Physics, Introduction to the theory, Springer Verlag (2007) p213.

Electron phonon coupling

coupling strength

α/2=Δ/ħω_L

where Δ is the deformation energy.

Added name

name added 06/27/2008