CrO Minimal Structure-Derived Tight-Binding Formulas

This project is a minimal effective model extracted from the CIF geometry. It does not claim to be a fully fitted electronic-structure Hamiltonian. The rules used here are: one effective orbital per crystallographic site, Cr/O-dependent onsite energies, and nearest Cr-O hoppings that decay exponentially with bond length.

1. Basis and Bloch State

Basis size: N = 8.
|ψ_n(k)⟩ = ∑_α=1⁸ c_αn(k) |α, k&rangle

Ordered basis:
Cr1_1 (Cr), Cr1_2 (Cr), Cr1_3 (Cr), Cr1_4 (Cr), O1_1 (O), O1_2 (O), O1_3 (O), O1_4 (O)

2. Onsite Term

H_onsite = diag(ε₁, ..., ε₈)
ε_Cr = 1.200000, ε_O = -1.200000

3. Distance-Dependent Cr-O Hopping

t_ij(d_ij) = t₀ exp[-β (d_ij / d₀ - 1)]

d₀ = 2.028152 Å (shortest Cr-O bond from the CIF)
β = 8.000000

4. 8x8 Bloch Hamiltonian

H_αβ(k) = ε_α δ_αβ + ∑_m t_αβ^(m) exp[i k · (Δr_αβ^(m))] + h.c.

Δr_αβ^(m) = r_β + R^(m) - r_α

5. Half-Filled Valence / Conduction Split

E₁(k) ≤ ... ≤ E₄(k) ≤ E₅(k) ≤ ... ≤ E₈(k)
E_v(k) = E₄(k), E_c(k) = E₅(k)

E_gap^dir = min_k [E_c(k)] - max_k [E_v(k)]

6. Mid-Gap Energy Reference Used in the Plots

E_ref = 1/2 [max_k E_v(k) + min_k E_c(k)] = 0.000000

˜E_n(k) = E_n(k) - E_ref

7. Brillouin-Zone Sampling and DOS

k(u,v,w) = u b₁ + v b₂ + w b₃, u,v,w ∈ [0,1)

DOS(E) ≈ 1/N_k ∑_k,n exp[-(E - ˜E_n(k))² / 2σ²] / (σ √(2π))

8. k_z = 0 Reciprocal-Space Slice

k = (k_x, k_y, 0), -|b₁|/2 ≤ k_x ≤ |b₁|/2, -|b₂|/2 ≤ k_y ≤ |b₂|/2

The static and interactive reciprocal-space figures show E_v(k_x,k_y,0) and E_c(k_x,k_y,0).

9. Full 3D Reciprocal-Space Volume

For the interactive 3D reciprocal-space plot, the code samples the full orthorhombic reciprocal box (k_x, k_y, k_z) and assigns E_v(k_x,k_y,k_z) and E_c(k_x,k_y,k_z) to each sampled point.

k_x ∈ [-|b₁|/2, |b₁|/2], k_y ∈ [-|b₂|/2, |b₂|/2], k_z ∈ [-|b₃|/2, |b₃|/2]

10. Bond Table Used in This Run

Left site	Right site	Lattice translation [n1,n2,n3]	Distance (Å)	Hopping
Cr1_1 (Cr)	O1_2 (O)	[0, -1, -1]	2.028152	1.000000
Cr1_1 (Cr)	O1_2 (O)	[0, 0, -1]	2.041289	0.949500
Cr1_1 (Cr)	O1_4 (O)	[0, -1, 0]	2.028152	1.000000
Cr1_1 (Cr)	O1_4 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_2 (Cr)	O1_1 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_2 (Cr)	O1_1 (O)	[0, 1, 0]	2.028152	1.000000
Cr1_2 (Cr)	O1_3 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_2 (Cr)	O1_3 (O)	[0, 1, 0]	2.028152	1.000000
Cr1_3 (Cr)	O1_1 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_3 (Cr)	O1_1 (O)	[1, 0, 0]	2.028152	1.000000
Cr1_3 (Cr)	O1_2 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_3 (Cr)	O1_2 (O)	[1, 0, 0]	2.028152	1.000000
Cr1_4 (Cr)	O1_3 (O)	[-1, 0, 0]	2.028152	1.000000
Cr1_4 (Cr)	O1_3 (O)	[0, 0, 0]	2.041289	0.949500
Cr1_4 (Cr)	O1_4 (O)	[-1, 0, 0]	2.028152	1.000000
Cr1_4 (Cr)	O1_4 (O)	[0, 0, 0]	2.041289	0.949500

Current numerical result
Sampled direct gap = 2.400000