YAMBO Formulas Quick Reference

Quick lookup guide for main theoretical formulas in YAMBO

Table of Contents

1. Basis Sets
2. GW Formulas
3. BSE Formulas
4. Symmetry Operations
5. Code Variables

1. Basis Sets

Plane-Wave Expansion

ψ_{n,k}(r) = (1/√Ω) Σ_G c_{n,k}(G) e^{i(k+G)·r}

Code: WF%c(ig,ib,ik) in mod_wave_func.F

G-Vector Cutoff

|k + G|² / 2 ≤ E_cut

Code: wf_ng, wf_ncx, wf_igk(ig,ik)

Oscillator Strength

ρ_{nm}(k,q,G) = ⟨n,k|e^{-i(q+G)·r}|m,k+q⟩
              = Σ_G' c*_{n,k}(G') c_{m,k+q}(G'+G)

Code: scatter_Bamp → isc%rhotw(ig)

2. GW Formulas

Self-Energy Operator

Σ(r,r',ω) = i/(2π) ∫ G(r,r',ω+ω') W(r,r',ω') e^{iω'η} dω'

Components: - G(r,r',ω) = Green's function - W(r,r',ω) = Screened Coulomb interaction

Screened Interaction

W(r,r',ω) = ∫ ε^{-1}(r,r'',ω) v(r'',r') dr''

Code: X_par(iq)%blc(ig1,ig2,iw) stores ε^{-1} - 1

Matrix Element Form

⟨n,k|Σ(ω)|n,k⟩ = Σ_{m,q,G,G'} ρ*_{nm}(k,q,G) W_{GG'}(q,ω-ε_m) ρ_{nm}(k,q,G')

Code: QP_ppa_cohsex.F, lines 650-750

Plasmon-Pole Approximation

W_{GG'}(q,ω) = δ_{GG'} v_G(q) + Ω²_{GG'}(q) / [ω² - ω²_{GG'}(q) + iη]

Pole Determination:

ω_{GG'}(q) = √[Ω²_{GG'}(q) / ε^{-1}_{GG'}(q,0) - 1]

Code: - X_par(iq)%blc(ig1,ig2,1) = Ω² (residue) - X_par(iq)%blc(ig1,ig2,2) = ω (pole energy)

Correlation Self-Energy (PPA)

Σ^c_{n,k}(ω) = Σ_{m,q,G,G'} ρ*_{nm}(k,q,G) Ω²_{GG'}(q) / [ω - ε_m - ω_{GG'}(q) + iη] ρ_{nm}(k,q,G') f_m

Code:

W_i = X_blc_p(ig1,ig2,1) / (Sc_W(i_qp)%p(iw) - E_kmq - X_blc_p(ig1,ig2,2) + cI*QP_G_damp)

Exchange Self-Energy

Σ^x_{n,k} = -Σ_{m,q,G} ρ*_{nm}(k,q,G) v_G(q) ρ_{nm}(k,q,G) f_m

Code: XCo_Hartree_Fock.F

COHSEX Approximation

SEX (Screened Exchange):

Σ^{SEX}_{n,k} = -Σ_{m,q,G,G'} ρ*_{nm}(k,q,G) W_{GG'}(q,0) ρ_{nm}(k,q,G') f_m

COH (Coulomb Hole):

Σ^{COH}_{n,k} = (1/2) Σ_{m,q,G,G'} ρ*_{nn}(k,0,G) [W_{GG'}(q,0) - v_G(q)δ_{GG'}] ρ_{nn}(k,0,G')

Code: QP_ppa_cohsex.F, lines 507-620

Quasiparticle Energy

First-Order:

E^{QP}_{n,k} = ε^{DFT}_{n,k} + Z_{n,k} [⟨n,k|Σ(E^{QP}_{n,k})|n,k⟩ - V^{xc}_{n,k}]

Linearized:

E^{QP}_{n,k} ≈ ε^{DFT}_{n,k} + ⟨n,k|Σ(ε^{DFT}_{n,k})|n,k⟩ - V^{xc}_{n,k}

Renormalization Factor:

Z_{n,k} = [1 - ∂Σ/∂ω|_{ω=E^{QP}}]^{-1}

Code: QP_Sc(i_qp,i_w) stores self-energy

3. BSE Formulas

BSE Eigenvalue Problem

(E_c - E_v) A^S_{vck} + Σ_{v'c'k'} K_{vck,v'c'k'} A^S_{v'c'k'} = Ω^S A^S_{vck}

Indices: - v, v' = Valence bands - c, c' = Conduction bands - k, k' = k-points - S = Exciton state index

Code: BS_K_dim = dimension, BS_bands(1:4) = band ranges

BSE Kernel

K_{vck,v'c'k'} = K^{exch}_{vck,v'c'k'} + K^{dir}_{vck,v'c'k'}

Exchange Term (Attractive)

K^{exch}_{vck,v'c'k'} = -δ_{kk'} Σ_G v_G(0) ρ_{vc}(k,0,G) ρ*_{v'c'}(k',0,G)

Physical Meaning: Direct Coulomb attraction between electron and hole

Code: K_exchange_kernel.F

H_x = Σ_ig conjg(O_x(ig,i_Tp)) * O_x(ig,i_Tk) / bare_qpg(iq,ig)**2

Direct Term (Screened)

K^{dir}_{vck,v'c'k'} = Σ_{G,G'} ρ_{vc}(k,q,G) W_{GG'}(q,0) ρ*_{v'c'}(k',q,G')

Physical Meaning: Screened Coulomb repulsion (reduces binding)

Code: K_correlation_kernel_std.F

K_corr = Vstar_dot_V_gpu(BS_n_g_W, O2, O_times_W) * 4._SP * pi

Resonant-Antiresonant Form

┌           ┐ ┌   ┐     ┌   ┐
│  R    C   │ │ A │     │ A │
│           │ │   │ = Ω │   │
│  C*  -R*  │ │ B │     │ B │
└           ┘ └   ┘     └   ┘

Resonant Block:

R_{vck,v'c'k'} = (E_c - E_v) δ_{vv'} δ_{cc'} δ_{kk'} + K_{vck,v'c'k'}

Coupling Block:

C_{vck,v'c'k'} = K^{coupling}_{vck,v'c'k'}

Code: BS_K_coupling enables coupling

Tamm-Dancoff Approximation (TDA)

Neglect coupling (C = 0):

R A = Ω A

Code: Default unless BSEmod="coupling"

Optical Absorption

ε_M(ω) = 1 - lim_{q→0} v(q) Σ_S |⟨S|ρ(q)|0⟩|² / (ω - Ω^S + iη)

Oscillator Strength:

|⟨S|ρ(q)|0⟩|² = |Σ_{vck} A^S_{vck} ρ_{vc}(k,q)|²

Code: K_observables.F

Haydock Recursion

ε_M(ω) = 1 - |d|² / [ω - a_0 - b²_1/(ω - a_1 - b²_2/(ω - a_2 - ...))]

Recursion:

|ψ_{n+1}⟩ = H|ψ_n⟩ - a_n|ψ_n⟩ - b_n|ψ_{n-1}⟩

Code: K_Haydock.F

4. Symmetry Operations

Symmetry Group

{S_i | τ_i},  i = 1, ..., nsym

• S_i = Rotation matrix (3×3)
• τ_i = Fractional translation

Code: - nsym = Total symmetries - dl_sop(3,3,is) = Real-space operations - rl_sop(3,3,is) = Reciprocal-space operations - dl_tra(3,is) = Translations

k-Point Mapping

S_i k = k' + G

Star of k:

Star(k) = {S_i k | i = 1, ..., nsym}

Code: - nstar(ik) = Size of star - star(ik,is) = Symmetry mapping - sstar(ikbz,1:2) = (IBZ index, symmetry)

Wavefunction Transformation

ψ_{n,Sk}(r) = e^{iSk·τ} ψ_{n,k}(S^{-1}r)

Reciprocal Space:

c_{n,Sk}(G) = e^{iSk·τ} e^{iG·τ} c_{n,k}(S^{-1}G)

Code: - g_rot(ig,is) = Rotated G-vector index - g_phs(ig,is) = Phase factor e^{iG·τ} - WF_phases(ig,is,ib,ik,isp) = Wavefunction phases

Time-Reversal Symmetry

ψ_{n,-k}(r) = ψ*_{n,k}(r)

Reciprocal Space:

c_{n,-k}(G) = c*_{n,k}(-G)

Code: - i_time_rev = 1 if present - Symmetries with is > nsym/(i_time_rev+1) include TR

Spatial Inversion

I: r → -r,  k → -k,  G → -G

Code: - i_space_inv = 1 if present - inv_index = Symmetry index of -I

Small Group of k

Small_Group(k) = {S_i | S_i k = k + G}

Code: - grp_nsym(ik) = Number in small group - grp_table(ik,is_grp) = Mapping to full list

5. Code Variables

Wavefunctions

Reciprocal Lattice

k-Points

Symmetries

GW Variables

BSE Variables

Coulomb Interaction

Quick Formula Lookup

Need to compute...

Quasiparticle energy? → E^{QP} = ε^{DFT} + ⟨Σ(E^{QP})⟩ - V^{xc}

Optical absorption? → ε_M(ω) = 1 - Σ_S |⟨S|ρ|0⟩|²/(ω - Ω^S)

Oscillator strength? → ρ_{nm}(k,q,G) = ⟨n,k|e^{-i(q+G)·r}|m,k+q⟩

Screened interaction? → W = ε^{-1} v

BSE kernel? → K = K^{exch} + K^{dir}

Exchange term? → K^{exch} = -Σ_G v_G ρ ρ*

Direct term? → K^{dir} = Σ_{GG'} ρ W ρ*

Normalization Factors

Common Approximations

Units and Constants

Constants: - HA2EV = 27.21138602 (Ha to eV) - pi = 3.14159265358979323846 - cI = (0, 1) (imaginary unit)

File Locations

End of Quick Reference