TOMBO Ver.2 Manual

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2.5 The GW approximation 17 Vertex function: ∫ Γ(1,2,3) = δ(1,2)δ(1,3) + Dielectric function: d(4567) ∂Σ(1,2) G(4,6)G(7,5)Γ(6,7,3) (2.22) ∂G(4,5) ε = 1 − νP (2.23) The screened potential W can also be written as ∫ W(1,2) = d(3)ε −1 (1,3)ν(3,2) (2.24) From Hedin’s equations, we know how to calculate the self-energy. Although the GWΓ method is implemented in a future version of <strong>TOMBO</strong>, the following GW approximation is needed for the moment. 2.5 The GW approximation A practical approximation to calculate Σ is the GW approximation proposed by Hedin [17]. In this GW approximation, the vertex function Γ is approximated in its zeroth-order form, Γ(1,2,3) = δ(12)δ(13). So the Hedin’s equations become: Σ(1,2) = iG(1,2)W(1 + ,2) (2.25) ∫∫ G(1,2) = G 0 (1,2) + d(3)d(4)G 0 (1,3)Σ(3,4)G(4,2) (2.26) P(1,2) = −iG(1,2)G(2,1) (2.27) ∫∫ W(1,2) = ν(1,2) + d(3)d(4)ν(1,3)P(3,4)W(4,2) (2.28) This self-consistent GW calculation is possible in <strong>TOMBO</strong> by setting the first character of iApp as “G” in INPUT.inp, although crystal calculation and numerical ω integration (which will be explained below) are not available now. The explicit form of the self-energy Σ in the GW approximation is given by Σ GW (r,r ′ ;ε) = i ∫ 2π dωe −i0+ω G(r,r ′ ;ε − ω)W(r,r ′ ;ω) (2.29)
2.5 The GW approximation 18 The GW self-energy can be separated into two terms ( Σ GW = Σx + Σ c ). The exchange part is given by Σ x (r,r ′ ) = i ∫ 2π dωe iω0+ G(r,r ′ ;ω)ν(r − r ′ ) = − occ ∑ nk ψ nk (r)ψ ∗ nk (r) |r − r ′ | (2.30) In equation (2.30), the symbol occ in the sum means that the summation is taken over the occupied states only. The diagonal matrix elements of this exchange part of the self-energy become ⟨ n,k ∣ ∣Σx (r,r ′ ) ∣ ∣ n,k ⟩ = − occ ∑ ∑ n ′ q ∑ G 4π ⟨ ΩG 2 ∣ n,k∣e i(q+G)·r∣ ⟩⟨ ∣ ∣n ′ ,k − q n ′ ∣ k − q∣e −i(q+G)·r′∣ ⟩ ∣ ∣n,k (2.31) The correlation part, Σ c can be evaluated by using generalized plasmon-pole (GPP) model [19]. Σ c can also be evaluated by using the full ω integration [20]: Σ c (r,r ′ ;ω) = i ∫ 2π dω ′ e −iω0+ G(r,r ′ ;ω − ω ′ ) [ W(r,r ′ ;ω ′ ) − ν(r − r ′ ) ] (2.32) In Equation (2.32), it is difficult to perform the ω ′ integral along the real axis, since W and G have a strong structure on this axis. In order to avoid this difficulty, Godby et al.[21–24] restricted the values of ω to small imaginary numbers and changed the contour of the ω ′ integral from real axis to imaginary axis [16]. Then, by analytic continuation, the resulting Taylor series is used to estimate the matrix elements for real values of ω. Ishii and Ohno et al. [20, 25] suggested that this intergration method employed by Godby et al. [16] can be extended rather easily to real number of ω by slightly modifying the contour. The contour along the real ω ′ axis from −∞ to +∞ for the integral Equation (2.32) can be replaced by the contour C shown in Fig.2.2. Here we further use the symmetry W(ω) = W(−ω) to reduce the contour to the positive real and imaginary parts only. The diagonal matrix elements of the correlation part of the self-energy become ⟨ n,k ∣ ∣Σc (r,r ′ ;ω) ∣ ∣ n,k ⟩ = ∑ ∑ n ′ q × i 2π ⟨ ∑ G,G ′ ∫ ∣ n,k∣e i(q+G)·r∣ ⟩⟨ ∣ ∣n ′ ,k − q n ′ ∣ k − q∣e −i(q+G′ )·r ′∣ ⟩ ∣ ∣n,k dω ′ [ W G,G ′(q,ω ′ ) − (4π/ΩG 2 ] )δ G,G ′ 1 × ( ω + ω ′ + − ε k−q,n − iδ k−q,n 1 ω − ω ′ − ε k−q,n − iδ k−q,n ) (2.33)
Page 1 and 2: TOMBO Ver.2 Manual for LDA and GW C
Page 3 and 4: Table of contents 1 Introduction 1
Page 5 and 6: Table of contents v 3.2.42 Mcheb .
Page 7 and 8: 1.2 Mixed basis formulation 2 to th
Page 9 and 10: 1.2 Mixed basis formulation 4 Many-
Page 11 and 12: 1.3 All-electron charge density and
Page 13 and 14: 1.3 All-electron charge density and
Page 15 and 16: 1.4 Calculation flow 10 1.4 Calcula
Page 17 and 18: 1.5 TDDFT dynamics simulation 12 of
Page 19 and 20: 2.2 The Green function 14 2.2 The G
Page 21: 2.4 Hedin’s equations (GWΓ metho
Page 25 and 26: 2.6 One-shot GW approximation 20 or
Page 27 and 28: 3.1 COORDINATES.inp 22 3.1 COORDINA
Page 29 and 30: 3.1 COORDINATES.inp 24 10. The Atom
Page 31 and 32: 3.2 INPUT.inp 26 3.1.1 rct "XX_rct"
Page 33 and 34: 3.2 INPUT.inp 28 An example of INPU
Page 35 and 36: 3.2 INPUT.inp 30 3.2.5 Ulevel, Llev
Page 37 and 38: 3.2 INPUT.inp 32 calculation of the
Page 39 and 40: 3.2 INPUT.inp 34 Whether the subtra
Page 41 and 42: 3.2 INPUT.inp 36 3.2.30 smixSCF Cha
Page 43 and 44: 3.3 KPOINT.inp 38 3.2.42 Mcheb Numb
Page 45 and 46: 3.4 QPOINT.inp 40 10, line 15, and
Page 47 and 48: 3.5 SPOINT.inp 42 Table 3.9 QPOINT.
Page 49 and 50: Chapter 4 Output Files Note: 1. In
Page 51 and 52: 4.8 WaveF_HOMO-1.cube, WaveF_HOMO-1
Page 53 and 54: 4.9 GWA.out 48 Table 4.2 GWA.out of
Page 55 and 56: Chapter 5 Examples 5.1 LDA calculat
Page 57 and 58: 5.1 LDA calculation of isolated dim
Page 59 and 60: 5.2 Molecular Dynamics 54 Fig. 5.4
Page 61 and 62: 5.4 LDA calculation of rutile TiO 2
Page 63 and 64: 5.4 LDA calculation of rutile TiO 2
Page 65 and 66: 5.5 Wave function calculation of ru
Page 67 and 68: References 62 [12] M. S. Bahrany, M
Page 69 and 70: A.1 Build crystal model 64 A.1 Buil
Page 71 and 72: A.3 *.cell file 66 Fig. A.4 1 st Br
Page 73 and 74:
A.3 *.cell file 68 Step 4 In "CASTE
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