Dmitrii Nabok Humboldt-Universität zu Berlin. August 8th, 2016
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1 Dmitrii Nabok Humboldt-Universität zu Berlin August 8th, 2016
2 Outline Introduction G0W0 approximation Implementation Program workflow Product basis representation Matrix form of GW equations Usage Input file Output Convergence control
3 Introduction
4 GW approximation G=G0+G0ΣG Σ=iGW P=-iGG W=v+vPW L. Hedin, Phys. Rev. 139, A796 (1965) L. Hedin and B. I. Lundqvist, Solid State Phys. 23, 1 (1969)
5 G0W0 approximation Kohn-Sham (KS) eigenvalues DFT exchange-correlation potential KS eigenfunctions M. Hybertsen and S. Louie, PRB 34, 5390 (1986)
6 G0W0 approximation G Non-interacting Green's function
7 G0W0 approximation P Polarizability in the random phase approximation (RPA) x Fermi distribution function
8 G0W0 approximation Dynamically screened Coulomb potential W Dielectric function (bare) Coulomb potential
9 G0W0 approximation Exchange self-energy: Σ = Correlation self-energy:
10 GW approximation G0W0 Think of self-consistency? - Partial: G0W, GW0 - Full (e.g. QSGW):
11 Implementation
12 Program flowchart Groundstate LAPW basis functions Kohn-Sham eigenvalues/ eigenfunctions G0W0
13 Product-basis representation Matrix form:
14 LAPW basis
15 Mixed basis MT region Use only with Neglect = : Form orthonormal basis + eliminate linearly dependent products Take in account the translational symmetry F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 49, (1994) T. Kotani and M. van Schilfgaarde, Solid State Comm. 121, 461 (2002)
16 Mixed basis Interstitial region Overlap matrix Diagonalize Build an orthonormal basis set: G MB max, where
17 Mixed basis
18 Mixed basis Completeness test: MT region Si, k=(0,0,0),k-q=(0,0,1), n=5, m=4
19 Mixed basis Completeness test: Interstitial Si, k=(0,0,0),k-q=(0,0,1), n=5, m=4
20 G0W0 equations in matrix form Coulomb potential Special treatment for the case q 0 Exchange part of the self-energy
21 G0W0 equations in matrix form Polarizability Generalized tetrahedron method (LIBBZINT library): + /input/gw/@vqloff
22 G0W0 equations in matrix form Dielectric function Correlation term of the screened Coulomb potential
23 G0W0 equations in matrix form Correlation part of the self-energy /input/gw/freqgrid Integral over imaginary frequency: Analytic continuation onto the real frequency axis
24 G0W0 quasi-particle energy KS energies QP renormalization factor Diagonal matrix elements of the exchange-correlation potential
25 Usage
26 Input file Groundstate <groundstate rgkmax= 8.0 ngridk= gmaxvr= 14 xctype= LDA_PW > </groundstate> Quality of the starting ( zero order ) potential, eigenvalues, eigenvectors
27 Input file GW <gw taskname= g0w0... <mixbasis lmaxmb= 3 epsmb= 1.0d-4 gmb= 1.0 ></mixbasis>... <barecoul... ></barecoul>... <selfenergy... ></selfenergy>... <freqgrid... ></freqgrid> Type of calculations: ''band'' - QP bandstructure plot ''dos'' - QP density of states... </gw>
28 Input file GW <gw taskname= g0w0... <mixbasis lmaxmb= 3 epsmb= 1.0d-4 gmb= 1.0 ></mixbasis>... <barecoul... ></barecoul>... <selfenergy... ></selfenergy>... <freqgrid... ></freqgrid> </gw> Quality of the product basis
29 Input file GW <gw taskname= g0w0... <mixbasis lmaxmb= 3 epsmb= 1.0d-4 gmb= 1.0 ></mixbasis>... Parameters for more advanced usage <barecoul... ></barecoul>... <selfenergy... ></selfenergy>... <freqgrid... ></freqgrid> </gw>
30 Output GW_INFO.OUT ================================================================================ = Main GW output file = ================================================================================ ******************************************************************************** * GW input parameters * ******************************************************************************** GW taskname: g0w0 - G0W0 run Frequency integration parameters: Number of frequencies: 16 Cutoff frequency: Grid type: gaule2 - Grid for double Gauss-Legendre quadrature, from 0 to freqmax and from freqmax to infinity Convolution method: imfreq : weights calculated for imaginary frequecies Correlation self-energy parameters:...
31 Output EVALQP.DAT k-point # 1: state E_KS E_HF E_GW Sx Sc Vxc DE_HF DE_GW Znk k-point # 2: state E_KS E_HF E_GW Sx Sc Vxc DE_HF DE_GW Znk
32 Convergence control Product basis nempty= 17, ngridk= gmb= 1.0 lmaxmb lmaxmb= 2 gmb
33 Convergence control k / q -point grids: /input/groundstate/@ngridk lmaxmb= 3, gmb= 1.0, nempty= 17
34 Convergence control Number of unoccupied states: ngridk= ngridk= ngridk= 6 6 6
35 <gw taskname= skip </gw> <xs> xstype= BSE ngridk= ngridq= nempty= <BSE />... </xs> bsetype="singlet" nstlbse=" " nstlbsemat=" " Signal to use QP spectrum in XS calculations
36 How exciting?
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