01 How To Model Crystals.pdf - How to Model Crystals Periodic Boundary Conditions Silvia Casassa Universit` a degli Studi di Torino September 4 2016 Ab

# 01 How To Model Crystals.pdf - How to Model Crystals...

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How to Model Crystals: Periodic Boundary Conditions Silvia Casassa Universit` a degli Studi di Torino September 4, 2016 Ab initio Modelling of Solids (UniTo) Symmetry in Space September 4, 2016 1 / 51 Introduction Outline 1 Introduction 2 How to model Real Systems 3 Symmetry 4 The Periodic Model 5 Symmetry exploitation in Reciprocal Space 6 SCF iterative procedue Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 2 / 51 Introduction CRYSTAL code The Target Solve the Schr¨odinger Equation for macroscopic systems ˆ H Ψ( x 1 , .. x n ) = E Ψ( x 1 , .. x n ) (1) ˆ H : Born-Oppenheimer apx., electrostatic contributions x i ( r i , σ ), cartesian and electrons spin coordinates States superposition Wave function as a linear combination of different electronic configurations Ψ( x 1 , .. x n ) = r c r Ψ r ( x 1 , .. x n ) Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 3 / 51 Introduction The Single Determinantal Approximation Wave function as an antisimmetrized product of spin-orbitals { ϕ i ( x 1 ) } Ψ( x 1 , .. x n ) Ψ 0 = ˆ A Φ 0 ˆ A | ..ϕ i ( x 1 ) ...ϕ n ( x n ) | ˆ H Ψ 0 ( x 1 , .. x n ) = E Ψ 0 ( x 1 , .. x n ) The Variational Theorem E [Ψ] E exact (2) E [Ψ] E (Ψ[ { ϕ i } ]) minimize E : E ( { ϕ i } ) ∂ϕ i = 0 Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 4 / 51 Introduction Hartree-Fock (HF)/Kohn-Sham (KS) equations (depending on ˆ f i , the one-electron operator) .... .. ˆ f i ϕ i ( r i ) = i ϕ i ( r i ) .... .. linearization, i.e.: from a set of n -differential equations in { ϕ i } to a matrix representation in the atomic orbitals (AO) basis set: ϕ i ( r i ) = X μ c μ i χ μ ( r ) FC = SCE (3) Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 5 / 51 Introduction initial information geometry basis set, { χ μ ( r ) } guess for the density matrix, P μν calculation of F μν ( P ) diagonalize F FC = SCE update the new P matrix P μν = 2 occ j c * j μ c j ν E n - E n - 1 < TOL ? stop no yes Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 6 / 51 How to model Real Systems Outline 1 Introduction 2 How to model Real Systems 3 Symmetry 4 The Periodic Model 5 Symmetry exploitation in Reciprocal Space 6 SCF iterative procedue Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 7 / 51 How to model Real Systems a car Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 8 / 51 How to model Real Systems description of a car using a basis set of localized type orbitals Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 9 / 51 How to model Real Systems effect of the basis set on the Fock matrix dimension Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 10 / 51 How to model Real Systems The Model Ab initio Modelling of Solids (UniTo) Symmetry in Space Torino - September 2016 11 / 51 How to model Real Systems CRYSTAL code The Challenge USE OF A LOCAL BASIS SET to describe a virtually infinite system!  • • • 