Translation Symmetry, Space Groups, Bloch functions, Fermi energy

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1 Translation Symmetry, Space Groups, Bloch functions, Fermi energy Roberto Orlando and Silvia Casassa Università degli Studi di Torino July 20, 2015 School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions July 20, / 42

2 Introduction Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

3 Introduction CRYSTAL code The Target Solve the Schrödinger Equation for macroscopic systems Ĥ Ψ(x 1,..x n ) = E Ψ(x 1,..x n ) (1) Ĥ: 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 ) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

4 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 = Â Φ 0 Â..ϕ i(x 1 )...ϕ n (x n ) Ĥ Ψ 0 (x 1,..x n ) = E Ψ 0 (x 1,..x n ) The Variational Theorem minimize E: E[Ψ] E exact (2) E[Ψ] E(Ψ[{ϕ i }]) E({ϕ i }) ϕ i = 0 School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

5 Introduction Hartree-Fock (HF)/Kohn-Sham (KS) equations (depending on ˆf i, the one-electron operator) linearization, i.e.:..... ˆf i ϕ i (r i ) = ɛ i ϕ i (r i )..... 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 ) = µ c µi χ µ (r) FC = SCE (3) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

6 Introduction geometry initial information basis set, {χ µ (r)} guess for the density matrix, P µν calculation of F µν (P) P µν = 2 occ j cjµ c jν update the new P matrix diagonalize F FC = SCE no E n E n 1 < TOL? yes stop School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

7 Introduction CRYSTAL code The Challenge USE OF A LOCAL BASIS SET to describe a virtually infinite system! How to build and diagonalize the Fock matrix of a macroscopic system? in a basis set of atomic orbitals? School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

8 Model of a Crystal Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

9 Model of a Crystal The Model of a Perfect Crystal A real crystal: a macroscopic finite array of a very large number n of atoms/ions with surfaces the fraction of atoms at the surface is proportional to n 1/3 (very small) if the surface is neutral, the perturbation due to the boundary is limited to a few surface layers a real crystal mostly exhibits bulk features and properties A macro-lattice of N = N 1 N 2 N 3 unit cells is good model of such a crystal as N is very large and surface effects are negligible in the bulk, the macro-lattice can be repeated periodically under the action of translation vectors T N = N 1 a 1, N 2 a 2, N 3 a 3 without affecting its properties School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

10 Model of a Crystal thus, our model of a perfect crystal coincides with the crystallographic model of an infinite array of cells containing the same group of atoms School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

11 Model of a Crystal SYMMETRY EXPLOITATION Symmetry is not a human construction but a matter fundation MOLECULES point symmetry (when presents) can help in the interpretation of the results (normal modes, transition rules, etc.) make the calculation faster (symmetry adapted molecular orbitals and block diagonal Fock matrix in terms of the IRREP of the point group) but it is not pivotal! School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

12 Model of a Crystal SYMMETRY EXPLOITATION Symmetry is not a human construction but a matter fundation CONDENSED MATTER Symmetry is exploited at many levels: Direct Lattice Metric: translational invariance 1 build up the infinite system from few geometrical data (space group, asymmetric unit, see the case of nanotubes!); 2 Fock matrix construction: all the contributions are calculated with respect to the reference cell Reciprocal Lattice Metric 1 Fock matrix becomes block diagonal! School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

13 Symmetry Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

14 Symmetry Point Symmetry POINT SYMMETRY The symmetry operators During the action of a point symmetry operation, performed by the symmetry operator, ˆR, of the point group of the molecule, at least one point stays put. School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

15 Symmetry Point Symmetry POINT SYMMETRY The symmetry operators During the action of a point symmetry operation, performed by the symmetry operator, ˆR, of the point group of the molecule, at least one point stays put. chool Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

16 Symmetry Point Symmetry POINT SYMMETRY The 32 crystallographic point group School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

+ t 3 a 3 A lattice is translation invariant: translations along lattice vectors are symmetry operations, { ˆT

17 Symmetry Translation Symmetry Translation Symmetry The Lattice Given a set of basis vectors, a 1, a 2, a 3 Every lattice vector g is a linear combination of the basis vectors (t i integer number): g 1,2,3 = t 1 a 1 + t 2 a 2 + t 3 a 3 A lattice is translation invariant: translations along lattice vectors are symmetry operations, { ˆT }: ˆT g (4) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

18 Symmetry Translation Symmetry The Unit Cell Unit cell: a volume in space that fills space entirely when translated by all lattice vectors. The obvious choice: a parallelepiped defined by three basis vectors with a 1, a 2, a 3 as orthogonal as possible the cell is as symmetric as possible School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

19 Symmetry Translation Symmetry The 7 crystal systems School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

20 Symmetry Translation Symmetry Primitive and Centered lattice The Primitive Cell are unit cell containing one lattice point is called primitive cell. School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

21 Symmetry Translation Symmetry The 14 Bravais lattice School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

22 Symmetry Translation Symmetry New symmetry operators School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

International Tables for Crystallography Classification and details about space groups can be

23 Symmetry Translation Symmetry The 230 Space Group All possible combinations of symmetry operations compatible with the 14 Bravais lattices originate the 230 space groups. International Tables for Crystallography Classification and details about space groups can be found in the School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

24 Symmetry Exploitation in Crystal construction Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

The infinite crystal of quartz can be generated from two atoms, Si O, by the application of

25 Symmetry Exploitation in Crystal construction How do we generate a Crystal from a structural motif? The infinite crystal of quartz can be generated from two atoms, Si O, by the application of symmetry operations.... exploiting point and translational symmetry! ((i.e. INVARIANCE)) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

26 Symmetry Exploitation in Crystal construction α-quartz School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

27 Symmetry Exploitation in Crystal construction α-quartz School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

28 Symmetry Exploitation in Crystal construction α-quartz School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

29 Symmetry Exploitation in Crystal construction α-quartz School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

30 Symmetry Exploitation in Fock matrix construction Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the

31 Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the CRYSTAL: N cells Square matrix N 2 blocks of size M < χ g i µ (r) ˆF χ g j ν (r) > = < χ 0 µ (r) ˆF χ (g j g i ) ν (r) > F g k µν School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

32 Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the CRYSTAL: N cells Square matrix N 2 blocks of size M < χ g i µ (r) ˆF χ g j ν (r) > = < χ 0 µ (r) ˆF χ (g j g i ) ν (r) > F g k µν School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

33 Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the CRYSTAL: N cells Square matrix N 2 blocks of size M < χ g i µ (r) ˆF χ g j ν (r) > = < χ 0 µ (r) ˆF χ (g j g i ) ν (r) > F g k µν School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

34 Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the CRYSTAL: N cells Square matrix N 2 blocks of size M < χ g i µ (r) ˆF χ g j ν (r) > = < χ 0 µ (r) ˆF χ (g j g i ) ν (r) > F g k µν School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

35 Symmetry Exploitation in Fock matrix construction Direct Space Fock matrix in direct Space Representation in the AO basis set a set of M AO: {χ 0 µ (r)} M in the reference cell dimension of the CRYSTAL: N cells Square matrix N 2 blocks of size M < χ g i µ (r) ˆF χ g j ν (r) > = < χ 0 µ (r) ˆF χ (g j g i ) ν (r) > F g k µν School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

36 Symmetry Exploitation in Fock matrix construction Direct Space How do we solve the HF/KS equation for a Crystal? Fock matrix diagonalization is still a dreadful task... Exploiting the periodic boundary condition! chool Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

A new metric is defined: g (t 1, t 2, t 3 ) ˆTg = t 1 a 1 + t 2 a 2 + t 3 a 3 ˆT N = N 1 a 1 + N 2 a 2 + N 3 a 3 Periodic (or cyclic)

37 Symmetry Exploitation in Fock matrix construction Reciprocal Space Born-Von Karman boundary conditions Our model of a crystal consists of an infinite array of macro-lattices. Every macro-lattice contains N unit cells. A new metric is defined: g (t 1, t 2, t 3 ) ˆTg = t 1 a 1 + t 2 a 2 + t 3 a 3 ˆT N = N 1 a 1 + N 2 a 2 + N 3 a 3 Periodic (or cyclic) Boundary Conditions the translational symmetry group becomes a (abelian) cyclic group; chool Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

38 Symmetry Exploitation in Fock matrix construction Reciprocal Space for a monodimensional crystal, N 1, t 1, ˆT g = ta 1 given a mother operator ˆT 1 = 1 a 1 ˆT N 1 = ( ˆT 1 ) N 1 = Î A( ˆT 1 ) = A(Î ) 1 N 1 any other operator, ˆTg i g i = t i a 1 can be written as: ˆTg ˆT ti = ˆT 1 ˆT 1... = ( ˆT 1 ) t i then the character of the t 1 -operator in the k 1 -irreducible representation (IR) becomes one of the N 1 -row of the unity A( Tˆ t1 ) k 1 = exp (2πi k 1 N 1 t 1 ) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

39 Symmetry Exploitation in Fock matrix construction Reciprocal Space The Reciprocal lattice A reciprocal lattice with lattice basis vectors b 1, b 2, b 3 corresponds to every real (or direct) lattice with lattice basis vectors a 1, a 2, a 3. Any lattice point can be written (K 1, K 2, K 3 are integer): K = K 1 b 1 + K 2 b 2 + K 3 b 3 (5) Basis vectors obey the following orthogonality rules: b i a j = 2πδ ij < K T > = 2πn (6) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

40 Symmetry Exploitation in Fock matrix construction Reciprocal Space The Reciprocal lattice from real to reciprocal space: metrics School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

Symmetry Exploitation in Fock matrix construction Reciprocal Space The first Brillouin zone Wigner-Seitz cell The portion of space closer to one

41 Symmetry Exploitation in Fock matrix construction Reciprocal Space The first Brillouin zone Wigner-Seitz cell The portion of space closer to one lattice point than to anyone else... Brillouin zone School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

42 Symmetry Exploitation in Fock matrix construction Reciprocal Space The Pack-Monkhorst net The complete set of k vectors in the first Brillouin zone define the Pack-Monkhorst net.. where each point can be written as (k i = k i N i are real) k = k 1 N 1 b 1 + k 2 N 2 b 2 + k 3 N 3 b 3 = k 1 b 1 + k 2 b 2 + k 3 b 3 and then because < k g > = 2π(k 1 t 1 + k 2 t 2 + k 3 t 3 ) (7) the character of the translational operators can be written as: A( Tg ˆ ) k = exp (2πi k 1 N 1 t 1 ) exp (ikg ) (8) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

43 Symmetry Exploitation in Fock matrix construction Reciprocal Space Block function it is possibile to write the Wigner projection operators ˆP k = 1 N N exp (ikg ) ˆTg g that applied to the AO basis set yield a set of well-behavied function, the Bloch functions, BF: Φ k µ (r) = ˆP k χ 0 µ (r) = 1 N exp (ikg ) ˆTg χ 0 µ (r) = N g = 1 N exp (ikg ) χ g µ (r) (9) N g School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

Symmetry Exploitation in Fock matrix construction Reciprocal Space Consequences of Bloch Theorem 1 due to the orthogonality between functions belonging to different IR, elements of the Fock matrix in

44 Symmetry Exploitation in Fock matrix construction Reciprocal Space Consequences of Bloch Theorem 1 due to the orthogonality between functions belonging to different IR, elements of the Fock matrix in the BF basis set are orthogonal F µν (k 1, k 2 ) = < Φ k 1 µ (r) ˆF Φ k 2 ν (r) >= δ k1, k 2 (10) 2 continuity of E(k) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

45 SCF iterative procedue Outline 1 Introduction 2 Model of a Crystal 3 Symmetry Point Symmetry Translation Symmetry 4 Symmetry Exploitation in Crystal construction 5 Symmetry Exploitation in Fock matrix construction Direct Space Reciprocal Space 6 SCF iterative procedue School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

46 SCF iterative procedue A Fock matrix in the BF basis set 1 Representation of on-electron operator in the basis set of the AOs: calculation of (M 2 xn) elements of F g µν 2 at every point k of the Pack-Monkhorst net (k BZ) perform the Fourier tranformation of F g µν: F µν (k) = N g exp (ikg ) F g µν 3 At every k the eigenvalue equation of size M is solved: F (k)c(k) = S(k)C(k)E(k) 4 M crystalline orbitals (CO) Ψ j (k, r) associated with M eigenvalues ɛ j (k) are obtained: Ψ j (k, r) = M µ c µj(k)φ k µ (r) School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

47 SCF iterative procedue Calculation of CO with SCF approx 5 Aufbau principle in populating electronic bands Fermi Energy, ɛ F 6 Calculation of density matrix in the AO basis from occupied CO in reciprocal space occ P µν (k) = 2 c µj (k)cνj(k) and then transformed in direct space P g µν = 1 N j N exp ( ikg ) P µν (k) k 7 ready for the calculation of F in the AO basis for the next iterative step School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

48 SCF iterative procedue geometry initial information basis set, {χ µ (r)} guess for the density matrix, P µν calculation of F µν (P) P µν = 2 occ j cjµ c jν update the new P matrix diagonalize F FC = SCE no E n E n 1 < TOL? yes stop School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

49 SCF iterative procedue geometry initial information basis set, {χ 0 µ(r)} k-grid in the first BZ guess for the density matrix, P g µν calculation of F g µν(p) F (k) µν = g exp(ikg ) F g µν P g µν = k exp(ikg ) P(k) µν P(k) µν = 2 occ j c (k) jµ c(k) jν diagonalize F (k) matrices F (k)c(k) = S(k)C(k)E(k) no E n E n 1 < TOL? yes stop School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

50 SCF iterative procedue Band structure Representation α-quartz School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

51 SCF iterative procedue Fermi energy isolated systems n states = 2 N M j N θ[ɛ F ɛ j (k)] k θ(x) = 0 θ(x) = 1 x < 0 ɛ j (k) > ɛ F x > 0 ɛ j (k) < ɛ F School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

52 SCF iterative procedue Fermi energy isolated systems n states = 2 N M j N θ[ɛ F ɛ j (k)] k metallic systems n states = 2 Ω BZ M j θ[ɛ F ɛ j (k)]dk θ(x) = 0 θ(x) = 1 x < 0 ɛ j (k) > ɛ F x > 0 ɛ j (k) < ɛ F School Ab initio Modelling of Solids (UniTo) Symmetry and Bloch Functions Regensburg - July / 42

Wavefunction and electronic struture in solids: Bloch functions, Fermi level and other concepts.

Wavefunction and electronic struture in solids: Bloch functions, Fermi level and other concepts. Silvia Casassa Università degli Studi di Torino July 12, 2017 Minnesota Workshop on ab initio MSC Symmetry