108 W.Hergert,M.Dane,and D.Kodderitzsch (6.4) i=1 The character system of a group is called the character table.The character table can be generated automatically using CharacterTable [c4vs].We will get: C1C22C32C42C5 1111 11 C1=(E) 3111-1-1 C2=(C2z) r411-1 1-1 C3=(C4,Cz) 211-1-11 C4=(IC2x,IC2x) 52-2100 Cs=(IC2a,IC2b) The character table represents a series of character theorems,which are im- portant for later use:i)the number of inequivalent irreducible representations of a group g is equal to the number of classes of g,ii)the characters of the group elements in the same class are equal,iii)the sum of the squares of the dimensions of all irreducible representations is equal to the order of the group,iv)two representations are equivalent if their character systems are equivalent.Especially important are the following theorems: -A representation I is irreducible,if: ∑1xT)P=9 (6.5) Teg (The sum of the squares of the characters of the rotation matrices of Cv exceeds the group order g =8,indicating that this representation cannot be irreducible. -The number n,how often an irreducible representation I,or a repre- sentation equivalent to I,is contained in the reduction of the reducible representation I,is given by: n=1∑xT)xT) (6.6) 9 Teg (g-order of the group,x(T),x'(T)-character of the group element in the representation I and I) Orthogonality theorems for characters ∑x(Ck)'x(Ck)Nk=gi (6.7) ∑X'(Ck)'X(C)N=g (6.8)
108 W. Hergert, M. D¨ane, and D. K¨odderitzsch χ(A) = l j=1 Γ(A)jj (6.4) The character system of a group is called the character table. The character table can be generated automatically using CharacterTable[c4vs]. We will get: C1 C2 2 C3 2C4 2C5 Γ1 11 1 1 1 Γ3 1 1 1 -1 -1 Γ4 1 1 -1 1 -1 Γ2 1 1 -1 -1 1 Γ5 2 -2 1 0 0 C1 = (E) C2 = (C2z) C3 = (C4z, C−1 4z ) C4 = (IC2x,IC2x) C5 = (IC2a,IC2b) The character table represents a series of character theorems, which are important for later use: i) the number of inequivalent irreducible representations of a group G is equal to the number of classes of G, ii) the characters of the group elements in the same class are equal, iii) the sum of the squares of the dimensions of all irreducible representations is equal to the order of the group, iv) two representations are equivalent if their character systems are equivalent. Especially important are the following theorems: – A representation Γ is irreducible, if: T∈G | χ(T) | 2= g . (6.5) (The sum of the squares of the characters of the rotation matrices of C4v exceeds the group order g = 8, indicating that this representation cannot be irreducible.) – The number n, how often an irreducible representation Γi , or a representation equivalent to Γi , is contained in the reduction of the reducible representation Γ, is given by: n = 1 g T∈G χ(T)χi (T) ∗ (6.6) (g- order of the group, χ(T), χi (T) - character of the group element in the representation Γ and Γi ) – Orthogonality theorems for characters k χi (Ck) ∗χj (Ck)Nk = g δij (6.7) i χi (Ck) ∗χi (Cl)Nk = g δkl (6.8)
6 Symmetry Properties of Electronic and Photonic Band Structures 109 (x(C),x(Ck)characters of elements in class Ck in irreducible represen- tations I and I,N&-number of elements in Ck.In the first relation the summation runs over all classes of the group.In the second relation it runs over all inequivalent irreducible representations of the group.) The character theorems can be easily checked for all the point groups using Mathematica. 6.4.2 Basis Functions of Irreducible Representations Our final goal is the investigation of the symmetry properties of the solu- tions of Maxwell's equation.Therefore we have to associate the symmetry expressed by the symmetry group,i.e.the point group Cav in our example, with the symmetry properties of scalar and vector fields. We define a transformation operator to a symmetry element T as P(T). For a scalar function it holds: P(T)f(r)=f(T-r) (6.9) The operators P(T)form a group of linear unitary operators.If we discuss Maxwell's equations,we have to deal with vector fields in general.Instead of(6.9)the following transformation has to be applied to the vector field F: P(T)F(r)=TF(T-r) (6.10) Now we define the basis functions of an irreducible representation (IR) and concentrate on scalar fields.If a set of l-dimensional matrices I(T)forms a representation of the group g and (r),...,o(r)is a set of linear inde- pendent functions such that P(T)pn(r)=∑T(T)mnm(r）n=l,2,,l (6.11) m=1 then functions on(r)are called partners in a set of basis functions of the rep- resentation T.on is said to transforms like the nth row of the representation. We can write any function o(r),which can be normalized,as a sum of basis functions of the irreducible representations I'P of the group. p (r)= (6.12) p n=l Note,that this is not an expansion like the Fourier series.We don't expand a function with respect to an orthonormal complete set of functions here.The set of functions o(r)depends on o(r)itself.The functions op(r)in (6.12) can be found by means of the so called projection operators
6 Symmetry Properties of Electronic and Photonic Band Structures 109 (χi (Ck), χj (Ck) characters of elements in class Ck in irreducible representations Γi and Γj , Nk - number of elements in Ck. In the first relation the summation runs over all classes of the group. In the second relation it runs over all inequivalent irreducible representations of the group.) The character theorems can be easily checked for all the point groups using Mathematica. 6.4.2 Basis Functions of Irreducible Representations Our final goal is the investigation of the symmetry properties of the solutions of Maxwell’s equation. Therefore we have to associate the symmetry expressed by the symmetry group, i.e. the point group C4v in our example, with the symmetry properties of scalar and vector fields. We define a transformation operator to a symmetry element T as Pˆ(T). For a scalar function it holds: Pˆ(T)f(r) = f(T −1r) (6.9) The operators Pˆ(T) form a group of linear unitary operators. If we discuss Maxwell’s equations, we have to deal with vector fields in general. Instead of (6.9) the following transformation has to be applied to the vector field F: Pˆ(T)F(r) = TF(T −1r) (6.10) Now we define the basis functions of an irreducible representation (IR) and concentrate on scalar fields. If a set of l-dimensional matrices Γ(T) forms a representation of the group G and φ1(r),...,φl(r) is a set of linear independent functions such that Pˆ(T)φn(r) = l m=1 Γ(T)mnφm(r) n = 1, 2,...,l (6.11) then functions φn(r) are called partners in a set of basis functions of the representation Γ. φn is said to transforms like the nth row of the representation. We can write any function φ(r), which can be normalized, as a sum of basis functions of the irreducible representations Γp of the group. φ(r) = p lp n=1 φp n(r) (6.12) Note, that this is not an expansion like the Fourier series. We don’t expand a function with respect to an orthonormal complete set of functions here. The set of functions φp n(r) depends on φ(r) itself. The functions φp n(r) in (6.12) can be found by means of the so called projection operators
110 W.Hergert,M.Dane,and D.Kodderitzsch >(T)mP(T),Pn9(r)=6p(r)(6.13) T∈G ∑x(T)*P(T) (6.14) TeG The projection operators are implemented in the package. (cf.CharacterProjectionOperator[classes_,chars_,func_].)As an ex- ample,we investigate the symmetry properties of a function o(r)=(ax+ by)g(r)(a and b are constants).We apply the projection operators connected to the group Cav,to the function and get the following result (r)=Pio(r)=az g(r)Pi2o(r)=bz g(r) (6.15) Pio(r)=ayg(r) (r)=P2(r）)=byg(r） It is impossible to project out parts of (r)transforming like other IRs of Cav.Therefore our trial function o(r)=(ax+by)g(r)is a sum of functions, transforming like the representation I'5 (E). If we consider three-dimensional photonic crystals we cannot resort to scalar fields anymore.We have to take into account the full vectorial nature of the fields also in symmetry considerations.A more detailed discussion of the subject can be found in 6.14-6.16]. 6.5 Symmetry Properties of Schrodinger's Equation and Maxwell's Equations Here we want to investigate the symmetry properties of scalar or vector fields, which we get as solutions of Schrodinger's equation or Maxwell's equations. The operators P(T)form a group of linear unitary operators.This group is isomorphic to the group of symmetry elements T.The Hamilton-Operator H(r)of the time-independent Schrodinger equation H(r)=E(r)is given by h202 H(r)=-2m8m2+V(r). (6.16) For an arbitrary transformation T the transformation behavior of the Hamil- tonian is given by (r)=P(T)(Tr)P(T)-1. (6.17) For transformations which leave产invariant,.i.e.a(Tr）=H(r）we get: [a,P(T1=0. (6.18)
110 W. Hergert, M. D¨ane, and D. K¨odderitzsch Pp mn = lp g T∈G Γp(T) ∗ mnPˆ(T), Pp mnφq i (r) = δpqδni φp m(r) (6.13) Pp = lp g T∈G χp(T) ∗Pˆ(T) (6.14) The projection operators are implemented in the package. (cf. CharacterProjectionOperator[classes_,chars_,func_].) As an example, we investigate the symmetry properties of a function φ(r)=(a x + b y)g(r) (a and b are constants). We apply the projection operators connected to the group C4v, to the function and get the following result φ5 1(r) = P5 11 φ(r) = axg(r) P5 12 φ(r) = bxg(r) P5 21 φ(r) = ay g(r) φ5 2(r) = P5 22 φ(r) = by g(r) (6.15) It is impossible to project out parts of φ(r) transforming like other IRs of C4v. Therefore our trial function φ(r)=(a x + b y)g(r) is a sum of functions, transforming like the representation Γ5 (E). If we consider three-dimensional photonic crystals we cannot resort to scalar fields anymore. We have to take into account the full vectorial nature of the fields also in symmetry considerations. A more detailed discussion of the subject can be found in [6.14–6.16]. 6.5 Symmetry Properties of Schr¨odinger’s Equation and Maxwell’s Equations Here we want to investigate the symmetry properties of scalar or vector fields, which we get as solutions of Schr¨odinger’s equation or Maxwell’s equations. The operators Pˆ(T) form a group of linear unitary operators. This group is isomorphic to the group of symmetry elements T. The Hamilton-Operator Hˆ (r) of the time-independent Schr¨odinger equation Hψˆ (r) = E ψ(r) is given by Hˆ (r) = − 2 2m ∂2 ∂r2 + V (r) . (6.16) For an arbitrary transformation T the transformation behavior of the Hamiltonian is given by Hˆ (r) = Pˆ(T)Hˆ (Tr)Pˆ(T) −1 . (6.17) For transformations which leave Hˆ invariant, i.e. Hˆ (Tr) = Hˆ (r) we get: [H, ˆ Pˆ(T)] = 0 . (6.18)