Lecture 9. operators
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- Chantal Maus
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1 Lecture 9 9 operators set of operators can also build a basis for a representation O(g)Ôµ mo(g )= X m 0 [D(g)] µ m 0 môµ m 0 transformed operator matrix 20
2 invariant operators O(g)ÔO(g )=Ô r 2 is invariant under rotations and what about the gradient? O(g)rO(g )=R r R = cos sin sin cos 2 operators can be scalar operator: invariant under all rotations r 2 vector operator: it transforms as a polar vector tensor operator: it transforms as a tensor of rank r 3 r components r polar vector: all components of rank tensor L axial vector: three components of rank 2 tensor 22
3 some of the most relevant groups 23 O(3) it is the group of transformations r 0 = Ar where A is an orthogonal matrix det A=: proper rotations det A=-: improper rotations det A=: subgroup SO(3) or O(3) O(3) = SO(3) C i 24
4 classes of O(3) E I R k ( ) S k ( ) example: atom in free space 25 Euclidian Group the group of transformations r 0 = Ar a where A is an orthogonal matrix and a is a translation " : g = {R a} group elements of the group convention: the translation follows the rotation other possible notation: g=ar O(3) is a subgroup of the euclidian group the group of the translations is also a subgroup of the euclidian group and it is abelian 26
5 operations translations (R=E) proper and improper rotations with the origin of coordinates as fixed point (a=0) proper rotations with a generic fixed point R k ( )r a, a? k r r =Rra r r O r O=ROa 27 operations front screw axes back k R k ( )r a, a k k a C2 28
6 operations improper rotations with generic fixed point S k ( )r a, a? k improper rotation with shifted plane S k ( )r a, a k k reflection wrt translated plane kr a, a k k 29 operations glide planes kr a, a? k k a k 30
7 stereographic projections 3 molecules and crystals uniaxial groups Cn only one rotation axis pyramid with n sides C2 ( =23 4 )istdiesymm C6 abelian and cyclic n elements n classes C3 C4 32
8 dihedral groups Dn Cn one binary axis u perpendicular to the n-fold axis (and all their products) u2 D3 u4 D4 u3 u2 u u prism with n sides u3 D6 O O if there is a binary axis u, there are n 2n elements n even: n/2 3 classes n odd: (n 3)/2 classes O O O O O 33 O Lecture 0 34
9 some definitions binary axis= 2-fold axis = rotation of 80 o bilateral axis= axis of order n such that Cn n-k and Cn k are conjugated 35 tetrahedron group T 4C3 3C2 octahedron group O 3C44C3 6C2 4C3 6C2 3C2 4C3 3C4 2 elements 24 elements and 5 classes 4 classes 36
10 Icosahedron Group I A regular icosahedron has 20 equilateral triangle faces with five meeting at each of its twelve vertices. 6 C5 0 C3 5 C2 C60 molecule 2 pentagons, 20 hexagons C2 C3 C S2n all powers of S(2n) 2n elements, Cn is a subgroup direct product of Cn and Ci S4 Cnh direct product of Cn and Ch C4h Cnv full group of a regular pyramid isomorph to Dn C4v Dnh full group of a prism direct product of Dn and Ch D4h Dnd full group of a a solid made by two regular prisms joint direct product of Dn and Ci D2d 38
11 Td full group of a tetrahedron Th direct product of T and Ci Oh full group of a cube direct product of O and Ci 39 Illustrative Examples of Point Groups I Shapes Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory 40
12 Assign a point group Source: Shriver & Atkins, Inorganic Chemistry, 3 rd Edition. The symmetry elements can be considered operators. 4 Bravais lattice Infinite set of points generated by the translations T=n an2 b2n3 b3 42
13 crystals: 2d Bravais lattices a α a2 oblique a2 a2 a α rectangular α =90 o a α centered rectangular a α a2 square α =90 o a=a2 a2 α a hexagonal α =60 o a=a2 43 crystals: 3d Bravais lattices cubic P I F c α β γ b tetragonal P I a rhombic P C I F hexagonal monoclinic P P C 5 trigonal trigonal P 7 triklin P 4 Arten von Einheitszelle P=primitive I= body centered F: face centered C: base centered P = primitiv I = raumzentriert F = flächenzentriert C = basiszentriert triclinic 44
14 space groups all operations defined by {R 0} {E T} {R T} {R f} R: point group operation T: lattice translation f: fraction of lattice translation 3 dimensions 32 point groups 230 space groups symmorphic groups: no {R f} operations (73 in 3 dim) 45 cubic: T Th Td O Oh tetragonal: C4 S4 C4h S4h C4v D4h D2d orthorhombic: C2v D2 D2h monoclinic: C2 Ch C2h triclinic: C Ci trigonal: C3 S6 D3 C3v D3d hexagonal: C6 C3h C6h D6 D6v D3h D6h non crystallographic: C5 D5 C5v C5h D4d D5d D5h D6d I Ih.. 46
15 International tables International Tables for Crystallography (2006). Vol. A, Chapter.4, pp Graphical symbols for symmetry elements in one, two and three dimensions BY TH. HAHN.4.. Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions) Symmetry plane or symmetry line Reflection plane, mirror plane Reflection line, mirror line (two dimensions) Graphical symbol Glide vector in units of lattice translation vectors parallel and normal to the projection plane None Printed symbol m Axial glide plane 2 lattice vector along line in projection plane a, b or c Glide line (two dimensions) 2 lattice vector along line in figure plane g Axial glide plane 2 lattice vector normal to projection plane a, b or c Double glide plane* (in centred cells only) Diagonal glide plane Diamond glide plane (pair of planes; in centred cells only) Two glide vectors: 2 along line parallel to projection plane and 2 normal to projection plane One glide vector with two components: 2 along line parallel to projection plane, 2 normal to projection plane 4 along line parallel to projection plane, combined with 4 normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) e n d *Forfurtherexplanationsofthe double glideplanee see Note (iv) below and Note (x) in Section.3.2. See footnote x to Section International tables.4. GRAPHICAL SYMBOLS FOR SYMMETRY ELEMENTS.4.5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure Symmetry axis or symmetry point Graphical symbol* Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis Printed symbol (partial elements in parentheses) Identity None None Twofold rotation axis None 2 Twofold rotation point (two dimensions) Twofold screw axis: 2 sub 2 2 Threefold rotation axis Threefold rotation point (two dimensions) None 3 Threefold screw axis: 3 sub Threefold screw axis: 3 sub Fourfold rotation axis None 4 (2) Fourfold rotation point (two dimensions) Fourfold screw axis: 4 sub Fourfold screw axis: 4 sub 2 2 Fourfold screw axis: 4 sub Sixfold rotation axis Sixfold rotation point (two dimensions) 42 2 None 6 (3,2) Sixfold screw axis: 6 sub 6 6 3,2 Sixfold screw axis: 6 sub ,2 Sixfold screw axis: 6 sub , 2 Sixfold screw axis: 6 sub ,2 5 Sixfold screw axis: 6 sub ,2 Centre of symmetry, inversion centre: bar Reflection point, mirror point (one dimension) Inversion axis: 3 bar None 3 3, Inversion axis: 4 bar None 4 2 Inversion axis: 6 bar None 6 3=m None Twofold rotation axis with centre of symmetry None 2=m Twofold screw axis with centre of symmetry 2 2=m Fourfold rotation axis with centre of symmetry None 4=m 4, 2, 4 sub 2 screw axis with centre of symmetry 2 42=m 4, 2, Sixfold rotation axis with centre of symmetry None 6=m 6, 3, 3, 2, 6 sub 3 screw axis with centre of symmetry 2 63=m 6, 3, 3, 2, *Notesonthe heights h of symmetry points, 3, 4 and6: () Centres of symmetry and3, as well as inversion points 4 and6 on4 and6 axes parallel to [00], occur in pairs at heights h and h 2. In the space-group diagrams, only one fraction h is given, e.g. 4 stands for h ˆ 4 and 3 4.Nofractionmeansh ˆ 0and 2.Incubic space groups, however, because of their complexity, both fractions are given for vertical 4 axes,includingh ˆ 0and 2. (2) Symmetries 4=m and 6=m contain vertical 4 and6 axes; their4 and6 inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams. (3) Symmetries 42=m and 63=m also contain vertical 4 and6 axes,buttheir4 and6 inversion points alternate with the centres of symmetry; i.e. pointsath and h 2 interleave with 4 or6 pointsath 4 and h 3 4. In the tetragonal and hexagonal space-group diagrams, only one fraction for andonefor4 or6 is given. In the cubic diagrams, all four fractions are listed for 42=m; e.g. Pm3n (No. 223): : 0, 2 ; 4: 4,
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