Advanced Inorganic Chemistry. نیم سال اول Ferdowsi University of Mashhad

Size: px

Start display at page:

Download "Advanced Inorganic Chemistry. نیم سال اول Ferdowsi University of Mashhad"

Darren Ramsey
6 years ago
Views:

1 Advanced Inorganic Chemistry

2 ماتریس matrix ماتریس آرایه ای مستطیلی از اعداد یا عالئم می باشد عضو های قطری 2

3 ماتریس واحد 3

4 جمع و تفریق ماتریس ها 4

5 جمع و تفریق ماتریس ها 5

6 نکته 6

7 تمرین جمع و تفریق ماتریس ها 7

8 ضرب ماتریس ها دو ماتریسی که در هم ضرب می شوند باید همساز باشند )تعداد ستون ماتریس اول با ردیف ماتریس دوم برابر باشد( a a 2 a 2 a 22 b b 2 b 3 b c 4 c 2 c 3 c 4 a 3 a 32 b 2 b 22 b 23 b c 24 2 c 22 c 23 c 24 c 3 c 32 c 33 c 34 c = a b + a 2 b 2 c 2 = a b 2 + a 2 b 22 c 3 = a b 3 + a 2 b 23 c 4 = a b 4 + a 2 b 24 c 2 = a 2 b + a 22 b 2 c 22 = a 2 b 2 + a 22 b 22 c 23 = a 2 b 3 + a 22 b 23 c 24 = a 2 b 4 + a 22 b 24 c 3 = a 3 b + a 32 b 2 c 32 = a 3 b 2 + a 32 b 22 c 33 = a 3 b 3 + a 32 b 23 c 34 = a 3 b 4 + a 32 b 24 8

9 مثال 9

10 حاصل جمع قطر ماتریس را ماهیت ماتریس می گویند. 25

11 ساده کردن ضرب ماتریس ها ماتریس ها ضرب ماتریس تعویض پذیر نیست مگر AB=-BA

12 U=A/B=AB - تقسیم ماتریس ها عمل یکسانی 2

13 جابجایی Transport ماتریس 3

14 دترمینان تعیین یک مقدار مرتبط با مربعی ماتریس 4

15 تمرین 5

16 بردار 6

17 7

18 B y A y B x A x 8

19 x 2 = r 2 Cosβ Cosα - r 2 Sinβ Sinα طول بردار در چرخش ثابت است یعنی r r= 2 x 2 = r 2 Cosβ Cosα - r 2 Sinβ Sinα = x Cosα y Sinα x y y 2 = r 2 Sinβ Cosα + r 2 Cosβ Sinα = x Sinα + y Cosα r y بین x برای و داریم رابطه های زیر بر قرار است. x = r Cos β y = r Sin β x 2 = r 2 Cos ( β + α ) x 2 Cosα Sinα Sinα Cosα 9

20 اگر چرخش خالف جهت عقربه های ساعت باشد داریم Cosα Sinα -Sinα Cosα 2

21 شرایط متعامد بودن 2

22 نمایش ماتریسی گروه های نقطه ای About the Template ماتریسی که بر نقاط x و y و z اعمال شده و یک عمل تقارنی حاصل گردد. ماتریس واحد = نمایش عمل یکسانی 22

23 فقط ماتریس صفحه xy در صفحه هستند ماتریس صفحه yz ) y x جابجا شده ( و z مرکز تقارن 23

24 ماتریس محور دوران ثابت نمایید 24

25 ماتریس محور دوران مرکب 25

26 لوا لاس مین ' ' ' z y x z y x Original coordinates Transformation matrix Results of transformation. =

27 MATRIX MULTIPLICATION To get an element of the product vector a row in the operation square matrix is multiplied by the original vector matrix. ' ' ' z y x z y x = V M V j j j i i V V M, ' For example V 2 = y = M 2, * V + M 2,2 * V 2 + M 2,3 * V 3 y = * x + * y + * z = y

The matrices record how the x,y,z coordinates are modified as a result of an operation.

28 Character Tables - 2 The matrix representation of the symmetry operations of a point group is the set of matrices corresponding to all the symmetry operations in that group. The matrices record how the x,y,z coordinates are modified as a result of an operation. For example, the C 2v point group consists of the following operations E: do nothing. Unchanged. C 2 : rotate 8 degrees about the z axis: x becomes x; y becomes y and z unchanged. s v (xz): y becomes y s v (yz): x becomes -x E C 2 s v (xz): s v (yz):

29 Consider f(x) = x 2 s v (f(x)) = s v (x 2 ) = (-x) 2 = x 2 = f(x) or s v (f(x)) = * f(x) f(x) is an eigenfunction of this reflection operator with an eigenvalue of +. This is called a symmetric eigenfunction. Similarly OPERATIONS APPLIED TO FUNCTIONS - f(x) = x 3 s v (f(x)) = - * f(x) f(x) is an eigenfunction of this reflection operator with an eigenvalue of -. This is called a antisymmetric eigenfunction.

30 PLOTS OF FUNCTIONS 25 2 Reflection yields Series 5 Series 5 5 Here f(x) is x 2. It can be seen to be a symmetric function for reflection at x = because of mirror plane. The reflection carries out the mapping shown with the red arrows.

31 Multiplying two matrices (a reminder) a a 2 a 2 a 22 a 3 a 32 b b 2 b 3 b 2 b 22 b 23 = c c 2 c 3 c 2 c 22 c 23 c 3 c 32 c 33 c = a b + a 2 b 2 c 2 = a b 2 + a 2 b 22 c 3 = a b 3 + a 2 b 23 c 2 = a 2 b + a 22 b 2 c 22 = a 2 b 2 + a 22 b 22 c 23 = a 3 b 3 + a 32 b 23 c 3 = a 3 b + a 32 b 2 c 32 = a 3 b 2 + a 32 b 22 c 33 = a 3 b 3 + a 32 b 23 Most of the transformation matrices we use have the form ± ± ±

32 x - x x y = - y = - y = z z z E C 2 s v (xz) s v (yz) What is the inverse of C 2? C x- y = - x = y = z z What is the inverse of s v? s v - x y - = x = y = x y z z z

33 What of the products of operations? x - x x y = - y = - y = z z z E C 2 s v (xz) s v (yz) - x x - x E * C 2 =? C 2 y = - y= = - y = s v * C 2 =? s v z z z x- x - y = - y = = z z

34 34

35 ماتریس متعامد 35

36 تمرین 36

37 نمایش گروه ها 37

38 38

39 نمایش کاهش ناپذیر 39

40 4

41 R متعامد بودن نمایش ها Irreducible representations are orthogonal i (R) j (R) = i j 4

42 n Γ =/h Σn g χ R χ Γ کاهش نمایش های کاهش پذیر 42

43 Properties of irreducible representations. The total number of symmetry operations in the group is called the order, h 2. The number of irreducible representations equals the number of classes 3. The sum of the squares of the dimensions of each irreducible equals h 43

44 4. For any irreducible representation h = [ i (R)] 2 R 5. Irreducible representations are orthogonal R i (R) j (R) = 6. Totally symmetric representations have all operations with characters of 44

45 مثال برای جدول زیر نمایش Γa را کاهش دهید 45

46 46

47 تمرین 47

48 تمرین 48

49 پایان 5

50 52

51 Reflection yields. 5 5 PLOTS OF FUNCTIONS - 2 Series Series Here f(x) is x 3. It can be seen to be a antisymmetric function for reflection at x =. The reflection carries out the mapping shown with the red arrows.

52 Reflection yields. 5 5 PLOTS OF FUNCTIONS - 2 Series Series Here f(x) is x 3. It can be seen to be a antisymmetric function for reflection at x =. The reflection carries out the mapping shown with the red arrows.

53 Now consider f(x) = (x-2) 2 = x 2 4x + 4 s v (f(x)) = s v (x-2) 2 = (-x-2) 2 = x 2 + 4x + 4 f(x) = (x-2) 2 is not an eigenfunction of this reflection operator because it does not return a constant times f(x). Series Reflection yields this function, not an eigenfunct ion OPERATIONS APPLIED TO FUNCTIONS - 2 Series Neither symmetric nor antisymmetric for reflection thru x =.

54 Reflection ATOMIC ORBITALS s orbital z Get the same orbital back, multiplied by +, an eigenfunction of the reflection, symmetric with respect to the reflection. The s orbital forms the basis of an irreducible representation of the operation

55 Reflection ATOMIC ORBITALS p orbital z Get the same orbital back, multiplied by -, an eigenfunction of the reflection, antisymmetric with respect to the reflection. The p orbital behaves differently from the s orbital and forms the basis of a different irreducible representation of the operation

56 RENT WAYS THAT OBJEC TS CAN BEHAV E FOR A GROU P CONSI STING OF E AND THE REFLE CTION C s E s h A A - x, y,r z z, R x,r y x 2,y 2,z 2,xy yz, xz s orbital is spherical behaves as x 2 + y 2 + z 2. s orbital is A p z orbital has a multiplicative factor of z times a spherical factor. Behaves as A.

57 HYBRIDS Reflection hybrid z Do not get the same hybrid back. The two hybrids form the basis of a reducible representation of the operation Recall: the hybrid can be expressed as the sum of an s orbital and a p orbital. = + Reduction: expressing a reducible representation as a combination of irreducible representations.

58 REDUCIBL E REPRESEN TATIONS Use the two sp z hybrids as the basis of a representation h h 2 E operation. s h operation. 2 h h = 2 h h 2 h h = 2 h h The hybrids are unaffected by the E operation. The reflection operation interchanges the two hybrids. Proceed using the trace of the matrix representation. + = 2 + = h becomes h ; h 2 becomes h 2. h becomes h ; h 2 becomes h 2.

59 C s E s h TATIONS REPRESEN A A - x, y,r z z, R x,r y x 2,y 2,z 2,xy yz, xz The representation derived from the two hybrids can be attached to the table. G 2 (h, h 2 ) Note that G = A + A,, h h 2 h - h 2 h + h 2

60 Character Tables - 3 Irreducible representations are not linear combinations of other representation (Reducible representations are) # of irreducible representations = # of classes of symmetry operations Instead of the matrices, the characters are used (traces of matrices) A character Table is the complete set of irreducible representations of a point group

61 Character Table Point group Symmetry operations Mülliken symbols Characters + symmetric behavior - antisymmetric Each row is an irreducible representation

LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad

LINEAR CONTROL SYSTEMS Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Lecture Stability analyi Topic to be covered include: Stability of linear control ytem. Bounded input bounded output tability