Appendixes POINT GROUPS AND THEIR CHARACTER TABLES

Transcription

1 Appendixes APPENDIX I. POINT GROUPS AND THEIR CHARACTER TABLES The following are the character tables of the point groups that appear frequently in this book. The species (or the irreducible representations) of the point group are labeled according to the following rules: A and B denote nondegenerate species (onedimensional representation). A represents the symmetric species (character ¼þ) with respect to rotation about the principal axis (chosen as z axis), whereas B represents the antisymmetric species (character ¼) with respect to rotation about the principal axis; E and F denote doubly degenerate (two-dimensional representation) and triply degenerate species (three-dimensional representation) respectively. If two species in the same point group differ in the character of C (other than the principal axis), they are distinguished by subscripts,, 3,... If two species differ in the character to s (other than s v ), they are distinguished by 0 and 00. If two species differ in the character of i, they are distinguished by subscripts g and u. If these rules allow several different labels, g and u take precedence over,, 3,..., which in turn take precedence over 0 and 00. The labels of species of point groups C n and D h (linear molecules) are exceptional and are taken from the notation for the component of the electronic orbital angular momentum along the molecular axis. C I A Infrared and Raman Spectra of Inorganic and Coordination Compounds, Sixth Edition, Part A: Theory and Applications in Inorganic Chemistry, by Kazuo Nakamoto Copyright Ó 009 John Wiley & Sons, Inc. 355

2 356 APPENDIXES A þ þ T Z,R Z a xx, a yy, a zz, a xy B þ T x,t y,r x,r y a yz, a xz C I C (z) A 0 þ þ T x, T y, R Z a xx, a yy, a zz, a xy B 00 þ T z, R x,r y a yz, a xz C s I s(xy) A g þ þ R x, R y, R z All components of a A u þ T x, T y, T x C i S I i A þ þ þ þ T z a xx, a yy, a zz A þ þ R z a xy B þ þ T x, R y a xz B þ þ T y, R x a yz C v I C (z) s v (xz) s v (yz) A g þ þ þ þ R z a xx, a yy, a zz, a xy A u þ þ T z B g þ þ R x, R y a yz, a xz B u þ þ T x, T y C h I C (z) s h (xy) i A þ þ þ þ a xx, a yy, a zz B þ þ T z, R z a xy B þ þ T y, R y a xz B 3 þ þ T x, R x a yz D V I C (z) C (y) C (x) A g þ þ þ þ þ þ þ þ a xx, a yy, a zz A u þ þ þ þ B g þ þ þ þ R z a xy B u þ þ þ þ T z B g þ þ þ þ R y a xz B u þ þ þ þ T y D h V h I s(xy) s(xz) s(yz) i C(z) C(y) C(x) B 3g þ þ þ þ R x a yz B 3u þ þ þ þ T x

3 APPENDIXES 357 A T z, R z a xx þ a yy, a zz C 3 E ¼ e pi/3 I C 3 C 3 (T x, T y ), (R x, R y ) (a xx a yy, a xy ), (a yz, a xz ) A T z, R z a xx þ a yy, a zz C 4 I C 4 C C 3 4 B a xx a yy, a xy i i E (T i i x, T y ), (R x, R y ) (a yz, a xz ) A T z, R z a xx þ a yy, a zz C 5 E E I C 5 C 5 C 3 5 C 4 5 (T x, T y ), (R x, R y ) (a yz, a xz ) (a xx a yy, a xy ) ¼ e pi/5 A T z, R z a xx þ a yy, a zz C 6 I C 6 C 3 C C 3 C 5 6 B E E (T x, T y ), (R x, R y ) (a xz, a yz ) (a xx a yy, a xy ) ¼ e pi/6 A R z a xx þ a yy, a zz S 4 I S 4 C S 3 4 B T z a xx a yy, a xy i i E (T i i x, T y ), (R x, R y ) (a xz, a yz )

4 358 APPENDIXES A g R z a xx þ a yy, a zz S 6 C 3i E g I C 3 C 3 i S 5 6 S 6 A u T z E u (T x, T y ) ¼ e pi/3 (R x, R y ) (a xx a yy, a xy ), (a xz, a yz ) A þ þ þ a xx þ a yy, a zz A þ þ T z, R z E þ 0 (T x, T y ), (R x, R y ) (a xx a yy, a xy ), (a yz, a xz ) D 3 I C 3 (z) 3C A a xx þ a yy, a zz A T z, R z B a xx a yy B a xy E (T x, T y ), (R x, R y ) (a xz, a yz ) D 4 I C 4 C ð¼ C 4 Þ C0 C 00 A a xx þ a yy, a zz D 5 I C 5 C 5 5C A T z, R z E c p c 4p 0 (T x, T y ), (R x, R y ) (a xz, a yz ) 5 5 E c 4p 5 C ¼ cosine c p 5 0 (a xx a yy, a xy ) A a xx þ a yy, a zz A T z, R z B B E 0 0 (T x, T y ), (R x, R y ) (a xz, a yz ) E 0 0 (a xx a yy, a xy ) D 6 I C 6 C 3 C 3C 0 3C 00

5 APPENDIXES 359 A þ þ þ T z a xx þ a yy, a zz A þ þ R z E þ 0 (T x, T y ), (R x, R y ) (a xx a yy, a xy ), (a yz, a xz ) C 3v I C (z) 3s v C 4v I C 4 (z) C4 C00 s v s d A þ þ þ þ þ T z a xx þ a yy, a zz A þ þ þ R z B þ þ þ a xx a yy B þ þ þ a xy E þ (T x, T y ), (R x, R y ) (a yz, a xz ) C 5v I C 5 C 5 5s v A T z a xx þ a yy, a zz A R z E c p c 4p (a xx a yy, a xy ) E c 4p 5 c ¼ cosine c p 5 0 (T x, T y ), (R x, R y ) (a xz, a yz ) A T z a xx þ a yy, a zz A R z B B E 0 0 (T x, T y ), (R x, R y ) (a xz, a yz ) E 0 0 (a xx a yy, a xy ) C 6v I C 6 C 3 C 3s v 3s d C v I C f C f C 3f... sv F þ cos 3 cos 3 cos S þ þ þ þ þ... þ T z a xx þ a yy, a zz S þ þ þ þ... R z P þ cos cos cos (T x, T y ), (R x, R y ) (a yz, a xz ) D þ cos cos cos (a xx a yy, a xy ) ¼ arbitrary angle

6 360 APPENDIXES A 0 R z a xx þ a yy, a zz C 3h I C 3 C 3 s h S 3 S 5 3 B 00 T z E 0 (T x, T y ) (a xx a yy, a xy ) E 00 (R x, R y ) (a xz, a yz ) ¼ e pi/3 C 4h I C 4 C C 3 4 i S 3 4 s h S 4 A g R z a xx þ a yy, a zz A u T z B g (a xx a yy, a xy ) i i i i E g i i i i i i i i E u (T i i i i x, T y ) B u (R x, R y ) (a xz, a yz ) C 5h I C 5 C 5 C 3 5 C 4 5 s h S 5 S 7 5 S 3 5 S 9 5 A 0 R z a xx þ a yy, a zz A 00 T z E 0 (T x, T y ) E 00 E 0 (a xx a yy, a xy ) (R x, R y ) (a xz, a yz ) E 00 ¼ e pi/5

7 C 6h I C 6 C 3 C C 3 C 5 6 i S 5 3 S 5 6 s h S 6 S 3 (R x, R y ) (a xz, a yz ) A g R z a xx þ a yy, a zz A u T z B g B u E g E u E g E u APPENDIXES 36 (T x, T y ) (a xx a yy, a xy ) ¼ e pi/6 A þ þ þ þ þ a xx þ a yy, a zz A þ þ þ R z B þ þ þ a xx a yy D d V d I S4(z) S 4 C00 C s d B þ þ þ T z a xy E þ (T x, T y ), (R x, R y ) (a yz, a xz ) D 3d I S6(z) S6 C 3 3 S6 S i 3C 3s d A g þ þ þ þ þ þ a xx þ a yy, a zz A u þ þ þ A g þ þ þ þ R z A u þ þ þ T z E g þ þ 0 0 (R x, R y ) (a xx a yy, a xy ), (a yz, a xz ) E u þ þ 0 0 (T x, T y ) D 4d I S8(z) S8 C S8 S8 C00 4C 4sd A þ þ þ þ þ þ þ a xx þ a yy, a zz A þ þ þ þ þ R z B þ þ þ þ p E þ þ ffiffiffi p 0 ffiffiffi 0 0 (T x, T y ) E þ 0p 0p þ 0 0 (a xx a yy, a xy ) B þ þ þ þ T z E 3 þ ffiffiffi 0 þ ffiffiffi 0 0 (R x, R y ) (a yz, a xz )

8 36 APPENDIXES D 5d I C 5 C 5 5C i S 3 0 S 0 5s d 0 (R x, R y ) (a xz, a yz ) A g a xx þ a yy, a zz A u A g R z A u T z E g c p 5 E u c p 5 E g c 4p 5 E u c 4p 5 c 4p 5 c 4p 5 c p 5 c p 5 0 c p 5 0 c p 5 0 c 4p 5 0 c 4p 5 c 4p 5 c 4p 5 c p 5 c p 5 0 (T x, T y ) 0 (a xx a yy, a xy ) c ¼ cosine D 3h I C 3 (z) 3C s h S 3 3s v A 0 þ þ þ þ þ þ a xx þ a yy, a zz A 00 þ þ þ A 0 þ þ þ þ R z A 00 þ þ þ T z E 0 þ 0 þ 0 (T x, T y ) (a xx a yy, a xy ) E 00 þ 0 þ 0 (R x, R y ) (a yz, a xz ) D 4h I C4(z) C C C sh sv sd S4 S i A g þ þ þ þ þ þ þ þ þ þ a xx þ a yy, a zz A u þ þ þ þ þ A g þ þ þ þ þ þ R z A u þ þ þ þ þ T z B g þ þ þ þ þ þ a xx a yy B u þ þ þ þ þ B g þ þ þ þ þ þ a xy B u þ þ þ þ þ E g þ þ (R x, R y ) (a yz, a xz ) E u þ þ (T x, T y ) D 5h I C 5 C 5 5C s h S 5 S 3 5 5s v A 0 a xx þ a yy, a zz A 00 A 0 R z A 00 T z E 0 c p c 4p 0 c p c 4p 0 (T x, T y ) E 00 c p c 4p 0 c p c 4p 0 (R x, R y ) (a xz, a yz ) E 0 c 4p c p 0 c 4p c p 0 (a xx a yy, a xy ) E 00 c 4p c p 0 c 4p c p c ¼ cosine 0

9 D6h I C6(z) C 6 C 3 C 3 6 C 3C 0 3C 00 sh 3sv 3sd S6 S3 S 3 6 S i A g þ þ þ þ þ þ þ þ þ þ þ þ a xx þ a yy, a zz Au þ þ þ þ þ þ Ag þ þ þ þ þ þ þ þ Rz A u þ þ þ þ þ þ T z B g þ þ þ þ þ þ Bu þ þ þ þ þ þ Bg þ þ þ þ þ þ B u þ þ þ þ þ þ E g þ þ þ þ (R x, R y ) (a yz, a xz ) Eu þ þ 0 0 þ 0 0 þ (Tx, Ty) Eg þ þ 0 0 þ 0 0 þ (axx ayy, a xy ) Eu þ þ þ þ 363

10 D h I C f C f C 3f... sh C sv S f S f... S i S þ g þ þ þ þ... þ þ þ þ þ... þ axx þ ayy, azz S þ u þ þ þ þ... þ... T z S g þ þ þ þ... þ þ þ... þ R z S u þ þ þ þ... þ... Pg þ cos cos cos cos cos... þ (Rx, Ry) (ayz, axz) P u þ cos cos cos 3... þ 0 0 þ cos þ cos... (T x, T y ) Dg þ cos cos 4 cos 6... þ 0 0 þ cos þ cos 4... þ (axx ayy, axy) Du þ cos cos 4 cos cos cos 4... Fg þ cos 3 cos 6 cos cos 3 cos 4... þ F u þ cos 3 cos 6 cos 9... þ 0 0 þ cos 3 þ cos ¼ arbitrary angle 364

11 APPENDIXES 365 A a xx þ a yy þ a zz T E I 4C 3 4C 3 3C (a zz a xx a yy, a xx a yy ) F (R x, R y, R z ), (T x, T y, T z ) (a xy, a xz, a yz ) ¼ e pi/3 A g a xx þ a yy þ a zz T h I 4C 3 4C 3 3C i 4S 6 4S 5 6 3s v A u E g E u F g (R x, R y, R z ) (a xy, a yz, a xz ) F u (T x, T y, T z ) ¼ e pi/3 (a zz a xx a yy, a xx a yy ) A þ þ þ þ þ a xx þ a yy þ a zz A þ þ þ E þ 0 0 þ (a xx þ a yy a zz, a xx a yy ) F þ3 0 þ (R x, R y, R z ) F þ3 0 þ (T x, T y, T z ) (a xy, a yz, a xz ) T d I 8C 3 6s d 6S 4 3S 4 3C A a xx þ a yy þ a zz A E 0 0 (a zz a xx a yy, a xx a yy ) F 3 0 (R x, R y, R z ), (T x, T y, T z ) F 3 0 (a xy, a yz, a xz ) O I 6C 4 3C ð¼ C 4 Þ 8C 3 6C

12 O h I 8C 3 6C 6C 4 3C 4 3C00 S i 6S 4 8S 6 3s h 6s d Ag þ þ þ þ þ þ þ þ þ þ axx þ ayy þ azz A u þ þ þ þ þ A g þ þ þ þ þ þ Au þ þ þ þ þ Eg þ 0 0 þ þ 0 þ 0 (axx þ ayy azz, axx ayy) E u þ 0 0 þ 0 þ 0 Fg þ3 0 þ þ3 þ 0 (Rx, Ry,Rz) Fu þ3 0 þ 3 0 þ þ (Tx, Ty, Tz) Fg þ3 0 þ þ3 0 þ (axy, ayz, axz) F u þ3 0 þ 3 þ 0 þ 366

13 I C 5 C 5 0C 3 5C i S 0 S 3 0 0S 6 5s v Ag axx þ ayy þ azz A u Fg 3 c p 5 Fu 3 c p 5 Fg 3 c 3p 5 Fu 3 c 3p 5 c 3p 5 c 3p 5 c p c 3p c 3p c p 5 c p 5 c p 5 c 3p 5 0 (Rz, Ry, Rz) 0 (Tx, Ty, Tz) 0 c p 0 3 c p c 3p Gg G u Hg (azz axx ayy, axx ayy, axy, ayz, axz) Hu c ¼ cosine; G and H denote four- and five-fold degenerate species, respectively I h 367

14 368 APPENDIXES APPENDIX II. MATRIX ALGEBRA II.. Definition of a Matrix A matrix is an array of numbers or symbols for numbers. In general, it is written as: 3 a a a 3 a n a a a 3 a n A ¼ a 3 a 3 a 33 a 3n ¼½a i Š a m a m a m3 a mn Here, the square brackets indicate that these elements constitute a matrix. A and [a i ] indicate the same matrix in abbreviated forms. In the latter, a i denotes the element in the ith row and th column. If m ¼ n, it is called a square matrix. Ina square matrix, the set of elements a i with i ¼ are called the diagonal elements. If all the diagonal elements are one and all the off-diagonal elements are zero, such a matrix is called a unit (or identity) matrix, and expressed as E (or I). A diagonal matrix, D, is similar to the unit matrix except that the diagonal elements are not necessarily equal. Thus d E ¼ D ¼ 4 0 d d 33 A matrix is called symmetric if the relationship a i ¼ a i holds for all off-diagonal elements. If a i ¼a i, it is called antisymmetric. If m n, the matrix is called a rectangular matrix. Among them, the one-column matrix and one-row matrix shown below are important. X ¼ 4 x x x ~X ¼ ½ x x x 3 Š Sometimes, the former is called a vector. The tilde sign over X in the latter indicates a transpose of a matrix in which the elements are interchanged across the diagonal. In the case of a matrix, we have A ¼ a a a a ~A ¼ a a a a

15 II.. Addition and Subtraction When two matrices are of same dimensions, they can be added or subtracted by the rule that u i ¼ a i b i.ifb is B ¼ b b b b APPENDIXES 369 then U ¼ A B ¼ a b a b a b a b II.3. Multiplication Two matrices can be multiplied only if the number of columns of the first matrix B is equal to the number of rows of the second matrix A. Each element of the resulting matrix, U ¼ BA, is given by u i ¼ Xn k¼ b ik a k For example BA ¼ b a þ b a b a þ b a b a þ b a b a þ b a If X is a column matrix and ~X is its transpose, then " #" # " a a x AX ¼ ¼ a # x þ a x a a x a x þ a x " # a x þ a x ~XAX ¼ ½x x Š ¼ a x a x þ a x þ a x x þ a x x þa x It should be noted that matrix multiplication is generally not commutative. Namely, BA is not necessarily equal to AB. For example " #" # " 0 ¼ 5 # " #" # " 0 ¼ #

16 370 APPENDIXES However, the associative law holds for matrix multiplication. Thus AðB þ CÞ ¼AB þ AC and ðabþc ¼ AðBCÞ If a constant l is multiplied to A, each element of A is multiplied by l: la ¼ la la la la II.4. Division Division by a matrix is accomplished as multiplication of its reciprocal (or inverse) matrix. For example, the division of B by A is written as U ¼ B=A ¼ BA Here, A is the reciprocal matrix of A, which is defined as AA ¼ A A ¼ E Only the square matrix can have its reciprocal matrix. The reciprocal of the product matrix is given by ðbaþ ¼ A B If A ¼ ~A (i.e., ~AA ¼ E), such a square matrix is called an orthogonal matrix. For example p = ffiffi p 3 = ffiffiffi p 3 = ffiffi 3 3 p 3 = ffiffi pffiffi pffiffi cos f sin f = p6 = 6 0 = ffiffiffi pffiffi 5; 4 sin f cos f 0 5 = 0 0 II.5. Determinant The determinant A of a square matrix A is defined as a a a 3 a n a a a 3 a n A ¼ a 3 a 3 a 33 a 3n.... ¼ X ðþ h a a a b a nk a n a n a n3 a nn

17 APPENDIXES 37 Here, h is the number of exchanges necessary to bring the sequence a, b,..., k back to the natural order,,,..., n, and the summation is taken over all permutations of a, b,..., k. For example Here, the dotted lines indicate how to obtain positive terms. Three different lines were used to show how the products on the right side were obtained. Likewise, the negative terms can be obtained by changing the direction of the dotted lines by 90. As shown above, vertical lines are used to express the determinant. It can be shown that AB ¼AB II.6. Eigenvalues (Characteristic Values) If A is a square matrix of dimension n and E is the unit matrix of the same dimension, then A le ¼0 is called the characteristic equation of the matrix A. For example, the characteristic equation for A given below is written as A ¼ a b al b b a b al ¼ 0 Expansion of the characteristic equation gives The two eigenvalues of this equation are l al þ a b ¼ 0 l ¼ a þ b; l ¼ a b More generally, the characteristic equation of a matrix A having the dimension n n is written as l n þ c l n þ c l n þ c n l þ c n ¼ 0 There are two simple relationships between the coefficients c, c,..., c n and eigenvalues: a þ a þ þa nn ¼c ¼ l þ l þ þl n A ¼c n ¼ l l l n ðþfor even n; for odd nþ

18 37 APPENDIXES These relationships are readily confirmed by the example above, given for a matrix. II.7. Eigenvectors If l a is an eigenvalue of A, a vector, l a, which satisfies the relation Al a ¼ l a l a is called an eigenvector of A. As an example, consider the matrix mentioned above. The l for l is a b l ¼ l ða þ bþ b a l l Then, we obtain l ¼ l. Their absolute values can be determined only by using the normalization condition: l þ l ¼ Then p l ¼ l ¼ = ffiffiffi Using the same procedure, we obtain p l ¼ = ffiffi and p l ¼= ffiffi for l. If we assemble these two results by columns, we have pffiffiffi p a b = = ffiffi pffiffi p p b a = ffiffiffi pffiffi = = ffiffi ¼ p = = ffiffi pffiffiffi aþb 0 = 0 ab More generally, this is written as AL ¼ LL By multiplying L on both sides, we obtain L AL ¼ L LL ¼ L It is seen that the L matrix can transform the A matrix into a diagonal matrix with its eigenvalues as the diagonal elements. As shown above, the L matrix can be calculated once the l values of the A matrix are obtained.

19 APPENDIXES 373 APPENDIX III. GENERAL FORMULAS FOR CALCULATING THE NUMBER OF NORMAL VIBRATIONS IN EACH SPECIES Most of these tables were reproduced from G. Herzberg, Molecular Spectra and Molecular Structure, Vol. II (Ref. of Chapter ). For the derivations of these tables, see K. Nakamoto and M. A. McKinney, J. Chem. Educ. 77, 775 (000). TABLE A. Point Groups Including Only Nondegenerate Vibrations Point a Group Total Number of Atoms Species Number of Vibrations C m þ m 0 A 3m þ m 0 B 3m þ m 0 4 C s m þ m 0 A 0 3m þ m 0 3 A 00 3m þ m 0 3 C i S m þ m 0 A g 3m 3 A u 3m þ 3m 0 3 C v 4m þ m xz þ m yz þ m 0 A 3m þ m xz þ m yz þ m 0 A 3m þ m xz þ m yz B 3m þ m xz þ m yz þ m 0 B 3m þ m xz þ m yz þ m 0 C h 4m þ m h þ m þ m 0 A g 3m þ m h þ m A u 3m þ m h þ m þ m 0 B g 3m þ m h þ m B u 3m þ m h þ m þ m 0 D V 4m þ m x þ m y A 3m þ m x þ m y þ m z þ m z þ m 0 B 3m þ m x þ m y þ m z þ m 0 B 3m þ m x þ m y þ m z þ m 0 B 3 3m þ m x þ m y þ m z þ m 0 D h V h 8m þ 4m xy þ 4m xz þ 4m yz A g 3m þ m xy þ m xz þ m yz þ m x þ m y þ m z þ m x þ m y þ m z þ m 0 A u 3m þ m xy þ m xz þ m yz B g 3m þ m xy þ m xz þ m yz þ m x þ m y B u 3m þ m xy þ m xz þ m yz þ m x þ m y þ m z þ m 0 B g 3m þ m xy þ m xz þ m yz þ m x þ m z B u 3m þ m xy þ m xz þ m yz þ m x þ m y þ m z þ m 0 B 3g 3m þ m xy þ m xz þ m yz þ m y þ m z B 3u 3m þ m xy þ m xz þ m yz þ m x þ m y þ m z þ m 0 a Note that m is always the number of sets of equivalent nuclei not on any element of symmetry; m 0 is the number of nuclei lying on all symmetry elements present; m xy, m xz,m yz are the numbers of sets of nuclei lying on the xy, xz, yz plane, respectively, but not on any axes going through these planes; m is the number of sets of nuclei on a twofold axis but not at the point of intersection with another element of symmetry; m x, m y, m z are the numbers of sets of nuclei lying on the x, y,orz axis if they are twofold axes, but not on all of them; m h is the number of sets of nuclei on a plane s h but not on the axis perpendicular to this plane.

20 374 APPENDIXES TABLE B. Point Groups Including Degenerate Vibrations Point Group Total Number of Atoms Species Number of Vibrations a D 3 6m þ 3m þ m 3 þ m 0 A 3m þ m þ m 3 A 3m þ m þ m 3 þ m 0 E 6m þ 3m þ m 3 þ m 0 C 3v 6m þ 3m v þ m 0 A 3m þ m v þ m 0 A 3m þ m v E 6m þ 3m v þ m 0 C 4v 8m þ 4m v þ 4m d þ m 0 A 3m þ m v þ m d þ m 0 A 3m þ m v þ m d B 3m þ m v þ m d B 3m þ m v þ m d E 6m þ 3m v þ 3m d þ m 0 C v m 0 S þ m 0 S 0 P m 0 D, F,... 0 A 3m þ m þ m þ m D d V d 8m þ 4m d þ 4m þ m 4 þ m 0 A d 4 3m þ m d þ m B 3m þ m d þ m B 3m þ m d þ m þ m 4 þ m 0 E 6m þ 3m d þ 3m þ m 4 þ m 0 A 3m þ m þ m þ m D 3d m þ 6m d þ 6m þ m 6 þ m 0 g A u d 6 3m þ m d þ m A g 3m þ m d þ m A u 3m þ m d þ m þ m 6 þ m 0 E g 6m þ 3m d þ 3m þ m 6 E u 6m þ 3m d þ 3m þ m 6 þ m 0 A 3m þ md þ m þ m D 4d 6m þ 8m d þ 8m þ m 8 þ m 0 A 8 3m þ m d þ m B 3m þ m d þ m B 3m þ m d þ m þ m 8 þ m 0 E 6m þ 3m d þ 3m þ m 8 þ m 0 E 6m þ 3m d þ 3m E 3 6m þ 3m d þ 3m þ m 8 A 0 3m þ m þ m þ m þ m D 3h m þ 6m v þ 6m h þ 3m þ m 3 þm 0 A 00 v h 3 3m þ m v þ m h A 0 3m þ m v þ m h þ m A 00 3m þ m v þ m h þ m þ m 3 þ m 0 E 0 6m þ 3m v þ 4m h þ m þ m 3 þ m 0 E 00 6m þ 3m v þ m h þ m þ m 3

21 APPENDIXES 375 TABLE B. (Continued ) Point Group Total Number of Atoms Species Number of Vibrations a D 4h 6m þ 8m v þ 8m d þ 8m h þ 4m þ 4m 0 þ m 4 þ m 0 A g 3m þ m v þ m d þ m h þ m þ m 0 þ m 4 A u 3m þ m v þ m d þ m h A g 3m þ m v þ m d þ m h þ m þ m 0 A u 3m þ m v þ m d þ m h þ m þ m 0 þ m 4 þ m 0 B g 3m þ m v þ m d þ m h þ m þ m 0 B u 3m þ m v þ m d þ m h þ m 0 B g 3m þ m v þ m d þ m h þ m þ m 0 B u 3m þ m v þ m d þ m h þ m E g 6m þ 3m v þ 3m d þ m h þ m þ m 0 þ m 4 E u 6m þ 3m v þ 3m d þ 4m h þ m þ m 0 þ m 4 þ m 0 D 5h 0m þ 0m v þ 0m h A 0 3m þ m v þ m h þ m þ m 5 þ 5m þ m 5 þ m 0 A 00 3m þ m v þ m h A 0 3m þ m v þ m h þ m A 00 3m þ m v þ m h þ m þ m 5 þ m 0 E 0 6m þ 3m v þ 4m h þ m þ m 5 þ m 0 E 00 6m þ 3m v þ m h þ m þ m 5 E 0 6m þ 3m v þ 4m h þ m E 00 6m þ 3m v þ m h þ m D 6h 4m þ m v þ m d þ m h þ 6m þ 6m 0 þ m 6 þ m 0 A g 3m þ m v þ m d þ m h þ m þ m 0 þ m 6 A u 3m þ m v þ m d þ m h A g 3m þ m v þ m d þ m h þ m þ m 0 A u 3m þ m v þ m d þ m h þ m þ m 0 þ m 6 þ m 0 B g 3m þ m v þ m d þ m h þ m 0 B u 3m þ m v þ m d þ m h þ m þ m 0 B g 3m þ m v þ m d þ m h þ m B u 3m þ m v þ m d þ m h þ m þ m 0 E g 6m þ 3m v þ 3m d þ m h þ m þ m 0 þ m 6 E u 6m þ 3m v þ 3m d þ 4m h þ m þ m 0 þ m 6 þ m 0 E g 6m þ 3m v þ 3m d þ 4m h þ m þ m 0 E u 6m þ 3m v þ 3m d þ m h þ m þ m 0 D h m þ m 0 S þ g m S þ u m þ m 0 S g ; S u 0 P g m P u m þ m 0 D g, D u, 0 F g, F u,... 0 (Continued )

22 376 APPENDIXES TABLE B. (Continued ) Point Group Total Total Number of Atoms of Atoms Species Number Number of Vibrations of Vibrations a a A 3m þ m þ m þ m T d 4m þ m d þ 6m þ 4m 3 þ m 0 A d 3 3m þ m d E 6m þ 3m d þ m þ m 3 F 9m þ 4m d þ m þ m 3 F 9m þ 5m d þ 3m þ m 3 þ m 0 O h 48m þ 4m h þ 4m d A g 3m þ m h þ m d þ m þ m 3 þ m 4 þ m þ 8m 3 þ 6m 4 þ m 0 A u 3m þ m h þ m d A g 3m þ m h þ m d þ m A u 3m þ m h þ m d þ m þ m 3 E g 6m þ 4m h þ 3m d þ m þ m 3 þ m 4 E u 6m þ m h þ 3m d þ m þ m 3 F g 9m þ 4m h þ 4m d þ m þ m 3 þ m 4 F u 9m þ 5m h þ 5m d þ 3m þ m 3 þ m 4 þ m 0 F g 9m þ 4m h þ 5m d þ m þ m 3 þ m 4 F u 9m þ 5m h þ 4m d þ m þ m 3 þ m 4 A 3m þ m þ m þ m T h 4m þ m v þ 8m 3 þ 6m þ m 0 g A u v 3 3m þ m v þ m 3 E g 3m þ m v þ m 3 þ m E u 3m þ m v þ m 3 F g 9m þ 4m v þ 3m 3 þ m F u 9m þ 5m v þ 3m 3 þ 3m þ m 0 A 3m þ m þ m þ m þ m I h 0m þ 60m v þ 30m þ 0m 3 þ m 5 þ m 0 g A u v 3 5 3m þ m v F g 9m þ 4m v þ m þ m 3 þ m 5 F u 9m þ 5m v þ 3m þ m 3 þ m 5 þ m 0 F g 9m þ 4m v þ m þ m 3 F u 9m þ 5m v þ 3m þ m 3 þ m 5 G g m þ 6m v þ 3m þ m 3 þ m 5 G u m þ 6m v þ 3m þ m 3 þ m 5 H g 5m þ 8m v þ 4m þ 3m 3 þ m 5 H u 5m þ 7m v þ 3m þ m 3 þ m 5 a Note that m is the number of sets of nuclei not any element of symmetry; m 0 is the number of nuclei on all elements of symmetry; m, m 3, m 4,...are the numbers of sets of nuclei on a twofold, threefold, fourfold, and so on, axis but not on any other element of symmetry that does not wholly coincide with that axis; m 0 is the number of sets of nuclei on a twofold axis called C 0 in the previous character tables; m v,m d,m h, are the numbers of sets of nuclei on planes s v, s d, s h, respectively, but not on any other element of symmetry.

23 APPENDIX IV. DIRECT PRODUCTS OF IRREDUCIBLE REPRESENTATIONS As shown in Sec..0, the characters for direct products can be obtained by multiplying the corresponding characters of two representations and resolving the result into those of the irreducible representations by using Eq..74. This procedure, however, can be greatly simplified if we use the following rules (Ref. 3 of Chapter ).. General Rules A A ¼ A; B B ¼ A; A B ¼ B; A E ¼ E B E ¼ E; A F ¼ F; B F ¼ F g g ¼ g; u u ¼ g; u g ¼ u ¼ 0 ; ¼ 0 ; ¼ 00 A E ¼ E ; A E ¼ E ; B E ¼ E ; B E ¼ E. Subscripts on A or B APPENDIXES 377 ¼ ; ¼ ; ¼ except for D and D h where ¼ 3; 3 ¼ ; 3 ¼ 3. Doubly Degenerate Representations For C 3, C 3h, C 3v, D 3, D 3h, D 3d, C 6, C 6h, C 6v, D 6, D 6h, S 6, O, O h, T, T d,t h. E E ¼ E E ¼ A þa þe E E ¼ B þb þe For C 4, C 4v, C 4h,D d, D 4, D 4h, and S 4 E E ¼ A þ A þ B þ B For groups in the lists above that have symbols A, B, or E without subscripts, read A ¼ A ¼ A, and so on. 4. Triply Degenerate Representations For T d, O, O h : E F ¼ E F ¼ F þ F F F ¼ F F ¼ A þ E þ F þ F F F ¼ A þ E þ F þ F For T and T h, drop subscripts and from A and F.

24 378 APPENDIXES 5. Linear Molecules (C v and D h ) S þ S þ ¼ S S ¼ S þ ; S þ P ¼ S P ¼ P P P ¼ S þ þ S þ D D D ¼ S þ þ S þ G P D ¼ P þ F S þ S ¼ S S þ D ¼ S D ¼ D Using rule 3, in C 3v, we find that E E ¼ A þ A þ E (Sec..0). Similarly, using rules and 3, in D 4h, we find that E u E u ¼ A g þ A g þ B g þ B g (Sec..3). APPENDIX V. NUMBER OF INFRARED- AND RAMAN-ACTIVE STRETCHING VIBRATIONS FOR MX n Y m -TYPE MOLECULES Compound Structure Point Group IR or Raman MX Stretching MY Stretching MX 6 Octahedral O h IR F u R A g, E g MX 5 Y Octahedral C 4v IR A, E A R A, B, E A trans-mx 4 Y Octahedral D 4h IR E u A u R A g, B g A g cis-mx 4 Y Octahedral C v IR A, B, B A, B R A, B, B A, B mer-mx 3 Y 3 Octahedral C v IR A, B A, B R A, B A, B fac-mx 3 Y 3 Octahedral C 3v IR A, E A, E R A, E A, E MX 5 Trigonal bipyramidal D 3h IR R A 00, E 0 A 0, E 0 MX 5 Tetragonal pyramidal C 4v IR R A, E A, B, E MX 4 Tetrahedral T d IR F R A, F MX 3 Y Tetrahedral C 3v IR A, E A R A, E A MX Y Tetrahedral C v IR A, B A,B R A, B A,B Polymeric MX Y a Octahedral C i IR R A u A g MX 4 Square planar D 4h IR E u R A g, B g MX 3 Y Planar C v IR A, B A R A, B A A u A g

25 APPENDIXES 379 APPENDIX V. (Continued ) Compound Structure Point Group IR or Raman MX Stretching MY Stretching trans-mx Y Planar D h IR B 3u B u R A g A g cis-mx Y Planar C v IR A, B A, B R A, B A, B MX 3 Pyramidal C 3v IR A, E R A, E MX 3 Planar D 3h IR E 0 a Bridging through X atoms. R A 0,E 0 APPENDIX VI. DERIVATION OF EQ..3 Using the rectangular coordinates, we write the kinetic energy as T ¼ ~_XM _X ða:þ where x y z x X ¼ and m m M ¼ 6 4 m m z N m N By definition, the momentum p x conugated with x is given by p x ¼ qt q_x ¼ m _x p y p zn take similar forms. Using the conugate momenta, we write T as T ¼ m p x þ m p y þþ m N P zn ¼ ~P x M P x ða:þ

26 380 APPENDIXES where p x p y P x ¼ p zn 3 m 7 5 and m M ¼ m N The column matrix P x can be expressed as P x ¼ M _X ða:3þ Define a set of conugate momenta P associated with internal coordinates, R.As is shown at the end of this appendix, we have P x ¼ ~BP ða:4þ Equations A.3 and A.4 give M _X ¼ ~BP ða:5þ Equation.8 in Chapter gives By inserting Eq. A.5 into Eq. A.6, we obtain R ¼ BX and _R ¼ B _X ða:6þ _R ¼ BM ~BP ða:7þ Using Eq. A.4, we write Eq. A. as T ¼ ~PBM ~BP ða:8þ If we define G ¼ BM ~B ðeq: :7 in ChapterÞ Eq. A.8 is written as T ¼ ~PGP ða:9þ If Eq..7 is combined with Eq. A.7, we obtain _R ¼ GP

27 APPENDIXES 38 or G _R ¼ G GP ¼ P ða:0þ Using Eq. A.0, Eq. A.9 can be written T ¼ ~_R~G GG _R ¼ ~_RG _R ðeq..3 in chapter Þ VI.. Derivation of Eq. A.4 The momentum p Rk conugated with the internal coordinate R k is given by p Rk ¼ qt ; k ¼ ; ;...; s q _R k If we denote the coordinates corresponding to the translational and rotational motions of the molecule by R 0 and its conugate momentum by p 0 R p 0 R ¼ qt q _R 0 ; ¼ ; ;...; 6 then the momentum p x in terms of rectangular coordinates is written as p x ¼ qt q_x ¼ Xs k qt q _R k qr k qx þ X6 qr 0 qt q _R 0 qx ¼ Xs k p Rk B k;x þ X6 p 0 R qr 0 qx The second term becomes zero since the momenta corresponding to the translational and rotational motions are zero. Thus, we have p x ¼ Xs k p Rk B k;x p y ¼ X p Rk B k;y.. In matrix form, this is written as.. p zn ¼ X p Rk B k;zn P x ¼ ~BP ða:4þ

28 38 APPENDIXES APPENDIX VII. MOLECULES THE G AND F MATRIX ELEMENTS OF TYPICAL In the following tables, F represents F matrix elements in the GVF field, whereas F* denotes those in the UBF field. In the latter, F 0 ¼ 0 F was assumed for all cases, and the molecular tension (Refs of Chapter ) was ignored. VII.. Bent XY Molecules (C v ) A species infrared- and Raman-active: G ¼ m y þm x ð þ cosaþ pffiffiffi G ¼ r m x sina G ¼ r m y þ m x ð cosaþ F ¼ f r þ f rr pffiffi F ¼ ð Þrfra F ¼ r f a F ¼ K þ F sin a F ¼ ð0:9þð pffiffiffi a ÞrFsin cos a 8 93 < F ¼ r H þ F cos a a = 4 : þð0:þsin 5 ; B species infrared- and Raman-active: G ¼ m y þ m x ð cos aþ F ¼ f r f rr F ¼ Kð0:ÞF cos a

29 APPENDIXES 383 VII.. Pyramidal XY 3 Molecules (C 3v ) A species infrared- and Raman-active: G G ¼ m y þ m x ð þ cos aþ ¼ ðþcos aþð cos aþ m r sina x 0 G ¼ þ cos A r m þ cos a y þ m x ð cos aþ F F F ¼ f r þ f rr ¼ rðf ra þ f 0 raþ ¼ r ðf a þ f aa Þ F ¼ K þ 4Fsin a F ¼ ð:8þrfsin a cos a 0 3 F ¼ r 4H þ F@ cos a a þð0:þsin A5 E species infrared- and Raman-active: G ¼ m y þm x ðcos aþ G ¼ ðcos aþ m r sin a x G F F F ¼ h i r ðþcos aþ ðþcos aþm yþðcos aþ m x ¼ f r f rr ¼ rðf ra þf 0 raþ ¼ r ðf a f aa Þ 0 F ¼ Kþ@ sin a a ð0:3þcos AF F ¼ ð0:9þrfsin a cos a 0 3 F ¼ r 4HþF@ cos a a þð0:þsin A5

30 384 APPENDIXES Here f ra denotes interaction between Dr and Da having a common bond (e.g., Dr and Da or Da 3 ); f 0 ra denotes interaction between Dr and Da having no common bonds (e.g., Dr and Da 3 ); see Fig..0c. VII.3. Planar XY 3 Molecules (D 3h ) A 0 species Raman-active: A 00 species infrared-active: G ¼ m y E 0 species infrared- and Raman-active: F ¼ f r þf rr F ¼ Kþ3F G ¼ 9 4r ðm yþ3m x Þ F ¼ F ¼ r f y G ¼ m y þ 3 m x G ¼ 3 p ffiffi 3 r m x G ¼ 3 r ðm yþ3m x Þ F F F F F F ¼ f r f rr ¼ rðf 0 raf ra Þ ¼ r ðf a f aa Þ ¼ Kþ0:675F pffiffiffi 3 ¼ ð0:9þ 4 rf ¼ r ðhþ0:35fþ The symbols f ra and f 0 ra are defined in VII.; f y denotes the force constant for the out-of-plane mode (see Fig..0f).

31 APPENDIXES 385 VII.4. Tetrahedral XY 4 Molecules (T d ) A species Raman-active: G ¼ m y E species Raman-active: F ¼ f r þ 3f rr F ¼ Kþ 4F G ¼ 3m y r F ¼ r ðf a f aa þ f 0 aaþ F ¼ r ðh þ 0:4FÞ F species infrared- and Raman-active: G ¼ m y þ 4 3 m x G ¼ 8 3r m x 0 G 6 r 3 m xþm y A F F F F ¼ f r f rr pffiffi ¼ ð Þrðfra f 0 raþ ¼ r ðf a f 0 aaþ ¼ Kþ 6 5 F F F ¼ 3 5 rf ¼ r ðh þ 0:4FÞ where f aa denotes interaction between two Da having a common bond; f 0 aa denotes interaction between two Da having no common bond. VII.5. Square Planar XY 4 Molecules (D 4h ) A g species Raman-active: G ¼ m y F ¼ f r þ f rr þ f 0 rr F ¼ K þ F

32 386 APPENDIXES B g species Raman-active: G ¼ m y B g species Raman-active: F ¼ f r f rr þ f 0 rr F ¼ K 0:F E u species infrared-active: G ¼ 4m y r F ¼ r ðf a f aa þ f 0 aaþ F ¼ r ðh þ 0:55FÞ G ¼ m x þm y G ¼ p ffiffiffi r m x G ¼ r ðm yþ m x Þ F ¼ f r f 0 rr pffiffiffi F ¼ ð Þrðfra f 0 raþ F ¼ r ðf a f 0 aaþ F ¼ K þ 0:9F pffiffiffi ¼ ð Þrð0:45ÞF F F ¼ r ðh þ 0:55FÞ The symbol f rr denotes interaction between two Dr perpendicular to each other; f 0 rr denotes interaction between two Dr on the same straight line. In addition, a square-planar XY 4 molecule has two out-of-plane vibrations in the A u and B u species. VII.6. Octahedral XY 6 Molecules (O h ) A g species Raman-active: G ¼ m y F ¼ f r þ 4f rr þ f rr 0 F ¼ Kþ 4F

33 APPENDIXES 387 E g species Raman-active: G ¼ m y F ¼ f r f rr þ f 0 rr F ¼ K þ 0:7F F u species infrared-active: G ¼ m y þm x G ¼ 4 r m x G ¼ r ðm yþ4m x Þ F g species Raman-active: F u species inactive: F ¼ f r f 0 rr F ¼ rf ra F ¼ r ðf a þf aa Þ F ¼ Kþ:8F F ¼ 0:9rF F ¼ r ðh þ 0:55FÞ G ¼ 4m y r F ¼ r ðf a f 0 aaþ F ¼ r ðhþ 0:55FÞ G ¼ m y r F ¼ r ðf a f aa Þ F ¼ r ðhþ 0:55FÞ The symbol f rr denotes interaction between two Dr perpendicular to each other, whereas f 0 rr denotes those between two Dr on the same straight line; f aa denotes interaction between two Da perpendicular to each other, whereas f 0 aa denotes those between two Da on the same plane. Only the interaction between two Da having a common bond was considered.

34 APPENDIX VIII. GROUP FREQUENCY CHARTS 388

35 389

36 390 APPENDIXES

37 39

38 39

39 APPENDIXES 393 APPENDIX IX. CORRELATION TABLES The following tables were reproduced with permission from the book by W. G. Fateley, F. R. Dollish, N. T. McDevitt, and F. F. Bentley (Ref. 5 in Chapter ). In some cases, more than one correlation is available between a pair of point groups. Then, it is necessary to specify the choice of symmetry operation from a larger group. For example, in the table of C 4v! C s correlation, s v and s d are written above C s to show two different possibilities: the first (s v ) and second (s d ) columns are used when the s v and s d planes, respectively, of the parent molecule become the (sole) s plane of the C s molecule. In the D 6h! C v correlation, two different correlations exist depending upon whether the C 0 or C00 axis of the D 6h molecule becomes the (sole) C axis of the C v molecule. In using the correlation tables for the purpose of the correlation method (Sec..7), the rules given below must be followed. Those species of the point groups C 3h, C 4h, C 5h, C 6h, C 6, S 4, S 6, S 8, T, and T h marked with an asterisk (*) will not use the coefficient of the E i species in this correlation procedure only. Also, for those species of the point group T and T h marked with a double dagger (z) a coefficient will be added to the E i term related to the F i species of the point group.

40 394 APPENDIXES A g A A 0 A g A A A 0 A 0 B g B A 00 A g A A A 00 A 00 A u A A 00 A u B B A 0 A 00 B u B A 0 A u B B A 00 A 0 C h C C s C i C v s(zx) C C s C s s(yz) C! C (z) C C 0 D C z C y C x D d S 4 D C v C C C s A A A A A A A A A A A 0 B A B B A A B A A B A 00 B B A B B B A A A A A 00 B 3 B B A B B B A A B A 0 E E B þ B 3 B þ B B Aþ B A 0 þ A 00 D h D C (z) C (y) C (x) C (z) C (y) C (x) C (z) C (y) C (x) s(xy) s(zx) s(yz) Cv Cv Cv Ch Ch Ch C C C Ci Cs Cs Ci A g A A A A A g A g A g A A A A 0 A 0 A 0 A g B g B A B B A g B g B g A B B A 0 A 00 A 00 A g B g B B A B B g A g B g B A B A 00 A 0 A 00 A g B 3g B 3 B B A B g B g A g B B A A 00 A 00 A 0 A g A u A A A A A u A u A u A A A A 00 A 00 A 00 A u B u B A B B A u B u B u A B B A 00 A 0 A 0 A u B u B B A B B u A u B u B A B A 0 A 00 A 0 A u B 3u B 3 B B A B u B u A u B B A A 0 A 0 A 00 A u A 0 A A 0 A A A A 0 A A A E 0 E A 0 * A* A A A 00 A A B A 00 A A 00 A E E A 0 þ A 00 E E Aþ B E 00 E A 00 * A* C 3h C 3 C s C C 3v C 3 C s D 3 C 3 C A g A A A g A A g A A 0 A g A g A A A g A B g B A 00 A g E g E E E g E A g þ B g A þ B A 0 þ A 00 A g A u A A A u A A u A A 00 A u A u A A A u A B u B A 0 A u E u E E E u E A u þ B u A þ B A 0 þ A 00 A u D 3d D 3 C 3v S 6 C 3i C 3 C h C C s C i

41 APPENDIXES 395 A A 0 A A A A A A A 0 D 3h C 3h D 3 C 3v s h!s v ðzyþ fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} C v C 3 C s h C s A A 0 A A B A B A 0 A 00 E 0 E 0 E E A þ B E Aþ B A 0 A 0 þ A 00 A 00 A 00 A A A A A A 00 A 00 A 00 A 00 A A B A B A 00 A 0 E 00 E 00 E E A þ B E Aþ B A 00 A 0 þ A 00 A A A g A A A g A A 0 A g A B A B g B B A g A A 0 A g A E B E g E E B * u B* A 00 * A * g A* A u A B A u A A 00 A u A B u B A A u A A 00 A u A E u E E B * u B* A 0 * A * u A* C 4 C C4h s v C s C 4 S 4 C h C C s C i C s v s d s v s d C 4v C 4 C v C v C C s C s A A A A A A 0 A 0 A A A A A A 00 A 00 B B A A A A 0 A 00 B B A A A A 00 A 0 E E B þ B B þ B B A 0 þ A 00 A 0 þ A 00 A A A A A A A A B B A A B B B A B B A A B B B A B A B A E B þ B 3 B þ B 3 E B Aþ B Aþ B D 4 C 0 C 00 D D C 4 C C C C 0 C 00 A A A A A A A A A 0 A A A A A A A B A 00 B A A B A A A A A 00 B A A B A A A B A 0 E E E E E B þ B B Aþ B A 0 þ A 00 E B þ B B þ B E B A þ A A Aþ B A 0 þ A 00 E 3 E E E 3 E B þ B B Aþ B A 0 þ A 00 D 4d C C 0 D 4 C 4v S 8 C 4 C v C C C s

42 D4h C 0 C 00 C 0 D4 Dd Dd C4v C4h Dh Dh C4 S4 D D C 00 A g A A A A A g A g A g A A A A A A A g A A A A A g B g B g A A B B A A Bg B B B B Bg Ag Bg B B A B A A Bg B B B B Bg Bg Ag B B B A A A E g E E E E E g B g þ B 3g B g þ B 3g E E B þ B 3 B þ B 3 B þ B B þ B Au A B B A Au Au Au A B A A A A Au A B B A Au Bu Bu A B B B A A B u B A A B B u A u B u B A A B A A B u B A A B B u B u A u B A B A A A Eu E E E E Eu Bu þ B3u Bu þ B3u E E B þ B3 B þ B3 B þ B B þ B C 0 C 00 C; sv fflffl{zfflffl} Cv C; sd fflffl{zfflffl} Cv 396

43 D4h C 0 C 00 C C 0 C 00 C C 0 C 00 s h s v s d Cv Cv Ch Ch Ch C C C Cs Cs Cs Ci A g A A A g A g A g A A A A 0 A 0 A 0 A g Ag B B Ag Bg Bg A B B A 0 A 00 A 00 Ag Bg A B Ag Ag Bg A A B A 0 A 0 A 00 Ag B g B A A g B g A g A B A A 0 A 00 A 0 A g E g A þ B A þ B B g A g þ B g A g þ B g B Aþ B Aþ B A 00 A 0 þ A 00 A 0 þ A 00 A g Au A A Au Au Au A A A A 00 A 00 A 00 Au Au B B Au Bu Bu A B B A 00 A 0 A 0 Au B u A B A u A u B u A A B A 00 A 00 A 0 A u B u B A A u B u A u A B A A 00 A 0 A 00 A u Eu A þ B A þ B Bu Au þ Bu Au þ Bu B Aþ B Aþ B A 0 A 0 þ A 00 A 0 þ A 00 Au 397

44 398 APPENDIXES A A A A 0 A A 0 A B A A E 0 E A 0 * A* E B* A* E 0 E A 0 * A* A 00 A A 00 A E 0 E A 00 A* E 00 E A 00 * A* S 4 C C C 5h C 5 C s C A A A 0 A A A A A A 00 A A B E E A 0 þ A 00 E E A þ B E E A 0 þ A 00 E E A þ B C 5v C 5 C s D 5 C 5 C A g A A A A A 0 A g A g A A A B A 00 A g E g E E E A þ B A 0 þ A 00 A g E g E E E A þ B A 0 þ A 00 A g A u A A A A A 00 A u A u A A A B A 0 A u E u E E E A þ B A 0 þ A 00 A u E u E E E A þ B A 0 þ A 00 A u D 5d D 5 C 5v C 5 C C s C i D 5h s h!sðzxþ fflfflfflfflfflffl{zfflfflfflfflfflffl} s h s v D 5 C 5v C 5h C 5 C v C C s C s A 0 A A A 0 A A A A 0 A 0 A 0 A A A 0 A B B A 0 A 00 E 0 E E E 0 E A þ B A þ B A 0 A 0 þ A 00 E 0 E E E 0 E A þ B A þ B A 0 A 0 þ A 00 A 00 A A A 00 A A A A 00 A 00 A 00 A A A 00 A B B A 00 A 0 E 00 E E E 00 E A þ B A þ B A 00 A 0 þ A 00 E 00 E E E 00 E A þ B A þ B A 00 A 0 þ A 00

45 APPENDIXES 399 A A A A B A B A E E B* A* E E A* A* C 6 C 3 C C C 6h C 6 C 3h S 6 : C 3i C h C 3 C C s C i C A g A A 0 A g A g A A A 0 A g A B g B A 00 A g B g A B A 00 A g A E g E E 00 E g B * g E B* A 0 * A * g A* E g E E 0 E g A * g E A* A 0 * A * g A* A u A A 00 A u A u A A A 00 A u A B u B A 0 A u B u A B A 0 A u A E u E E 0 E u B * u E B* A 0 * A * u A* E u E E 00 E u A * u E A* A 00 * A * u A* C 6v s v s d s v!sðzxþ fflfflfflfflfflffl{zfflfflfflfflfflffl} s v s d C 6 C 3v C 3v C C 3 C C s C s v A A A A A A A A 0 A 0 A A A A A A A A 00 A 00 B B A A B A B A 0 A 00 B B A A B A B A 00 A 0 E E E E B þ B E B A 0 þ A 00 A 0 þ A 00 E E E E A þ A E A A 0 þ A 00 A 0 þ A 00 D 6 C 0 C 00 C C 0 C 00 C 6 D 3 D 3 D C 3 C C C A A A A A A A A A A A A A B A A B B B B A A B A B A B B B A A B 3 A B B A E E E E B þ B 3 E B Aþ B Aþ B E E E E Aþ B E A Aþ B Aþ B

46 D6d D6 C6v C6 Dd D3 C3v D Cv S4 C3 C C C 00 C Cs A A A A A A A A A A A A A A 0 A A A A A A A B A A A A B A 00 B A A A B A A A A B A A A A 00 B A A A B A A B A B A A B A 0 E E E E E E E B þ B3 B þ B E E B Aþ B A 0 þ A 00 E E E E B þ B E E Aþ B A þ A B E A Aþ B A 0 þ A 00 E 3 B þ B B þ B B E A þ A A þ A B þ B 3 B þ B E A B Aþ B A 0 þ A 00 E4 E E E A þ A E E Aþ B A þ A A E A Aþ B A 0 þ A 00 E5 E E E E E E B þ B3 B þ B E E B Aþ B A 0 þ A

47 D 6h C 0 C 00 C 0 C 00 D 6 D 3h D 3h C 6v C 6h D 3d D 3d sh!sðxyþ C 6 C 3h D 3 sv!sðyzþ fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} C 00 sv sd Dh D 3 C 3v C 3v S 6 D C 0 Ag A A 0 A 0 A Ag Ag Ag Ag A A 0 A A A A Ag A A g A A 0 A 0 A A g A g A g B g A A 0 A A A A A g B B g B A 00 A 00 B B g A g A g B g B A 00 A A A A A g B Bg B A 00 A 00 B Bg Ag Ag B3g B A 00 A A A A Ag B3 Eg E E 00 E 00 E Eg Eg Eg Bg þ B3g E E 00 E E E E Eg B þ B3 E g E E 0 E 0 E E g E g E g A g þ B g E E 0 E E E E E g A þ B Au A A 00 A 00 A Au Au Au Au A A 00 A A A A Au A Au A A 00 A 00 A Au Au Au Bu A A 00 A A A A Au B Bu B A 0 A 0 B Bu Au Au Bu B A 0 A A A A Au B B u B A 0 A 0 B B u A u A u B 3u B A 0 A A A A A u B 3 Eu E E 0 E 0 E Eu Eu Eu Bu þ B3u E E 0 E E E E Eu B þ B3 Eu E E 00 E 00 E Eu Eu Eu Au þ Bu E E 00 E E E E Eu A þ B 40

48 D6h C Cv C 0 Cv C 00 Cv C Ch C 0 Ch C 00 Ch C3 C C C 0 C C 00 C s h Cs s d Cs s v Cs Ci A g A A A A g A g A g A A A A A 0 A 0 A 0 A g A g A B B A g B g B g A A B B A 0 A 00 A 00 A g Bg B A B Bg Ag Bg A B A B A 00 A 0 A 00 Ag Bg B B A Bg Bg Ag A B B A A 00 A 00 A 0 Ag E g B þ B A þ B B þ A B g A g þ B g A g þ B g E B Aþ B Aþ B A 00 A 0 þ A 00 A 0 þ A 00 A g E g A þ A A þ B A þ B A g A g þ B g A g þ B g E A Aþ B Aþ B A 0 A 0 þ A 00 A 0 þ A 00 A g Au A A A Au Au Au A A A A A 00 A 00 A 00 Au Au A B B Au Bu Bu A A B B A 00 A 0 A 0 Au B u B A B B u A u B u A B A B A 0 A 00 A 0 A u Bu B B A Bu Bu Au A B B A A 0 A 0 A 00 Au Eu B þ B A þ B B þ A Bu Au þ Bu Au þ Bu E B Aþ B Aþ B A 0 A 0 þ A 00 A 0 þ A 00 Au E u A þ A A þ B A þ B A u A u þ B u A u þ B u E A Aþ B Aþ B A 00 A 0 þ A 00 A 0 þ A 00 a u 40

49 APPENDIXES 403 C 3i : S 6 C 3 C i C S 8 C 4 C C A g A A g A A A A A E g E A * g A* B A A A A u A A u A E E B A E u E A * u A* E B A A E 3 E B A A :S þ A A A A A :S A A A A E :P E E E B þ B E :D E B þ B E A þ A E 3 :F B þ B E A þ A B þ B E 4 :G E A þ A E A þ A... C v C 6v C 4v C 3v C v S þ g A g A A A g A A S þ : A S g A g A A A g A A S : A P g E g E E E g E B þ B P:E D g E g E E B g þ B g B þ B A þ B D:E... S þ u A u A A A u A A S þ : A S u A u A A A u A A S : A P u E u E E E u E B þ B P:E D u E u E E B u þ B u B þ B A þ A D:E... D h D 6h C 6v C 3v D 4h C 4v C v C v The z axis of C v and D h groups must coincide with z axis of the point group. A A A A A E A* E A* A* F B þ B þ B 3 A þ Ez A þ B 3A T D C 3 C C A A A A A A A A A A 0 A A B A B A A A A A 00 E E A þ B E Aþ B A A þ A E A A 0 þ A 00 F F A þ E A þ E Aþ E B þ B þ B 3 A þ B þ B A þ E Aþ B A 0 þ A 00 F F B þ E A þ E Bþ E B þ B þ B 3 A þ B þ B A þ E Aþ B A 0 þ A 00 T d T D d C 3v S 4 D C v C 3 C C s

50 T h T D h S 6 : C 3i D C v C h C 3 C C s C i C Ag A Ag Ag A A Ag A A A 0 Ag A E g E A* g E g A* A * A * g E A* A 0 * A * g A* F g F B g þ B g þ B 3g A g þ E g z B þ B þ B 3 A þ B þ B A þ B g A þ Ez A þ B A 0 þ A 00 3A g 3A Au A Au Au A A Au A A A 00 Au A Eu E A * u Eu A* A * A * u E A* A 00 * A * u A* F u F B u þ B u þ B 3u A u þ E u z B þ B þ B 3 A þ B þ B A u þ B u A þ Ez A þ B A 0 þ A 00 3A u 3A 404

51 APPENDIXES C C {z} ; C C C 0 fflfflfflffl{zfflfflfflffl} O T D 4 D 3 C 4 D C D 3 C C A A A A A A A A A A A A B A B A B A A B E E A þ B E Aþ B A Aþ B E A Aþ B F F A þ E A þ E Aþ E B þ B þ B 3 B þ B þ B 3 A þ E Aþ B Aþ B F F B þ E A þ E Bþ E B þ B þ B 3 A þ B þ B 3 A þ E Aþ B A þ B A g A A A g A A g A g A A A g A g A A A g A A g B g A A A g E g E E E g E E g A g þ B g E E E g F g F F F g F A g þ E g A g þ E g A þ E A þ E A g þ E g F g F F F g F A g þ E g B g þ E g A þ E A þ E A g þ E g A u A A A u A A u A u A A A u A u A A A u A A u B u A A A u E u E E E u E E u A u þ B u E E E u F u F F F u F A u þ E u A u þ E u A þ E A þ E A u þ E u F u F F F u F A u þ E u B u þ E u A þ E A þ E A u þ E u O h O T d T h T D 3d D 4h C 3v D 3 C 3i : S 6 O h C, s d C 0, s h C 3 D d D d C 4v D 4 C 4h S 4 C 4 A g A A A A A A g A A A g A B B B B B g B B E g E A þ B A þ B A þ B A þ B A g þ B g A þ B Aþ B F g A þ E A þ E A þ E A þ E A þ E A g þ E g A þ E Aþ E F g A þ E B þ E B þ E B þ E B þ E B g þ E g B þ E Bþ E A u A B B A A A u B A A u A A A B B B u A B E u E A þ B A þ B A þ B A þ B A u þ B u A þ B Aþ B F u A þ E B þ E B þ E A þ E A þ E A u þ E u B þ E Bþ E F u A þ E A þ E A þ E B þ E B þ E B u þ E u A þ E Bþ E

52 Oh 3C C; C 0 C, s h C, s d C 0 ; sh 3C C; C 0 Dh Dh Cv Cv Cv D D A g A g A g A A A A A A g A g B g A A B A B Eg Ag Ag þ Bg A A þ A A þ B A Aþ B Fg Bg þ Bg þ B3g Bg þ Bg þ B3g A þ B þ B A þ B þ B A þ B þ B B þ B þ B3 B þ B þ B3 F g B g þ B g þ B 3g A g þ B g þ B 3g A þ B þ B A þ B þ B A þ A þ B B þ B þ B 3 A þ B þ B 3 A u A u A u A A A A A Au Au Bu A A B A B Eu Au Au þ Bu A A þ A A þ B A Aþ B F u B u þ B u þ B 3u B u þ B u þ B 3u A þ B þ B A þ B þ B A þ B þ B B þ B þ B 3 B þ B þ B 3 Fu Bu þ Bu þ B3u Au þ Bu þ B3u A þ B þ B A þ B þ B A þ A þ B B þ B þ B3 A þ B þ B3 406

53 APPENDIXES 407 C, s h C 0 ; s h s h s d C C 0 O h C h C h C s C s C C C i C A g A g A g A 0 A 0 A A A g A A g A g B g A 0 A 00 A B A g A E g A g A g þ B g A 0 A 0 þ A 00 A Aþ B A g A F g A g þ B g A g þ B g A 0 þ A 00 A 0 þ A 00 A þ B Aþ B 3A g 3A F g A g þ B g A g þ B g A 0 þ A 00 A 0 þ A 00 A þ B A þ B 3A g 3A A u A u A u A 00 A 00 A A A u A A u A u B u A 00 A 0 A B A u A E u A u A u þ B u A 00 A 0 þ A 00 A Aþ B A u A F u A u þ B u A u þ B u A 0 þ A 00 A 0 þ A 00 A þ B Aþ B 3A u 3A F u A u þ B u A u þ B u A 0 þ A 00 A 0 þ A 00 A þ B A þ B 3A u 3A I h I C 5 C 3 C C A g A A A A A A u A A A A A F g F A þ E A þ E Aþ B 3A F u F A þ E A þ E Aþ B 3A F g F A þ E A þ E Aþ B 3A F u F A þ E A þ E Aþ B 3A G g G E þ E A þ E A þ B 4A G u G E þ E A þ E A þ B 4A H g H Aþ E þ E A þ E 3A þ B 5A H u H Aþ E þ E A þ E 3A þ B 5A APPENDIX X. SITE SYMMETRY FOR THE 30 SPACE GROUPS The tables of site symmetry for the 30 space groups shown below were reproduced with permission from the book of J. R. Ferraro and J. S. Ziomek (Ref. 9 of Chapter I). The number in front of the point group notation represents the number of distinct sets of sites, and those in parentheses indicates the number of equivalent sites for each distinct set. Since the number of sites for C p, C pv (p ¼,, 3,...) and C s is infinite, no coefficients are given in front of these point group notations. Space group Site symmetries P C C () P C i 8C i (); C () 3 P C 4C (); C () 4 P C C () 5 B or C C 3 C (); C () 6 Pm C C s (); C () 7 Pb or Pc C s C () 8 Bm or Cm C 3 s C s (); C () 9 Bb or Cc C 4 C () 0 P/m C h 8C h (); 4C (); C s (); C (4) P /m C h 4C i (); C s (); C (4) (Continued )

54 408 APPENDIXES (Continued ) Space group Site symmetries B/m or C/m C 3 h 4C h (); C i (); C (); C s (); C (4) 3 P/b or P/c C 4 h 4C i (); C (); C (4) 4 P /b or P /c C 5 h 4C i (); C (4) 5 B/b or C/c C 6 h 4C i (); C (); C (4) 6 P D 8D (); C (); C (4) 7 P D 4C (); C (4) 8 P D 3 C (); C (4) 9 P D 4 C (4) 0 C D 5 C (); C (4) C D 6 4D (); 7C (); C (4) F D 7 4D (); 6C (); C (4) 3 I D 8 4D (); 6C (); C (4) 4 I D 9 3C (); C (4) 5 Pmm C v 4C v (); 4C s (); C (4) 6 Pmc C v C s (); C (4) 7 Pcc C 3 v 4C (); C (4) 8 Pma C 4 v C (); C s (); C (4) 9 Pca C 5 v C (4) 30 Pnc C 6 v C (); C (4) 3 Pmn C 7 v C s (); C (4) 3 Pba C 8 v C (); C (4) 33 Pna C 9 v C (4) 34 Pnn C 0 v C (); C (4) 35 Cmm C v C v (); C (); C s (); C (4) 36 Cmc C v C s (); C (4) 37 Ccc C 3 v 3C (); C (4) 38 Amm C 4 v C v (); 3C s (); C (4) 39 Abm C 5 v C (); C s (); C (4) 40 Ama C 6 v C (); C s (); C (4) 4 Aba C 7 v C (); C (4) 4 Fmm C 8 v C v (); C (); C s (); C (4) 43 Fdd C 9 v C (); C (4) 44 Imm C 0 v C v (); C s (); C (4) 45 Iba C v C (); C (4) 46 Ima C v C (); C s (); C (4) 47 Pmmm D h 8D h (); C v (); 6C s (4); C (8) 48 Pnnn D h 4D (); C i (4); 6C (4); C (8) 49 Pccm D 3 h 4C h (); 4D (); 8C (4); C s (4); C (8) 50 Pban D 4 h 4D (); C i (4); 6C (4); C (8) 5 Pmma D 5 h 4C h (); C v (); C (4); 3C s (4); C (8) 5 Pnna D 6 h C i (4); C (4); C (8) 53 Pmna D 7 h 4C h (); 3C (4); C s (4); C (8)

55 APPENDIXES 409 (Continued ) Space group Site symmetries 54 Pcca D 8 h C i (4); 3C (4); C (8) 55 Pbam D 9 h 4C h (); C (4); C (4); C (8) 56 Pccn D 0 h C i (4); C (4); C (8) 57 Pbcm D h C i (4); C (4); C s (4); C (8) 58 Pnnm D h 4C h (); C (4); C s (4); C (8) 59 Pmmm D 3 h C v (); C i (4); C s (4); C (8) 60 Pbcn D 4 h C i (4); C (4); C (8) 6 Pbca D 5 h C i (4); C (8) 6 Pnma D 6 h C i (4); C s (4); C (8) 63 Cmcm D 7 h C h (); C v (); C i (4); C (4); C s (4); C (8) 64 Cmca D 8 h C h (); C i (4); C (4); C s (4); C (8) 65 Cmmm D 9 h 4D h (); C h (); 6C v (); C (4); 4C s (4); C (8) 66 Cccm D 0 h D (); 4C h (); 5C (4); C s (4); C (8) 67 Cmma D h D (); 4C h (); C v (); 5C (4); C s (4); C (8) 68 Ccca D h D (); C i (4); 4C (4); C (8) 69 Fmmm D 3 h D h (); 3C h (); D (); 3C v (); 3C (4); 3C s (4); C (8) 70 Fddd D 4 h D (); C i (4); 3C (4); C (8) 7 Immm D 5 h 4D h (); 6C v (); C i (4); 3C s (4); C (8) 7 Ibam D 6 h D (); C h (); C i (4); 4C (4); C s (4); C (8) 73 Ibca D 7 h C i (4); 3C (4); C (8) 74 Imma D 8 h 4C h (); C v (); C (4); C s (4); C (8) 75 P4 C 4 C 4 (); C (); C (4) 76 P4 C 4 C (4) 77 P4 C 3 4 3C (); C (4) 78 P4 3 C 4 4 C (4) 79 I4 C 5 4 C 4 (); C (); C (4) 80 I4 C 6 4 C (); C (4) 8 P4 S 4 4S 4 (); 3C (); C (4) 8 I4 S 4 4S 4 (); C (); C (4) 83 P4/m C 4h 4C 4h (); C h (); C 4 (); C (4); C s (4); C (8) 84 P4 /m C 4h 4C h (); S 4 (); 3C (4); C s (4); C (8) 85 P4/n C 3 4h S 4 ();C 4 (); C i (4); C (4); C (8) 86 P4 /n C 4 4h S 4 (); C i (4); C (4); C (8) 87 I4/m C 5 4h C 4h ();C h (); S 4 (); C 4 (); C i (4); C (4); C s (4); C (8) 88 I4 /a C 6 4h S 4 (); C i (4); C (4); C (8) 89 P4 D 4 4D 4 (); D (); C 4 (); 7C (4); C (8) 90 P4 D 4 D ();C 4 (); 3C (4); C (8) 9 P4 D 3 4 3C (4); C (8) 9 P4 D 4 4 C (4); C (8) 93 P4 D 5 4 6D (); 9C (4); C (8) 94 P4 D 6 4 D (); 4C (4); C (8) 95 P4 3 D 7 4 3C (4); C (8) (Continued )

56 40 APPENDIXES (Continued ) Space group Site symmetries 96 P4 3 D 8 4 C (4); C (8) 97 I4 D 9 4 D 4 (); D (); C 4 (); 5C (4); C (8) 98 I4 D 0 4 D (); 4C (4); C (8) 99 P4mm C 4v C 4v ();C v (); 3C s (4); C (8) 00 P4bm C 4v C 4 ();C v (); C s (4); C (8) 0 P4 cm C 3 4v C v ();C (4); C s (4); C (8) 0 P4 nm C 4 4v C v ();C (4); C s (4); C (8) 03 P4cc C 5 4v C 4 ();C (4); C (8) 04 P4nc C 6 4v C 4 ();C (4); C (8) 05 P4 mc C 7 4v 3C v (); C s (4); C (8) 06 P4 bc C 8 4v C (4); C (8) 07 I4mm C 9 4v C 4v ();C v (); C s (4); C (8) 08 I4cm C 0 4v C 4 ();C v (); C s (4); C (8) 09 I4 md C 4v C v ();C s (4); C (8) 0 I4 cd C 4v C (4);C (8) P4m D d 4D d (); D (); C v (); 5C (4); C s (4); C (8) P4c D d 4D (); S 4 (); 7C (4); C (8) 3 P4 m D 3 d S 4 (); C v (); C (4); C s (4); C (8) 4 P4 c D 4 d S 4 (); C (4); C (8) 5 P4m D 5 d 4D d (); 3C v (); C (4); C s (4); C (8) 6 P4c D 6 d D (); S 4 (); 5C (4); C (8) 7 P4b D 7 d S 4 (); D (); 4C (4); C (8) 8 P4n D 8 d S 4 (); D (); 4C (4); C (8) 9 I4m D 9 d 4D d (); C v (); C (4); C s (4); C (8) 0 I4c D 0 d D (); S 4 (); D (); 4C (4); C (8) I4m D d D d (); D (); S 4 (); C v (); 3C (4); C s (4); C (8) I4d D d S 4 (); C (4); C (8) 3 P4/mmm D 4h 4D 4h (); D h (); C 4v (); 7C v (4); 5C s (8); C (6) 4 P4/mcc D 4h D 4 (); C 4h (); D 4 (); C 4h (); C h (4); D (4); C 4 (4); 4C (8); C s (8); C (6) 5 P4/nbm D 3 4h D 4 (); D d (); C h (4); C 4 (4); C v (4); 4C (8); C s (8); C (6) 6 P4/nnc D 4 4h D 4 (); D (4); S 4 (4); C 4 (4); C i (8); 4C (8); C (8) 7 P4/mbm D 5 4h C 4h (); D h (); C 4 (4); 3C v (4); 3C s (8); C (6) 8 P4/mnc D 6 4h C 4h (); C h (4); D (4); C 4 (4); C (8); C s (8); C (6) 9 P4/nmm D 7 4h D d (); C 4v (); C h (4); C v (4); C (8); C s (8); C (6) 30 P4/ncc D 8 4h D (4); S 4 (4); C 4 (4); C i (8); C (8); C (6) 3 P4 /mmc D 9 4h 4D h (); D d (); 7C v (4); C (8); 3C s (8); C (6) 3 P4 /mcm D 0 4h D h (); D d (); D h (); D d (); D (4); C h (4); 4C v (4); 3C (8); C s (8); C (6) 33 P4 /nbc D 4h 3D (4); S 4 (4); C i (8); 5C (8); C (6)

57 APPENDIXES 4 (Continued ) Space group Site symmetries 34 P4 /nnm D 4h D d (); D (4); C h (4); C v (4); 5C (8); C s (8); C (6) 35 P4 /mbc D 3 4h C h (4); S 4 (4); C h (4); D (4); 3C (8); C s (8); C (6) 36 P4 /mnm D 4 4h D h (); C h (4); S 4 (4); 3C v (4); C (8); C s (8); C (6) 37 P4 /nmc D 5 4h D d (); C v (4); C i (8); C (8); C s (8); C (6) 38 P4 /ncm D 6 4h D (4); S 4 (4); C h (4); C v (4); 3C (8); C s (8); C (6) 39 I4/mmm D 7 4h D 4h (); D h (); D d (); C 4v (); C h (4); 4C v (4); C (8); 3C s (8); C (6) 40 I4/mcm D 8 4h D 4 (); D d (); C 4h (); D h (); C h (4); C 4 (4); C v (4); C (8); C s (8); C (6) 4 I4 /amd D 9 4h D d (); C h (4); C v (4); C (8); C s (8); C (6) 4 I4 /acd D 0 4h S 4 (4); D (4); C i (8); 3C (8); C (6) 43 P3 C 3 3C 3 (); C (3) 44 P3 C 3 C (3) 45 P3 C 3 3 C (3) 46 R3 C 4 3 C 3 (); C (3) 47 P3 C 3i C 3i (); C 3 (); C i (3); C (6) 48 R3 C 3i C 3i (); C 3 (); C i (3); C (6) 49 P3 D 3 6D 3 (); 3C 3 (); C (3); C (6) 50 P3 D 3 D 3 (); C 3 (); C (3); C (6) 5 P3 D 3 3 C (3); C (6) 5 P3 D 4 3 C (3); C (6) 53 P3 D 5 3 C (3); C (6) 54 P3 D 6 3 C (3); C (6) 55 R3 D 7 3 D 3 (); C 3 (); C (3); C (6) 56 P3ml C 3v 3C 3v (); C s (3); C (6) 57 Pm C 3v C 3v (); C 3 (); C s (3); C (6) 58 P3cl C 3 3v 3C 3 (); C (6) 59 P3c C 4 3v C 3 (); C (6) 60 R3m C 5 3v C 3v (); C s (3); C (6) 6 R3c C 6 3v C 3 (); C (6) 6 P3m D 3d D 3d (); D 3 (); C 3v (); C h (3); C 3 (4); C (6); C s (6); C () 63 P3c D 3d D 3 (); C 3i (); D 3 (); C 3 (4); C i (6); C (6); C () 64 P3m D 3 3d D 3d (); C 3v (); C h (3); C (6); C s (6); C () 65 P3c D 4 3d D 3 (); C 3i (); C 3 (4); C i (6); C (6); C () 66 P3m D 5 3d D 3d (); C 3v (); C h (3); C (6); C s (6); C () 67 R3c D 6 3d D 3 (); C 3i (); C 3 (4); C i (6); C (6); C () 68 P6 C 6 C 6 (); C 3 (); C (3); C (6) 69 P6 C 6 C (6) 70 P6 5 C 3 6 C (6) 7 P6 C 4 6 C (3); C (6) (Continued )

58 4 APPENDIXES (Continued ) Space group Site symmetries 7 P6 4 C 5 6 C (3); C (6) 73 P6 3 C 6 6 C 3 (3); C (6) 74 P6 C 3h 6C 3h (); 3C 3 (); C s (3); C (6) 75 P6/m C 6h C 6h (); C 3h (); C 6 (); C h (); C 3 (4); C (6); C s (6); C () 76 P6 3 /m C 6h C 3h (); C 3i (); C 3h (); C 3 (4); C i (6); C s (6); C () 77 P6 D 6 D 6 (); D 3 (); C 6 (); D (3); C 3 (4); 5C (6); C () 78 P6 D 6 C (6); C () 79 P6 5 D 3 6 C (6); C () 80 P6 D 4 6 4D (3); 6C (6); C () 8 P6 4 D 5 6 4D (3); 6C (6); C () 8 P6 3 D 6 6 4D 3 (); C 3 (4); C (6); C () 83 P6mm C 6v C 6v (); C 3v (); C v (3); C s (6); C () 84 P6cc C 6v C 6 (); C 3 (4); C (6); C () 85 P6 3 cm C 3 6v C 3v (); C 3 (4); C s (6); C () 86 P6 3 mc C 4 6v C 3v (); C s (6); C () 87 P6m D 3h 6D 3h (); 3C 3v (); C v (3); 3C s (6); C () 88 P6c D 3h D 3 (); C 3h (); D 3 (); C 3h (); D 3 (); C 3h (); 3C 3 (4); C (6); C s (6); C () 89 P6m D 3 3h D 3h (); C 3h (); C 3v (); C v (3); C 3 (4); 3C s (6); C () 90 P6c D 4 3h D 3 (); 3C 3h (); C 3 (4); C (6); C s (6); C () 9 P6/mmm D 6h D 6h (); D 3h (); C 6v (); D h (3); C 3v (4); 5C v (6); 4C s (); C (4) 9 P6/mcc D 6h D 6 (); C 6h (); D 3 (4); C 3h (4); C 6 (4); D (6); C h (6); C 3 (8); 3C (); C s (); C (4) 93 P6 3 /mcm D 3 6h D 3h (); D 3d (); C 3h (4); D 3 (4); C 6 (4); C h (6); C v (6); C 3 (8); C (); C s (); C (4) 94 P6 3 /mmc D 4 6h D 3d (); 3D 3h (); C 3v (4); C h (6); C v (6); C (); C s (); C (4) 95 P3 T T(); D (3); C 3 (4); 4C (6); C () 96 F3 T 4T(); C 3 (4); C (6); C () 97 I3 T 3 T(); D (3); C 3 (4); C (6); C () 98 P 3 T 4 C 3 (4); C () 99 I 3 T 5 C 3 (4); C (6); C () 00 Pm3 T h T h (); D h (3); 4C v (6); C 3 (8); C s (); C (4) 0 Pn3 T h T(); C 3i (4); D (6); C 3 (8); C (); C (4) 0 Fm3 T 3 h T h (); T(); C h (6); C v (6); C 3 (8); C (); C s (); C (4) 03 Fd3 T 4 h T(); C 3i (4); C 3 (8); C (); C (4) 04 Im3 T 5 h T h (); D h (3); C 3i (4); C v (6); C 3 (8); C s (); C (4) 05 Pa3 T 6 h C 3i (4); C 3 (8); C (4) 06 Ia3 T 7 h C 3i (4); C 3 (8); C (); C (4)