CHAPTER 4: SYMMETRY AND GROUP THEORY


 Randolf Phelps
 3 years ago
 Views:
Transcription
1 hapter 4 Symmetry and Group Theory 33 APTER 4: SYMMETRY AND GROUP TEORY 4. a. Ethane in the staggered conformation has 2 3 axes (the line), 3 perpendicular 2 axes bisecting the line, in the plane of the two s and the s on opposite sides of the two s. No h, 3 d, i, S 6. D 3d. b. Ethane in eclipsed conformation has two 3 axes (the line), three perpendicular 2 axes bisecting the line, in the plane of the two s and the s on the same side of the two s. Mirror planes include h and 3 d. D 3h. c. hloroethane in the staggered conformation has only one mirror plane, through both s, the l, and the opposite on the other. s. d.,2dichloroethane in the trans conformation has a 2 axis perpendicular to the bond and perpendicular to the plane of both l s and both s, a h plane through both l s and both s, and an inversion center. 2h. 4.2 a. Ethylene is a planar molecule, with 2 axes through the s and perpendicular to the bond both in the plane of the molecule and perpendicular to it. It also has a h plane and two d planes (arbitrarily assigned). D 2h. b. hloroethylene is also a planar molecule, with the only symmetry element the mirror plane of the molecule. s. c.,dichloroethylene has a 2 axis coincident with the bond, and two mirror planes, one the plane of the molecule and one perpendicular to the plane of the molecule through both s. 2v. cis,2dichloroethylene has a 2 axis perpendicular to the bond, and in the plane of the molecule, two mirror planes (one the plane of the molecule and one perpendicular to the plane of the molecule and perpendicular to the bond). 2v. trans,2dichloroethylene has a 2 axis perpendicular to the bond and perpendicular to the plane of the molecule, a mirror plane in the plane of the molecule, and an inversion center. 2h. 4.3 a. Acetylene has a axis through all four atoms, an infinite number of perpendicular 2 axes, a h plane, and an infinite number of d planes through all four atoms. D h. b. luoroacetylene has only the axis through all four atoms and an infinite number of mirror planes, also through all four atoms. v. c. Methylacetylene has a 3 axis through the carbons and three v planes, each including one hydrogen and all three s. 3v. d. 3hloropropene (assuming a rigid molecule, no rotation around the bond) has no rotation axes and only one mirror plane through l and all three atoms. s. e. Phenylacetylene (again assuming no internal rotation) has a 2 axis down the long axis of the molecule and two mirror planes, one the plane of the benzene ring and the other perpendicular to it. 2v
2 34 hapter 4 Symmetry and Group Theory 4.4 a. Napthalene has three perpendicular 2 axes, and a horizontal mirror plane (regardless of which 2 is taken as the principal axis), making it a D 2h molecule. b.,8dichloronaphthalene has only one 2 axis, the bond joining the two rings, and two mirror planes, making it a 2v molecule. c.,5dichloronaphthalene has one 2 axis perpendicular to the plane of the molecule, a horizontal mirror plane, and an inversion center; overall, 2h. d.,2dichloronaphthalene has only the mirror plane of the molecule, and is a s molecule. 4.5 a.,dichloroferrocene has a 2 axis parallel to the rings, perpendicular to the l e l h mirror plane. It also has an inversion center. 2h. b. Dibenzenechromium has collinear 6, 3, and 2 axes perpendicular to the rings, six perpendicular 2 axes and a h plane, making it a D 6h molecule. It also has three v and three d planes, S 3 and S 6 axes, and an inversion center. c. Benzenebiphenylchromium has a mirror plane through the r and the biphenyl bridge bond and no other symmetry elements, so it is a s molecule. d. 3 O + has the same symmetry as N 3 : a 3 axis, and three v planes for a 3v molecule. e. O 2 2 has a 2 axis perpendicular to the O O bond and perpendicular to a line connecting the fluorines. With no other symmetry elements, it is a 2 molecule. f. ormaldehyde has a 2 axis collinear with the =O bond, a mirror plane including all the atoms, and another perpendicular to the first and including the and O atoms. 2v. g. S 8 has 4 and 2 axes perpendicular to the average plane of the ring, four 2 axes through opposite bonds, and four mirror planes perpendicular to the ring, each including two S atoms. D 4d. h. Borazine has a 3 axis perpendicular to the plane of the ring, three perpendicular 2 axes, and a horizontal mirror plane. D 3h. i. Tris(oxalato)chromate(III) has a 3 axis and three perpendicular 2 axes, each splitting a bond and passing through the r. D 3. j. A tennis ball has three perpendicular 2 axes (one through the narrow portions of each segment, the others through the seams) and two mirror planes including the first rotation axis. D 2d. 4.6 a. yclohexane in the chair conformation has a 3 axis perpendicular to the average plane of the ring, three perpendicular 2 axes between the carbons, and three v planes, each including the 3 axis and one of the 2 axes. D 3d. b. Tetrachloroallene has three perpendicular 2 axes, one collinear with the double bonds and the other two at 45 to the l l planes. It also has two v planes, one defined by each of the l l groups. Overall, D 2d. (Note that the ends of tetrachlorallene are staggered.)
3 hapter 4 Symmetry and Group Theory 35 c. The sulfate ion is tetrahedral. T d. d. Most snowflakes have hexagonal symmetry (igure 4.2), and have collinear 6, 3, and 2 axes, six perpendicular 2 axes, and a horizontal mirror plane. Overall, D 6h. (or high quality images of snowflakes, including some that have different shapes, see K. G. Libbrecht, Snowflakes, Voyageur Press, Minneapolis, MN, 2008.) e. Diborane has three perpendicular 2 axes and three perpendicular mirror planes. D 2h. f.,3,5tribromobenzene has a 3 axis perpendicular to the plane of the ring, three perpendicular 2 axes, and a horizontal mirror plane. D 3h.,2,3tribromobenzene has a 2 axis through the middle Br and two perpendicular mirror planes that include this axis. 2v,2,4tribromobenzene has only the plane of the ring as a mirror plane. s. g. A tetrahedron inscribed in a cube has T d symmetry (see igure 4.6). h. The left and right ends of B 3 8 are staggered with respect to each other. There is a 2 axis through the borons. In addition, there are two planes of symmetry, each containing four atoms, and two 2 axes between these planes and perpendicular to the original 2. The point group is D 2d. i. A mountain swallowtail butterfly has only a mirror that cuts through the head, thorax, and abdomen. s j. The Golden Gate Bridge has a 2 axis and two perpendicular mirror planes that include this axis. 2v 4.7 a. A sheet of typing paper has three perpendicular 2 axes and three perpendicular mirror planes. D 2h. b. An Erlenmeyer flask has an infinitefold rotation axis and an infinite number of v planes, v. c. A screw has no symmetry operations other than the identity, for a classification. d. The number 96 (with the correct type font) has a 2 axis perpendicular to the plane of the paper, making it 2h. e. Your choice the list is too long to attempt to answer it here. f. A pair of eyeglasses has only a vertical mirror plane. s. g. A fivepointed star has a 5 axis, five perpendicular 2 axes, one horizontal and five vertical mirror planes. D 5h. h. A fork has only a mirror plane. s. i. Wilkins Micawber has no symmetry operation other than the identity..
4 36 hapter 4 Symmetry and Group Theory j. A metal washer has a axis, an infinite number of perpendicular 2 axes, an infinite number of v mirror planes, and a horizontal mirror plane. D h. 4.8 a. D 2h f. 3 b. D 4h (note the four knobs) g. 2h c. s h. 8v d. 2v i. D h e. 6v j. 3v 4.9 a. D 3h f. s (note holes) b. D 4h g. c. s h. 3v d. 3 i. D h e. 2v j. 4.0 ands (of identical twins): 2 Baseball: D 2d Atomium: 3v Eiffel Tower: 4v Dominoes: 6 6: 2v 3 3: 2 5 4: s Bicycle wheel: The wheel shown has 32 spokes. The point group assignment depends on how the pairs of spokes (attached to both the front and back of the hub) connect with the rim. If the pairs alternate with respect to their side of attachment, the point group is D 8d. Other arrangements are possible, and different ways in which the spokes cross can affect the point group assignment; observing an actual bicycle wheel is recommended. (If the crooked valve is included, there is no symmetry, and the point group is a much less interesting.) 4. a. Problem 3.4 * : a. VOl 3 : 3v b. Pl 3 : 3v c. SO 4 : 2v d. SO 3 : D 3h e. Il 3 : 2v f. S 6 : O h g. I 7 : D 5h h. XeO 2 4 : D 4h i. 2 l 2 : 2v j. P 4 O 6 : T d * Incorrectly cited as problem 4.30 in first printing of text.
5 hapter 4 Symmetry and Group Theory 37 b. Problem 3.42 * : a. P 3 : 3v b. 2 Se: 2v c. Se 4 : 2v d. P 5 : D 3h e. I 5 : 4v f. XeO 3 : 3v g. B 2 l: 2v h. Snl 2 : 2v i. Kr 2 : D h j. IO : D 5h 4.2 a. igure 3.8: a. 2 : D h b. SO 3 : D 3h c. 4 : T d d. Pl 5 : D 3h e. S 6 : O h f. I 7 : D 5h g. Ta 8 3 : D 4d b. igure 3.5: a. 2 : D h b. 2 : 2v c. NO 2 : 2v d. SO 3 : D 3h e. SN 3 : 3v f. SO 2 l 2 : 2v g. XeO 3 : 3v h. SO 4 2 : T d i. SO 4 : 2v j. lo 2 3 : 2v k. XeO 3 2 : D 3h l. IO 5 : 4v 4.3 a. p x has v symmetry. (Ignoring the difference in sign between the two lobes, the point group would be D h.) b. d xy has D 2h symmetry. (Ignoring the signs, the point group would be D 4h.) c. d x 2 y 2 has D 2h symmetry. (Ignoring the signs, the point group would be D 4h.) d. d z 2 has D h symmetry. e. f xyz has T d symmetry a. The superimposed octahedron and cube show the matching symmetry elements. The descriptions below are for the elements of a cube; each element also applies to the octahedron. 2, 4 E Every object has an identity operation. Diagonals through opposite corners of the cube are 3 axes. Lines bisecting opposite edges are 2 axes. Lines through the centers of opposite faces are 4 axes. Although there are only three such lines, there are six axes, counting the 4 3 operations. (= 4 2 ) The lines through the centers of opposite faces are 4 axes as well as 2 axes. * Incorrectly cited as problem 3.4 in first printing of text.
6 38 hapter 4 Symmetry and Group Theory i 6S 4 8S 6 3 h 6 d The center of the cube is the inversion center. The 4 axes are also S 4 axes. The 3 axes are also S 6 axes. These mirror planes are parallel to the faces of the cube. These mirror planes are through two opposite edges. b. O h c. O 4.5 a. There are three possible orientations of the two blue faces. If the blue faces are opposite each other, a 3 axis connects the centers of the blue faces. This axis has 3 perpendicular 2 axes, and contains three vertical mirror places (D 3d ). If the blue faces share one vertex of the octahedron, a 2 axis includes this vertex, and this axis includes two vertical mirror planes ( 2v ). The third possibility is for the blue faces to share an edge of the octahedron. In this case, a 2 axis bisects this shared edge, and includes two vertical mirror planes ( 2v ). b. There are three unique orientations of the three blue faces. If one blue face is arranged to form edges with each of the two remaining blue faces, the only symmetry operations are identity and a single mirror plane ( s ). If the three blue faces are arranged such that a single blue face shares an edge with one blue face, but only a vertex with the other blue face, the only symmetry operation is a mirror plane that passes through the center of the blue faces, and the point group is s. If the three blue faces each share an edge with the same yellow face, a 3 axis emerges from the center of this yellow face, and this axis includes three vertical mirror planes ( 3v ). c. If there are four different colors, and each pair of opposite faces has the identical color, the only symmetry operations are identity and inversion ( i ). 4.6 our point groups are represented by the symbols of the chemical elements. Most symbols have a single mirror in the plane of the symbol ( s ), for example, s! Two symbols have D 2h symmetry (, I), and two more (, S) have 2h. Seven exhibit 2v symmetry (B,, K, V, Y, W, U). In some cases, the choice of font may affect the point group. or example, the symbol for nitrogen may have 2h in a sans serif font () but otherwise s (N). The symbol of oxygen has D h symmetry if shown as a circle but D 2h if oval.
7 hapter 4 Symmetry and Group Theory a. ondeck circle D h e. home plate 2v b. batter s box D 2h f. baseball D 2d (see igure 4.) c. cap s g. pitcher d. bat v 4.8 a. D 2h d. D 4h g. D 2h b. 2v e. 5h h. D 4h c. 2v f. 2v i. 2 y N 2 S 4.9 SN 3 3 N (top view) x Symmetry Operations: 2 3 N N 3 N 3 2 after E after 3 after v (xz) Matrix Representations (reducible): 2 E: : cos 2 sin sin 2 cos v (xz): haracters of Matrix Representations: 3 0 (continued on next page)
8 40 hapter 4 Symmetry and Group Theory Block Diagonalized Matrices: Irreducible Representations: E v oordinates Used 2 0 (x, y) z haracter Table: 3v E v Matching unctions A z x 2 + y 2, z 2 A 2 R z E 2 0 (x, y), (R x, R y ) (x 2 y 2, xy)(xz, yz) l 4.20 a. 2h molecules have E, 2, i, and h operations. l b. E: 2 : i: h : c. These matrices can be block diagonalized into three matrices, with the representations shown in the table. (E) ( 2 ) (i) ( h ) B u from the x and y coefficients A u from the z coefficients The total is = 2B u +A u. d. Multiplying B u and A u : + ( ) + ( ) ( ) + ( ) = 0, proving they are orthogonal.
9 hapter 4 Symmetry and Group Theory a. D 2h molecules have E, 2 (z), 2 (y), 2 (x), i, (xy), (xz), and (yz) operations. b. E: (z): (y): (x): i: (xy): (xz): (yz): c. E 2 (z) 2 (y) 2 (x) i (xy) (xz) (yz) 3 3 d. matching B 3u 2 matching B 2u 3 matching B u e. 2 = + ( ) ( ) + ( ) + ( ) + ( ) ( ) + + ( ) + ( ) = 0 3 = + ( ) + ( ) ( ) + ( ) + ( ) ( ) + ( ) + + ( ) = = + ( ) + ( ) + ( ) ( ) + ( ) ( ) + ( ) + ( ) + = a. h = 8 (the total number of symmetry operations) b. A E = ( 2) = 0 A 2 E = ( 2) + 2 ( ) ( ) 0 = 0 B E = ( ) 0 + ( 2) ( ) 0 = 0 B 2 E = ( ) 0 + ( 2) + 2 ( ) = 0 c. E: = 8 A : = 8 A 2 : = 8 B : = 8 B 2 : = 8
10 42 hapter 4 Symmetry and Group Theory d. = 2A + B + B 2 + E: A : /8[ ] = 2 A 2 : /8[ ( ) ( ) 2] = 0 B : /8[ ( ) ( ) 2] = B 2 : /8[ ( ) ( ) ] = E: /8[ ( 2) ] = 2 = 3 A + 2A 2 + B : A : /8[ ] = 3 A 2 : /8[ ( ) ( ) 0] = 2 B : /8[ ( ) ( ) 0] = B 2 : /8[ ( ) ( ) ] = 0 E: /8[ ( 2) ] = v = 3A + A 2 + E: A : /6[ ] = 3 A 2 : /6[ ( ) 2] = E: /6[ ( ) ] = 2 = A 2 + E: A : /6[ ( ) + 3 ( )] = 0 A 2 : /6[ ( ) + 3 ( ) ( )] = E: /6[ ( ) ( ) ( )] = 2 O h 3 = A g + E g + T u : A g : /48[ ] = A 2g : /48[ ] = 0 E g : /48[ ] = T g : /48[ ] = 0 T 2g : /48[ ] = 0 A u : /48[ ] = 0 A 2u : /48[ ] = 0 E u : /48[ ] = 0 T u : /48[ ] = T 2u : /48[ ] = 0
11 hapter 4 Symmetry and Group Theory The d xy characters match the characters The d x2y2 characters match the characters of the B 2g representation: of the B g representation: d xy d x 2 y 2 E E 4 4 y 2 y x 2 x 2 d xy i S 4 d x2 y 2 i S 4 h h v v d d 4.25 hiral: 4.5: O 2 2, [r( 2 O 4 ) 3 ] 3 4.6: none 4.7: screw, Wilkins Micawber 4.8: recycle symbol 4.9: set of three wind turbine blades, lying Mercury sculpture, coiled spring 4.26 a. Point group: 4v O 4v E v 2 d A z A 2 R z B B 2 E (x, y), (R x, R y ) Xe b. = 4 A + A 2 + 2B + B 2 + 5E c. Translation: A + E (match x, y, and z) Rotation: A 2 + E (match R x, R y, and R z ) Vibration: all that remain: 3 A + 2B + B 2 + 3E O d. The character for each symmetry operation for the Xe O stretch is +. This corresponds to the A irreducible representation, which matches the function z and is therefore IRactive. Xe
12 44 hapter 4 Symmetry and Group Theory 4.27 or S 6, the axes of the sulfur should point at three of the fluorines. The fluorine axes can be chosen in any way, as long as one from each atom is directed toward the sulfur atom. There are seven atoms with three axes each, for a total of 2. O h E i 6S 4 8S 6 3 h 6 d T u (x,y,z) T g (R x, R y, R z ) A g A 2u E g (2z 2 x 2 y 2, x 2 y 2 ) T 2u T 2g (xy, xz, yz) Reduction of gives = 3T u + T g + A g + E g + T 2g + T 2u. T u accounts for translation and also infrared active vibrational modes. T g is rotation. The remainder are infraredinactive vibrations a. cise() 4 l 2 has 2v symmetry. The vectors for stretching have the representation : 2v E 2 v (xz) v (yz) A z A 2 B x B 2 y l l O e O O O n(a ) = /4[ ] = 2 n(a 2 ) = /4[ ( ) + 2 ( )] = 0 n(b ) = /4[4 + 0 ( ) ( )] = n(b 2 ) = /4[4 + 0 ( ) + 2 ( ) + 2 ] = = 2 A + B + B 2, all four IR active. l b. transe() 4 l 2 has D 4h symmetry. O O e O l D 4h E i 2S 4 h 2 v 2 d A 2u z E u (x,y)
13 hapter 4 Symmetry and Group Theory 45 Omitting the operations that have zeroes in : n(a 2u ) = /6[ ( ) + 4 ( ) ] = 0 n(e u ) = /6[ ] = (IR active) Note: In checking for IRactive bands, it is only necessary to check the irreducible representations having the same symmetry as x, y, or z, or a combination of them. c. e() 5 has D 3h symmetry. The vectors for O stretching have the following representation : D 3h E h 2S 3 3 v E (x, y) A 2 z O O O e O n(e) = /2 [ (5 2) + (2 2 ) + (3 2)] = n(a 2 ) = /2 [(5 ) + (2 2 ) + (3 ) + (3 ) + (3 3 )] = There are two bands, one matching Eand one matching A 2. These are the only irreducible representations that match the coordinates x, y, and z In 4.28a, the symmetries of the stretching vibrations of cise() 4 l 2 ( 2v symmetry) are determined as 2 A + B + B 2. Each of these representations matches Ramanactive functions: A (x 2, y 2, z 2 ) ; A 2 (xy), B (xz); and B 2 (yz), so all are Ramanactive. In 4.28b, the symmetries of the stretching vibrations of transe() 4 l 2 (D 4h symmetry) are A g + B g + E u. Only A g (x 2 + y 2, z 2 ) and B g (x 2 y 2 ) match Raman active functions; this complex exhibits two Ramanactive stretching vibrations. In 4.28c, the symmetries of the stretching vibrations of e() 5 (D 3h symmetry) are 2 A + E + A 2. Only A (x 2 + y 2, z 2 ) and E(x 2 y 2, xy) match Ramanactive functions; this complex exhibits four Ramanactive stretching vibrations a. The point group is 2h. b. Using the Si I bond vectors as a basis generates the representation: 2h E 2 i h A g x 2 + y 2, z 2 B g xz, yz A u z B u x, y = A g + B g + A u + B u The A u and B u vibrations are infrared active. c. The A g and B g vibrations are Raman active.
14 46 hapter 4 Symmetry and Group Theory 4.3 trans isomer (D 4h ): cis isomer ( 2v ): The simplest approach is to consider if the number of infraredactive I O stretches is different for these structures. (Alternatively, one could also determine the number of IRactive I stretches, a slightly more complicated task.) trans: D 4h E i 2S 4 h 2 v 2 d A g A 2u z There is only a single IRactive I O stretch (the antisymmetric stretch), A 2u. cis: 2v E 2 (xz) (yz) A z B x There are two IRactive I O stretches, the A and B (symmetric and antisymmetric). Infrared spectra should therefore be able to distinguish between these isomers. (Reversing the x and y axes would give A + B 2. Because B 2 matches y, it would also represent an IRactive vibration.) Because these isomers would give different numbers of IRactive absorptions, infrared spectra should be able to distinguish between them. The reference provides detailed IR data a. One way to deduce the number of Ramanactive vibrations of AsP 3 is to first P determine the symmetries of all the degrees of freedom. This complex exhibits P P 3 symmetry, with the 3 axis emerging from the As atom. The (E) is 2; the x, y, and z axes of the four atoms do not shift when the identity operation is carried out. Only the As atom contributes to the character of the 3 transformation matrix; the P atoms shift during rotation about the 3 axis. The general transformation matrix for rotation about the z axis (Section 4.3.3) affords 0 as ( 3 ) for The As atom and one P atom do not shift when a v reflection is carried out, and ( v ) = 2 (see the v(xz) transformation matrix in Section 4.33 for the nitrogen atom of N 3 as a model of how the two unshifted atoms of AsP 3 will contribute to the character of the v transformation matrix). As 3v E 3 v A z x 2 + y 2, z 2 A 2 R x E 2 0 (x, y), (R x, R y ) (x 2 y 2, xy), (xz, yz)
15 hapter 4 Symmetry and Group Theory 47 Reduction of the reducible representation affords 3 A + A E. On the basis of the character table, the translational modes of AsP 3 have the symmetries A + E, and the rotational modes have the symmetries A 2 + E. The six vibrational modes of AsP 3 subsequently have the symmetries 2 A + 2 E. These vibrational modes are Ramanactive, and four absorptions are expected (and observed) since the sets of E modes are degenerate. B. M. ossairt, M.. Diawara,.. ummins, Science 2009, 323, 602 assigns the bands as: 33 (a ), 345 (e), 428 (a ), 557 (e) cm . Alternatively, the set of six bonds may be selected as the basis for a representation focused specifically on stretches of these bonds. This approach generates the following representation: 3v E 3 v This representation reduces to 2 A + 2 E, the same result as obtained by first considering all degrees of freedom, then subtracting the translational and rotational modes. b. As 2 P 2 exhibits 2v symmetry, and (like AsP 3 ) will have six vibrational modes (3N 6). Inspection of the 2v character table indicates that all vibrational P modes will be Raman active. Since each irreducible representation has a dimension of, the number of Raman absorptions expected is 6 (that is, there will be no degenerate vibrational modes). This prediction can be confirmed via deduction of the symmetries of the vibrational modes. As in part a, we will first determine the symmetries of all the degrees of freedom. The (E) of the transformation matrix is 2. In As 2 P 2, the 2 axis does not pass through any atoms, and all four atoms shift upon rotation; ( 2 ) = 0. Two atoms do not shift upon reflection through each of the v planes. The contribution to the character of the transformation matrix for each of these unshifted atoms is, and ( v (xz)) = ( v (yz)) = 2. As P As 2v E 2 v (xz) v (yz) A z x 2, y 2, z 2 A 2 R z xy B x, R y xz B 2 y, R x yz Reduction of the reducible representation affords 4 A + 2 A B + 3 B 2. On the basis of the above character table, the translational modes of As 2 P 2 have the symmetries A + B + B 2, and the rotational modes have the symmetries A 2 + B + B 2. The six anticipated Ramanactive vibrational modes of As 2 P 2 subsequently have the symmetries 3 A + A 2 + B + B 2. (continued on next page)
16 48 hapter 4 Symmetry and Group Theory As in part a, an alternative approach is to select the set of six bonds as the basis for a representation focused specifically on stretching vibrations. This approach generates the following representation: 2v E 2 v (xz) v (yz) This representation reduces to 3 A + A 2 + B + B 2, the same result as obtained by first considering all degrees of freedom, then subtracting the translational and rotational modes. c. The issue here is whether or not P 4 (T d ) exhibits 4 Ramanactive vibrations as does AsP 3. We will first determine the symmetries of all the degrees of freedom. The (E) of the transformation matrix is 2. Only one P 4 atom is fixed upon rotation about each 3 axis; ( 3 ) = 0. All four atoms are shifted upon application of the 2 and S 4 axes; ( 2 ) = (S 4 ) = 0. Two atoms do not shift upon reflection through each of the d planes. The contribution to the character of the transformation matrix for each of these unshifted atoms is, and ( d ) = 2. P P P P T d E S 4 6 d A x 2 + y 2 + z 2 A 2 E (2z 2 x 2 y 2, x 2 y 2 ) T 3 0 (R x, R y, R z ) T (x, y, z) (xy, xz, yz) Reduction of the reducible representation affords A + E + T + 2 T 2. Since the symmetries of the translational modes and rotational modes are T 2 and T, respectively, the symmetries of the vibrational modes are A + E + T 2, all of these modes are Ramanactive so three Raman absorptions are expected for P 4, and P 4 could potentially be distinguished from AsP 3 solely on the basis of the number of Raman absorptions. The alternative approach, using the set of six P P bonds as the basis for a representation focused specifically on bond stretches, generates the following representation: T d E S 4 6 d This representation reduces to A + E + T 2, the same result as obtained by first considering all degrees of freedom, then subtracting the translational and rotational modes.
17 hapter 4 Symmetry and Group Theory The possible isomers are as follows, with the triphenylphosphine ligand in either the axial (A) or equatorial (B) sites. Ph 3 P O e O e PPh 3 (A) 3v (B) 2v Note that the triphenylphosphine ligand is approximated as a simple L ligand for the sake of the point group determination. Rotation about the e P bond in solution is expected to render the arrangement of the phenyl rings unimportant in approximating the symmetry of these isomers in solution. The impact of the phenyl rings would likely be manifest in the IR ν() spectra of these isomers in the solidstate. or A, we consider each bond as a vector to deduce the expected number of carbonyl stretching modes. The irreducible representation is as follows: 3v E 3 v 4 2 A z x 2 + y 2, z 2 A 2 R x E 2 0 (x, y), (R x, R y ) (x 2 y 2, xy), (xz, yz) Reduction of the reducible representation affords 2 A + E. These stretching modes are IRactive and three ν() absorptions are expected for A. or B, a similar analysis affords the following irreducible representation: 2v E 2 v (xz) v (yz) A z x 2, y 2, z 2 A 2 R z xy B x, R y xz B 2 y, R x yz Reduction of the reducible representation affords 2 A + B + B 2. These stretching modes are IRactive, and four ν() absorptions are expected for A. The reported ν() IR spectrum is consistent with formation of isomer A, with the triphenylphosphine ligand in the axial site.
18 50 hapter 4 Symmetry and Group Theory 4.34 As in 4.33, we consider the triphenylphosphine ligand as a simple L group for point group determination. The point groups for isomers A, B, and are as follows: PPh 3 PPh 3 O e PPh 3 PPh 3 O e PPh 3 O e PPh 3 (A) 2v (B) D 3h () s or A, the set of irreducible representations for the three stretching vibrational modes is 2 A + B. These modes are all IRactive in the 2v character table, and three ν() IR absorptions are expected for isomer A. or B, the set of irreducible representations for the three stretching vibrational modes is A + E. Only the E mode is IRactive in the D 3h point group, and one ν() IR absorption is expected for isomer B. or, the set of irreducible representations for the three stretching vibrational modes is 2 A These modes are all IRactive in the s point group, and three ν() IR absorptions are expected for isomer. The single ν() IR absorption reported for e() 3 (PPh 3 ) 2 supports the presence of the D 3h isomer B. The trans isomer B is reported in R. L. Keiter, E. A. Keiter, K.. ecker,. A. Boecker, Organometallics, 988, 7, 2466, and the authors observe splitting of the absorption associated with the Emode in l 3. The forbidden A stretching mode was observed as a weak absorption The IR spectrum exhibits two ν() absorptions. The two proposed metal carbonyl fragments will be considered independently for analysis. O O Ti O O Ti D 4h 4v The reducible representation for the four vectors associated with the bonds in the 4v fragment is as follows: 4v E v 2 d Reduction of this representation affords A + B + E. The A and E modes are IRactive, and a titanium complex with a square pyramidal titanium tetracarbonyl fragment is supported by the IR spectral data.
19 hapter 4 Symmetry and Group Theory 5 or the D 4h fragment, the reducible representation for the set of vectors associated with the bonds is: D 4h E i 2S 4 h 2 v 2 d Reduction of the reducible representation affords A g + B g + E u. Only the E modes are IRactive, and a complex with a square planar titanium tetracarbonyl fragment is expected to exhibit a single IR ν() stretching absorption. This D 4h possibility can therefore be ruled out on the basis of the spectrum A reasonable product is the 4v molecule Mo() 5 (P(OPh) 3 ), with replacing a triphenylphosphite ligand. The reducible representation for the five vectors associated with the bonds in this molecule is: O O (OPh) 3 P Mo O 4v E v 2 d 5 3 Reduction of this representation affords 2 A + B + E. The A and E modes are IRactive, and three IR ν() stretching absorptions are expected. The reported IR spectrum features three strong ν() absorptions, and one very weak absorption attributed to the forbidden B mode in D. J. Darensbourg, T. L. Brown, Inorg. hem., 968, 7, I has 2 symmetry, with a 2 axis running right to left, perpendicular to the l, N, l, N and l, P, l, P faces. II also has 2 symmetry, with the same 2 axis as I. (Lower left corner occupied by l, not.) III has only an inversion center and i symmetry An example for each of the five possible point groups: T d : 3v : 2v : s : : Br Br Br l 4.39 a. The S portion is linear, so the molecule has a 3 axis along the line of these three atoms, three v planes through these atoms and an atom on each end, but no other symmetry elements. 3v b. The molecule has only an inversion center, so it is i. The inversion center is equivalent to an S 2 axis perpendicular to the average plane of the ring. c. M 2 l 6 Br 4 is i. d. This complex has a 3 axis, splitting the three N atoms and the three P atoms (almost as drawn), but no other symmetry elements. 3
20 52 hapter 4 Symmetry and Group Theory e. The most likely isomer has the less electronegative l atoms in equatorial positions. Point group: 2v 4.40 The structures on the top row are D 2d (left) and s (right). Those on the bottom row are 2h (left) and 4v (right). l l P 4.4 a. 3v b. D 5h c. Square structure: D 2d (bottom ligand on lower left Re should be instead of L); orner structure: s d. D 3d 4.42 Web of Science and Sciinder Scholar should be helpful, but simply searching for these symmetries using a general Internet search should provide examples of these point groups. Some examples: a. S 6 : Mo 2 (S 6 2 Me 3 ) 6 (M.. hisholm, J.. orning, and J.. uffman, J. Am. hem. Soc., 983, 05, 5924) Mo 2 (NMe 2 ) 6 (M.. hisholm, R. A. otton, B. A. Grenz, W. W. Reichert, L. W. Shive, and B. R. Stults, J. Am. hem. Soc., 976, 98, 4469) [Nae 6 (OMe) 2 (dbm) 6 ] + (dbm = dibenzoylmethane, ) (. L. Abbati, A. ornia, A.. abretti, A. aneschi, and D. Garreschi, Inorg. hem., 998, 37, 430) b. T Pt( 3 ) 4, 44 c. I h 20, 80 d. T h [o(no 2 ) 6 ] 3, Mo(NMe 2 ) 6 In addition to examples that can be found using Web of Science, Sciinder, and other Internet search tools, numerous examples of these and other point groups can be found in I. argittai and M. argittai, Symmetry Through the Eyes of a hemist, as listed in the General References section.
Chapter 5. Molecular Symmetry
Chapter 5. Molecular Symmetry Symmetry is present in nature and in human culture An introduction to symmetry analysis Symmetry perations and Elements Definitions Symmetry peration = a movement of a body
More informationCH6. Symmetry Symmetry elements and operations Point groups Character tables Some applications. Symmetry elements
CH6. Symmetry Symmetry elements and operations Point groups Character tables Some applications 1 Symmetry elements symmetry element: an element such as a rotation axis or mirror plane indicating a set
More informationSolution Key, Problemset 2
Solution Ke, Problemset 2 3.1 Use VSEPR theor to obtain the molecular structures. After ou have the shape of the molecule, if necessar, use Pauling electronegativit values to obtain the bond dipole moments
More informationThe Unshifted AtomA Simpler Method of Deriving Vibrational Modes of Molecular Symmetries
Est. 1984 ORIENTAL JOURNAL OF CHEMISTRY An International Open Free Access, Peer Reviewed Research Journal www.orientjchem.org ISSN: 0970020 X CODEN: OJCHEG 2012, Vol. 28, No. (1): Pg. 189202 The Unshifted
More informationChapter Draw sketches of C, axes and a planes? (a) NH3? (b) The PtC1 24 ion? 4.2 S4 or i : (a) C02? (b) C2H2? (c) BF3? (d) SO 24?
Chapter 4 4.1 Draw sketches of C, axes and a planes? (a) NH 3? In the drawings below, the circle represents the nitrogen atom of ammonia and the diamonds represent the hydrogen atoms. The mirror plane
More informationElements and Operations. A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed.
Elements and Operations A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed. A symmetry operation is a movement of an object about a symmetry element
More informationSymmetry. Using Symmetry in 323
Symmetry Powerful mathematical tool for understanding structures and properties Use symmetry to help us with: Detecting optical activity and dipole moments Forming MO s Predicting and understanding spectroscopy
More informationMolecular Symmetry. Symmetrical: implies the species possesses a number of indistinguishable configurations.
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) used with group theory to predict vibrational spectra
More informationCHAPTER 5: MOLECULAR ORBITALS
Chapter 5 Molecular Orbitals 5 CHAPTER 5: MOLECULAR ORBITALS 5. There are three possible bonding interactions: p z d z p y d yz p x d xz 5. a. Li has a bond order of. (two electrons in a bonding orbital;
More informationNotes pertinent to lecture on Feb. 10 and 12
Notes pertinent to lecture on eb. 10 and 12 MOLEULAR SYMMETRY Know intuitively what "symmetry" means  how to make it quantitative? Will stick to isolated, finite molecules (not crystals). SYMMETRY OPERATION
More informationSymmetry and group theory
Symmetry and group theory or How to Describe the Shape of a Molecule with two or three letters Natural symmetry in plants Symmetry in animals 1 Symmetry in the human body The platonic solids Symmetry in
More informationSymmetry elements, operations and point groups ( in the molecular world )
ymmetry elements, operations and point groups ( in the molecular world ) ymmetry concept is extremely useful in chemistry in that it can help to predict infrared spectra (vibrational spectroscopy) and
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infrared spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More informationSymmetry and Molecular Structures
Symmetry and Molecular Structures Some Readings Chemical Application of Group Theory F. A. Cotton Symmetry through the Eyes of a Chemist I. Hargittai and M. Hargittai The Most Beautiful Molecule  an Adventure
More informationDeriving character tables: Where do all the numbers come from?
3.4. Characters and Character Tables 3.4.1. Deriving character tables: Where do all the numbers come from? A general and rigorous method for deriving character tables is based on five theorems which in
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationGroup Theory and Molecular Symmetry
Group Theory and Molecular Symmetry Molecular Symmetry Symmetry Elements and perations Identity element E  Apply E to object and nothing happens. bject is unmoed. Rotation axis C n  Rotation of object
More informationModeling Molecular Structure
PRELAB: Reading: Modeling Molecular Structure Make sure to have read Chapters 8 and 9 in Brown, Lemay, and Bursten before coming to lab. Bring your textbook to lab and a pencil. INTRDUCTIN: Chemists often
More information3. Assign these molecules to point groups a. (9) All of the dichlorobenzene isomers Cl
CE 610 Spring 2007 Problem Set 1 key Total points = 200 1. (10 points) What distinguishes transition metals from the other elements in the periodic table? List five physical or chemical features that are
More informationDodecahedron Faces = 12 pentagonals with three meeting at each vertex Vertices = 20 Edges = 30
1 APTER Introduction: oncept of Symmetry, Symmetry Elements and Symmetry Point Groups 1.1 SMMETR Symmetry is one idea by which man through the ages has tried to understand and create order, periodicity,
More information15 Molecular symmetry
5 Molecular symmetry Solutions to exercises Discussion questions E5.(b) Symmetry operations Symmetry elements. Identity, E. The entire object 2. nfold rotation 2. nfold axis of symmetry, C n 3. Reflection
More information18 electron rule : How to count electrons
18 electron rule : How to count electrons The rule states that thermodynamically stable transition metal organometallic compounds are formed when the sum of the metal d electrons and the electrons conventionally
More informationAll About the Chair Conformation
All About the Chair Conformation Background Before we begin, here are some terms to know: 1. Conformation: the shape that a molecule can adopt due to rotation around one or more single bonds 2. Angle strain:
More informationEffect of unshared pairs on molecular geometry
Chapter 7 covalent bonding Introduction Lewis dot structures are misleading, for example, could easily represent that the electrons are in a fixed position between the 2 nuclei. The more correct designation
More informationAn introduction to molecular symmetry
Chapter 3 An introduction to molecular symmetry TOPICS & & & & & Symmetry operators and symmetry elements Point groups An introduction to character tables Infrared spectroscopy Chiral molecules 3.1 Introduction
More information2. Distinguish between the concepts of (a) symmetry operation, (b) symmetry element, and (c) symmetry point group.
Chemistry 3820 Answers to Problem Set #2 1. The symmetry of the following molecules (with C models): C 2v C 3v T d D 2h D 3d C 2 D 2d C 2v 2. Distinguish between the concepts of (a) symmetry operation,
More informationCopyright Page 1
1 Introduction to Bonding: TwoDimensional Lewis tructures Key Terms: abbreviated electron configuration combines the inert, noble core electrons (abbreviated via a noble element) with the outermost or
More information1. Find all the possible symmetry operations for 1,2propadiene:
CHEM 352: Examples for chapter 4. 1. Find all the possible symmetry operations for 1,2propadiene: The operations are: E, C 2, C 2, C 2, S 4, S4 3, σ v, σ v. The C 2 axis is along C = C = C. The C 2 and
More informationPolyatomic Molecular Orbital Theory
Polyatomic Molecular Orbital Theory Transformational properties of atomic orbitals When bonds are formed, atomic orbitals combine according to their symmetry. Symmetry properties and degeneracy of orbitals
More informationENGINEERING DRAWING. UNIT I  Part A
ENGINEERING DRAWING UNIT I  Part A 1. Solid Geometry is the study of graphic representation of solids of  dimensions on plane surfaces of  dimensions. 2. In the orthographic projection,
More informationCHAPTER 9 COVALENT BONDING: ORBITALS. Questions
APTER 9 VALET BDIG: RBITALS Questions 9. In hybrid orbital theory, some or all of the valence atomic orbitals of the central atom in a molecule are mixed together to form hybrid orbitals; these hybrid
More informationSymmetric Stretch: allows molecule to move through space
BACKGROUND INFORMATION Infrared Spectroscopy Before introducing the subject of IR spectroscopy, we must first review some aspects of the electromagnetic spectrum. The electromagnetic spectrum is composed
More informationVSEPR Model. The ValenceShell Electron Pair Repulsion Model. Predicting Molecular Geometry
VSEPR Model The structure around a given atom is determined principally by minimizing electron pair repulsions. The ValenceShell Electron Pair Repulsion Model The valenceshell electron pair repulsion
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationPROTEINS THE PEPTIDE BOND. The peptide bond, shown above enclosed in the blue curves, generates the basic structural unit for proteins.
Ca 2+ The contents of this module were developed under grant award # P116B001338 from the Fund for the Improvement of Postsecondary Education (FIPSE), United States Department of Education. However, those
More informationIn part I of this twopart series we present salient. Practical Group Theory and Raman Spectroscopy, Part I: Normal Vibrational Modes
ELECTRONICALLY REPRINTED FROM FEBRUARY 2014 Molecular Spectroscopy Workbench Practical Group Theory and Raman Spectroscopy, Part I: Normal Vibrational Modes Group theory is an important component for understanding
More informationAN INTRODUCTION TO SPARTAN
AN INTRODUCTION TO SPARTAN OBJECTIVE This tutorial introduces a number of basic operations in Spartan Student required for molecule manipulation, property query and spectra and graphics display. Specifically
More informationChapter 9. Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure
Chapter 9 Molecular Geometry & Bonding Theories I) Molecular Geometry (Shapes) Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure Molecular
More informationChapter 1 Symmetry of Molecules p. 1  Symmetry operation: Operation that transforms a molecule to an equivalent position
Chapter 1 Symmetry of Molecules p. 11. Symmetry of Molecules 1.1 Symmetry Elements Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation
More informationLaboratory 20: Review of Lewis Dot Structures
Introduction The purpose of the laboratory exercise is to review Lewis dot structures and expand on topics discussed in class. Additional topics covered are the general shapes and bond angles of different
More informationThe Lewis electron dot structures below indicate the valence electrons for elements in Groups 12 and Groups 1318
AP EMISTRY APTER REVIEW APTER 7: VALENT BNDING You should understand the nature of the covalent bond. You should be able to draw the Lewis electrondot structure for any atom, molecule, or polyatomic ion.
More informationSYMMETRY M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier
Mathematics Revision Guides Symmetry Page 1 of 12 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier SYMMETRY Version: 2.2 Date: 20112013 Mathematics Revision Guides Symmetry Page
More informationSolving Spectroscopy Problems
Solving Spectroscopy Problems The following is a detailed summary on how to solve spectroscopy problems, key terms are highlighted in bold and the definitions are from the illustrated glossary on Dr. Hardinger
More informationSymmetry and Molecular Spectroscopy
PG510 Symmetry and Molecular Spectroscopy Lecture no. 2 Group Theory: Molecular Symmetry Giuseppe Pileio 1 Learning Outcomes By the end of this lecture you will be able to:!! Understand the concepts of
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationPRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT
UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,
More information7.14 Linear triatomic: ABC. Bond angles = 180 degrees. Trigonal planar: Bond angles = 120 degrees. B < B A B = 120
APTER SEVEN Molecular Geometry 7.13 Molecular geometry may be defined as the threedimensional arrangement of atoms in a molecule. The study of molecular geometry is important in that a molecule s geometry
More information5.04 Principles of Inorganic Chemistry II
MIT OpenourseWare http://ocw.mit.edu 5.4 Principles of Inorganic hemistry II Fall 8 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.4, Principles of
More informationSuggested solutions for Chapter 3
s for Chapter PRBLEM Assuming that the molecular ion is the base peak (00% abundance) what peaks would appear in the mass spectrum of each of these molecules: (a) C5Br (b) C60 (c) C64Br In cases (a) and
More informationORGANIC COMPOUNDS IN THREE DIMENSIONS
(adapted from Blackburn et al., Laboratory Manual to Accompany World of hemistry, 2 nd ed., (1996) Saunders ollege Publishing: Fort Worth) Purpose: To become familiar with organic molecules in three dimensions
More informationTRANSITION METALS AND COORDINATION CHEMISTRY
CHAPTER TWENTYONE TRANSITION METALS AND COORDINATION CHEMISTRY For Review 1. Chromium ([Ar]:4s 0 3d 5 ) and copper [Ar]:4s 1 3d 10 ) have electron configurations which are different from that predicted
More informationChemical Crystallography Lab #1
Tutorials on Point Group Symmetry Chemical Crystallography Lab #1 "It is just these very simple things which are extremely likely to be overlooked." Sir Arthur ConanDoyle, "The sign of four" (1) Bring
More informationBegin recognition in EYFS Age related expectation at Y1 (secure use of language)
For more information  http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices
More informationInorganic Chemistry with Doc M. Day 3. Covalent bonding: Lewis dot structures and Molecular Shape.
Inorganic Chemistry with Doc M. Day 3. Covalent bonding: Lewis dot structures and Molecular Shape. Topics: 1. Covalent bonding, Lewis dot structures in review, formal charges 2. VSEPR 6. Resonance 3. Expanded
More informationReporting Category 1: Matter and Energy
Name: Science Teacher: Reporting Category 1: Matter and Energy Atoms Fill in the missing information to summarize what you know about atomic structure. Name of Subatomic Particle Location within the Atom
More informationWhat Are the Shapes of Molecules?
Lab 7 Name What Are the Shapes of Molecules? PreLab Assignment Read the entire lab handout. There is no written prelab assignment for this lab. Learning Goals Derive the Lewis structure of a covalent
More informationOrganic Chemistry Tenth Edition
Organic Chemistry Tenth Edition T. W. Graham Solomons Craig B. Fryhle Welcome to CHM 22 Organic Chemisty II Chapters 2 (IR), 9, 320. Chapter 2 and Chapter 9 Spectroscopy (interaction of molecule with
More informationBonding Models. Bonding Models (Lewis) Bonding Models (Lewis) Resonance Structures. Section 2 (Chapter 3, M&T) Chemical Bonding
Bonding Models Section (Chapter, M&T) Chemical Bonding We will look at three models of bonding: Lewis model Valence Bond model M theory Bonding Models (Lewis) Bonding Models (Lewis) Lewis model of bonding
More informationMODERN ATOMIC THEORY AND THE PERIODIC TABLE
CHAPTER 10 MODERN ATOMIC THEORY AND THE PERIODIC TABLE SOLUTIONS TO REVIEW QUESTIONS 1. Wavelength is defined as the distance between consecutive peaks in a wave. It is generally symbolized by the Greek
More informationCommunicated March 31, 1951 CHR CHR CHR. *H* zz '" *H _ 0....H / k C,.. CHR CNR CHR CHR CHR *HN/' N 'H_N/' H_./  H(H.
VOL. 37, 1951 CHEMISTR Y: PA ULING AND COREY 251 THE PLEATED SHEET, A NEW LAYER CONFIGURATION OF POL YPEPTIDE CHAINS BY LINUS PAULING AND ROBERT B. COREY GATES AND CRELLIN LABORATORIES OF CHEMISTRY,* CALIFORNIA
More informationMolecular Symmetry 1
Molecular Symmetry 1 I. WHAT IS SYMMETRY AND WHY IT IS IMPORTANT? Some object are more symmetrical than others. A sphere is more symmetrical than a cube because it looks the same after rotation through
More informationPSS 27.2 The Electric Field of a Continuous Distribution of Charge
Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight ProblemSolving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.
More informationMolecular Models Experiment #1
Molecular Models Experiment #1 Objective: To become familiar with the 3dimensional structure of organic molecules, especially the tetrahedral structure of alkyl carbon atoms and the planar structure of
More information1 Electrons and Chemical Bonding
CHAPTER 1 1 Electrons and Chemical Bonding SECTION Chemical Bonding BEFORE YOU READ After you read this section, you should be able to answer these questions: What is chemical bonding? What are valence
More informationLaboratory 11: Molecular Compounds and Lewis Structures
Introduction Laboratory 11: Molecular Compounds and Lewis Structures Molecular compounds are formed by sharing electrons between nonmetal atoms. A useful theory for understanding the formation of molecular
More informationSymmetry & Group Theory
ymmetry & Group Theory MT Chap. Vincent: Molecular ymmetry and group theory ymmetry: The properties of selfsimilarity 1 Re(CO)10 W(CO)6 C C60 ymmetry: Construct bonding based on atomic orbitals Predict
More information2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?
MATH 206  Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of
More informationMolecular Geometry and VSEPR We gratefully acknowledge Portland Community College for the use of this experiment.
Molecular and VSEPR We gratefully acknowledge Portland ommunity ollege for the use of this experiment. Objectives To construct molecular models for covalently bonded atoms in molecules and polyatomic ions
More informationThe Clar Structure of Fullerenes
The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25
More informationMolecular Structures. Chapter 9 Molecular Structures. Using Molecular Models. Using Molecular Models. C 2 H 6 O structural isomers: .. H C C O..
John W. Moore onrad L. Stanitski Peter. Jurs http://academic.cengage.com/chemistry/moore hapter 9 Molecular Structures Stephen. oster Mississippi State University Molecular Structures 2 6 structural isomers:
More informationC has 4 valence electrons, O has six electrons. The total number of electrons is 4 + 2(6) = 16.
129 Lewis Structures G. N. Lewis hypothesized that electron pair bonds between unlike elements in the second (and sometimes the third) row occurred in a way that electrons were shared such that each element
More informationSHAPE, SPACE AND MEASURES
SHPE, SPCE ND MESURES Pupils should be taught to: Understand and use the language and notation associated with reflections, translations and rotations s outcomes, Year 7 pupils should, for example: Use,
More informationCHEM 2323 Unit 1 General Chemistry Review
EM 2323 Unit 1 General hemistry Review I. Atoms A. The Structure of the Atom B. Electron onfigurations. Lewis Dot Structures II. Bonding A. Electronegativity B. Ionic Bonds. ovalent Bonds D. Bond Polarity
More information10. Geometry Hybridization Unhybridized p atomic orbitals
APTER 14 VALET BDIG: RBITALS The Localized Electron Model and ybrid rbitals 9. The valence orbitals of the nonmetals are the s and p orbitals. The lobes of the p orbitals are 90E and 180E apart from each
More informationCovalent Bonding & Molecular Orbital Theory
Covalent Bonding & Molecular Orbital Theory Chemistry 754 Solid State Chemistry Dr. Patrick Woodward Lecture #16 References  MO Theory Molecular orbital theory is covered in many places including most
More informationAP CHEMISTRY CHAPTER REVIEW CHAPTER 6: ELECTRONIC STRUCTURE AND THE PERIODIC TABLE
AP CHEMISTRY CHAPTER REVIEW CHAPTER 6: ELECTRONIC STRUCTURE AND THE PERIODIC TABLE You should be familiar with the wavelike properties of light: frequency ( ), wavelength ( ), and energy (E) as well as
More informationIntroduction and Symmetry Operations
Page 1 of 9 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Introduction and Symmetry Operations This page last updated on 27Aug2013 Mineralogy Definition of a Mineral A mineral is a naturally
More informationB) atomic number C) both the solid and the liquid phase D) Au C) Sn, Si, C A) metal C) O, S, Se C) In D) tin D) methane D) bismuth B) Group 2 metal
1. The elements on the Periodic Table are arranged in order of increasing A) atomic mass B) atomic number C) molar mass D) oxidation number 2. Which list of elements consists of a metal, a metalloid, and
More informationSYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING
Daniel DOBRE, Ionel SIMION SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING Abstract: This article discusses the relation between the concept of symmetry and its applications
More informationVisualizing Molecular Orbitals: A MacSpartan Pro Experience
Introduction Name(s) Visualizing Molecular Orbitals: A MacSpartan Pro Experience In class we have discussed Lewis structures, resonance, VSEPR, hybridization and molecular orbitals. These concepts are
More informationDigital Image Processing. Prof. P.K. Biswas. Department of Electronics & Electrical Communication Engineering
Digital Image Processing Prof. P.K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture  27 Colour Image Processing II Hello, welcome
More informationNotes: Most of the material presented in this chapter is taken from Bunker and Jensen (2005), Chap. 3, and Atkins and Friedman, Chap. 5.
Chapter 5. Geometrical ymmetry Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (005), Chap., and Atkins and Friedman, Chap. 5. 5.1 ymmetry Operations We have already
More informationIntroduction to Group Theory with Applications in Molecular and Solid State Physics
Introduction to Group Theory with Applications in Molecular and Solid State Physics Karsten Horn FritzHaberInstitut der MaxPlanckGesellschaft 3 84 3, email horn@fhiberlin.mpg.de Symmetry  old concept,
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationCh a p t e r s 8 a n d 9
Ch a p t e r s 8 a n d 9 Covalent Bonding and Molecular Structures Objectives You will be able to: 1. Write a description of the formation of the covalent bond between two hydrogen atoms to form a hydrogen
More informationABERLINK 3D MKIII MEASUREMENT SOFTWARE
ABERLINK 3D MKIII MEASUREMENT SOFTWARE PART 1 (MANUAL VERSION) COURSE TRAINING NOTES ABERLINK LTD. EASTCOMBE GLOS. GL6 7DY UK INDEX 1.0 Introduction to CMM measurement...4 2.0 Preparation and general hints
More informationCrystals are solids in which the atoms are regularly arranged with respect to one another.
Crystalline structures. Basic concepts Crystals are solids in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry elements.
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More information5. Structure, Geometry, and Polarity of Molecules
5. Structure, Geometry, and Polarity of Molecules What you will accomplish in this experiment This experiment will give you an opportunity to draw Lewis structures of covalent compounds, then use those
More informationAn introduction to molecular symmetry
44 An introduction to molecular symmetry 4.2 igure 4.1 or answer 4.2: the principal axis of rotation, and the two mirror pianes in H^O. (a) E is the identity operator. It effectively identifies the molecular
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationEXPERIMENT ONE COVALENT BONDING AND MOLECULAR MODELS. In order to draw proper Lewis structures chemists use two rules,
EXPERIMENT ONE COVALENT BONDING AND MOLECULAR MODELS Today you will use ballandstick molecular model kits to better understand covalent bonding. You will figure out the structures of several different
More informationDrawing, Representation, and Projection of Chemical Formulae.
Drawing, Representation, and Projection of hemical Formulae. In order to learn and communicate organic chemistry you must learn the language of organic chemistry. Pictures play an important role in this
More information12B The Periodic Table
The Periodic Table Investigation 12B 12B The Periodic Table How is the periodic table organized? Virtually all the matter you see is made up of combinations of elements. Scientists know of 118 different
More informationBASIC GEOMETRY GLOSSARY
BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that
More informationTwo hydrogen atoms join together to attain the helium Noble gas configuration by sharing electrons and form a molecule.
Lecture Simple Molecular Orbitals  Sigma and Pi Bonds in Molecules 1 An atomic orbital is located on a single atom. When two (or more) atomic orbitals overlap to make a bond we can change our perspective
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationTHE STRUCTURE OF MOLECULES AN EXPERIMENT USING MOLECULAR MODELS 2009 by David A. Katz. All rights reserved.
THE STRUCTURE OF MOLECULES AN EXPERIMENT USING MOLECULAR MODELS 2009 by David A. Katz. All rights reserved. In a footnot to a 1857 paper, Friedrich August Kekulé suggested that carbon was tetratomic, that
More information