L9_Group_Theory-I

L9_Group_Theory-I - Group theory: exploita0on of...

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Unformatted text preview: Group theory: exploita0on of symmetry Symmetry Why using symmetry in Chemistry ? 1.  Understand proper0es of electronic orbitals 2.  Predict and interpret IR and UV ­Vis spectra 3.  Predict and interpret op0cal ac0vity of a molecule Symmetry Elements and Opera7ons –  Deﬁni0ons 1.  Symmetry Element = geometrical en0ty such as a line, a plane, or a point, with respect to which one or more symmetry opera0ons can be carried out 2.  Symmetry Opera7on = a movement of a body such that the appearance of the body (e.g. molecule) aLer the opera0on is indis0nguishable from the original appearance (if you can tell the diﬀerence, it wasn’t a symmetry opera0on) The Symmetry Opera7ons 1. E (Iden0ty Opera0on) = no change in the object a.  Needed for mathema0cal completeness b.  Every molecule has at least this symmetry opera0on 2.  Cn (Rota0on Opera0on) = rota0on of the object 360/n degrees about an axis 3.  σ (Reﬂec0on Opera0on) = exchange of points through a plane to an opposite and equidistant point 4.  i (Inversion Opera0on) = each point moves through a common central point to a posi0on opposite and equidistant 5.  Sn (Improper Rota0on Opera0on) = rota0on about 360/n axis followed by reﬂec0on through a plane perpendicular to axis of rota0on Cn (Rota7on Opera7on) = rota0on of the object 360/n degrees about an axis a.  The symmetry element is a line b.  Counterclockwise rota0on is taken as posi0ve c.  Principal axis = axis with the largest possible n value d.  Examples: C23 = two C3’s C33 = E C17 axis Rota7onal axes of BF3 principal axis (highest value of Cn) C3 C3 C2 C2 . three ­fold axis three ­fold axis two ­fold axis two ­fold axis viewed from viewed from viewed from viewed from above the side the side above Note: there are 3 C2 axes σ (Reﬂec7on Opera7on) = exchange of points through a plane to an opposite and equidistant point a.  Symmetry element is a plane b.  Human Body has an approximate σ opera0on c.  Linear objects have inﬁnite σ‘s d.  σ h = plane perpendicular to principle axis e.  σ v = plane includes the principal axis f.  σ d = plane includes the principal axis, but not the outer atoms Mirror planes (σ) of BF3: Mirror planes can contain the principal axis (σv) or be at right angles to it (σh). BF3 has one σh and three σv planes: (v = ver0cal, h = horizontal) σv mirror plane C3 principal axis σh mirror plane C3 principal axis σv mirror plane contains the C3 axis σh mirror plane is at right angles to the C3 axis Rota7onal axes and mirror planes of the water molecule: C2 principal axis C2 σv mirror plane C2 σv mirror plane The water molecule has only one rota0onal axis, its C2 axis, which is also its principal axis. It has two mirror planes that contain the principal axis, which are therefore σv planes. It has no σh mirror plane, and no center of symmetry. i (Inversion Opera7on) = each point moves through a common central point to a posi0on opposite and equidistant a.  Symmetry element is a point b.  Some0mes diﬃcult to see, some0mes not present when you think you see it c.  Ethane has i, methane does not d.  Tetrahedra, triangles, pentagons do not have i e.  Squares, parallelograms, rectangular solids, octahedra do Sn (Improper Rota7on Opera7on) = rota0on about 360/n axis followed by reﬂec0on through a plane perpendicular to axis of rota0on a.  Methane has 3 S4 opera0ons (90 degree rota0on, then reﬂec0on) b.  2 Sn opera0ons = Cn/2 (S24 = C2) c.  nSn = E, S2 = i, S1 = σ d.  Snowﬂake has S2, S3, S6 axes Rota0onal axes and mirror planes of benzene C6 principal axis C2 C2 C6 σh C2 σv C2 σv C6 C6 principal axis principal axis Rota0onal axes and mirror planes of boron triﬂuoride C3 principal axis C2 C2 C2 σh σh σv σv boron triﬂuoride has a C3 principal axis and three C2 axes, a σh mirror plane three σv mirror planes, but no center of inversion C3 principal axis Point Groups –  Deﬁni0ons: 1.  Point Group = the set of symmetry opera0ons for a molecule 2.  Group Theory = mathema0cal treatment of the proper0es of the group which can be used to ﬁnd proper0es of the molecule –  Assigning the Point Group of a Molecule: b. High Symmetry Groups Finding the Point Group •  Determine whether the molecule belongs to one of the special cases of low or high symmetry. –  Low symmetry •  C1 (only E), Cs (E and σh), and Ci (E and i) –  High symmetry •  Linear with inversion will be D∞h; without will be C∞v. •  Other point groups; Td, Oh, and Ih •  Find the rota0on axis with the highest n. –  This will be the principal axis. Finding the Point Group •  Does the molecule have any C2 axes ⊥ to the Cn axis? –  If so, the molecule is in the D set of groups. –  If not, the molecule is in the C or S set. –  If so, the molecule is Cnh or Dnh. –  If not, con0nue with other mirror planes. •  Does the molecule have a mirror plane (σh). •  Does the molecule contain any mirror planes that contain the Cn axis? –  If so, the molecule is Cnv or Dnd. –  If not and in the D group, the molecule is Dn. –  If not and in the C group, con0nue to next. Finding the Point Group •  Is there any S2n axis collinear with the Cn axis? –  If so, the molecule is S2n. –  If not, the molecule is Cn. •  This assignment is very rare. Ver0cal planes contain the highest order Cn axis. In the Dnd case, the planes are dihedral because they are between the C2 axes. Purely rota0on groups of Ih, Oh, and Td are I, O, and T, respec0vely (only other symmetry opera0on is E). These are rare. The Th point group is derived by adding inversion symmetry to the T point group. These are rare. Assign point groups of molecules Rota0on axes of “normal” symmetry molecules Perpendicular C2 axes Horizontal Mirror Planes Ver0cal or Dihedral Mirror Planes and S2n Axes D. Proper0es of Point Groups 1.  Symmetry opera0on of NH3 a.  Ammonia has E, 2C3 (C3 and C23) and 3σv b. Point group = C3v 2.  Proper0es of C3v (any group) a.  Must contain E b.  Each opera0on must have an inverse; doing both gives E (right to leL) c.  Any product equals another group member d.  Associa0ve property Proper0es of Groups •  Proper0es of a group –  Each group must have an iden0ty opera0on. –  Each opera0on in the group must have an inverse: –  The product of any two group opera0ons must also be a member of the group. –  The associa0ve property holds: –  Number of symmetry opera0on in a group is the order of the group •  A symmetry opera0on can be expressed as a transforma0on matrix. –  [new coordinates]=[transforma0on matrix][old coordinates] Transforma0on matrices Consider a water molecule and a reference system centered on the oxygen atom. Consider the eﬀect of all symmetry opera0ons of the symmetry group of the molecule on an arbitrary vector The matrix C2 represents the operator C2 Transforma0on matrices Consider a water molecule and a reference system centered on the oxygen atom. Consider the eﬀect of all symmetry opera0ons of the symmetry group of the molecule on an arbitrary vector Transforma0on matrices Consider a water molecule and a reference system centered on the oxygen atom. Consider the eﬀect of all symmetry opera0ons of the symmetry group of the molecule on an arbitrary vector Representa7on of the group : set of matrices represen0ng the symmetry opera0ons of the group; there are an inﬁnite number of representa0ons ; matrices in any representa0on are not required to be diagonal or of any speciﬁc order. Irreducible representa7ons: all other group representa0ons can be expressed in terms of so called irreducible representa0ons. 1D irreducible representa0on are designed by A if they symmetrical under rota0on about the principal axis; they are designed by B if they are an0symmetric. Irreducible representa0on of C2v are 1D Irreducible representa0ons Irreducible representa0on of C2v are 1D h = order of group d = Dimension of irreducible representa0on Group characters: traces of the irreducible representa0on matrices ...
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