Group theory in action: molecular vibrations We will follow the following steps: 1. Decide on a basis to describe our molecule 2. Assign the point group of the molecule in question 3. Generate a reducible representation of our basis 4. Generate irreducible representations form the reducible representation 5. Examine the irreducible representations in terms of the molecular properties Molecular vibrations A basis to describe vibrations Molecular vibrations: the relative motion of atoms with respect to one another These vibrations are an energy state of the molecule We should be able to use group theory to figure out which vibrations the molecule can actually have. 1 Step 1. We use displacement coordinates. Step 2. The water molecule is C2v point group. Step 3. Generating a reducible representation Two steps are required to generate a reducible representation for the sets of orthogonal Cartesian coordinates 1: Figure out the number of unshifted atoms on for each symmetry operation ' C2v E C2 v (xZ) V (yz) ＃of unshifted atoms 3 1 1 3 2: Use the following equation to calculate the contribution to the character per unshifted atom of the representations 360 360 E = 3, i 3, S =-1+2 cos( ) , 1, C =1+ 2cos( ) n n n n ' C2v E C2 V V (yz) Contribution to character 3 -1 1 1 2 3: The reducible representation for the displacement coordinates is the multiplication of the number of unshifted atoms and the contribution to character ' C2v E C2 V V (yz) ＃of unshift atoms 3 1 1 3 Contributions to character 3 -1 1 1 Γ 3N 9 -1 1 3 Γ 3N : The symbol of the reducible representation Step 4. Reducing the representation 1 ((9 aA1= 4 ×1×1)+(-1×1×1)+(1×1×1)+(3×1×1)) = 3, aA2= 1 , aB1= 2 , aB2= 3 Γ 3N 3A 1 + A2 + 2B1 +3 B2 Exercise Use the reduction formula to work out how many times each of the irreducible representations appears in Γ 3N of H2O 3 What have we just done? ．Generate a reducible representation of our description of molecule. ．The reducible representation has then be reduced into its individual irreducible representation components ．These irreducible representation components are of interest ．They describe the possible energy states of the molecule according to our basis Step 5. Examining the irreducible representations Displacement coordinates tell us how each atom moves relative to the others in molecule atoms move in the same direction at once translation along an axis A 1 , B 1 , B 2 rotation A 2 , B 1 , B 2 4 In the 3N representation, six of the irreducible representations correspond to translations and rotations of the molecule. every non-linear molecule has 3N-6 vibrations , where N is the number of atoms. The irreducible representations of vibrations vib vib =3N - T - R =(3A 1 +A 2 +2B 1 +3B 2 ) - (A 1 +A 2 +2B 1 +2B 2 ) =2A 1 + B 2 ．each irreducible representation corresponds to a single vibration ．The water molecule have three distinct vibration ．all other fundamental vibrations do not exist, because they are not solution to the Schrödinger equation NH3 Step 1, Use displacement coordinates. Step 2, What is the point group of NH3 Step 3, Generate a reducible representation Exercise Using displacement coordinates of each atom come out reducible representation for each 5 symmetry operation of NH3 molecule. C 3V E 2C 3 3 V # of unshifted atoms 4 1 2 Contribution to character 3 0 1 3N 12 0 2 Step 4, Reducing the representation Exercise Use the reduction formula to work out how many times each of the irreducible representations appears in Γ 3N of NH3 3N =3A 1 +A 2 +4E 6 Step 5, Examining the irreducible representations (Determine vib ) T =A 1 +E, R =A 2 +E RT =A 1 +A 2 +2E *Each E representation corresponds to a doublely degenerate state, 2E represents four energy states vib =3N - T R =(3A 1 +A 2 +4E 3 )-(A 1 +A 2 +2E) = 2A 1 +2E (3N - 6= 3 4 – 6 = 6) 2A 1 Two energy states; 2E four energy states How can we use these results to help explain the properties of molecules? What do the vibrations actually look like? Can group theory help us to the examine the vibration in more detail?[Yes] 7 Internal Coordinates Vibration of water molecule: vib =2A 1 +B 2 Which are bond-stretching vibration & which are bending vibrations? Internal coordinate r : the displacement of one atom with respect to another along the line of the bond ' E C2 V V # of unshifted coordinates 2 0 0 2 2 0 0 2 8 1 1 aA1 = 4 [(211) + (211)] = 1; aA2 = 4 [(211) + (2(-1)1)] = 0; 1 1 aB1 = 4 [(211) + (2(-1)1)] = 1; aB2 = 4 [(211) + (211)] = 1 str =A1+B2 vib = str + bend 2A1+B1=A1+B1+bend bend =A1 NH3 E 2C3 3 v # of unshifted coordinate 3 0 1 1 aA1= 6 [(311)+(113)]=1; 1 aA2= 6 [(311)+(1(-1)3)]=0 1 aE = 6 (321) =1 str =A1+E vib =str +bend 2A1+2E = A1+E+ bend bend =A1+ E 9 Can we use internal coordinates to determine bend E 2C3 3 v # of unshifted angles 3 0 1 bend = A1 + E The same answer as we got from the difference method ( bend =vib - str ) ＊Sometime, generating bend from the angles of the molecule, does NOT give the same answer as difference method. A redundant coordinate is found in the bend representation, which is often A1. When this occurs it can be simply removed from the bend representation. 10 Exercise What is the point group of the PCl5 molecule. [D3h] Exercise Using displacement coordinates as a basis, generate the reducible representation of 3N in PCl5 molecule. [E(18), 2C3(0), 3C2(-2), h(4), 2S3(-2), 3v(4)] Exercise How many times does each of the irreducible representations appear in 3N. [2A1’+A2’+4E’+3A2”+2E”] Exercise Calculate vib for the PCl5 molecule. [2A1’ + 3E’ + 2A2” + E”] Exercise Using internal coordinate as a basis, derive a reducible representation for stretching vibrations in PCl5 molecule. [E(5), 2C3(2), 3C2(1), h(3), 2S3(0), 3v(3)] Exercise Reduce the representation for stretching vibrations in PCl5 molecule. [2A1’ + E’ + A2”] Exercise Determine bend. [2E’ + A2” + E”] Exercise Using internal coordinate as a basis, derive a reducible representation for bending vibrations in PCl5 molecule. [2A1’ + 2E’ + A2” + E”] 11 Projection Operators Q: Can we use group theory to analise the vibrations further, and also provide a link with experiment? A: We can use these results and use group theory to let us “see” what the vibrations look like Water molecule has two possible stretching vibration, A1 and B2 The linear combination of the coordinates also have A1 and B2 symmetry Projection operators Px [ (R)R]x (R )x (R )x ... (R )x 1 2 h R P: the projection operator x: the generating function, coordinate or vector (R): the character of the irreducible representation R: the symmetry operator 12 Consider the integral coordinates, r1 and r2 , and choose one of these to be generating coordinates C2v E C2 v v’ r1 r1 r2 r2 r1 C2v E C2 v v’ A1 1 1 1 1 C2v E C2 v v’ 1× r1 1× r2 1× r2 1× r1 1 P r1 =1× r1 ＋1× r2 +1× r2 +1× r1 = 2 r1 +2 r2 normalized ( r1 + r2 ) 2 A1 representation has r1 + r2 as a basis r r A 1 stretch has 1 and 2 increasing or decreasing at the same time. 13 r1 + r2 is know as a symmetry adopted linear combination (SALC) Exercise Please use r2 as generating coordinate and find out the SALC for A1. [ r1 + r2 ] Show the motion of the atoms in the B2 vibration in water. C2v E C2 v v’ r1 r1 r2 r2 r1 C2v E C2 v v’ B2 1 -1 -1 1 14 C2v E C2 v v’ r1 - r2 - r2 r1 1 P r1 = r1 - r2 - r2 + r1 = 2 r2 - 2 r2 normalize ( r1 - r2 ) 2 B2 vibration in water is the one where the r1 coordinates is reducing whilst the r2 coordinates is increase, and vice versa. Exercise Please use r2 as generating coordinate and find out the SALC for A1. [ r1 - r2 ] 15 NH 3 (C3v) 2 C3v E C3 C3 v v’ v” r1 r1 r2 r r2 r r2 3 3 2 C3v E C3 C3 v v’ v” A1 1 1 1 1 1 1 E 2 -1 -1 0 0 0 For A1 Pr1 =( r1 + r2 + r3 ); For E P r1 =2 r1 - r2 - r3 The letters A and B are used for the symmetry species of one-dimensional irreducible representation E is used for 2-D T is used for 3-D. The E vibration is doubly degenerate and should have different types of vibrational motions. Exercise Please use r2 , r3 as generating coordinate and find out the SALC for E. 16 Use r 2 and r3 P r2 =2 r 2 - r1 - r3 ; P r3 =2 r3 - r 2 - r1 These three linear combinations are not linearly independent. In fact, their vibrations are the same. Try to find a linear combinations of the three combination P( r2 - r3 ) = P r2 - P r3 = 2 r 2 - r1 - r3 -2 r3 + r 2 + r1 = 3 r 2 -3 r3 r2 - r3 is orthogonal to the first one. It is the second E vibration. Use the projection operator method to determine the SALC for the bending vibration of NH 3 2 C3v E C3 C3 v v’ v” 1 1 2 3 2 1 3 17 2 C3v E C3 C3 v v’ v” A1 1 1 1 1 1 1 E 2 -1 -1 0 0 0 For A1 1 + 2 +3 + 2 +1 +3 =21 +2 2 +23 For E 21 - 2 -3 P( 2 +3 ) = P 2 +P3 = 2 -3 [與(21 - 2 -3 )正交] Exercise Please use, 2 and 3 as generating coordinate and find out the SALC for A1 and E.