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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. 3N 9 1 1 3, The symbol of the reducible representation. Step 4 Reducing the representation, aA1 4 9 1 1 1 1 1 1 1 1 3 1 1 3 aA2 1 aB1 2 aB2 3. 3A A2 2B1 3 B2, Exercise Use the reduction formula to work out how many times each of the irreducible representations.
appears in 3N of H2O, 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 B B. rotation A B B 2 1 2, In the 3 N 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 3 N 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 Schr dinger equation. 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. symmetry operation of NH3 molecule, C 3V E 2C 3 3 V.
of unshifted atoms 4 1 2, Contribution to character 3 0 1. 3 N 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 3N of NH3, 3 N 3A 1 A 2 4E, Step 5 Examining the irreducible representations Determine vib. T A 1 E R A 2 E R T A 1 A 2 2E, Each E representation corresponds to a doublely degenerate state 2E represents four energy states. vib 3 N T R 3A 1 A 2 4E 3 A 1 A 2 2E 2A 1 2E 3N 6 3 4 6 6. 2A 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. Internal Coordinates, Vibration of water molecule vib 2A B 1 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. of unshifted coordinates 2 0 0 2, aA1 4 2 1 1 2 1 1 1 aA2 4 2 1 1 2 1 1 0. aB1 4 2 1 1 2 1 1 1 aB2 4 2 1 1 2 1 1 1, str A1 B2 vib str bend 2A1 B1 A1 B1 bend bend A1.
of unshifted coordinate 3 0 1, aA1 6 3 1 1 1 1 3 1. aA2 6 3 1 1 1 1 3 0, aE 6 3 2 1 1, str A1 E vib str bend 2A1 2E A1 E bend bend A1 E. Can we use internal coordinates to determine bend, of unshifted angles 3 0 1. 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. 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 3 v 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 3 v 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, 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 R1 x R2 x Rh x, P the projection operator. x the generating function coordinate or vector, R the character of the irreducible representation.
R the symmetry operator, 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, P r1 1 r1 1 r2 1 r2 1 r1 2 r1 2 r2 normalized r1 r2. A1 representation has r1 r2 as a basis, A stretch has r1 and r2 increasing or decreasing at the same time. 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.
C2v E C2 v v, r1 r2 r2 r1, P r1 r1 r2 r2 r1 2 r2 2 r2 normalize r1 r2. 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. C3v E C3 C32 v v v, r1 r1 r2 r3 r2 r3 r2, C3v E C3 C32 v v v. A1 1 1 1 1 1 1, E 2 1 1 0 0 0, For A1 P r1 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, 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, C3v E C3 C32 v v v, 1 1 2 3 2 1 3. C3v E C3 C32 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 2 1 2 2 2 3. For E 2 1 2 3, P 2 3 P 2 P 3 2 3 2 1 2 3, Exercise Please use 2 and 3 as generating coordinate and find out the SALC for A1 and E.
D 4h E 2C 4 C 2 2C, 2 i 2S 4 h 2 v 2 d, of unshifted coordinates 4 0 0 2 0 0 0 4 2 0. aA1g 1 aA2g 0 aB1g 1 aB2g 0 aEg 0 aA1u 0 aA2u 0 aB1u 0 aB2u 0 aEu 1. st A1g B1g Eu, E C41 C43 C2 C2 x C2 y C2 1 C2 2 i S41 S43 h v v d d. For A1g P r1 r1 r2 r3 r4 For B1g P r1 r1 r2 r3 r4 For Eu P r1 r1 r3. For the remaining Eu SALC use r2 as a generating coordinate. P r2 r2 r4 r1 r3, Exercise Please use r2 r3 as generating coordinate and find out the SALC for Eu. Exercise For PCl5 determine the SALCs for the stretching vibrations of the molecule Use the. projection operation method to view the SALCs sketch the results. Spectroscopy and symmetry selection rules, Whether a transition is allowed or not can be determined by solving the following equation. i wave function of the ground state, f wave function of the excited state.
T transition moment operator, d small space, A transition is allowed if the equation is not zero. We can use the results of group theory to simplify the use of this daunting equation greatly. Infra red spectroscopy, In infra red spectroscopy a molecule can be excited from its ground vibration state to a fundamental. vibration using infra red radiation This all occurs if there is a dipole moment change in the molecule. upon absorption, For infra red spectroscopy, i T f d i f d. qr dipole moment of the molecule, For H2O vib 2A B 1 2. 1 f A1 or B2, 3 character table x B1 y B2 z A1, i T f d i f d A1 B2 f d.
All we have to decide is whether the symmetry of the appropriate parts of the equation all multiply. together to give a non zero result without having to do the integration. In group theory language all we need to decide is whether the multiplication of the symmetries gives the. totally symmetric representation e g A1 A1g or not. For f A1 we need to figure out is the results of the following. To multiply irreducible representations together we need to use the Direct Product Rules. A1 B1 A1 B1, A1 B2 A1 B 2, A1 A1 A1 A1, A1 is included in one of the answer and we can conclude that absorbance due to transition from the. ground state to A1 vibrations in H2O will be observed in the infra red spectrum. Exercise Will the B2 vibration in H2O be observable by infra red spectroscopy. Raman spectroscopy, T x y z xy yz xz, Consider H 2O. x 2 y 2 z 2 A1 xy A2 xz B1 yz B2, A1 A1 A1 A1, One equation contains the A1 representation hence we expect to see bands in the Raman spectrum. corresponding to the A1 vibrations, Exercise Will the B2 vibration in H2O be observable by Raman spectroscopy. If a band due to a vibration appears in the infra red spectrum the vibration is known as infra red active. If a band due to a vibration appears in the Raman spectrum the vibration is known as Raman active. In H 2 O we expect to see three fundamental vibration bands all three of which are Raman and infra red. To assign vibration spectra of molecule we need one rule and one rule of thumb. Rule polarized bands can only come from A1 vibrations. Rule of thumb stretching vibration usually appear at higher frequency. Any molecule which possesses a centre of symmetry i is subject to something know as the mutual. exclusion rule This rule states that any vibration in such a molecule cannot be both IR active and. Raman active, Tetra chloromethane CCl 4, Infra red spectrum cm 1 Raman spectrum cm.
218 depolarized, 305 324 depolarized, 458 polarized. 768 762 depolarized, 1 point group Td, 2 consider displacement coordinates. a A1 15 1 1 1 1 3 1 1 6 3 1 6 1, a A 0 aE 1 aT 1 aT 3. 3N A1 E T1 3T2 vib 3v T R A1 E 2T2, Use internal coordinates. Td E 8 C3 3 C2 6 S4 6 d, of unshift coordinate 4 1 0 0 2.
str 4 1 0 0 2, a A1 1 a A2 0 aE 0 aT1 0 aT2 1, str A1 T2 bend E T2. Use internal coordinates, E 8C 3 3C 2 6S 4, of unshifted coordinate 6 0 2 0 2. bend 6 0 2 0 2, a A1 1 a A2 0 aE 1 aT1 0 aT2 1, bend A1 E T2. The extra A1 is redundant coordinate and can be ignored str A1 E T2 bend E T2. for f A1 A1 T2 T1 T2, f T2 A1 T2 T2 A1 E T1 T1, f E A1 T2 E T1 T2. Two IR bands one T2 stretch one T2 bend, xy yz zx T2 x y z E A1.
For f A1 A1 A1 A1 A1 Raman active A1 E A1 E A1 T2 A1 T2. For f T2 A1 A1 T2 T2 A1 E A1 T2 A1 T2 T2 A1 E T2 Raman active. For f E A1 A1 E E A1 E E A1 E Raman active A1 T2 E T1 T2. Four bands two stretches and two bends to be Raman active. Polarized band in the Raman is A1 stretch, IR spectrum Raman spectrum. 218 depolarized E bend, 305 324 depolarized T2 bend. 458 polarized A1 stretch, 768 762 depolarized T2 strrtch. 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

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