The heart of group theory

Transcription

1 The heart of group theory. We can represent a molecule in a mathematical way e.g. with the coordinates of its atoms. This mathematical description of the molecule forms a basis for symmetry operation. 2. Using this basis, we can generate mathematical representations of the symmetry operations with simple rules. 3. They are either reducible or irreducible representations. 4. The representations can be expressed using characters (a number). 5. The irreducible representation for all the common point group have been work out. The representations are tabulated in character table. Character table. C 2v 2. E C 2 v (xz) v (yz) 3. A z x 2, y 2, z 2 A R z xy B - - x,r y xz B y,r x yz. Point group symbols 2. symmetry operations 3. Mulliken symbols (Each irreducible representation is given a shorthand symbol to describe it) 4. Irreducible representation (irreducible representations do not necessarily only contain s and -s) 5. Common bases for representation 2

2 Each reducible representation can be broken down into a combination of irreducible representations. g nr The reduction formula ai R ir R a i = the number of times a particular irreducible representation appears in this reducible representation g = the number of symmetry operation in the point group R the character of the reducible representation for a particular symmetry element R the character of the particular irreducible representation for a particular symmetry i element n the numbers of symmetry operations in that particular class R 3 C 3v E 2C 3 3 v RR 4 0 RR: reducible representation Number of times A appears in reducible representation C 3v E 2C 3 3 v A z x 2 +y 2, z 2 A 2 - R z x 2 -y 2, xy E 2-0 (x, y)( R x, R y ) xz, yz Exercise Using the reduction formula, figure out the number of times that the A 2 and E irreducible representations appear in the reducible representation. [, ] Reducible representation = a A A +a A2 A 2 +a E E The answers, a i must be integers. We can reduce the reducible representation shown above to A +A 2 +E 4

3 Exercise Suppose the reducible representation of a C 3v point group is : E(4), 2C 3 (), and 3 v (0). Using the reduction formula, figure out the number of times that the A 2 and E irreducible representations appears in the reducible representation [, ] Schrödinger equation and group theory H = E: mathematical function which describes the molecule; H : a mathematical operator which changes the into E, where E is the energy of only certain types of mathematical functions are solutions to this equation (eigen functions) What are these functions? Q: What is the effect on a molecule s energy if a symmetry operation is carried on it? A: the new molecule must have the same energy Symmetry operation Os, H s s must be a basis for the symmetry operations of the point group of the molecule we can eliminate all but a few possibilities 5 Mathematical representations of O s will tell us the possible solutions to the Schrödinger equation It is the irreducible representations which have the most important meaning. The solution of Schrödinger equation must be a basis for irreducible representation of the point group of the molecule Take the water molecule and consider the 2s and 2p atomic orbitals on the oxygen atom.. The point group of the water molecule is C 2v 2. How the orbitals change upon the symmetry operation of the group 2s 2p x - - 2p y - - 2p z Character table of C 2v 6

4 C 2v A A B - - B All of these representations are the same ad irreducible representations are the same as irreducible representations of C 2v Each orbitals is a basis for the representations and a possible solution to the Schrödinger equation which describe these orbitals in water. What about H atom? Consider s orbital of just one of the hydrogen atoms. s 0 0 This representation does not match any irreducible representation of C 2v point group. 7 We can not use reduction formula to reduce this representation into anything smaller. The single H s orbital in H 2 O does not form a basis for a representation of the point group and can not be used to calculate the energy of the molecule with Schrödinger equation. Exercise Consider the s orbital of just one of the hydrogen atoms in NH 3. Write down the reducible representation. Does this representation match any of the irreducible representation of the C 3v point group? Try to use reduction formula to reduce this representation into smaller representation. Can you reduce it into anything smaller? [E() 2C 3 (0) v () v (0) v (0), No] Consider both s orbitals of the hydrogen atoms, but as a linear combination. s a +s b s a +s b s b +s a s b +s a s a +s b s a +s b s a +s b = s b +s a 8

5 Combination of the two orbitals does from an irreducible representation of the point group. The combination of H s orbitals is a solution to Schrödinger equation for these orbital in H 2 O. We must consider the two H atoms together. The other combination s a -s b s a -s b s a -s b s b -s a s b -s a s a -s b s a -s b - - The out-of-phase combination of s a -s b also forms a basis for irreducible representation of the point group and a possible energy state for these orbitals in the water molecule. s a +s b A combination, s a -s b B 2 combination Exercise Using the reduction formula, figure out the number of times that the A, A 2, B, and B 2 9 irreducible representations appear in the reducible representation for s orbitals of H atoms in H 2 O. [, 0, 0, ] The atom which lie on the point of the group The atomic orbitals which gives rise to particular irreducible representations are written in the columns at the right-hand side of the character table Ex. p x orbital of oxygen. x B symmetry. Exercise What symmetry does the p orbitals of nitrogen atom in NH 3 have? [E, A ] 0