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3. Symmetry and Group Theory READING: Chapter 6 3.1. Point groups • Each object can be assigned to a point group according to its symmetry elements:

Source: Shriver & Atkins, Inorganic Chemistry, 3rd Edition. • The symmetry elements can be considered operators. Operator Instruction for an operation to be performed on a function or object that

follows it:

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3.2. Group theory in a nut-shell • “Group Theory” is the mathematical treatment of the properties of “groups”

(e.g., point groups!). • Powerful mathematical tool → used to simplify quantum mechanical calculations

using molecular symmetry. • Ultimately, it can help generate molecular orbitals and predict the spectroscopic

characteristics of a molecule using “back-of-an-envelope” calculations. • There are entire textbooks devoted to chemical applications of Group Theory, but we

will just look at a brief introduction to the subject … the rest you can learn in a spectroscopy course or in grad school!

Group theory is used to construct character tables, e.g. C2V

(See Resource Section 4 of your Textbook for a complete set of character tables.) A quick walk around a character table • The symbol in the top left shows the point group to which the character table applies

(C2v in this case). • The symbols across the top show the symmetry operations. A molecule with C2v

symmetry (e.g., H2O) will have a C2 axis of rotation and two mirror planes (σv and σv’), but it will not have a center of inversion (no i operation shows up in this table).

• The symbols down the left side (A1, A2, B1, B2) are Mulliken symbols, denoting

irreducible representations (we’ll get to that…) • The symbol in the bottom left corner (Γx,y,z) is a reducible representation (we’ll get

to that, too…) • The numbers in the table provide specific information about the relationship between

the representations and the symmetry operations (we’ll see this shortly).

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So let’s have a look at how symmetry operations can be treated as “groups”… Consider the C3v point group: e.g., NH3 • C3v has six different symmetry operators {E, C3

1, C32, σv, σv′, σv″}

These operators can be combined into products, e.g.:

C31 x C3

1 = C32 σv x σv′ = C3

2 • For the set of operators we can define a multiplication table.

The multiplication table describes all products of any two operators in our sample set:

E C31 C3

2 σv σv′ σv″ E E C3

1 C32 σv σv′ σv″

C31 C3

1 C32 E σv′ σv″ σv

C32 C3

2 E C31 σv″ σv σv′

σv σv σv″ σv′ E C32 C3

1 σv′ σv′ σv σv″ C3

1 E C32

σv″ σv″ σv′ σv C32 C3

1 E • The above operator multiplication table fulfills the four criteria that define a group … and so do the symmetry operators that define ANY point group! The four criteria that define a “group” are: 1) In each group there exists an operator that commutes with all other operators in the

group and leaves them unchanged:

E x C32 = C3

2 x E = C32

E is the identity operator (neutral element) of the group (from the German Einheitsoperator = unity operator) 2) The product of two group elements must also be an element of the group.

No new operators mysteriously appear! i.e. the group is closed

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C3

2 x σv″ = σv′ σv′ x σv′ = E etc. (see table)

3) The products are associative, i.e. (X x Y) x Z = X x (Y x Z) for all elements

C31 x (σv′ x σv″) = (C3

1 x σv′) x σv″ C3

1 x C32 = σv″ x σv″

E = E 4) For each element Z a reciprocal element Z-1 exists.

An element and its reciprocal commute. i.e., Z x Z-1 = Z-1Z = E

There must be a reciprocal element for C31 such that: C3

1 x [C31]-1 = E

See multiplication table to find: C3

1 x C32 = E

Therefore C3

2 is the reciprocal element of C31: C3

2 = [C31]-1

Question: Which is the reciprocal element of σv″ ? Def.: A group in which the order of multiplication of the elements is irrelevant is called an Abelian group. Cyclic point groups are typically Abelian, others are usually not. The C3v point group is not Abelian, because

C32 x σv ≠ σv x C3

2 Def.: If a subset of a group is a group by itself, it is called a subgroup. e.g., The following table is a subgroup of our example multiplication table:

E C31 C3

2 E E C3

1 C32

C31 C3

1 C32 E

C32 C3

2 E C31

Def.: The number of elements in a (sub)group is called it order. - The order of the C3v point group is 6. - The order of the above subgroup is 3. Def.: Two group elements X and Y are conjugate, if the following equality holds for any X and Z:

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Z-1 x X x Z = Y We say that X and Y are conjugate. We say that Y is the similarity transform of X by Z.

This equality is called a similarity transformation.

[C31]-1 x σv″ x C3

1 C3

2 x σv″ x C31

σv′ x C31

= σv

So, σv″ and σv are conjugate and σv is the similarity transform of σv″ by C3

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Def.: A class is a complete set of operators that are conjugated to each other

e.g. {E},{ C31, C3

2} and { σv, σv′, σv″ } in the C3v point group. The number of elements in a class is an integral divider of the group order. HOMEWORK: Prove to yourself that ALL POSSIBLE similarity transforms of σv are members of the { σv, σv′, σv″ } class. HOMEWORK: For boric acid (shown below), determine all applicable symmetry operations and create a multiplication table. Prove that these symmetry operations meet all four criteria to form a group. To which point group does boric acid belong? Is this an Abelian point group?

Boric acid

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3.3. Matrix Representations of Symmetry Operations • Mathematically, the best way to understand how the symmetry operations of a

molecule influence its properties is to study the sets of matrices which mirror, by their group table, those same operations.

• Such a set of matrices are said to form a representation of the point group.

• Essentially, when we introduce a matrix representation, we are replacing the

geometry of symmetry operations with the algebra of matrices.

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Brief review of matrix algebra Matrix: Rectangular array of numbers – here we will only need quadratic matrices.

i.e., # rows = # columns:

• Multiplication of matrix with a vector x (Dot products!)

€

ym

= amjxj

j

∑ (row x column)

A • x = Y

i.e., y 1 = a11x1 + a12x2 + … + a1nxn

e.g.:

• Product of two matrices A • X = Y

i.e., y11 = a11x11 + a12x21 + a13x31 + … + a1nxn1

e.g.:

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Any symmetry operation σ, i, Cn, etc. can mathematically be represented by a matrix. • Consider point group C2h, which consists of the operations E, σxy, i, C2

What do these operations do to a general vector (x, y, z)?

Let’s operate on the vector r = (x, y, z)

Identity operator E:

Applying the identity operator E to a vector (x,y,z) should give us back the vector (x,y,z). The same result is achieved by multiply the vector by the above 3x3 matrix. CONCLUSION: The identity operator E can be mathematically represented by the

above 3x3 matrix!

Inversion center i:

Mirror plane σxy:

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Rotation C2: … a bit more complicated. Let’s derive the matrix for a general rotation of vector r1 of length l into vector r2 first: Recall your high school trigonometry:

sin α = opposite/hypotenuse cos α = adjacent/hypotenuse x1 = l cos α y1 = l sin α x2 = l cos (α + θ) y2 = l sin (α + θ) Using the trigonometric addition theorems:

cos (α + θ) = cos θ cos α - sin θ sin α

sin (α + θ) = sin θ cos α + cos θ sin α x2 = l cos θ cos α - l sin θ sin α = x1 cos θ - y1 sin θ y2 = l sin θ cos α + l cos θ sin α = x1 sin θ + y1 cos θ Written as a matrix:

For C2 θ = 180 ° → cos θ = -1 and sin θ = 0 In all three dimensions the matrix representing C2 therefore is:

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• The matrices that represent the symmetry operations that define C2h form a group.

• The matrices and symmetry operators are isomorphous (have the same form). Multiplication table for C2h:

E C2 σh i E E C2 σh i C2 C2 E i σh σh σh i E C2

i i σh C2 E

e.g.: C2 x σh = i:

Some more general properties of matrices we need to know: • The product of two diagonal matrices (all elements aij with i ≠ j = 0) is also a diagonal

matrix, i.e.:

• A block diagonal matrix consists of submatrices of quadratic shape; the product of

two block diagonalized matrices is again a block diagonalized matrix of the same form:

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Def.: For any cooperative matrix there exists an inverse matrix such that

Q x Q-1 = 1 where 1 = unit matrix

→ inverse elements in multiplication tables !!! • Using inverse matrices we can perform similarity transformations with matrices:

B = Q-1 x A x Q

We say matrices A and B are conjugated. Reducible and irreducible representations • Consider a set of matrices {A, B, C} that represent the symmetry operations of a

point group.

• Similarity transformations can be performed on them until we arrive at a set of block diagonalized matrices {A’, B’, C’}, i.e.:

A’ = Q-1 x A x Q

B’ = Q-1 x B x Q

C’ = Q-1 x C x Q

• These matrices {A’, B’, C’} are still representations of the same operations

represented by {A, B, C}, e.g. if A x B = C then A’ x B’ = C’ must also be true. Proof (direct):

A’ x B’ = (Q-1 x A x Q) x (Q-1 x B x Q) = Q-1 x A x (Q x Q-1) x B x Q = Q-1 x A x B x Q = Q-1 x C x Q = C’

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The block-diagonalized matrices look like this …

… which means A’1 x B’1 = C’1 ; A’2 x B’2 = C’2 ; A’3 x B’3 = C’3

… which means that the individual blocks of the block diagonalized matrix behaves just like the entire matrix itself !!! • Each small block An’, Bn’, Cn’ of the matrices A’, B’, C’ is a complete new reduced

representation of the same operation as represented by the entire matrices A’, B’, C’ (or A, B, C) – the overall information content is the same !

• An’, Bn’, Cn’ fulfill the same multiplication table as the A’, B’, C’ (or A, B, C), i.e., they also are a valid representation of the group.

• If the representations An’, Bn’, Cn’ cannot be further simplified by similarity

transformations they are called irreducible representations of the particular symmetry operation to which they relate.

• For each point group there exists an infinite number of reducible representations, but

only one finite set of irreducible representations.

• The irreducible representation are denoted by Mulliken symbols:

Symbol Property A symmetric under n-fold rotation B anti-symmetric under n-fold rotation E two-dimensional T three-dimensional

Index Position Property 1 below symmetric under σv or C2 ⊥ Cn

2 below anti-symmetric under σv or C2 ⊥ Cn g below symmetric under i u below anti-symmetric under i ‘ above symmetric under σh if i is not present ‘ ‘ above anti-symmetric under σh if i is not present + above symmetric under σv in D∞h

- above anti-symmetric under σv in D∞h