Question

What is the symmetry span (or the linear combination of symmetry species) that represents the atomic motions of H₂O? Show how you solved the problem step by step in quantum mechanic conditions. Please show all work.

Answer 1

Method uses the known symmetries of the stretching modes of the molecule. To investigate this method, the construction of the SALCs of water is examined. Water has three vibrational states, 2a₁+b₂. Two of these vibrations are stretching modes. One is symmetric with the symmetry A₁, and the other is antisymmetric with the symmetry B₂. vectors represented by b_j as the basis set. The vectors demonstrate the irreducible representations of molecular vibrations.

The SALCs of water can be composed by creating a linear combination of the stretching vectors.

ϕ(A1)=b1+b2

ϕ(A1)=b1+b2

and

ϕ(B1)=b1−b2

ϕ(B1)=b1−b2

Normalization

The final step in constructing the SALCs of water is to normalize expressions. To normalize the SALC, multiply the entire expression by the normalization constant that is the inverse of the square root of the sum of the squares of the coefficients within the expression.

ϕi=N∑jcijbj

ϕi=N∑jcijbj

ϕ(A1)=12–√(b1+b2)

ϕ(A1)=12(b1+b2)

and

ϕ(B1)=12–√(b1−b2)

ϕ(B1)=12(b1−b2)

Normalizing the SALCs ensures that the magnitude of the SALC is unity, and therefore the dot product of any SALC with itself will equal one.