The vibration of polyatomic molecules involves stretching, bending and torsional motions.

The classical kinetic energy of a vibrating -nuclei molecule is

where , and are the displacements of the -th nucleus from its equilibrium position of (see diagram below).

Eq85 can be simplified by changing the Cartesian displacement coordinates to * mass-weighted displacement coordinates* , where

which gives

where and is a column vector with elements .

The potential energy of a vibrating -nuclei molecule is also a function of and hence a function of . We can approximate the multivariable function by expanding it in a Maclaurin series around the equilibrium positions:

which can be abbreviated to

When the nuclei are at their equilibrium positions, the slope of is zero, i.e. . For small vibrations, where is close to , we ignore all the higher power terms, giving

or in matrix notation

where is the column vector with elements and is a matrix, called the ** Hessian matrix**, with elements .

###### Question

Show that .

###### Answer

We want to derive an eigenvalue equation that resembles the harmonic oscillator equation so that we can easily solve for the vibrational energies of a polyatomic molecule. However, the Hamiltonian formed by combining eq86 and eq87 doesn’t resemble the Hamiltonian of a harmonic oscillator. Therefore, we need to further simply eq86 and eq87 with a change of variables.

With regard to eq88, the potential energy of a system is a real-valued physical property. This implies that must consists of real-valued entries . Since , the matrix is both real and symmetric.

###### Question

i) Prove that eigenvectors of a real and symmetric matrix with distinct eigenvalues are orthogonal.

ii) What if some of the eigenvalues are degenerate?

###### Answer

i) Let and , where . Multiply the first eigenvalue equation by , we have

where the 2^{nd} equality uses the identity mentioned in this article, while the 3^{rd} equality employs the property of a symmetrc matrix.

Since , we have , i.e. is orthogonal to .

ii) When some eigenvalues are degenerate, the corresponding eigenvectors belong to the same eigenspace. Since we can always select a set of linearly independent eigenvectors that span the eigenspace, these eigenvectors can be orthogonalised using the Gram-Schmidt process without changing the eigenvalue. So, for a real symmetric matrix, we can always find a basis of eigenvectors that are orthogonal to each other, regardless of whether the eigenvalues are distinct or degenerate.

Let the orthogonal eigenvectors of be the columns of the matrix , where . As shown in this article, we can express the eigenvalue problem as

where is the diagonal eigenvalue matrix with diagonal entries .

Computing the secular equation allows us to find , and solving the simultaneous equations of , which is equivalent to (where , enables us to find .

Consider the following linear combinations of mass-weighted displacement coordinates

where is the column vector with elements .

Since are the components of the -th orthogonal eigenvector of and each basis describes the displacement of a nucleus, , known as the ** normal coordinates**, is a set of orthogonal vectors that expresses independent motions (called degrees of freedom) of the molecule as a whole.

Substituting in eq88

Since , we have or and eq86 becomes

Combining eq90 and eq91, the classical total energy of the system is , where we have set . This is because we are usually interested in computing transition energies and the eigenvalue corresponding to is subtracted out when we calculate vibrational transition energies for a given electronic state. The Hamiltonian is derived by replacing the classical observables by quantum-mechanical operators:

###### Question

Show that the classical kinetic energy term can be replaced by the quantum-mechanical operator .

###### Answer

From eq89, . Substituting the momentum operator in , we have . Using the single-variable chain rule , we have , which when substituted in gives . We can find similar expressions for . So, . Since is a function of , and is a function of , the multi-variable chain rule gives . Relabelling as , we have and therefore, , which when substituted in gives . Substituting this last expression of in gives .

The kinetic energy term in eq92 involves a change in , which is a function of and hence a function of the displacement coordinates , and , which are relative to the equilibrium positions of the nuclei. In other words, is a function of the displacements of all the nuclei from their respective equilibrium positions. For the molecule as a whole, all the equilibrium positions of the nuclei can be taken as the center of mass of the molecule. Since translational and rotational motions of the molecule involve no displacement relative to the molecule’s centre of mass, for such motions. Furthermore, translational and rotational motions are purely kinetic with no potential energy component. Therefore, we can rewrite eq92 as:

where and for a non-linear molecule and a linear molecule respectively.

Each is called a ** normal mode** and corresponds to a unique vibrational motion of the molecule. The Schrodinger equation is

where the separation of variables technique allows us to approximate the total vibrational wavefunction as and hence, .

It follows that each describes a normal mode of the molecule. Comparing with eq4a, eq93 is the Schrodinger equation for a harmonic oscillator of unit mass and force constant . This implies that (see eq46)

where is the normalisation constant for the Hermite polynomials .

Therefore, eq93 can be solved by computing individual Schrodinger equations. With reference to eq21, each solution gives

where is the vibrational quantum number of the -th normal mode of the molecule.

The total vibrational energy is

The vibrational state of a polyatomic molecule is characterised by quantum numbers. For example the vibrational ground state of , where , is . The ground state vibrational wavefunction of any polyatomic molecule is

Eq97 is a Gaussian function that is symmetrical in . The implication of this in group theory is that the ground state vibrational wavefunction of any polyatomic molecule is totally symmetric under all symmetry operations of the point group that the molecule belongs to. In fact, group theory provides an easy way to determine the vibrational modes of a molecule and to analyse the possible transitions between different vibrational states.