Theory Normal Mode Decomposition

Decomposition of normal modes

The B-matrix

Given a molecule with $N$ atoms, there are $3N$ Cartesian coordinates or $\vec{X}$ or $S = 3N-6(5)$ internal coordinates $\vec{X}$.

• $\vec{X}^T = (x_1,x_2,z_1 ... x_N,y_N,z_N)$
• $\vec{R}^T = (r_1,r_2 ... r_S)$

Using the Wilson B-Matrix, the coordinate systems can be transformed into each other.

$\vec{R} = B \cdot \vec{X}$

Force constant matrixes

The potential energy of a given molecule can be approximated by the quadratic force constants, which are second derivatives of the energy E with respect to Cartesian or internal coordinates:

• $f_{ij} = \frac{\partial^2 E}{\partial x_i \partial x_k}$
• $F_{ij} = \frac{\partial^2 E}{\partial r_i \partial r_j}$

Now using the Wilson B-Matrix we defined bevore the matrices can be transformed into each other (Matrix-Transformation

• $F = (\tilde{B}^{-1})^T f \tilde{B}^{-1}$
• $f = \tilde{B}^T F \tilde{B}$

Here $\tilde{B}$ is the augmented B-matrix where additional rows are added in order to make the B-matrix quadratic $(3N \times 3N)$. This additional rows represent rotational and translational motions.

Normal Modes and the Description of normal modes using internal coordinates

The mass-weighted Cartesian force constant matrix

• $f' = (M^{-1/2})^T f M^{-1/2}$

can be diagonalized (Matrix-Diagonalization)

• $\Lambda = L^T f' L$

where the matrix $\Lambda$ is the diagonal matrix and contains the eigenvalues $\lambda_i$ on the diagonal usually denoted as harmonic frequencies. The $L$ matrix $(3N \times 3N)$ contains eigenvectors usually called normal modes. In the diagonal matrix $M^{-1/2}$ the elements are the reciprocal square roots of the atomic masses.

Now the diagonalization can be rewritten using the (not mass-weighted) Cartesian force constant matrix

• $\Lambda = l^T f l$

where one defined $l = M^{-1/2}L$. Now the $l$ matrix contains eigenvectors which can be denoted as mass-weighted normal modes in Cartesian coordinates. Using the B-Matrix transformation we can now transform the Cartesian force constant matrix $f$ into the internal force constant matrix $F$ and redefine the diagonalization

• $\Lambda = l^T f l = l^T(B^T F B) l$

now if we define the $D$-matrix as $D = Bl$ this matrix contains eigenvectors that describe the mass-weighted normal modes in internal coordinates. Thus, the diagonalization of the force constant matrix using the $D$-matrix can be written as

• $\Lambda = D^T F D$