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Vibrational frequency calculation: concept simplified?


Vibrational frequency calculation holds a key that can decide whether the obtained optimized structure is a minimum or a transition state or some thing else. I always wanted a simplistic adaptation with minimal understandable and required information that in a matter of minutes can help a new person understand the concept as easily as possible.

which makes it really easy for anybody. So, what we least need to know when you submit a frequency calculation.

In frequency calculation also called force calculation, we are solving a hessian matrix. What is a Hessian matrix?
The Hessian matrix is the matrix of second derivatives of the energy with respect to geometry.

What is the trouble in frequency calculation?

Although first derivatives are relatively easy to calculate, second derivatives are not. To overcome this problem, the double derivative is evaluated in terms of first derivatives. Here, initially for a given geometry, corresponding first derivative is evaluated. Then, the geometry is subjected to a finite displacement. After the SCF calculation for this displaced geometry, once again first derivatives are evaluated. The second derivatives are then calculated as a difference of the first derivatives for these two geometries divided by the step size. The unidirectional displacement based second derivative calculation is called single-sided derivative.

Owing to the sensitivity of the hessian towards geometry, they are calculated at stationary points only and using double-sided derivatives i.e. double derivatives calculated considering displacement in either direction w.r.t. initial geometry

The elements of the Hessian are defined as:
\begin{displaymath}H_{i,j} = \frac{\delta^2E}{\delta x_i\delta x_j}\end{displaymath}

Thus employing double sided derivative for the evaluation of each element is then done in two steps.

Each atomic coordinate xi is initially first incremented by a small amount and the gradient is calculated. Then the coordinate is decremented w.r.t. to the original position and again the gradient corresponding to this displacement is calculated. The second derivative is then obtained from the difference of the two derivatives and the step size.
\begin{displaymath}H_{i,j} = \frac{(\frac{\delta E}{\delta x_i})_{_{+0.5\Delta x......rac{\delta E}{\delta x_i})_{_{-0.5\Delta x_j}}}{\Delta x_j}.\end{displaymath}
This is done for all 3N Cartesian coordinates.

In order to calculate the vibrational frequencies, the Hessian matrix is first mass-weighted:
\begin{displaymath}H^m_{i,j} = \frac{H_{i,j}}{\sqrt{M_i*M_j}}.\end{displaymath}

Diagonalization of this matrix yields the force constants of the system.
Diagonalization of this matrix yields eigenvalues, e, (force constants ) which represent the quantities (k/m)1/2 , from which the vibrational frequencies can be calculated:
\begin{displaymath}\bar{\nu}_i = \frac{1}{2\pi c}\sqrt{\epsilon_i}.\end{displaymath}

P.S: I have used some part of the text from the webpage as it is because there was no better and simplified way to explain it.

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