# Normal-Mode Analysis¶

Normal-mode analysis 5456 can be performed using GROMACS, by diagonalization of the mass-weighted Hessian $$H$$:

(1)\begin{split}\begin{aligned} R^T M^{-1/2} H M^{-1/2} R &=& \mbox{diag}(\lambda_1,\ldots,\lambda_{3N}) \\ \lambda_i &=& (2 \pi \omega_i)^2\end{aligned}\end{split}

where $$M$$ contains the atomic masses, $$R$$ is a matrix that contains the eigenvectors as columns, $$\lambda_i$$ are the eigenvalues and $$\omega_i$$ are the corresponding frequencies.

First the Hessian matrix, which is a $$3N \times 3N$$ matrix where $$N$$ is the number of atoms, needs to be calculated:

(2)\begin{aligned} H_{ij} &=& \frac{\partial^2 V}{\partial x_i \partial x_j}\end{aligned}

where $$x_i$$ and $$x_j$$ denote the atomic x, y or z coordinates. In practice, this equation is not used, but the Hessian is calculated numerically from the force as:

(3)\begin{split}\begin{aligned} H_{ij} &=& - \frac{f_i({\bf x}+h{\bf e}_j) - f_i({\bf x}-h{\bf e}_j)}{2h} \\ f_i &=& - \frac{\partial V}{\partial x_i}\end{aligned}\end{split}

where $${\bf e}_j$$ is the unit vector in direction $$j$$. It should be noted that for a usual normal-mode calculation, it is necessary to completely minimize the energy prior to computation of the Hessian. The tolerance required depends on the type of system, but a rough indication is 0.001 kJ mol$$^{-1}$$. Minimization should be done with conjugate gradients or L-BFGS in double precision.

A number of GROMACS programs are involved in these calculations. First, the energy should be minimized using mdrun. Then, mdrun computes the Hessian. Note that for generating the run input file, one should use the minimized conformation from the full precision trajectory file, as the structure file is not accurate enough. gmx nmeig does the diagonalization and the sorting of the normal modes according to their frequencies. Both mdrun and gmx nmeig should be run in double precision. The normal modes can be analyzed with the program gmx anaeig. Ensembles of structures at any temperature and for any subset of normal modes can be generated with gmx nmens. An overview of normal-mode analysis and the related principal component analysis (see sec. Covariance analysis) can be found in 57.