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Here, the force constant matrix, or the hessian matrix H holds information about the orientation of the nodes, but not about the type of the interaction (such is whether the interaction is covalent or non-covalent, hydrophobic or non-hydrophobic, etc.). In addition, the distance between the interacting nodes is not considered directly. To account for the distance between the interactions we can weight each interaction between nodes i, j by the distance, sp. The new off-diagonal elements of the Hessian matrix take the below form, where p is an empirical parameter:<br />
H_{ij} = {- 1 \over {s_{ij}}^{p+2}} \begin{bmatrix} {(X_j - X_i)(X_j - X_i)} & {(X_j - X_i)(Y_j - Y_i)} & {(X_j - X_i)(Z_j - Z_i)} \\ {(Y_j - Y_i)(X_j - X_i)} & {(Y_j - Y_i)(Y_j - Y_i)} & {(Y_j - Y_i)(Z_j - Z_i)} \\{(Z_j - Z_i)(X_j - X_i)} & {(Z_j - Z_i)(Y_j - Y_i)} & {(Z_j - Z_i)(Z_j - Z_i)} \end{bmatrix}
The counterpart of the [[Kirchhoff matrix]] Γ of the GNM is simply (1/γ) Η in the ANM. Its decomposition yields 3N - 6 non-zero [[eigenvalues]], and 3N - 6 eigenvectors that reflect the respective frequencies and shapes of the individual modes. The inverse of Η, which holds the desired information about fluctuations is composed of N x N super-elements, each of which scales with the 3 x 3 matrix of correlations between the components of pairs of fluctuation vectors. The Hessian, however is not invertible, as its rank is 3N-6 (6 variables responsible to a rigid body motion). To obtain a pseudo inverse, a solution to the eigenvalue problem is obtained:<br />
H = UDU<sup>T</sup><br />
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