The Hessian Matrix

A single harmonic oscillator constrained to the a -axis has one force constant k that—stretching a point—we might think of A as a 1 x 1 force constant matr ix. Two oscillators that interact with one another lead to a 2 x 2 force constant matr ix 

In a molecule, these latter two force constants are equal to one another because the coupling of atom 1 for atom 2 is the same as the coupling of atom 2 for atom 1. 

The force constants k 2 and k2 are the off-diagonal elements of the matrix. If they are zero, the oscillators are uncoupled, but even if they are not zero, the K matrix takes the simple fomi of a symmetrical matrix because ki2 = k2. The matrix is symmetrical even though may not be equal to k22- 

The force constants are second derivatives of the potential energy with respect to infinitesimal displacements of mass 1 and mass 2. 

This kind of matr ix is called a Hessian matrix. The derivatives give the cmvatme of V(x[,X2) in a two-dimensional space because there are two masses, even though both masses are constrained to move on the -axis. As we have already seen, these derivatives are pari of the Taylor series expansion 

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The matrix M contains atomic masses on its diagonal, and the Hessian matrix F contains the second derivatives of the potential energy evaluated at Xq. 

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. 

Tlie eigenvalues are A = 4 and A = 8. Thus both eigenvalues are positive and the point is minimum. At the point (0,0) the Hessian matrix is... 

If there were three masses moving on the x-axis and interacting with one another, the Hessian matrix would be 3 x 3... 

A diva It MM3 wilh Ihe cumrnand mm3. Answer questions file etheiie.mm3, parameter file Enter (default) line number 1, option 2. The defaull parameter sel is Ihe MM3 parameler sel don t ehange il. The line number starts Ihe system reading on the first line of your input file, and option 2 is the block diagonal followed by full matrix minimi7 ation mentioned at the end of the section on the Hessian matrix. You will see intermediate atomic coordinates as the system minimises the geometry, followed by a final steiic eireigy, Kird with 0, output Enter, cooidinates Enter,... 

The second energy derivatives with respect to the x, y, and z directions of centers a and b (for example, the x, y component for centers a and b is Hax,by = (3 E/dxa3yb)o) form the Hessian matrix H. The elements of H give the local curvatures of the energy surface along the 3N cartesian directions. 

If a program is given a molecular structure and told to find a transition structure, it will first compute the Hessian matrix (the matrix of second derivatives... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

A more sophisticated version of the sequential univariate search, the Fletcher-Powell, is actually a derivative method where elements of the gradient vector g and the Hessian matrix H are estimated numerically. 

This equation determines a rank-1 matrix, and the eigenvector of its only one nonzero eigenvalue gives the direction dictated by the nonadiabatic couphng vector. In the general case, the Hamiltonian differs from Eq.(l), and the Hessian matrix has the form... 

The above formula is obtained by differentiating the quadratic approximation of S(k) with respect to each of the components of k and equating the resulting expression to zero (Edgar and Himmelblau, 1988 Gill et al. 1981 Scales, 1985). It should be noted that in practice there is no need to obtain the inverse of the Hessian matrix because it is better to solve the following linear system of equations (Peressini et al. 1988)... 

As seen by comparing Equations 5.6 and 5.12 the steepest-descent method arises from Newton s method if we assume that the Hessian matrix of S(k) is approximated by the identity matrix. 

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