Hessian matrices

Bofill J M 1994 Updated Hessian matrix and the restricted step method for locating transition structures J. Comput. Chem. 15 1... 

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. 

Pragmatically, the procedure considers only one atom at a lime, computiiig the 3x3 Hessian matrix associated with that atom and the 3 compon en IS of Ihe gradien t for that atom and then inverts the 3x3 matrix and obtains new coordinates for the atom accord-ingto the Newton-Raphson form u la above. It then goes on lothe next atom and moves it in the same way. using first and second derivatives for the second atom that include any previous nioiioii of atom s. 

Xk) is the inverse Hessian matrix of second derivatives, which, in the Newton-Raphson method, must therefore be inverted. This cem be computationally demanding for systems u ith many atoms and can also require a significant amount of storage. The Newton-Uaphson method is thus more suited to small molecules (usually less than 100 atoms or so). For a purely quadratic function the Newton-Raphson method finds the rniriimum in one step from any point on the surface, as we will now show for our function f x,y) =x + 2/. 

At each iteration k, the new positions are obtained from the current positions x, the gradient gj. and the current approximation to the inverse Hessian matrix... 

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... 

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... 

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,... 

These transformation eoeffieients Crj,k can be used to earry out a unitary transformation of the 9x9 mass-weighted Hessian matrix. In so doing, we need only form bloeks... 

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. 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

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