Hessian computation

Fig. 9 Eigenvalues of the energy-difference Hessian computed at the Franck-Condon point of benzene in the 28-dimensional space orthogonal to the pseudo-branching plane. The labels refer to the most similar normal modes of So benzene (Wilson s convention). The dominant local motions are indicated in boxes (reprinted with permission from [31])...

Hulsmann, M., Kopp, S., Huber, M., Reith, D. Utilization of efficient gradient and Hessian computations in the force field optimization process of molecular simulations. Comput. Sci. Disc. 6, 015005 (2013)... 

We now review the general approach to the gradient and Hessian computations using the methods of Almlof and Taylor [30] and apply it specifically to the Ehrenfest wavefunction. We also explain the approximations used in computing the gradient and Hessian. 

Fig. 2 Error in conservation of energy. The error in the total energy is compared for two different integration algorithms involving the approximate gradient computation only, or the approximate gradient and approximate Hessian computations using the fifth order polynomial fit. The mass-weighted step size is 0.03 amu bohr (about 0.3 fs). This illustrates how the polynomial fit performs significantly better (error below 2 x 10 kcal/mol for 50 fs)...

A Hessian computed by semi-empirical MO methods. Some empirical adjustment of the second derivatives is usually necessary, since the semi-empirical methods tend to overestimate some terms and underestimate others. 

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

Unconstrained optimization methods [W. II. Press, et. ah, Numerical Recipes The An of Scieniific Compulime.. Cambridge University Press, 1 9H6. Chapter 101 can use values of only the objective function, or of first derivatives of the objective function. second derivatives of the objective function, etc. llyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. TlyperChem does not use optimizers that compute the full set of second derivatives (th e Hessian ) because it is im practical to store the Hessian for mac-romoleciiles with thousands of atoms. A future release may make explicit-Hessian meth oils available for smaller molecules but at this release only methods that store the first derivative information, or the second derivatives of a single atom, are used. 

The full Newton-Raphson method computes the full Hessian A of second derivatives and then computes a new guess at the 3X coordinate vector X, according to... 

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/. 

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. 

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 ... 

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... 

The optimization of a transition structure will be much faster using methods for which the Hessian can be analytically calculated. For methods that incrementally compute the Hessian (i.e., the Berny algorithm), it is fastest to start with a Hessian from some simpler calculation, such as a semiempirical calculation. Occasionally, dilficulties are encountered due to these simpler methods giving a poor description of the Hessian. An option to compute the initial Hessian at the desired level of theory is often available to circumvent this problem at the expense of additional CPU time. 

The Eigenvector Following method is in some ways similar to the Newton-Raph son method. Instead of explicitly calculating the second derivatives, it uses a diagonalized Hessian matrix to implicitly give the second derivatives of energy with respect to atomic displacements. The initial guess is computed empirically. 

---