Hessian equation

The matrix in front of T is the partitioned orbital Hessian. Equation (4 37) is not very practical, since it involves the inverse of the Cl part of the Hessian. But suppose that we work in a configuration basis (I0>, K>), where a is diagonal, that is we start each iteration by solving the Cl problem to all orders. The matrix a is then diagonal with the matrix elements =(EK - Eq),... 

The solution of the linear equations (equation 66) is usually less time-consuming than the calculation of the two-electron contribution to the Hessian (equation 64), but is treated in somewhat more detail here in order to demonstrate some general principles. The number of orbital rotations is usually so large that the electronic Hessian cannot be constructed or stored explicitly. Instead, iterative techniques are used, where the key step is the evaluation of matrix-vector products such as... 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... 

The second tenn in equation B3.5.11 deserves connnent. This tenn shows that Hessian (second-derivative) matrices... 

This loop is iterated A i times to cover the interval At kiAT = At) to produce Note the Hessian/vector products in the second equation of... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

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... 

Minimization of this quantity gives a set of new coefficients and the improved instanton trajecotry. The second and third terms in the above equation require the gradient and Hessian of the potential function V(q)- For a given approximate instanton path, we choose Nr values of the parameter zn =i 2 and determine the corresponding set of Nr reference configurations qo(2n) -The values of the potential, first and second derivatives of the potential at any intermediate z, can be obtained easily by piecewise smooth cubic interpolation procedure. 

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. 

The ratio of the largest to the smallest eigenvalue of the Hessian matrix at the minimum is defined as the condition number. For most algorithms the larger the condition number, the larger the limit in Equation 5.5 and the more difficult it is for the minimization to converge (Scales, 1985). 

We are now able to obtain the Hessian matrix of the objective function S(k) which is denoted by H and is given by the following equation... 

The Gauss-Newton method arises when the second order terms on the right hand side of Equation 5.20 are ignored. As seen, the Hessian matrix used in Equation 5.11 contains only first derivatives of the model equations f(x,k). Leaving out the second derivative containing terms may be justified by the fact that these terms contain the residuals e, as factors. These residuals are expected to be small quantities. 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

Compute Akc,, ) by solving Equation 5.12b but in this case the Hessian matrix H(k) has been replaced by that given by Equation 5.22. 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

The condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

More variables are retained in this type of NLP problem formulation, but you can take advantage of sparse matrix routines that factor the linear (and linearized) equations efficiently. Figure 15.5 illustrates the sparsity of the Hessian matrix used in the QP subproblem that is part of the execution of an optimization of a plant involving five unit operations. 

The four blocks of V can be alternatively expressed in terms of the principal geometric derivatives defining the generalized Hessian of Equation 30.8. This can be accomplished first by expressing AQ as function of AN and AF, using the second Equation 30.9, and then by inserting the result into the first Equation 30.9 ... 

A reference to the second Equation 30.29 shows that the effective geometrical Hessian of an open molecular system differs from that of the closed system (Equation 30.8) by the extra CT contribution involving the geometrical softnesses and NTT. One finally identifies the corresponding blocks of G by comparing the general relations of Equation 30.28 with the explicit transformations of Equation 30.29,... 

The electronic-nuclear coupling in molecules is also detected in the other partial Legendre-transformed representation H(/r, Q), which defines the combined Hessian G of Equation 30.27. Its first diagonal derivative,... 

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