Second derivative matrix

A different approach comes from the idea, first suggested by Flelgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Flessian (second-derivative) matrix, so if... 

Th c Newton-Raph son block dingotial method is a second order optim izer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. I hese derivatives provide information ahont both the slope and curvature of lh e poten tial en ergy surface, Un like a full Newton -Raph son method, the block diagonal algorilh m calculates the second derivative matrix for one atom at a lime, avoiding the second derivatives with respect to two atoms. 

To fin d a first order saddle poiri t (i.e., a trail sition structure), a m ax-imiim must be found in on e (and on/y on e) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) bein g varied. So, a tran sition structu re is ch aracterized by th e poin t wh ere all th e first derivatives of en ergy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. 

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. 

There are some systems for which the default optimization procedure may not succeed on its own. A common problem with many difficult cases is that the force constants estimated by the optimization procedure differ substantially from the actual values. By default, a geometry optimization starts with an initial guess for the second derivative matrix derived from a simple valence force field. The approximate matrix is improved at each step of the optimization using the computed first derivatives. 

The simple-minded approach for minimizing a function is to step one variable at a time until the function has reached a minimum, and then switch to another variable. This requires only the ability to calculate the function value for a given set of variables. However, as tlie variables are not independent, several cycles through tlie whole set are necessary for finding a minimum. This is impractical for more than 5-10 variables, and may not work anyway. Essentially all optimization metliods used in computational chemistry tlius assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some metliods also assume that tlie second derivative matrix, the Hessian H, can be calculated. 

Besides the above-mentioned problems with step control, there are also other computational aspects which tend to make the straightforward NR problematic for many problem types. The true NR method requires calculation of the full second derivative matrix, which must be stored and inverted (diagonalized). For some function types a calculation of the Hessian is computationally demanding. For other cases, the number of variables is so large that manipulating a matrix the size of the number of variables squared is impossible. Let us address some solutions to these problems. 

A significant advantage is that the constrained optimization can usually be carried out using only the first derivative of tlie energy. This avoids an explicit, and computationally expensive, calculation of the second derivative matrix, as is nomially required by Newton-Raphson techniques. 

In the Atoms In Molecules approach (Section 9.3), the Laplacian (trace of the second derivative matrix with respect to the coordinates) of the electron density measures the local increase or decrease of electrons. Specifically, if is negative, it marks an area where the electron density is locally concentrated, and therefore susceptible to attack by an electrophile. Similarly, if is positive, it marks an area where the electron density is locally depleted, and therefore susceptible to attack by a... 

FIGURE 4.3. Illustrating the effectiveness of different minimization schemes. The steepest-deicent method requires many steps to reach the minimum, while the Newton-Raphson method locates the minimum in a few steps (at the expense, however, of evaluating the second derivative matrix). 

Equation (4.9) requires the evaluation of the second derivative matrix F, which is quite involved. Alternatively, one can use the conjugated gradient... 

The treatment described above (which was introduced in Ref. 1) is much simpler than the standard treatment (which uses internal coordinates b, 0, large molecules or small proteins, evaluating the second derivative matrix F numerically, using analytical first derivatives. 

The second derivative matrix elements give rise to three terms ... 

The second derivatives can be calculated numerically from the gradients of the energy or analytically, depending upon the methods being used and the availability of analytical formulae for the second derivative matrix elements. The energy may be calculated using quantum mechanics or molecular mechanics. Infrared intensities, Ik, can be determined for each normal mode from the square of the derivative of the dipole moment, fi, with respect to that normal mode. 

Solve the second equation for 5, which produces 8 = p (X X) 1X y = p b. Insert this implicit solution into the first equation to produce n/p = Z, y (py - px/b). By taking p outside the summation and multiplying the entire expression by p, we obtain n = p2 Z, y, (y, - x/b) or p2 = n/[Z y (y - x/b)]. This is an analytic solution for p that is only in terms of the data - b is a sample statistic. Inserting the square root of this result into the solution for 8 produces the second result we need. By pursuing this a bit further, you canshow that the solution for p2 is just n/e e from the original least squares regression, and the solution for 8 is just b times this solution for p. The second derivatives matrix is... 

Derive the second derivatives matrix and show that the asymptotic covariance matrix for the maximum likelihood estimators is... 

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