Topic
Second derivative
About: Second derivative is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 57765 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: In this paper, a modified conjugate gradient algorithm for geometry optimization is presented for use with ab initio MO methods, where the second derivative matrix rather than its inverse is updated employing the gradients.
Abstract: A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.
3,740 citations
TL;DR: A procedure is derived for extracting the observed information matrix when the EM algorithm is used to find maximum likelihood estimates in incomplete data problems and a method useful in speeding up the convergence of the EM algorithms is developed.
Abstract: A procedure is derived for extracting the observed information matrix when the EM algorithm is used to find maximum likelihood estimates in incomplete data problems. The technique requires computation of a complete-data gradient vector or second derivative matrix, but not those associated with the incomplete data likelihood. In addition, a method useful in speeding up the convergence of the EM algorithm is developed. Two examples are presented.
2,357 citations
1 Jan 2009
TL;DR: Most changes to the variables are an approximate solution to a trust region subproblem, using the current quadratic model, with a lower bound on the trust region radius that is reduced cautiously, in order to keep the interpolation points well separated until late in the calculation, which lessens damage from computer rounding errors.
Abstract: BOBYQA is an iterative algorithm for finding a minimum of a function F(x), x2R n , subject to bounds axb on the variables, F being specified by a "black box" that returns the value F(x) for any feasible x. Each iteration employs a quadratic approximation Q to F that satisfies Q(y j )= F(y j ), j =1 ,2,...,m, the interpolation points y j being chosen and adjusted automatically, but m is a prescribed constant, the value m =2 n+1 being typical. These conditions leave much freedom in Q, taken up when the model is updated by the highly successful technique of minimizing the Frobenius norm of the change to the second derivative matrix of Q. Thus no first derivatives of F are required explicitly. Most changes to the variables are an approximate solution to a trust region subproblem, using the current quadratic model, with a lower bound on the trust region radius that is reduced cautiously, in order to keep the interpolation points well separated until late in the calculation, which lessens damage from computer rounding errors. Some other changes to the variables are designed to improve the model without reducing F. These techniques are described. Other topics include the starting procedure that is given an initial vector of variables, the value of m and the initial trust region radius. There is also a new device called RESCUE that tries to restore normality if severe loss of accuracy occurs in the matrix calculations of the updating of the model. Numerical results are reported and discussed for two test problems, the numbers of variables being between 10 and 320.
1,462 citations
TL;DR: A numerical method, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented and the efficiency and applicability of the method are shown.
987 citations
TL;DR: The theory of analytic second derivatives for the coupled-cluster (CC) model is presented in this paper, where analytic expressions for the second derivative of the energy are given in terms of the response (or relaxeded) density, discussed in part I, and the first-derivative t amplitudes for efficient evaluation.
Abstract: Second derivatives of the energy correspond to second‐order response properties and molecular force constants. Currently, both the theory and application of analytic second derivatives in many‐body methods are limited to second‐order perturbation theory. The general theory of analytic second derivatives for the coupled‐cluster (CC) model is presented. The analytic expressions for the second derivative of the energy are given in terms of the response (or ‘‘relaxed’’) density, discussed in part I, and the first‐derivative t amplitudes for efficient evaluation. Explicit expressions for the second derivatives of the coupled‐cluster singles, doubles, and linearized triples model (CCSDT‐1) are presented. Analytic derivatives for the finite‐order MBPT(3) and MBPT(4) models are derived as special cases of the theory.
559 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 5 |
| 2024 | 7 |
| 2023 | 19 |
| 2022 | 30 |
| 2021 | 95 |
| 2020 | 120 |