Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces

ABSTRACT

The theory of calculus of variations is a mathematical tool which is widely used in different scientific areas in particular in physics and chemistry. This theory is strongly related with optimisation. In fact the former seeks to optimise an integral related with some physical magnitude over some space to an extremum by varying a function of the coordinates. On the other hand, reaction paths and potential energy surfaces, in particular their stationary points, are the basis of many chemical theories, in particular reactions rate theories. We present a review where it is gathered together the variational nature of many types of reaction paths: steepest descent, Newton trajectories, artificial force induced reaction (AFIR) paths, gradient extremals, and gentlest ascent dynamics (GAD) curves. The variational basis permits to select the best optimisation technique in order to locate important theoretical objects on a potential energy surface.     

An Isosteric Triaza Analogue of a Polycyclic Aromatic Hydrocarbon Monkey Saddle

Since a few years, the interest in negatively-curved fused polycyclic aromatic hydrocarbons (PAHs) has significantly increased. Recently, the first chiral negatively-curved PAH with the topology of a monkey saddle was introduced. Herein the synthesis of its triaza congener is reported. The influence of this CH↔N exchange on photophysical and electrochemical properties is studied as well as the isomerization process of the enantiomers. The aza analogue has a significantly higher inversion barrier, which makes it easier to handle at room temperature. All experimental results are underpinned by theoretical DFT calculations.     

Rotation-Inversion Isomerization of Tertiary Carbamates: Potential Energy Surface Analysis of Multi-Paths Isomerization Using Boltzmann Statistics

Potential energy surface (PES) analyses at the SMD[MP2/6–311++G(d,p)] level and higher-level energies up to MP4(fc,SDTQ) are reported for the fluorinated tertiary carbamate N-ethyl-N-(2,2,2-trifluoroethyl) methyl carbamate ( VII ) and its parent system N,N-dimethyl methyl carbamate ( VI ). Emphasis is placed on the analysis of the rotational barrier about the CN carbamate bond and its interplay with the hybridization of the N-lone pair (NLP). All rotational transition state (TS) structures were found by computation of 1D relaxed rotational profiles but only 2D PES scans revealed the rotation-inversion paths in a compelling fashion. We found four unique chiral minima of VII , one pair each of E- and Z-rotamers, and we determined the eight unique rotational TS structures associated with every possible E/Z-isomerization path. It is a significant finding that all TS structures feature N-pyramidalization whereas the minima essentially contain sp²-hybridized nitrogen. We will show that the TS stabilities are affected by the synergetic interplay between NLP/CO₂ repulsion minimization, NLP→σ^*(CO) negative hyperconjugation, and two modes of intramolecular through-space electrostatic stabilization. We demonstrate how Boltzmann statistics must be applied to determine the predicted experimental rotational barrier based on the energetics of all eight rotamerization pathways. The computed barrier for VII is in complete agreement with the experimentally measured barrier of the very similar fluorinated carbamate N-Boc-N-(2,2,2-trifluoroethyl)-4-aminobutan-1-ol II . NMR properties of VII were calculated with a variety of density functional/basis set combinations and Boltzmann averaging over the E- and Z-rotamers at our best theoretical level results in good agreement with experimental chemical shifts δ(¹³C) and J(¹³C,¹⁹F) coupling constants of II (within 6 %).     

Distributed mirror descent method for saddle point problems over directed graphs

In this article, we consider a mini‐max multi‐agent optimization problem where multiple agents cooperatively optimize a sum of local convex–concave functions, each of which is available to one specific agent in a network. To solve the problem, we propose a distributed optimization method by extending classical mirror descent algorithms to the distributed setting. We obtain the convergence of the algorithm under wild conditions that the agent communication follows a directed graph and the related weighted matrices are row stochastic. In particular, when the weighted matrices are restricted to be doubly stochastic, we provide the explicit convergence rate of the algorithm by choosing the stepsize in a suitable way. The proposed algorithm can be viewed as a generalization of the subgradient projection methods since it utilizes a customized Bregman divergence instead of the usual Euclidean squared distance. Finally, some simulation results on a matrix game are presented to illustrate the performance of the algorithm. © 2016 Wiley Periodicals, Inc. Complexity 21: 178–190, 2016     

Preconditioning techniques for an image deblurring problem

In this paper, we consider the solution of a large linear system of equations, which is obtained from discretizing the Euler–Lagrange equations associated with the image deblurring problem. The coefficient matrix of this system is of the generalized saddle point form with high condition number. One of the blocks of this matrix has the block Toeplitz with Toeplitz block structure. This system can be efficiently solved using the minimal residual iteration method with preconditioners based on the fast Fourier transform. Eigenvalue bounds for the preconditioner matrix are obtained. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

求解广义鞍点问题的一个新的类SOR算法

本文提出了求解广义鞍点问题的一个新的类SOR迭代算法,并分析了新算法的收敛性.数值实验结果表明新算法是十分有效的.     

Finding the transition state without initial guess: the growing string method for Newton trajectory to isomerization and enantiomerization reaction of alanine dipeptide and poly(15)alanine

We report a new, high-dimensional application of a method for finding a transition state (TS) between a reactant and a product on the potential energy surface: the search of a growing string along a reaction path defined by any Newton trajectory in combination with the Berny method (Quapp, J Chem Phys (2005), 122, 174106; we have provided this algorithm on a web page). Two given minima are connected by a one-dimensional, but usually curvilinear reaction coordinate. It leads to the TS region. The application of the method to alanine dipeptide finds the TS of the isomerisation C(7 ax) --> C(5), some TSs of the enantiomerisation of C(7 ax) from L-form to quasi-D-form, and it finds the TS region of a transition of a partly unfolded, bent structure which turns back into a mainly alpha-helix in the Ac(Ala)(15)NHMe polyalanine (all at the quantum mechanical level B3LYP/6-31G: the growing string calculation is interfaced with the Gaussian03 package). The formation or dissolvation of some alpha- or 3(10)-hydrogen bonds of the helix are discussed along the TS pathway, as well as the case of an enantiomer at the central residue of the helix.     

一类三次系统的细奇点

本文研究了具有一个高阶奇点的系统（１,２）,通过在实系统中引入奇点量的概念使焦点量和鞍点量得到统一,并且计算了系统（１．２）的各阶奇点量,证明了这样的系统存在珐多三阶细奇点,而且同时存在细奇点的最大个数是四。当系统是四个细奇点时,它们可能是四个一阶细奇点或四个三阶细奇点。     

具有细链双曲无穷远鞍点的二次系统

本文得到：具有细链双曲无穷远鞍点和一个细焦点的二次系统至多存在一个极限环,若有细无穷远分界线环S,则其内部不存在极限环,其稳定性与它包围的奇点的稳定性相反．     

Loose saddle points of set-valued maps in topological vector spaces

We give a new existence theorem for loose saddle point of set-valued map having values in a partially ordered topological vector space which is based on continuity and quasiconvexity- quasiconcavity of its scalarized maps. Moreover, we prove a new saddle point theorem for vector-valued functions in locally convex topological vector spaces under weak condition that is the semicontinuity of two function scalarization.