Conical projection algorithms for linear programming

The Linear Programming Problem is manipulated to be stated as a Non-Linear Programming Problem in which Karmarkar's logarithmic potential function is minimized in the positive cone generated by the original feasible set. The resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non-linear programming algorithm. Several different procedures for the internal algorithm are proposed, giving priority either to the reduction of the potential function or of the actual cost. We show that Karmarkar's algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration. All variants of the new algorithm have the same complexity as Karmarkar's method, but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm.Research partly sponsored by CNPq-Brazilian National Council for Scientific and Technological Development, by National Science Foundation grant ECS-857362, Office of Naval Research contract N00014-86-K-0295, and AFOSR grant 86-0116.On leave from COPPE-Federal University of Rio de Janeiro, Cx. Postal 68511, 21941 Rio de Janeiro, RJ, Brasil.     

Preservation of persistence and stability under intersections and operations,part 1: Persistence

New results about convergence of sets and functions in possibly infinite-dimensional spaces are derived in a simple and unified way from two results dealing with the continuity with respect to a parameter of the intersection of two convex sets depending on this parameter.     

Isoperimetric problems and roses of neighborhood for stationary flat processes

The paper yields necessary conditions for the directional distributions of stationary k–flat processes in ?^d that maximize their intersection density of order 2, that is, the mean (2k – d)–volume of their self–intersections in an observation window of unit d–volume. The conditions are given in terms of the rose of intersections (i.e., the intensity of intersections of the flat process with test flats). The notion of the rose of neighborhood is introduced which is an analogue of the rose of intersections for lower dimensional flat processes. Some properties of the rose of neighborhood are studied and an asymptotically unbiased estimator is given.     

Comparison of algorithms for conical intersection optimisation using semiempirical methods

We present a comparison of three previously published algorithms for optimising the minimum energy crossing point between two Born–Oppenheimer electronic states. The algorithms are implemented in a development version of the MNDO electronic structure package for use with semiempirical configuration interaction methods. The penalty function method requires only the energies and gradients of the states involved, whereas the gradient projection and Lagrange–Newton methods also require the calculation of non-adiabatic coupling terms. The performance of the algorithms is measured against a set of well-known small molecule conical intersections. The Lagrange–Newton method is found to be the most efficient, with the projected gradient method also competitive. The penalty function method can only be recommended for situations where non-adiabatic coupling terms cannot be calculated. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Are Conical Intersections Responsible for the Ultrafast Processes of Adenine,Protonated Adenine,and the Corresponding Nucleosides?

Excited-state potential energy surfaces of adenine, protonated adenine, and their N9-methylated analogs are explored by means of a complete active space (CAS) and time-dependent density functional theory (TD-DFT) study to understand the dynamics associated with internal conversion. After photoexcitation of the ground-state molecules to the S(1) state, the nuclear motions that are responsible for taking the wavepacket out of the Franck-Condon region are either an H--N9/C--N9 stretch or a ring-puckering motion that leads to pyramidalization. These motions lead to accessible conical intersections with the ground-state surface. The results are used to successfully interpret previous measurements on the photodissociation of adenosine 5'-monophosphate nucleotide anions and cations, where the latter react in a highly nonstatistical manner.     

Transition state spectroscopy of open shell systems: Angle-resolved photodetachment spectra for the adiabatic singlet states of OHF

In this work three-dimensional potential energy surfaces of the first five singlet states of OHF are developed based on fits of more than 10,000 highly accurate ab initio points. An approximate treatment is presented for the calculation of the anisotropy parameter describing the electron angular distribution photodetached from a molecular anion. This method is used to calculate the angle-resolved photoelectron spectra in the photodetachment of OHF⁻. The wave packet formed in the neutral OHF system is placed at the transition state region, and yields the formation of OH + F and HF + O products. The results are compared with the recent experimental measurements published by Neumark [D.M. Neumark, Phys. Chem. Chem. Phys. 7 (2005) 433]. The intensity found at low electron kinetic energy including these five states and the three lower triplet states is found to be low. To analyze the effect of higher electronic states more excited ¹Σ⁻, ³Σ⁺ and ³Δ states are calculated at collinear geometry. The agreement with the experimental data improves, thus demonstrating that the correct simulation of the photodetachment spectrum at 213 nm involves at least 12 electronic states. All the structures of the experimental spectra are semiquantitatively reproduced finding an overall good agreement. It is concluded that the main problem of the simulation is in the intensity and anisotropy parameters. An alternative to their calculation would be to fit their values to reproduce the experimental results, but this would require to separate the contribution arising from different final electronic states.     

A criterion for the confluence of two seams of conical intersection in triatomic molecules

It is shown that the cross product t ^{I J} (R _x )≡g ^{I J} (R _x )×h ^{I J} (R _x ), where g _τ ^{I J} (R)=(c ^I (R _x )−c ^J (c ^I (R _x )+c ^J (R _x )), h _τ ^{I J} (R)=c ^I (R _x )^† c ^J (R _x ), τ is an internal nuclear coordinate, the c ^I (R) satisfy [H(R)−E _I (R)]c ^I (R)=0 and H(R) is the electronic Hamiltonian matrix, is a unique property of a conical intersection at R _x . t ^{I J} (R _x )=0 when R _x is located at the intersection of two (or more) seams of conical intersection. This criterion for an intersection of two seams of conical intersection has important implications for algorithms that seek to locate such points. Here it␣is␣used to analyze the trifurcation of a generic C_2v ^2S+1 A−^2S+1 B seam of conical intersection, analogous to those recently found in AlH₂ and CH₂. Received: 31 July 1997 / Accepted: 27 August 1997     

On the bound states of Schrödinger operators with δ-interactions on conical surfaces

In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a δ-interaction supported on a cone whose cross section is the sphere of codimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are nondecreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.     

Excited-State Proton Transfer and Decay in Hydrogen-Bonded Oxazole System:MS-CASPT2//CASSCF Study

Herein we have employed high-level multi-reference CASSCF and MS-CASPT2 electronic structure methods to systematically study the photochemical mechanism of intramolecularly hydrogen-bonded 2-(2'-hydroxyphenyl)-4-methyloxazole. At the CASSCF level, we have optimized minima, conical intersections, minimum-energy reaction paths relevant to the excited-state intramolecular proton transfer (ESIPT), rotation, photoisomerization, and the excited-state deactivation pathways. The energies of all structures and paths are refined by the MS-CASPT2 method. On the basis of the present results, we found that the ESIPT process in a conformer with the OH…N hydrogen bond is essentially barrierless process; whereas, the ESIPT process is inhibited in the other conformer with the OH…O hydrogen bond. The central single-bond rotation of the S₁ enol species is energetically unfavorable due to a large barrier. In addition, the excited-state deactivation of the S₁ keto species, as a result of the ultrafast ESIPT, is very efficient because of the existence of two easily-approached keto S₁/S₀ conical intersections. In stark contrast to the S₁ keto species, the decay of the S₁ enol species is almostly blocked. The present theoretical study contributes valuable knowledge to the understanding of photochemistry of similar intramolecularly hydrogen-bonded molecular and biological systems.     

UV photodissociation of N-methylpyrrole: The role of πσ states in non-hydride heteroaromatic systems

The photophysics of jet-cooled N-methylpyrrole molecules following excitation to their first excited singlet state (the ¹A₂ state, arising from a 3s/σ^*←π electron promotion) has been investigated by resonance enhanced multiphoton ionisation spectroscopy, by measurements of wavelength resolved ‘action’ spectra for forming CH₃ photoproducts, and by velocity map imaging studies of these CH₃ products (in their v = 0 and v₂ = 1 vibrational levels). CH₃ products are observed at all excitation wavelengths within the NMP absorption band. Direct dissociation on the ¹A₂ potential energy surface (PES) yields ‘fast’ CH₃ fragments, with an average total kinetic energy release (TKER) of 6500 cm⁻¹, but this product channel is only observed in a narrow wavelength range near the absorption band origin. All of the measured CH₃ images also show a broad component, peaking at lower TKER (1700 cm⁻¹); this component extends beneath the ‘fast’ feature in images recorded at wavelengths near the origin, and accounts for all of the CH₃ products observed at shorter photolysis wavelengths. These products are attributed to decay of highly vibrationally excited ground state molecules formed by radiationless transfer from the ¹A₂ state. Similarities and differences with the results of previous studies of the H + pyrrolyl products arising in the UV photodissociation of pyrrole are discussed in terms of the likely nuclear motions on the relevant ground and excited PESs (along R_N–CH3/R_N–H), and the possible couplings between these surfaces. The present study confirms that the proposed model of ¹πσ^* state induced bond fission in heteroaromatic molecules [A.L. Sobolewski, W. Domcke, Chem. Phys. 259 (2000) 181] is also applicable to non-hydride substituted heteroaromatics, but that mass effects can have an important influence on the subsequent nuclear dynamics.