Stability Properties of Feynman's Operational Calculus for Exponential Functions of Noncommuting Operators

Stability properties of Feynman's operational calculus are addressed in the setting of exponential functions of noncommuting operators. Applications of some of the stability results are presented. In particular, the time-dependent perturbation theory of nonrelativistic quantum mechanics is presented in the setting of the operational calculus and application of the stability results of this paper to the perturbation theory are discussed.     

Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

An explicit description of the fundamental unitary for SU(2) q

In this paper we study the Cauchy problem for the generalized equation of finite-depth fluids

A doubly deprotonated form of calix[6]arene: Crystal and molecular structure of (calix[6]arene-2H)2−·(HNEt 3 + )2

The doubly deprotonated form of calix[6]arene, with two protonated triethylamines as counter-ions, crystallizes in the monoclinic system: space groupP2₁/n,a=8.465(4),b=17.822(8),c=15.182(6) Å,=90.18(4)°,V=2291(2) ų,Z=2. Refinement led to a final conventionalR value of 0.063 for 1046 reflections. The macrocycle conformation is not apinched cone, usual for freeR-calix[6]arene, but a distorted 1,2,3-alternate cone, since the molecule lies on a symmetry center. Furthermore, one of the torsion angles defined by the methylene bridges is near to zero, which is unusual in calixarene structures. Supplementary Data relating to this article (atomic coordinates for hydrogen atoms, anisotropic displacement parameters for oxygen and nitrogen atoms, and observed and calculated structure factors) are deposited with the British Library as Supplementary Publication No. SUP 82182 (7 pages).     

Trapping H- bound to the nitrogenase FeMo-cofactor active site during H2 evolution: characterization by ENDOR spectroscopy

We here show that the iron-molybdenum (FeMo)-cofactor of the nitrogenase alpha-70(Ile) molybdenum-iron (MoFe) protein variant accumulates a novel S = (1)/(2) state that can be trapped during the reduction of protons to H(2). (1,2)H-ENDOR measurements disclose the presence of two protons/hydrides (H(+/)(-)) whose hyperfine tensors have been determined from two-dimensional field-frequency (1)H ENDOR plots. The two H(+/)(-) have large isotropic hyperfine couplings, A(iso)( )() approximately 23 MHz, which shows they are bound to the cofactor. The favored analysis for these plots indicates that the two H(+/)(-) have the same principal values, which indicates that they are chemically equivalent. The tensors are further related to each other by a permutation of the tensor components, which indicates an underlying symmetry of binding relative to the cofactor. At present, no model for the structure of the iron-molybdenum (FeMo)-cofactor in the S = (1)/(2) state trapped during the reduction of H(+) can be shown unequivocally to satisfy all of the constraints generated by the ENDOR analysis. The data disfavors any model that involves protonation of sulfides, and thus suggests that the intermediate instead contains two chemically equivalent bound hydrides; it appears unlikely that these are terminal monohydrides.     

Pairwise decomposition of residue interaction energies using semiempirical quantum mechanical methods in studies of protein-ligand interaction

Pairwise decomposition of the interaction energy between molecules is shown to be a powerful tool that can increase our understanding of macromolecular recognition processes. Herein we calculate the pairwise decomposition of the interaction energy between the protein human carbonic anhydrase II (HCAII) and the fluorine-substituted ligand N-(4-sulfamylbenzoyl)benzylamine (SBB) using semiempirical quantum mechanics based methods. We dissect the interaction between the ligand and the protein by dividing the ligand and the protein into subsystems to understand the structure-activity relationships as a result of fluorine substitution. In particular, the off-diagonal elements of the Fock matrix that is composed of the interaction between the ionic core and the valence electrons and the exchange energy between the subsystems or atoms of interest is examined in detail. Our analysis reveals that the fluorine-substituted benzylamine group of SBB does not directly affect the binding energy. Rather, we find that the strength of the interaction between Thr199 of HCAII and the sulfamylbenzoyl group of SBB affects the binding affinity between the protein and the ligand. These observations underline the importance of the sulfonamide group in binding affinity as shown by previous experiments (Maren, T. H.; Wiley: C. E. J. Med. Chem. 1968, 11, 228-232). Moreover, our calculations qualitatively agree with the structural aspects of these protein-ligand complexes as determined by X-ray crystallography.     

The electronic structure of FeV-cofactor in vanadium-dependent nitrogenase

The electronic structure of the active-site metal cofactor (FeV-cofactor) of resting-state V-dependent nitrogenase has been an open question, with earlier studies indicating that it exhibits a broad S = 3/2 EPR signal (Kramers state) having g values of ∼4.3 and 3.8, along with suggestions that it contains metal-ions with valencies [1V³⁺, 3Fe³⁺, 4Fe²⁺]. In the present work, genetic, biochemical, and spectroscopic approaches were combined to reveal that the EPR signals previously assigned to FeV-cofactor do not correlate with active VFe-protein, and thus cannot arise from the resting-state of catalytically relevant FeV-cofactor. It, instead, appears resting-state FeV-cofactor is either diamagnetic, S = 0, or non-Kramers, integer-spin (S = 1, 2 etc.). When VFe-protein is freeze-trapped during high-flux turnover with its natural electron-donating partner Fe protein, conditions which populate reduced states of the FeV-cofactor, a new rhombic S = 1/2 EPR signal from such a reduced state is observed, with g = [2.18, 2.12, 2.09] and showing well-defined ⁵¹V (I = 7/2) hyperfine splitting, a_iso = 110 MHz. These findings indicate a different assignment for the electronic structure of the resting state of FeV-cofactor: S = 0 (or integer-spin non-Kramers state) with metal-ion valencies, [1V³⁺, 4Fe³⁺, 3Fe²⁺]. Our findings suggest that the V³⁺ does not change valency throughout the catalytic cycle.

Active site FeV-cofactor of the V-nitrogenase and the EPR spectrum of the reduced cofactor showing ⁵¹V-hyperfine coupling.