Harmonic Surfaces Research Articles

Surfaces of constant curvature in [formula omitted] with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature K>0 in R3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in S2 with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R3, i.e. compact convex surfaces of constant curvature K>0 with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

Theoretical analysis of vibronic structure in absorption and fluorescence spectra of polyatomic molecules beyond the Condon approximation: Application to 11Ag ↔ 21Ag and 11Ag ↔ 11Bu electronic transitions in all-trans-1,3,5,7-octatetraene

Calculations of the vibronic structure in electronic absorption and fluorescence spectra of a large polyatomic molecule based on time- and frequency-domain approaches beyond the Condon approximation are developed. In time-domain approach, the molecular spectra are expressed in terms of Fourier transforms of appropriate time correlation functions, which are evaluated in closed form for displaced–distorted–rotated harmonic potential surfaces at finite temperature. In frequency-domain approach, use is made of the Franck–Condon factors which were evaluated in closed form for the same potential surfaces in our previous work (J. Molec. Spectrosc. 194 (1999) 179). The vibronic structure in dipole-forbidden (but vibronic-allowed) 11Ag↔21Ag and dipole-allowed 11Ag↔11Bu electronic transitions of all-trans-1,3,5,7-octatetraene have been calculated and compared with experimental data as illustrations of the approaches. Our vibronic structure calculations support the hypothesis that a sudden opening of a non-radiative decay channel occurs at 21Ag vibronic levels with energies larger than about 2100cm−1.

A Simple Coarse-Grained Model to Map the Transition Pathway Between Two Stable Conformations using the Anisotropic Elastic Network Model

Conformational transitions in biomolecules, especially proteins, play an important role in signaling and regulation of various biological processes. Here we propose a fast and simple method for constructing a transition pathway between two stable conformations of a protein. The protein is represented as a simplified coarse-grained (CG) model with a single site located at the C-alpha atom of each residue. The energy function of the two-state CG model is approximated by the anisotropic network model (ANM) harmonic energy surfaces of the end-states. The simple two-state energy surface comprises two local minima centered on the positions of the stable states and the system resides in one of these minima. There is a cusp with discontinuous first derivative in the multi-dimensional configuration space of the system, which acts as the transition state surface for the two-state potential. Given this simple prescription, two structures constrained to remain similar to one another but each residing on the opposite sides of the cusp, are optimized using energy minimization. This virtual “dumbbell” of two structures is then used as a crude approximation to the transition state. The pathway in the multi-dimensional space is constructed by letting these two structures slide down on their ANM surface using steepest descent energy minimization until the stable end-points are reached. This simple two-state ANM model was applied to explore the multi-dimensional and collective character of the conformational transition pathways in several systems of biological significance, including adenylate kinase, leucine transporter, sarcoplasmic reticulum Ca-ATPase and the glutamate transporter.

Hierarchical transformation of Hamiltonians with linear and quadratic couplings for nonadiabatic quantum dynamics: Application to the ππ*/nπ* internal conversion in thymine

We face with the general problem of defining a reduced number of effective collective coordinates to describe accurately the short-time nonadiabatic dynamics of large semirigid systems, amenable to a description in terms of coupled harmonic potential energy surfaces. We present a numeric iterative protocol to define a hierarchical representation of the Hamiltonian taking into account both linear and quadratic intra- and inter-state couplings (QVC, quadratic vibronic coupling model), thus generalizing the method introduced recently in the literature [E. Gindensperger, H. Köppel, and L. S. Cederbaum, J. Chem. Phys. 126, 034106 (2007)] for the linear vibronic coupling (LVC) model. This improvement allows to take into account the effect of harmonic frequency changes and Duschinsky mixings among the different electronic states, providing a route to upgrade the models for nonadiabatic harmonic systems to those nowadays routinely used for the simulation of vibronic spectra of adiabatic systems (negligible nonadiabatic couplings). We apply our method to the study of ππ∗ → nπ∗ internal conversion in thymine, analysing the differences in LVC and QVC predictions both for the absorption spectrum and the dynamics of electronic populations.

Parameterized Bases for Calculating Vibrational Spectra Directly from ab Initio Data Using Rectangular Collocation.

We compared different parametrized bases for computing anharmonic vibrational spectra using a new version of the rectangular collocation-optimization method of Manzhos and Carrington (Can. J. Chem. 2009, 87, 864; Chem. Phys. Lett. 2011, 511, 434). The method enables one to compute a small number of vibrational levels with an ultrasmall basis set without a potential function. To test the ideas, parametrized uncoupled and coupled Gaussian functions as well as direct-product and coupled Hermite basis sets are used to compute four low-lying vibrational energy levels of H2O on model harmonic and anharmonic uncoupled (polynomial) potential energy surfaces. In addition, we compute levels directly from ab initio points and thereby include all coupling and anharmonicity. We conclude that uncoupled parametrized Gaussian and Hermite functions are a good choice for anharmonic and coupled problems.

Quantum entanglement between electronic and vibrational degrees of freedom in molecules

We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.

Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales–Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.

Extending and correcting some research results on minimal and harmonic surfaces

In this article, we give necessary and beneficial extension for several important theorems by Monterde et al. on minimal and harmonic surfaces, so that they can be applied to all generic situations for n × m ( n ≠ m or n = m ) Bézier surfaces, instead of the special case of n × n Bézier surfaces. At the same time, we point out the errors in Monterde et al.ʼs six papers, and give the correct and precise answers.

Effective Time-Independent Calculations of Vibrational Resonance Raman Spectra of Isolated and Solvated Molecules Including Duschinsky and Herzberg-Teller Effects.

We present a method of modeling vibrational resonance Raman scattering (RRS) spectra of isolated and solvated systems with the inclusion of Franck-Condon (FC) and Herzberg-Teller (HT) effects and a full account for possible differences between the harmonic potential energy surfaces of the initial and resonant electronic states. It describes fundamentals, overtones, and combination bands and computes the RRS spectrum as a two-dimensional function of the incident and scattered frequencies. The theoretical foundations of the method are described and the differences with other currently available methodologies are outlined. Applications to the phenoxyl radical in the gas phase and indolinedimethine-malononitrile (IDMN) in acetonitrile and cyclohexane solution are reported, as well as comparisons with available experimental data.

Absorption and resonance Raman spectroscopy of the 1Bu state of trans‐1,3,5‐hexatriene including vibronic coupling

AbstractThe vibronic coupling between the first excited S1 (21Ag) and the second excited S2 (11Bu) singlet electronic states in spectroscopy of trans‐1,3,5‐hexatriene molecule is investigated on the basis of a model consisting of two electronic states coupled by two vibrational modes. Employing a perturbation theory that treats the intramolecular couplings in a perturbative manner, the absorption and resonance Raman cross sections and excitation profiles of this molecule are calculated using the time‐correlation function formalism. The non‐Condon corrections are included in evaluation of cross sections. The multidimensional time‐domain integrals that arise in these calculations have been evaluated for the case in which S0 (11Ag)S2 (11Bu) electronic transition takes place between displaced and distorted harmonic potential energy surfaces. The calculated spectra are in good agreement with the experimental ones. Copyright © 2011 John Wiley & Sons, Ltd.

Design for Rational Bézier Harmonic and Biharmonic Surfaces

Design for Rational Bézier Harmonic and Biharmonic Surfaces

Molecular simulations using spherical harmonics

Computer-aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the interacting part of molecules and they should reflex the complementarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The complementarity of lobe-hole is very important in protein-ligand interactions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a fingerprint to approximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined separately. This method can be used as a filter to eliminate candidates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.

PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces.

Bubble tree convergence for the harmonic sequence of harmonic surfaces in ℂℙ n

We show that any harmonic sequence determined by a harmonic map from a compact Riemannian surface M to ℂℙn has a terminating holomorphic (or anti-holomorphic) map from M to ℂℙn, or a “bubble tree limit” consisting of a harmonic map \( \hat f \): M → ℂℙn and a tree of bubbles hλµ: S2 → ℂℙn.

On Quasiconformal Harmonic Surfaces with Rectifiable Boundary

It is proved that every quasiconfomal harmonic mapping of the unit disk onto a surface with rectifiable boundary has absolutely continuous extension to the boundary as well as its inverse mapping has this property. In addition it is proved an isoperimetric type inequality for the class of these surfaces. These results extend some classical results for conformal mappings, minimal surfaces and surfaces with constant mean curvature treated by Kellogg, Courant, Nitsche, Tsuji, F. Riesz and M. Riesz, etc.

Numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces

This article studies the numerical approximation of harmonic maps mto surfaces, i.e., critical pomts for the Dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold. An iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised, and its global convergence to a discrete harmonic map is proved under general conditions. Weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations. Numerical simulations illustrate the performance of the algorithm for curved domains.

A Benchmark Study of Different Methods for Calculating One- And Two-Dimensional Optical Spectra

The accuracy and robustness of several approximate methods for computing linear and nonlinear optical spectra are considered. The analysis is performed in the context of a benchmark model that consists of a two-state chromophore with shifted harmonic potential surfaces that differ in frequency. The exact one- and two-dimensional spectra for this system are calculated and compared to spectra calculated via the following approximate methods: (1) The semiclassical forward-backward initial-value representation (FB-IVR) method; (2) the linearized semiclassical (LSC) method; (3) the standard second-order cumulant approximation which is based on the ground-state equilibrium frequency-frequency correlation function (2OC); (4) an alternative second-order cumulant approximation which is able to account for nonequilibrium dynamics on the excited-state potential surface (2OCa). All four approximate methods can be shown to reproduce the exact results when the frequencies of the ground and excited harmonic surfaces are identical. However, allowing for the ground and excited surfaces to differ in frequency leads to a more meaningful benchmark model for which none of the four approximate methods is exact. We present a comparison of one- and two-dimensional spectra calculated via the above-mentioned approximate methods to the corresponding exact spectra, as a function of the following parameters: (1) The ratio of excited state to ground-state frequencies; (2) Temperature; (3) The horizontal displacement of the excited-state potential relative to the ground-state potential; (4) The waiting time between the coherence periods in the case of two-dimensional spectra. The FB-IVR method is found to predict spectra which are practically indistinguishable from the exact ones over a wide region of parameter space. The LSC method is found to predict spectra which are in good agreement with the exact ones over the same region of parameter space. The 2OC and 2OCa are found to be highly inaccurate unless the frequencies of the ground and excited states are very similar. These observations give credence to the use of the LSC method for modeling spectra in complex systems, where exact or even FB-IVR-based calculations are prohibitively expensive.

Femtosecond stimulated Raman scattering for polyatomics with harmonic potentials: Application to rhodamine 6G

The perturbation theory of stimulated Raman scattering (SRS), with Raman pump on minus pump off and heterodyne detection along the probe direction, is reviewed. It has four third-order polarization terms, labeled as SRS or inverse Raman scattering (IRS): SRS(I), SRS(II), IRS(I), and IRS(II). These four polarizations have a wave packet interpretation. The polarizations, with homogenous and inhomogeneous broadening included, can be written as integrals over four-time correlation functions, and analytic formulas are derived for the latter for multidimensional harmonic potential surfaces with Franck-Condon displacements in the modes which facilitates the calculation of the SRS cross sections. The theory is applied to understand recent experimental results on the femtosecond SRS (FSRS) of a fluorescent dye, rhodamine 6G (R6G), where the Raman pump pulse is about 1 ps long, and the probe pulse is about 10 fs. The calculations compared very well with the R6G experimental results for off-resonance and resonance FSRS spectra spanning both Stokes and anti-Stokes bands, and for negative and positive pump-probe delay times on resonance.

Signatures of Nonequilibrium Solvation Dynamics on Multidimensional Spectra

Multidimensional electronic and vibrational spectroscopies have established themselves over the last decade as uniquely detailed probes of intramolecular structure and dynamics. However, these spectroscopies can also provide powerful tools for probing solute-solvent interactions and the solvation dynamics that they give rise to. To this end, it should be noted that multidimensional spectra can be expressed in terms of optical response functions that differ with respect to the chromophore's quantum state during the various time intervals separating light-matter interactions. The dynamics of the photoinactive degrees of freedom during those time intervals (that is, between pulses) is dictated by potential energy surfaces that depend on the corresponding state of the chromophore. One therefore expects the system to hop between potential surfaces in a manner dictated by the optical response functions. Thus, the corresponding spectra should reflect the system's dynamics during the resulting sequence of nonequilibrium solvation processes. However, the interpretation of multidimensional spectra is often based on the assumption that they reflect the equilibrium dynamics of the photoinactive degrees of freedom on the potential surface that corresponds to the chromophore's ground state. In this Account, we present a systematic analysis of the signature of nonequilibrium solvation dynamics on multidimensional spectra and the ability of various computational methods to capture it. The analysis is performed in the context of the following three model systems: (A) a two-state chromophore with shifted harmonic potential surfaces that differ in frequency, (B) a two-state atomic chromophore in an atomic liquid, and (C) the hydrogen stretch of a moderately strong hydrogen-bonded complex in a dipolar liquid. The following computational methods are employed and compared: (1) exact quantum dynamics (model A only), (2) the semiclassical forward-backward initial value representation (FB-IVR) method (models A and B only), (3) the linearized semiclassical (LSC) method (all three models), and (4) the standard ground-state equilibrium dynamics approach (all three models). The results demonstrate how multidimensional spectra can be used to probe nonequilibrium solvation dynamics in real time and with an unprecedented level of detail. We also show that, unlike the standard method, the LSC and FB-IVR methods can accurately capture the signature of solvation dynamics on the spectra. Our results also suggest that LSC and FB-IVR yield similar results in the presence of rapid dephasing, which is typical in complex condensed-phase systems. This observation gives credence to the use of the LSC method for modeling spectra in complex systems for which an exact or even FB-IVR-based calculation is prohibitively expensive.

Calculation of absorption and resonance Raman cross sections of ClO2 in the gas‐phase and in solution: including Duschinsky effects

AbstractThe time‐correlation function formalism has been used to calculate resonance Raman cross sections, excitation profiles, and electronic absorption spectra of the OClO molecule in the gas‐phase and in different solvents like cyclohexane, chloroform, and water. The multidimensional time domain integrals that arise in these calculations have been evaluated for the case in which an X2B1Ã2A1 electronic transition takes place between displaced‐distorted‐rotated harmonic potential energy surfaces. Ab initio calculations have been performed to provide the spectroscopic constants required for the evaluation of these integrals. The calculated absorption spectra and resonance Raman cross sections have been compared with the experimental results. Copyright © 2009 John Wiley & Sons, Ltd.