Branching Plane Research Articles

XMECP: Reaching State-of-the-Art MECP Optimization in Multiscale Complex Systems.

The Python-based program, XMECP, is developed for realizing robust, efficient, and state-of-the-art minimum energy crossing point (MECP) optimization in multiscale complex systems. This article introduces the basic capabilities of the XMECP program by theoretically investigating the MECP mechanism of several example systems including (1) the photosensitization mechanism of benzophenone, (2) photoinduced proton-coupled electron transfer in the cytosine-guanine base pair in DNA, (3) the spin-flip process in oxygen activation catalyzed by an iron-containing 2-oxoglutarate-dependent oxygenase (Fe/2OGX), and (4) the photochemical pathway of flavoprotein adjusted by the intensity of an external electric field. MECPs related to multistate reaction and multistate reactivity in large-scale complex biochemical systems can be well-treated by workflows suggested by the XMECP program. The branching plane updating the MECP optimization algorithm is strongly recommended as it provides derivative coupling vector (DCV) with explicit calculation and can equivalently evaluate contributions from non-QM residues to DCV, which can be nonadiabatic coupling or spin-orbit coupling in different cases. In the discussed QM/MM examples, we also found that the influence on the QM region by DCV can occur through noncovalent interactions and decay with distance. In the example of DNA base pairs, the nonadiabatic coupling occurs across the π-π stacking structure formed in the double-helix system. In contrast to general intuition, in the example of Fe/2OGX, the central ferrous and oxygen part contribute little to the spin-orbit coupling; however, a nearby arginine residue, which is treated by molecular mechanics in the QM/MM method, contributes significantly via two hydrogen bonds formed with α-ketoglutarate (α-KG). This indicates that the arginine residue plays a significant role in oxygen activation, driving the initial triplet state toward the productive quintet state, which is more than the previous knowledge that the arginine residue can bind α-KG at the reaction site by hydrogen bonds.

Photochemistry of Thymine in Protic Polar Nanomeric Droplets Using Electrostatic Embeding TD-DFT/MM.

Thymine photochemistry is important for understanding DNA photodamage. In the gas phase, thymine undergoes a fast non-radiative decay from S to S. In the S state, it gets trapped for several picoseconds until returning to the ground-state S. Here, we explore the electrostatic effects of nanomeric droplets of methanol and water on the excited states of thymine. For this purpose, we develop and implement an electrostatic embedding TD-DFT/MM method based on a QM/MM coupling defined through electrostatic potential fitting charges. We show that both in methanol and water, the mechanism is similar to the gas phase. The solvent molecules participate in defining the branching plane of S/S intersection and have a negligible effect on the S/S intersection. Despite the wrong topology of the ground/excited state intersections, electrostatic embedding TD-DFT/MM allows for a fast exploration of the potential energy surfaces and a qualitative picture of the photophysics of thymine in solvent droplets.

Solitons and cavitons in a nonlocal Whitham equation

Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one sheet represent smooth solutions of the equation but those which perform transitions through the branching plane represent solutions with jumps. They correspond to solutions with singularities of the fourth order differential equation – breaks of the first and third derivatives but continuous even derivatives. The Hamiltonian system can have two types of equilibria on different sheets, they can be saddle-centers or saddle-foci. Using analytic and numerical methods we found many types of homoclinic orbits to these equilibria both with a monotone asymptotics and oscillating ones. They correspond to solitons and cavitons of the initial equation. When we deal with homoclinic orbits to a saddle-center, the values of the second parameter (physical wave speed) are discrete but for the case of a saddle-focus they are continuous. The presence of multiplicity of such solutions displays the very complicated dynamics of the system.

Floral structure inThismia(Thismiaceae: Dioscoreales): new insights from anatomy, vasculature and development

AbstractThismia is characterized by an exceptionally complicated floral morphology that is currently not understood properly. In the taxonomic literature, descriptive rather than morphological terms are often applied to parts of the flower in Thismia, relating to the general appearance of the floral organs instead of their precise homologies. Precise understanding of the floral structure is complicated by the rarity of Thismia spp. and the paucity of appropriate material. Here we provide a comprehensive study of reproductive organs of three Thismia spp. (T. annamensis, T. javanica and T. mucronata) including the first investigation of inflorescence architecture and early floral development in Thismiaceae. We found a hitherto unknown diversity of the reproductive shoots in the genus, manifested in the number of floral prophylls (two or three, in contrast to a single prophyll in the vast majority of monocots) and in the branching plane resulting in two distinct inflorescence types, a drepanium and a bostryx. We report the non-acropetal sequence of initiation of floral whorls (with stamens being the last elements to initiate), never previously described in monocots, and the gynoecium composed of completely plicate carpels, also a rare feature for monocots. Floral vasculature is relatively uniform in Thismia, but significant interspecific differences are found in tepal innervation, including the number of tepal traces; some of these differences are not immediately related to the external tepal morphology. We argue that the annulus, which acts as a roof of the hypanthium, possesses an androecium nature and represents congenitally fused bases of stamen filaments. We describe the stamens as laminar structures, which are also shortly tubular in the distal part of the supraconnective with the adaxial tubular side forming a skirt-like appendage. Finally, the placentas, which are column-like when mature, are initially parietal, becoming secondarily similar to free-central placentas through schizogenous separation from the ovary wall.

Real-Time Tunneling Dynamics through Adiabatic Potential Energy Surfaces Shaped by a Conical Intersection.

Dynamic shaping of the adiabatic tunneling barrier in the S-H bond extension coordinate of several ortho-substituted thiophenols has been found to be mediated by low-frequency out-of-plane vibrational modes, which are parallel to the coupling vector of the branching plane comprising the conical intersection. The S-H predissociation tunneling rate (k) measured when exciting to the S1 zero-point level of 2-methoxythiophenol (44 ps)-1 increases abruptly, to k ≈ (22 ps)-1, at the energy corresponding to excitation of the 152 cm-1 out-of-plane vibrational mode and then falls back to k ≈ (40 ps)-1 when the in-plane mode is excited at 282 cm-1. Similar resonance-like peaks in plots of S1 tunneling rate versus internal energy are observed when exciting the corresponding low-frequency out-of-plane modes in the S1 states of 2-fluorothiophenol and 2-chlorothiophenol. This experiment provides clear-cut evidence for dynamical "shaping" of the lower-lying adiabatic potential energy surfaces by the higher-lying conical intersection seam, which dictates the multidimensional tunneling dynamics.

Theoretical Investigation of Excited-State Intramolecular Proton Transfer and Photoisomerization of 2-(Iminomethyl)phenol.

Time-dependent density functional theory and high-level ab initio calculations were performed to investigate the excited-state intramolecular proton transfer (ESIPT) and subsequent isomerization of 2-(iminomethyl)phenol (IMP). According to the results of the correlated theoretical methods, ESIPT is a barrierless process; subsequently, the isomerization (rotation of the torsion angle) of IMP also readily occurs. Fictitious intermediates are found due to an insufficient theoretical level. The molecular structure of the conical intersection (CI) during the isomerization process is optimized, and its branching plane is characterized. Both the gradient difference vector and the derivative coupling vector are significantly correlated to the C═O and H2N-C antiparallel stretching coordinates, and the dynamic electron correlation effect is crucial to optimize the molecular structure of the real CI of IMP. The relaxation pathway from the CI in the S0 state was examined; the dominant pathway proceeds to the trans-keto form of IMP. However, if the C═O and H2N-C antiparallel stretching mode is sufficiently populated, then the reaction proceeds to the cis-keto form of IMP and can eventually recover to the cis-enol form.

Crossing conditions in coupled cluster theory.

We derive the crossing conditions at conical intersections between electronic states in coupled cluster theory and show that if the coupled cluster Jacobian matrix is nondefective, two (three) independent conditions are correctly placed on the nuclear degrees of freedom for an inherently real (complex) Hamiltonian. Calculations using coupled cluster theory on a 21A'/31A' conical intersection in hypofluorous acid illustrate the nonphysical artifacts associated with defects at accidental same-symmetry intersections. In particular, the observed intersection seam is folded about a space of the correct dimensionality, indicating that minor modifications to the theory are required for it to provide a correct description of conical intersections in general. We find that an accidental symmetry allowed 11A″/21A″ intersection in hydrogen sulfide is properly described, showing no artifacts as well as linearity of the energy gap to first order in the branching plane.

On the applicability of a wavefunction-free, energy-based procedure for generating first-order non-adiabatic couplings around conical intersections.

The primal definition of first-order non-adiabatic couplings among electronic states relies on the knowledge of how electronic wavefunctions vary with nuclear coordinates. However, the non-adiabatic coupling between two electronic states can be obtained in the vicinity of a conical intersection from energies only, as this vector spans the branching plane along which degeneracy is lifted to first order. The gradient difference and derivative coupling are responsible of the two-dimensional cusp of a conical intersection between both potential-energy surfaces and can be identified to the non-trivial eigenvectors of the second derivative of the square energy difference, as first pointed out in Köppel and Schubert [Mol. Phys. 104(5-7), 1069 (2006)]. Such quantities can always be computed in principle for the cost of two numerical Hessians in the worst-case scenario. Analytic-derivative techniques may help in terms of accuracy and efficiency but also raise potential traps due to singularities and ill-defined derivatives at degeneracies. We compare here two approaches, one fully numerical, the other semianalytic, where analytic gradients are available but Hessians are not, and investigate their respective conditions of applicability. Benzene and 3-hydroxychromone are used as illustrative application cases. It is shown that non-adiabatic couplings can thus be estimated with decent accuracy in regions of significant size around conical intersections. This procedure is robust and could be useful in the context of on-the-fly non-adiabatic dynamics or be used for producing model representations of intersecting potential energy surfaces with complete obviation of the electronic wavefunctions.

Ultrafast Electronic Relaxation through a Conical Intersection: Nonadiabatic Dynamics Disentangled through an Oscillator Strength-Based Diabatization Framework.

We employ surface hopping trajectories to model the short-time dynamics of gas-phase and partially solvated 4-(N,N-dimethylamino)benzonitrile (DMABN), a dual fluorescent molecule that is known to undergo a nonadiabatic transition through a conical intersection. To compare theory vs time-resolved fluorescence measurements, we calculate the mixed quantum-classical density matrix and the ensemble averaged transition dipole moment. We introduce a diabatization scheme based on the oscillator strength to convert the TDDFT adiabatic states into diabatic states of La and Lb character. Somewhat surprisingly, we find that the rate of relaxation reported by emission to the ground state is almost 50% slower than the adiabatic population relaxation. Although our calculated adiabatic rates are largely consistent with previous theoretical calculations and no obvious effects of decoherence are seen, the diabatization procedure introduced here enables an explicit picture of dynamics in the branching plane, raising tantalizing questions about geometric phase effects in systems with dozens of atoms.

Estimating the Derivative Coupling Vector Using Gradients.

The difference gradient and derivative coupling vectors span the branching planes of conical intersections between electronic states. While gradients are commonly available in many electronic structure methods, the derivative coupling vectors are not always implemented and ready for use in characterizing conical intersections. This Letter shows how the derivative coupling vectors can be computed to high accuracy (direction and magnitude) using energy and gradient information. The new method is based on the combination of a linear-coupling two-state Hamiltonian and a finite-difference Davidson approach for computing the branching plane. Benchmark cases are provided showing these vectors can be efficiently computed near conical intersections.

Charge migration in polycyclic norbornadiene cations: Winning the race against decoherence.

The observation of electronic motion remains a key target in the development of the field of attoscience. However, systems in which long-lived oscillatory charge migration may be observed must be selected carefully, particularly because it has been shown that nuclear spatial delocalization leads to a loss of coherent electron density oscillations. Here we demonstrate electron dynamics in norbornadiene and extended systems where the hole density migrates between two identical chromophores. By studying the effect of nuclear motion and delocalization in these example systems, we present the physical properties that must be considered in candidate molecules in which to observe electron dynamics. Furthermore, we also show a key contribution to nuclear delocalization arises from motion in the branching plane of the cation. For the systems studied, the dephasing time increases with system size while the energy gap between states, and therefore the frequency of the density oscillation, decreases with size (obeying a simple exponential dependence on the inter-chromophore distance). We present a system that balances these two effects and shows several complete oscillations in the spin density before dephasing occurs.

Erratum: "Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules" [J. Chem. Phys. 141, 124122 (2014)

Erratum: "Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules" [J. Chem. Phys. 141, 124122 (2014)

Mechanical Forces Alter Conical Intersections Topology.

Photoreactivity can be influenced by mechanical forces acting over a reacting chromophore. Nevertheless, the specific effect of the external forces in the photoreaction mechanism remains essentially unknown. Conical intersections are key structures in photochemistry, as they constitute the funnels connecting excited and ground states. These crossing points are well known to provide valuable information on molecular photoreactivity, including crucial aspects as potential photoproducts which may be predicted by just inspection of the branching plane vectors. Here, we outline a general framework for understanding the effect of mechanical forces on conical intersections and their implications on photoreactivity. Benzene S1/S0 conical intersection topology can be dramatically altered by applying less than 1 nN force, making the peaked pattern of the intersection become a sloped one, also provoking the transition state in the excited state to disappear. Both effects can be related to an increase in the photostability as the conical intersection becomes more accessible, and its topology in this case favors the recovery of the initial reactant. The results indicate that the presence of external forces acting over a chromophore have to be considered as a potential method for photochemical reactivity control.

Landscapes of four-enantiomer conical intersections for photoisomerization of stilbene: CASSCF calculation.

The photoisomerization of cis- and trans-stilbene through conical intersections (CI) is mainly governed by four dihedral angles around central C═C double bonds. The two of them are C-C═C-C and H-C═C-H dihedral angles that are found to form a mirror rotation coordinate, and the mirror plane appears at the two dihedral angles equal to zeros with which the middle state is defined through partial optimization. There exist the first-type of hula-twist-CI enantiomers, the second-type of hula-twist-CI enantiomers, the first-type of one-bond-flip-CI enantiomers, and the second type of one-bond-flip-CI enantiomers as well as cis-enantiomers and trans-enantiomers with respect to this mirror plane. The complete active space self-consistent field method is employed to calculate minimum potential energy profile along the mirror rotation coordinate for each enantiomers, and it is found that the left-hand manifold and the right-hand manifold of potential energy surfaces can be energetically transferred via photoisomerization. Furthermore, two-dimensional potential energy surfaces in terms of the branching plane g-h coordinates are constructed at vicinity of each conical intersection, and the landscapes of conical intersections show distinct feature, and in excited-state four potential wells separated in different section of g-h plane related to different conical intersections which indicate different photoisomerization pathways.

Shape of Multireference, Equation-of-Motion Coupled-Cluster, and Density Functional Theory Potential Energy Surfaces at a Conical Intersection.

We report and characterize ground-state and excited-state potential energy profiles using a variety of electronic structure methods along a loop lying on the branching plane associated with a conical intersection (CI) of a reduced retinal model, the penta-2,4-dieniminium cation (PSB3). Whereas the performance of the equation-of-motion coupled-cluster, density functional theory, and multireference methods had been tested along the excited- and ground-state paths of PSB3 in our earlier work, the ability of these methods to correctly describe the potential energy surface shape along a CI branching plane has not yet been investigated. This is the focus of the present contribution. We find, in agreement with earlier studies by others, that standard time-dependent DFT (TDDFT) does not yield the correct two-dimensional (i.e., conical) crossing along the branching plane but rather a one-dimensional (i.e., linear) crossing along the same plane. The same type of behavior is found for SS-CASPT2(IPEA=0), SS-CASPT2(IPEA=0.25), spin-projected SF-TDDFT, EOM-SF-CCSD, and, finally, for the reference MRCISD+Q method. In contrast, we found that MRCISD, CASSCF, MS-CASPT2(IPEA=0), MS-CASPT2(IPEA=0.25), XMCQDPT2, QD-NEVPT2, non-spin-projected SF-TDDFT, and SI-SA-REKS yield the expected conical crossing. To assess the effect of the different crossing topologies (i.e., linear or conical) on the PSB3 photoisomerization efficiency, we discuss the results of 100 semiclassical trajectories computed by CASSCF and SS-CASPT2(IPEA=0.25) for a PSB3 derivative. We show that for the same initial conditions, the two methods yield similar dynamics leading to isomerization quantum yields that differ by only a few percent.

Designing Conical Intersections for Light-Driven Single Molecule Rotary Motors: From Precessional to Axial Motion

In the past, the design of light-driven single molecule rotary motors has been mainly guided by the modification of their ground-state conformational properties. Further progress in this field is thus likely to be achieved through a detailed understanding of light-induced dynamics of the system and the ways of modulating it by introducing chemical modifications. In the present theoretical work, the analysis of model organic chromophores and synthesized rotary motors is used for rationalizing the effect of electron-withdrawing heteroatoms (such as a cationic nitrogen) on the topography and branching plane of mechanistically relevant conical intersections. Such an analysis reveals how the character of rotary motion could be changed from a precessional motion to an axial rotational motion. These concepts are then used to design and build quantum chemical models of three distinct types of Schiff base rotary motors. One of these models, featuring the synthetically viable indanylidenepyrroline framework, has conical intersection structures consistent with an axial rotation not hindered by ground-state conformational barriers. It is expected that this type of motor should be capable of funneling the photon energy into specific rotary modes, thus achieving photoisomerization quantum efficiencies comparable to those seen in visual pigments.

Geometry deformation and mesomeric effect at the minimal-energy conical intersections and their relationship to the photoreactivity of indolylfulgides: A TD-DFT study

Minimum energy conical intersection (MECI) is the low-lying point on the CI hyperseam that facilitates ultrafast radiationless transition in photochromes. Time dependent density functional theory calculations using Maeda’s updated branching plane method were employed in the study of indolylfulgides’ structure at the MECI. Four criteria were used to effectively characterize the influence of the MECI structure on the photoreactivity of indolylfulgides; three of these criteria (molecular orbital, bond lengths and the height of C6 trigonal pyramid) were paramount to identifying one novel MECI structure and confirming 2 others previously reported.

Branching patterns emerge in a mathematical model of the dynamics of lung development

Recent experimental work has described an elegant pattern of branching in the development of the lung. Multiple forms of branching have been identified, including side branching and tip bifurcation. A particularly interesting feature is the phenomenon of 'orthogonal rotation of the branching plane'. The lung must fill 3D space with the essentially 2D phenomenon of branching. It accomplishes this by rotating the branching plane by 90° with each generation. The mechanisms underlying this rotation are not understood. In general, the programmes that underlie branching have been hypothetically attributed to genetic 'subroutines' under the control of a 'global master routine' to invoke particular subroutines at the proper time and location, but the mechanisms of these routines are not known. Here, we demonstrate that fundamental mechanisms, the reaction and diffusion of biochemical morphogens, can create these patterns. We used a partial differential equation model that postulates three morphogens, which we identify with specific molecules in lung development. We found that cascades of branching events, including side branching, tip splitting and orthogonal rotation of the branching plane, all emerge immediately from the model, without further assumptions. In addition, we found that one branching mode can be easily switched to another, by increasing or decreasing the values of key parameters. This shows how a 'global master routine' could work by the alteration of a single parameter. Being able to simulate cascades of branching events is necessary to understand the critical features of branching, such as orthogonal rotation of the branching plane between successive generations, and branching mode switch during lung development. Thus, our model provides a paradigm for how genes could possibly act to produce these spatial structures. Our low-dimensional model gives a qualitative understanding of how generic physiological mechanisms can produce branching phenomena, and how the system can switch from one branching pattern to another using low-dimensional 'control knobs'. The model provides a number of testable predictions, some of which have already been observed (though not explained) in experimental work.

A perfusion-independent role of blood vessels in determining branching stereotypy of lung airways

Blood vessels have been shown to play perfusion-independent roles in organogenesis. Here, we examined whether blood vessels determine branching stereotypy of the mouse lung airways in which coordinated branching of epithelial and vascular tubes culminates in their co-alignment. Using different ablative strategies to eliminate the lung vasculature, both in vivo and in lung explants, we show that proximity to the vasculature is indeed essential for patterning airway branching. Remarkably, although epithelial branching per se proceeded at a nearly normal rate, branching stereotypy was dramatically perturbed following vascular ablation. Specifically, branching events requiring a rotation to change the branching plane were selectively affected. This was evidenced by either the complete absence or the shallow angle of their projections, with both events contributing to an overall flat lung morphology. Vascular ablation also led to a high frequency of ectopic branching. Regain of vascularization fully rescued arrested airway branching and restored normal lung size and its three-dimensional architecture. This role of the vasculature is independent of perfusion, flow or blood-borne substances. Inhibition of normal branching resulting from vascular loss could be explained in part by perturbing the unique spatial expression pattern of the key branching mediator FGF10 and by misregulated expression of the branching regulators Shh and sprouty2. Together, these findings uncovered a novel role of the vasculature in organogenesis, namely, determining stereotypy of epithelial branching morphogenesis.

Updated Branching Plane for Finding Conical Intersections without Coupling Derivative Vectors.

The conical intersections (CIs) form a (f-2)-dimensional hyperspace on which two diabatic potential energy surfaces (PESs) belonging to the same symmetry cross, where f is the internal degree of freedom. The branching plane (BP) is a (two-dimensional) plane defined by the difference gradient vector (DGV) and the coupling derivative vector (CDV), and on the BP, the degeneracy of the two adiabatic PESs is lifted. The properties of the BP are often used in the exploration of the conical intersection hyperspace, such as determination of the minimum energy CI or the first-order saddle point in CI. Although both DGV and CDV are necessary to construct the BP in general, CDV is not always available depending on ab initio methods and programs. Therefore, we developed an approach for optimizing critical points on the CI hypersurface without CDV by using a BP updating method, which was shown to be accurate and very useful for minimum energy and saddle point optimization and for the minimum energy path following within the CI hypersurface in numerical tests for C6H6 and C5H8N(+).