Vibrational Coordinates Research Articles

Efficiency of energy funneling in the photosystem II supercomplex of higher plants.

The investigation of energy transfer properties in photosynthetic multi-protein networks gives insight into their underlying design principles. Here, we discuss the excitonic energy transfer mechanisms of the photosystem II (PS-II) C2S2M2 supercomplex, which is the largest isolated functional unit of the photosynthetic apparatus of higher plants. Despite the lack of a definite energy gradient in C2S2M2, we show that the energy transfer is directed by relaxation to low energy states. C2S2M2 is not organized to form pathways with strict energetically downhill transfer, which has direct consequences for the transfer efficiency, transfer pathways and transfer limiting steps. The exciton dynamics is sensitive to small changes in the energetic layout which, for instance, are induced by the reorganization of vibrational coordinates. In order to incorporate the reorganization process in our numerical simulations, we go beyond rate equations and use the hierarchically coupled equation of motion approach (HEOM). While transfer from the peripheral antenna to the proteins in proximity to the reaction center occurs on a faster time scale, the final step of the energy transfer to the RC core is rather slow, and thus the limiting step in the transfer chain. Our findings suggest that the structure of the PS-II supercomplex guarantees photoprotection rather than optimized efficiency.

Internal Conversion and Vibrational Energy Redistribution in Chlorophyll A.

We have computationally investigated the role of intramolecular vibrational modes in determining nonradiative relaxation pathways of photoexcited electronic states in isolated chlorophyll A (ChlA) molecules. To simulate the excited state relaxation from the initially excited Soret state to the lowest excited state Qy, the approach of nonadiabatic excited state molecular dynamics has been adopted. The intramolecular vibrational energy relaxation and redistribution that accompany the electronic internal conversion process is followed by analyzing the excited state trajectories in terms of the ground state equilibrium normal modes. The time dependence of the normal mode velocities is determined by projecting instantaneous Cartesian velocities onto the normal mode vectors. Our analysis of the time evolution of the average mode energies uncovers that only a small subset of the medium-to-high frequency normal modes actively participate in the electronic relaxation processes. These active modes are characterized by the highest overlap with the nonadiabatic coupling vectors (NACRs) during the electronic transitions. Further statistical analysis of the nonadiabatic transitions reveals that the electronic and vibrational energy relaxation occurs via two distinct pathways with significantly different time scales on which the hopping from Soret to Qx occurs thereby dictating the overall dynamics. Furthermore, the NACRs corresponding to each of the transitions have been consistently found to be predominantly similar to a set of normal modes that vary depending upon the transition and the identified categories. Each pathway exhibits a differential time scale of energy transfer and also a differential set of predominant active modes. Our present analysis can be considered as a general approach allowing identification of a reduced subset of specific vibrational coordinates associated with nonradiative relaxation pathways. Therefore, it represents an adequate prior strategy that can particularly facilitates mixed quantum-classical approaches.

Hybrid Optimized and Localized Vibrational Coordinates.

We present a new type of vibrational coordinates denoted hybrid optimized and localized coordinates (HOLCs) aiming at a good set of rectilinear vibrational coordinates supporting fast convergence in vibrational stucture calculations. The HOLCs are obtained as a compromise between the recently promoted optimized coordinates (OCs) and localized coordinates (LCs). The three sets of coordinates are generally different from each other and differ from standard normal coordinates (NCs) as well. In determining the HOLCs, we optimize the vibrational self-consistent field (VSCF) energy with respect to orthogonal transformation of the coordinates, which is similar to determining OCs but for HOLCs we additionally introduce a penalty for delocalization, by using a measure of localization similar to that employed in determining LCs. The same theory and implementation covers OCs, LCs, and HOLCs. It is shown that varying one penalty parameter allows for connecting OCs and LCs. The HOLCs are compared to NCs, OCs, and LCs in their nature and performance as basis for vibrational coupled cluster (VCC) response calculations of vibrational anharmonic energies for a small set of simple systems comprising water, formaldehyde, and ethylene. It is found that surprisingly good results can be obtained with HOLCs by using potential energy surfaces as simple as quadratic Taylor expansions. Quite similar coordinates are found for the already established OCs but obtaining these OCs requires much more elaborate and expensive potential energy surfaces and localization is generally not guaranteed. The ability to compute HOLCs for somewhat larger systems is demonstrated for coumarin and the alanine quadramer. The good agreement between HOLCs and OCs, together with the much easier applicability of HOLCs for larger systems, suggests that HOLCs may be a pragmatically very interesting option for anharmonic calculations on medium to large molecular systems.

Aspects of the Eckart frame ro-vibrational kinetic energy operator.

A most decisive part of developing the Eckart frame ro-vibrational kinetic energy operator (KEO) in terms of curvilinear internal coordinates is the introduction of vibrational displacements obeying the Eckart conditions. It is examined here, how such displacements may be constructed by orthogonal and nonorthogonal transformations, and how they can be employed in developing the KEO. In particular, it is shown analytically, by term by term comparison, that the Eckart KEOs obtained by employing Pickett's method and the gateway Hamiltonian method are equivalent. That is, although it may appear in the form of very different expressions, the general, exact ro-vibrational Eckart frame KEO is unique. As side result, it has been shown that the simple formula of the pseudo-potential derived by Watson for the exact KEO given in normal coordinates applies to any Eckart frame KEO, irrespective of the choice of the vibrational coordinates.

The vibrational Jahn–Teller effect in E⊗e systems

The Jahn–Teller theorem is applied in the vibrational context where degenerate high-frequency vibrational states (E) are considered as adiabatic functions of low-frequency vibrational coordinates (e). For CH3CN and Cr(C6H6)(CO)3, the global minimum of the non-degenerate electronic potential energy surface occurs at the C3v geometry, but in CH3OH, the equilibrium geometry is far from the C3v reference geometry. In the former cases, the computed spontaneous Jahn–Teller distortion is exceptionally small. In methanol, the vibrational Jahn–Teller interaction results in the splitting of the degenerate E-type CH stretch into what have been traditionally assigned as the distinct ν2 and ν9 vibrational bands. The ab initio vibrational frequencies are fit precisely by a two-state high-order Jahn–Teller Hamiltonian (Viel and Eisfeld, 2004). The presence of vibrational conical intersections, including 7 for CH3OH, has implications for spectroscopy, for geometric phase, and for ultrafast localized non-adiabatic energy transfer.

Equilibrium molecular structure of benzamide from gas-phase electron diffraction and theoretical calculations

The molecule structure of benzamide has been determined in the gas phase by electron diffraction using results from quantum chemical calculations for constraints in the analysis. The root-mean-square amplitudes of vibration and harmonic shrinkage corrections were calculated taking into account nonlinear relations between Cartesian and internal vibrational coordinates at the level of first-order perturbation theory. The corresponding r h1 structure agrees well with the results of B3LYP/cc-pVTZ calculations. To take into account vibrational effects, the corrections to the experimental r a bond lengths, the values of the differences r e − r a were obtained using quadratic and cubic force constants from ab initio calculations at the MP2/cc-pVTZ level of theory). The equilibrium structure from the experiment agrees well with the results of the ab initio calculations. The following structural parameters were obtained (bond lengths in Angstroms and bond angles in degrees with 3σ in parentheses): r(C=C)av = 1.392(1), r(C–C) = 1.502(4), r(C–N) = 1.367(1), r(C=O) = 1.224(3), r(C–H) = 1.084(4), ∠C6C1C2 = 119.8, ∠CCN = 116.8(10), ∠CCO = 120.8(11), ∠C2C1C7O = 18.4(27). The C=O bond is shorter by 0.02 A, and the C–N bond is longer by 0.03 A than the corresponding bonds in the crystal. These differences are ascribed to the effect of intermolecular hydrogen bonding on the structure in the crystal.

Understanding nuclear motions in molecules: Derivation of Eckart frame ro-vibrational Hamiltonian operators via a gateway Hamiltonian operator.

A new ro-vibrational Hamiltonian operator, named gateway Hamiltonian operator, with exact kinetic energy term, Tˆ, is presented. It is in the Eckart frame and it is of the same form as Watson's normal coordinate Hamiltonian. However, the vibrational coordinates employed are not normal coordinates. The new Hamiltonian is shown to provide easy access to Eckart frame ro-vibrational Hamiltonians with exact Tˆ given in terms of any desired set of vibrational coordinates. A general expression of the Eckart frame ro-vibrational Hamiltonian operator is given and some of its properties are discussed.

A hybrid variational-perturbation calculation of the ro-vibrational spectrum of nitric acid.

Rotation-vibration spectra of the nitric acid molecule, HNO3, are calculated for wavenumbers up to 7000 cm(-1). Calculations are performed using a Hamiltonian expressed in internal curvilinear vibrational coordinates employing a hybrid variational-perturbation method. An initial potential energy surface (PES) and dipole moment function (DMF) are calculated ab initio at the CCSD(T)/aug-cc-pVQZ level of theory. Parameters of the PES and DMF are varied to minimize differences between the calculated and experimental transition frequencies and intensities. The average, absolute deviation between calculated and experimental values is 0.2 cm(-1) for frequencies in the fundamental bands and 0.4 cm(-1) for those in the first overtone and lowest combination bands. For the intensities, the calculated and experimental values differ by 0.3% and 40% for the fundamentals and overtones, respectively. The optimized PES and DMF are used to calculate the room-temperature ro-vibrational spectrum. These calculation reproduce both the form of the absorption bands and fine details of the observed spectra, including the rotational structure of the vibrational bands and the numerous hot absorption band. Many of these hot bands are found to be missing from the compilation in HITRAN. A room temperature line list comprising 2 × 10(9) lines is computed.

Observation of the wavepacket dynamics on the (1)B2((1)Σ(u)(+)) state of CS2 by sub-20 fs photoelectron imaging using 159 nm probe pulses.

The wavepacket dynamics of CS2 after photoexcitation to the (1)B2((1)Σu(+)) state at 198 nm are studied by time-resolved photoelectron imaging using sub-20 fs 159 nm pulses, which enable single photon ionization from the entire region of the (1)B2 potential energy surface. The time-energy map of the photoelectron intensity reveals vibrational motions along the symmetric stretching and bending coordinates. The time-energy map of the photoelectron anisotropy parameter exhibits time-evolution within single oscillation periods of the ν1 and ν2 modes, which is attributed to variation of the excited state electronic character along these vibrational coordinates. The initially populated (1)B2 state evolves with two time constants of 107 and 394 fs.

Limitation on the method of strained coordinates for vibrations with weak grazing unilateral contact

The nonlinear normal modes have recently been investigated with the method of strained coordinates for spring-mass models with some springs with piecewise linear stiffness. The N d.o.f. case and the mathematical validity of the method for large time were rigorously proved. The time validity is related to the nature of contact. For grazing contact, this method and also the multiscale expansion lose nonlinear features. For a small piecewise linear stiffness, we show that this method is less precise for a weak unilateral grazing contact. Thus, the validity of the asymptotic expansion for large time cannot be improved and the method has to be modified.

The (E + A) × (e + a) Jahn-Teller and pseudo-Jahn-Teller Hamiltonian including spin-orbit coupling for trigonal systems.

The Hamiltonian describing E × e Jahn-Teller (JT) coupling and (E + A) × (e + a) pseudo-JT (PJT) coupling is developed beyond the standard JT theory for the example of XY3 systems, taking the bending modes of a and e symmetry into account. For the electrostatic (spin-free) Hamiltonian, the conventional Taylor expansion up to second order in symmetry-adapted displacements is replaced by an expansion in invariant polynomials up to arbitrarily high orders. The relevance of a systematic high-order expansion in the three large-amplitude bending modes is illustrated by the construction of an eighth-order three-sheeted three-dimensional ab initio potential-energy surface for PH3+. The theory of spin-orbit coupling in trigonal JT/PJT systems is extended beyond the standard model of JT theory by an expansion of the microscopic Breit-Pauli operator up to second order in symmetry-adapted vibrational coordinates. It is shown that a linear E × e JT effect of relativistic origin exists in C(3v) systems which vanishes at the planar (D(3h)) geometry. The linear relativistic 2E – 2A PJT coupling, on the other hand, persists at the planar geometry

Infrared and Raman spectra, theoretical calculations, conformations, and two-dimensional potential energy surface of 2-cyclopenten-1-one ethylene ketal.

The infrared and Raman spectra of the bicyclic spiro molecule 2-cyclopenten-1-one ethylene ketal (CEK) have been recorded. Density functional theory (DFT) calculations were used to compute the theoretical spectra, and these agree well with the experimental spectra. The structures and conformational energies for the two pairs of conformational minima, which can be defined in terms of ring-bending (x) and ring-twisting (τ) vibrational coordinates, have also been calculated. Utilizing the results from ab initio MP2/cc-PVTZ computations, a two-dimensional potential energy surface (PES) was calculated. The energy levels and wave functions for this PES were then calculated, and the characteristics of these were analyzed. At lower energies, all of the quantum states are doubly degenerate and correspond to either the lower-energy conformation L or to conformation H, which is 154 cm(-1) higher in energy. At energies above the barrier to interconversion of 264 cm(-1), the wave functions show that the quantum levels have significant probabilities for both conformations.

Reaction coordinates for elucidating reaction dynamics with anharmonic couplings

Saddle points can be found on the potential energy surface between the reactants and products regions in many chemical reactions. The dynamics occurring in the vicinity of a saddle point play an essential role in the study of reaction dynamics because the reactivity of a trajectory is primarily determined in the saddle region. There are many cases where a well‐defined boundary (called reactivity boundary here) between reacting and nonreacting trajectories can be located in the phase space. Locating the reactivity boundary, however, becomes challenging when anharmonic couplings between the reaction coordinate and the vibrational coordinates complicate the reaction dynamics. The aim of this article is to present a review of recent studies that use nonlinear coordinate transformations to disentangle the anharmonic couplings. The reaction coordinates constructed through such nonlinear coordinate transformations are decoupled from the other coordinates as much as possible, thus simplifying analysis of reaction dynamics. Several options are introduced for nonlinear coordinate transformations which should be appropriately chosen by examining the extent of the anharmonic couplings. Under certain conditions, special reaction coordinates constructed by a suitable nonlinear coordinate transformation reveals the existence of a clear reactivity boundary in the phase space even when there are strong anharmonic couplings. © 2014 Wiley Periodicals, Inc.

A time-dependent order parameter for ultrafast photoinduced phase transitions.

Strongly correlated electron systems often exhibit very strong interactions between structural and electronic degrees of freedom that lead to complex and interesting phase diagrams. For technological applications of these materials it is important to learn how to drive transitions from one phase to another. A key question here is the ultimate speed of such phase transitions, and to understand how a phase transition evolves in the time domain. Here we apply time-resolved X-ray diffraction to directly measure the changes in long-range order during ultrafast melting of the charge and orbitally ordered phase in a perovskite manganite. We find that although the actual change in crystal symmetry associated with this transition occurs over different timescales characteristic of the many electronic and vibrational coordinates of the system, the dynamics of the phase transformation can be well described using a single time-dependent 'order parameter' that depends exclusively on the electronic excitation.

ChemInform Abstract: Elementary Electron Transfer Reactions: From Basic Concepts to Recent Computational Advances

AbstractReview: 118 refs.

Rovibrational levels and wavefunctions of Cl(-)H2O.

In this paper, we report vibrational levels computed for Cl(-)H2O and compare with experimental observations and previous approximate calculations. In addition, we present rovibrational levels. The calculations are done using a symmetry adapted Lanczos algorithm and a product basis set and include all six vibrational coordinates. The basis functions have amplitude in both wells and enable us to determine tunnelling splittings. We use the potential energy surface of Rheinecker and Bowman [J. Chem. Phys. 125, 133206 (2006)]. Rovibrational levels are assigned vibrational labels using vibrational parent analysis. Our results reveal the effect of previous approximations, but are in fairly good agreement with prior calculations and experiments.

Vibrationally coherent crossing and coupling of electronic states during internal conversion in β-carotene.

Coupling of nuclear and electronic degrees of freedom mediates energy flow in molecules after optical excitation. The associated coherent dynamics in polyatomic systems, however, remain experimentally unexplored. Here, we combined transient absorption spectroscopy with electronic population control to reveal nuclear wave packet dynamics during the S2 → S1 internal conversion in β-carotene. We show that passage through a conical intersection is vibrationally coherent and thereby provides direct feedback on the role of different vibrational coordinates in the breakdown of the Born-Oppenheimer approximation.

Communication: The influence of vibrational parity in chiral photoionization dynamics

A pronounced vibrational state dependence of photoelectron angular distributions observed in chiral photoionization experiments is explored using a simple, yet realistic, theoretical model based upon the transiently chiral molecule H2O2. The adiabatic approximation is used to separate vibrational and electronic wavefunctions. The full ionization matrix elements are obtained as an average of the electronic dipole matrix elements over the vibrational coordinate, weighted by the product of neutral and ion state vibrational wavefunctions. It is found that the parity of the vibrational Hermite polynomials influences not just the amplitude, but also the phase of the transition matrix elements, and the latter is sufficient, even in the absence of resonant enhancements, to account for enhanced vibrational dependencies in the chiral photoionization dynamics.

Hydrogen bonded pyridine N-oxide/trichloroacetic acid complex in polar media: 2D potential energy surface and O–H⋯O vibration analysis using exact vibrational Hamiltonian

The analysis of O–H⋯O stretching vibrations of the pyridine N-oxide/trichloroacetic acid (PyO/TCA) H-bond complex in acetonitrile solution was carried out. 1D and 2D potential energy surfaces associated with the variation of valence coordinates of hydroxyl and hydrogen bonds were calculated for this purpose in the B3LYP/cc-pVTZ approximation. The exact form of kinetic energy operator was obtained using these coordinates and Wilson’s vectors. The numerical solution of 2D Schrödinger equation using Fourier series was realized and the wave numbers of O–H and O⋯H vibrations were calculated and compared with the results obtained using different sets of vibrational coordinates. The values of the O–H⋯O stretching frequencies obtained as a result of the matrix diagonalization were discussed and compared with the experimental data, the results of harmonic- and anharmonic computations as well as with the results of 3D computations of potential energy surfaces of the PyO/TCA complex using simplified form of kinetic energy operator.

Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).