The Electronic Adiabatic-Diabatic Transformation Matrix: A Theoretical and Numerical Study of a Three-State System

Quasi-Classical Trajectory Calculations on a Two-State Potential Energy Surface Including Nonadiabatic Coupling Terms as Friction for D+ + H2 Collisions

Akin to the traditional quasi-classical trajectory method for investigating the dynamics on a single adiabatic potential energy surface for an elementary chemical reaction, we carry out the dynamics on a 2-state ab initio potential energy surface including nonadiabatic coupling terms as friction terms for D+ + H2 collisions. It is shown that the resulting dynamics correctly accounts for nonreactive charge transfer, reactive non-charge transfer and reactive charge transfer processes. In addition, it leads to the formation of triatomic DH2+ species as well.

Coupled 3D (J ≥ 0) Time-Dependent Wave Packet Calculation for the F + H2 Reaction on Accurate Ab Initio Multi-State Diabatic Potential Energy Surfaces

We had calculated adiabatic potential energy surfaces (PESs), nonadiabatic, and spin-orbit (SO) coupling terms among the lowest three electronic states (12A', 22A', and 12A″) of the F + H2 system using the multireference configuration interaction (MRCI) level of theory, and the adiabatic-to-diabatic transformation equations were solved to formulate the diabatic Hamiltonian matrix [J. Chem. Phys. 2020, 153, 174301] for the entire region of the nuclear configuration space. The accuracy of such diabatic PESs is explored by performing scattering calculations to evaluate integral cross sections (ICSs) and rate constants. The nonadiabatic and SO effects are studied by utilizing coupled 3D time-dependent wave packet formalism with zero and nonzero total angular momentum on multiple adiabatic/diabatic surfaces calculation. We depict the convergence profiles of reaction probabilities for the reactive as well as nonreactive processes on various electronic states at different collision energies with respect to total angular momentum including all helicity quantum numbers. Finally, total ICSs are calculated as functions of collision energies for the initial rovibrational state (v = 0, j = 0) of the H2 molecule along with the temperature-dependent rate coefficient, where those quantities are compared with previous theoretical and experimental results.

Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?

While carrying out Beyond Born-Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of benzene radical cation (C6H6+) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results.

Beyond Born-Oppenheimer Constructed Diabatic Potential Energy Surfaces for HeH2. J Phys Chem A 2023;127:3832-3847. [PMID: 37098130 DOI: 10.1021/acs.jpca.3c01047]

First-principles based beyond Born-Oppenheimer theory has been employed to construct multistate global Potential-Energy Surfaces (PESs) for the HeH₂⁺ system by explicitly incorporating the Nonadiabatic Coupling Terms (NACTs). Adiabatic PESs and NACTs for the lowest four electronic states (1²A', 2²A', 3²A' and 4²A') are evaluated as functions of hyperangles for a grid of fixed values of the hyperradius in hyperspherical coordinates. Conical intersection between different states are validated by integrating the NACTs along appropriately chosen contours. Subsequently, adiabatic-to-diabatic (ADT) transformation angles are determined by solving the ADT equations to construct the diabatic potential matrix for the HeH₂⁺ system which are smooth, single-valued, continuous, and symmetric and are suitable for performing accurate scattering calculations for the titled system.

Beyond Born-Oppenheimer based diabatic surfaces of 1,3,5-C6H3F3+ to generate the photoelectron spectra using time-dependent discrete variable representation approach

In this article, Beyond Born-Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C₆H₃F₃⁺ over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes (Q₂, Q_9x, Q_9y, Q_13x, Q_13y, Q_18x, Q_18y, Q_10x, Q_10y, Q_12x and Q_12y) to include all possible nonadiabatic interactions among six coupled electronic states (X̃²E'', , B̃²E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys., 2019, 150, 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C₆H₃F₃. Our theoretically calculated spectra for X̃²E'', and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method.

The role of electron-nuclear coupling on multi-state photoelectron spectra, scattering processes and phase transitions

We present first principle based beyond Born-Oppenheimer (BBO) theory and its applications on various models as well as realistic spectroscopic and scattering processes, where the Jahn-Teller (JT) theory is brought in conjunction with the BBO approach on the phase transition of lanthanide complexes. Over one and half decades, our development of BBO theory is demonstrated with ab initio calculations on representative molecules of spectroscopic interest (NO2 radical, Na3 and K3 clusters, NO3 radical, C6H6+ and 1,3,5-C6H3F3+ radical cations) as well as triatomic reactive scattering processes (H+ + H2 and F + H2). Such an approach exhibits the effect of JT, Renner-Teller (RT) and pseudo Jahn-Teller (PJT) type of interactions. While implementing the BBO theory, we generate highly accurate diabatic potential energy surfaces (PESs) to carry out quantum dynamics calculation and find excellent agreement with experimental photoelectron spectra of spectroscopic systems and cross-sections/rate constants of scattering processes. On the other hand, such electron-nuclear couplings incorporated through JT theory play a crucial role in dictating higher energy satellite transitions in the dielectric function spectra of the LaMnO3 complex. Overall, this article thoroughly sketches the current perspective of the BBO approach and its connection with JT theory with various applications on physical and chemical processes.

Beyond Born-Oppenheimer constructed diabatic potential energy surfaces for F + H2 reaction

First principles based beyond Born-Oppenheimer theory has been implemented on the F + H₂ system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin-orbit (SO) coupling effect on the collision process of the F + H₂ reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (1²A', 2²A', and 1²A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn-Teller (JT) type conical intersections between the two A' states translate along C_2v and linear geometries in F + H₂. In addition, A' and A″ states undergo Renner-Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H₂, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H₂ system.

Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation

We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of non-adiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wave functions. ADT matrices eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrödinger equation to its diabatic form. Two real molecular systems NO₃ and C₆H₆⁺ (Bz⁺) are selectively chosen for the demonstration of workability of those equations. We examine the NACTs among the lowest five electronic states of the NO₃ radical [X̃²A₂^' (1²B₂), òE″ (1²A₂ and 1²B₁) and B̃²E' (1²A₁ and 2²B₂)], in which all types of non-adiabatic interactions, that is, Jahn-Teller (JT) interactions, Pseudo Jahn-Teller (PJT) interactions, and accidental conical intersections (CIs) are present. On the other hand, lowest five electronic states of Bz⁺ [X̃²E_1g (1²B_3g and 1²B_2g), B̃²E_2g (1²A_g and 1²B_1g), and C̃²A_2u (1²B_1u)] depict similar kind of complex feature of non-adiabatic effects. For NO₃ radical, the two components of degenerate in-plane asymmetric stretching mode are taken as a plane of nuclear configuration space (CS), whereas in case of Bz⁺, two pairs are chosen: One is the pair of components of degenerate in-plane asymmetric stretching mode, and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asymmetric stretching modes. We calculate ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chemistry package. Subsequently, the ADT is performed using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, symmetric, and continuous diabatic PESs for both the molecular systems.

Study of nonadiabatic effects in low-lying electronic states of HCNH with implication in its dissociation to HCN and HNC

Abnormal abundance of HNC in interstellar spaces has been the motivation of many experimental as well as theoretical studies of the branching ratio [HNC]/[HCN] in the dissociation of HCNH, right after its formation from electron capture by HCNH(+), available in the upper atmosphere. In the present work we were interested in nonadiabatic studies involving the dissociation channel of HCNH leading to the formation of HNC. This study reports for the first time that the conical intersection (CI) between the states 1(2)Σ(+) and 2(2)Σ(+) exists only in some bent geometry (and not in the collinear geometry) where both of these states have A' symmetry. This finding is important as this CI is crucial in the dissociation of HCNH. We further report that these two states strongly couple with 1(2)Π, the lowest electronic state of collinear HCNH. Hence we construct a three-state Hilbert subspace (HSS), comprising of the states 1(2)Π, 1(2)Σ(+), and 2(2)Σ(+), in a configuration space where these states interact very strongly and the adiabatic-to-diabatic transformation angles (mixing angle) yield meaningful values of topological (Berry) phase. This leads to the construction of the corresponding three-state diabatic potentials. We advocate that these diabatic potentials, considering both the linear as well as bent configurations, nicely elucidate the formation of the HNC molecule by the CH bond dissociation of HCNH molecule.

A tri-atomic Renner-Teller system entangled with Jahn-Teller conical intersections

The present study concentrates on a situation where a Renner-Teller (RT) system is entangled with Jahn-Teller (JT) conical intersections. Studies of this type were performed in the past for contours that surround the RT seam located along the collinear axis [see, for instance, G. J. Halász, Á. Vibók, R. Baer, and M. Baer, J. Chem. Phys. 125, 094102 (2006)]. The present study is characterized by planar contours that intersect the collinear axis, thus, forming a unique type of RT-non-adiabatic coupling terms (NACT) expressed in terms of Dirac-δ functions. Consequently, to calculate the required adiabatic-to-diabatic (mixing) angles, a new approach is developed. During this study we revealed the existence of a novel molecular parameter, η, which yields the coupling between the RT and the JT NACTs. This parameter was found to be a pure number η = 22/π (and therefore independent of any particular molecular system) and is designated as Renner-Jahn coupling parameter. The present study also reveals an unexpected result of the following kind: It is well known that each (complete) group of states, responsible for either the JT-effect or the RT-effect, forms a Hilbert space of its own. However, the entanglement between these two effects forms a third effect, namely, the RT/JT effect and the states that take part in it form a different Hilbert space.

Jahn-Teller intersections induced by introduction of bending in linear polyatomics: study with HCNH, a selected molecular system

Demonstration of onset of Jahn-Teller (JT) intersections, characterized by a topological phase of π, on introduction of bending in the collinear tetra-atomic C(2)H(2)(+) cation, originally a Renner-Teller (RT) system, has raised interest in the study of the generality of this phenomenon. This interest has initiated similar study by shifting one external light atom from the collinear molecular axes of systems such as HCNH and HC(2)O. Recent studies have revealed that slightly bent HCNH poses a specialty in this regard and thus demands focused attention. In the present work we performed a combined study using both the potential intersections and the angular nonadiabatic coupling terms (NACTs) and report two new results: (i) The first result is the appearance of the JT conical intersections (ci's) between the two lowest states of a slightly bent HCNH molecule (1(2)A' and 1(2)A", originated from collinear X(2)Π state) only with certain nonplanar configurations, in contrast to its appearance in C(S) configuration space (molecular plane) for slightly bent C(2)H(2)(+), HC(2)O, or some other tetra-atomics. This is also associated with the first time demonstration of Berry phase for such a single isolated ci for HCNH molecule. (ii) For energetically higher potentials of slightly bent HCNH, the present study reveals the existence of the intersection of the state 1(2)A" with the state 2(2)A' (originated from the collinear 1(2)Σ(+) state); in contrast to the (1,2) ci this appears in the molecular plane. The search for ci's has been performed by varying the distances of the two H-atoms (designated as H(C) and H(N)) from the C-N axis as well as the dihedral angle φ between the two planes (H(C),C,N) and (C,N,H(N)). Existence of these JT ci's results in deterioration of a two-state Hilbert subspace (HSS) model in diabatization, while the proper choice of an n (>2)-state HSS circumvents this problem.

Renner-Teller intersections along the collinear axes of polyatomic molecules: H2CN as a case study

The tetra-atomic C(2)H(2)(+) cation is known to form Renner-Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to 2pi, which is a result anticipated in the case of Renner-Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals pi (and not 2pi). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn-Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the H(2)CN molecule, which just like the C(2)H(2)(+) ion, is characterized by Renner-Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner-Teller intersections along the collinear axis, but in contrast to the C(2)H(2)(+) case a shift of one atom from the collinear arrangement did not form Jahn-Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner-Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected).

Curl condition for a four-state Born-Oppenheimer system employing the Mathieu equation

When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.

Direct UV Raman monitoring of 3(10)-helix and pi-bulge premelting during alpha-helix unfolding

We used UV resonance Raman (UVRR) spectroscopy exciting at approximately 200 nm within the peptide bond pi --> pi* transitions to selectively study the amide vibrations of peptide bonds during alpha-helix melting. The dependence of the amide frequencies on their Psi Ramachandran angles and hydrogen bonding enables us, for the first time, to experimentally determine the temperature dependence of the peptide bond Psi Ramachandran angle population distribution of a 21-residue mainly alanine peptide. These Psi distributions allow us to easily discriminate between alpha-helix, 3(10)-helix and pi-helix/bulge conformations, obtain their individual melting curves, and estimate the corresponding Zimm and Bragg parameters. A striking finding is that alpha-helix melting is more cooperative and shows a higher melting temperature than previously erroneously observed. These Psi distributions also enable the experimental determination of the Gibbs free energy landscape along the Psi reaction coordinate, which further allows us to estimate the free energy barriers along the AP melting pathway. These results will serve as a benchmark for the numerous untested theoretical studies of protein and peptide folding.

Fluorescence probe of Trp-cage protein conformation in solution and in gas phase

Measurements of protein unfolding in the absence of solvent, when combined with unfolding studies in solution, offer a unique opportunity to measure the effects of solvent on protein structure and dynamics. The experiments presented here rely on the fluorescence of an attached dye to probe the local conformational dynamics through interactions with a Trp residue and fields originating on charge sites. We present fluorescence measurements of thermal fluctuations accompanying conformational change of a miniprotein, Trp-cage, in solution and in gas phase. Molecular dynamics (MD) simulations are performed as a function of temperature, charge state, and charge location to elucidate the dye-protein conformational dynamics leading to the changes in measured fluorescence. The results indicate that the stability of the unsolvated protein is dominated by hydrogen bonds. Substituting asparagine for aspartic acid at position 9 results in a dramatic alteration of the solution unfolding curve, indicating that the salt bridge involving Lys8, Asp9, and Arg16 (+ - +) is essential for Trp-cage stability in solution. In contrast, this substitution results in minor changes in the unfolding curve of the unsolvated protein, showing that hydrogen bonds are the major contributor to the stability of Trp-cage in gas phase. Consistent with this hypothesis, the decrease in the number of hydrogen bonds with increasing temperature indicated by MD simulations agrees reasonably well with the experimentally derived enthalpies of conformational change. The simulation results display relatively compact conformations compared with NMR structures that are generally consistent with experimental results. The measured unfolding curves of unsolvated Trp-cage ions are invariant with the acetonitrile content of the solution from which they are formed, possibly as a result of conformational relaxation during or after desolvation. This work demonstrates the power of combined solution and gas-phase studies and of single-point mutations to identify specific noncovalent interactions which contribute to protein-fold stability. The combination of experiment and simulation is particularly useful because these approaches yield complementary information which can be used to deduce the details of structural changes of proteins in the gas phase.

Protein folding kinetics: barrier effects in chemical and thermal denaturation experiments

Recent experimental work on fast protein folding brings about an intriguing paradox. Microsecond-folding proteins are supposed to fold near or at the folding speed limit (downhill folding), but yet their folding behavior seems to comply with classical two-state analyses, which imply the crossing of high free energy barriers. However, close inspection of chemical and thermal denaturation kinetic experiments in fast-folding proteins reveals systematic deviations from two-state behavior. Using a simple one-dimensional free energy surface approach we find that such deviations are indeed diagnostic of marginal folding barriers. Furthermore, the quantitative analysis of available fast-kinetic data indicates that many microsecond-folding proteins fold downhill in native conditions. All of these proteins are then promising candidates for an atom-by-atom analysis of protein folding using nuclear magnetic resonance.1 We also find that the diffusion coefficient for protein folding is strongly temperature dependent, corresponding to an activation energy of approximately 1 kJ.mol-1 per protein residue. As a consequence, the folding speed limit at room temperature is about an order of magnitude slower than the approximately 1 micros estimates from high-temperature T-jump experiments. Our analysis is quantitatively consistent with the available thermodynamic and kinetic data on slow two-state folding proteins and provides a straightforward explanation for the apparent fast-folding paradox.

Thermal Denaturation of Polyalanine Peptide in Water by Molecular Dynamics Simulations and Theoretical Prediction of Infrared Spectra: Helix−Coil Transition Kinetics

Perspectives in the helix-coil transition kinetics of secondary structures are examined by temperature-jump molecular dynamics (T-jump MD) simulations and theoretically calculated infrared (IR) spectra. Homopolymeric polyalanine, Ac-(A)(21)-NHMe (A21), is unfolded in water by T-jumps from 273 to 300 K approximately 450 K using AMBER ff99 and ff03 force fields. MD simulation results provide in silico evidence that 3(10)-helix and type I beta-turn motifs are highly probable in both ff99 and ff03 results. Temperature-dependent difference IR spectra of A21 do not possess an isosbestic point in both results, and isotope-labeled difference IR spectra in ff03 results predict characteristic profiles observed in experiments. Unfolding rates obtained from simulated time-resoled IR spectra are in good agreement with those estimated by helical contents, but they are still an order of magnitude smaller than experimental values. We demonstrate that the conventional criteria such as single-exponential fit of transient amide I absorbance, sigmoidal fit of temperature-dependent amide I absorbance, and Arrhenius plot of relaxation rates cannot guarantee the validity of assuming a two-state mechanism. We suggest a way of determining T(m) by the temperature dependence of center frequency and full width at half-maximum of amide I band. Overall, both ff99 and ff03 force fields give consistent results in reproducing key aspects concerned experimentally, but are not predominantly satisfactory in quantitative aspects.

H3+ in the electronic triplet state: current status

We review the theoretical work carried out on the tri-hydrogen ion in the electronic triplet state 1(3)E', which is split into a3Sigma+u and 2(3)A' by vibronic interaction. We begin with an overview on analytical potential energy surfaces and calculations of rovibrational states by focusing on our own results, which are based on the most accurate potential energy surfaces available so far. This is followed by an examination of the selection rules and predictions of infrared transition frequencies. Finally, we discuss the Slonczewski resonance states supported by the upper sheet of the potential energy surface. Theoretical work reported here may be of interest for future experiments on the title ion.

Shedding Light on Biomolecule Conformational Dynamics Using Fluorescence Measurements of Trapped Ions

Biomolecule conformational change has been widely investigated in solution using several methods; however, much less experimental data about structural changes are available for completely isolated, gas-phase biomolecules. Studies of conformational change in unsolvated biomolecules are required to complement the interpretation of mass spectrometry measurements and in addition, can provide a means to directly test theoretical simulations of biomolecule structure and dynamics independent of a simulated solvent. In this Feature Article, we review our recent introduction of a fluorescence-based method for probing local conformational dynamics in unsolvated biomolecules through interactions of an attached dye with tryptophan (Trp) residues and fields originating on charge sites. Dye-derivatized biomolecule ions are formed by electrospray ionization and are trapped in a variable-temperature quadrupole ion trap in which they are irradiated with either continuous or short pulse lasers to excite fluorescence. Fluorescence is measured as a function of temperature for different charge states. Optical measurements of the dye fluorescence include average intensity changes, changes in the emission spectrum, and time-resolved measurements of the fluorescence decay. These measurements have been applied to the miniprotein, Trp-cage, polyproline peptides and to a beta-hairpin-forming peptide, and the results are presented as examples of the broad applicability and utility of these methods. Model fits to Trp-cage fluorescence data measured as a function of temperature provide quantitative information on the thermodynamics of conformational changes, which are reproduced well by molecular dynamics. Time-resolved measurements of the fluorescence decays of Trp-cage and small polyproline peptides definitively demonstrate the occurrence of fluorescence quenching by the amino acid Trp in unsolvated biomolecules.

Extended Born-Oppenheimer equation for a three-state system

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.

N-State Adiabatic-to-Diabatic Transformation Angle: Theory and Application

In this article is discussed a new diabatization procedure which is expected to be reliable and, also, relatively easy to implement. This procedure takes into account the two main ingredients related to diabatization: (1) The size N of the smallest (relevant) group of states that forms a Hilbert subspace (this fact enforces the dimension of the adiabatic-to-diabatic transformation matrix to be N). (2) The total energy E which determines the number of open states, p, within this group of N states. The main emphasis in this manuscript is on the case that N is arbitrary but p is equal to 2. The various derivations as well as the final results are accompanied by numerical examples extracted from three- to five-state ab initio calculations for the H + H2 system.

Experimental resolution of early steps in protein folding: testing molecular dynamics simulations

Time-resolved Tyr fluorescence spectroscopy coupled with a laser-induced temperature-jump (T-jump) was employed to follow the folding relaxation dynamics of the B-domain of Staphylococcal protein A. The single Tyr is located in helix 1 (H1) and is a sensitive probe of the structure of this helix and the overall helical bundle structure. The results from this study were compared to those from a complementary infrared T-jump study on this protein [Vu, D. M.; Myers, J. K.; Oas, T. G.; Dyer, R. B. Biochemistry 2004, 43, 3582]. Both methods detect a microsecond process that follows the cooperative relaxation of the helical bundle core. However, a fast process (10-7 s) that follows the relaxation of the individual helices was observed only with the infrared probe. Thus, fast formation of H1 is not observed, but rather H1 forms in the microsecond phase, concomitantly with the docking to (and stabilization by) the other two helices to form the helical bundle structure. This observation validates the results of several previous molecular dynamics simulations that predict H1 formation only in the final assembly of the helix bundle.

Discriminating the Helical Forms of Peptides by NMR and Molecular Dynamics Simulation

The HNCO NMR pulse sequence was applied to three selectively labeled (15)N and (13)C isotopic homologues of the peptide Ac-WAAAH(AAARA)(3)A-NH(2) to probe directly for hydrogen bonds between residues 8 and 11 (characteristic of a 3(10)-helix), 8 and 12 (alpha-helix), and 8 and 13 (pi-helix). The experiments demonstrate conclusively, and in agreement with circular dichroism studies, that the center of the peptide is alpha-helical; there is no discernible 3(10)- or pi-helix at these specific positions. Molecular dynamics simulations of the preceding peptide and Ac-(AAAAK)(3)A-NH(2) in water using the potential energy parameter set CHARMM22/CMAP correctly yield an alpha-helix, in contrast to simulations with the set CHARMM22, which result in a pi-helix.