The absorption and fluorescence spectra of 4-(3-methoxybenzylidene)-2-methyl-oxazalone interpreted by Franck-Condon simulation in various pH solvent environments.

The absorption and fluorescence spectra of 4-(3-methoxybenzylidene)-2-methyl-oxazalone (m-MeOBDI) dissolved in neutral, acidic, and basic solvent environments have been investigated and assigned by using Franck-Condon (FC) simulations at the quantum TDDFT level. Four different structures of m-MeOBDI in the ground and excited states are optimized and are found to be responsible for the observed absorption and fluorescence spectra. The (absorption) fluorescence of m-MeOBDI in pure methanol and neutral/basic methanol/water (1/9 vol) mixed solvent is found to arise from the (ground neutral N-I) excited neutral N-I* and cationic C-III* species, respectively. In acidic solvent, the absorption is found to arise from ground acidic C-II species, and the excited divalent cation DC-IV* is found to be formed in its excited state due to the excess H+ in the solution, and then it emits ∼560 nm fluorescence. FC simulations have also been employed to confirm our assignments as well as interpret the vibronic band profiles. As expected, the simulated emission spectrum of the divalent cationic species is in good agreement with the experimental observation. Therefore, within the present FC simulation, the observed absorption and fluorescence spectra have been reasonably interpreted and novel fluorescence mechanisms of m-MeOBDI in various pH solvent environments have been proposed.

Extremely solvent-enhanced absorbance and fluorescence of carbazole interpreted using a damped Franck-Condon simulation.

Extremely solvent-enhanced absorption and fluorescence spectra of carbazole were investigated by performing a generalized multi-set damped Franck-Condon spectral simulation. Experimental absorption and fluorescence spectra of carbazole in the gas phase were first well reproduced by performing an un-damped Franck-Condon simulation, but a one-set scaling damped Franck-Condon simulation severely underestimated the intensities of the peaks of experimental absorption and fluorescence spectra of carbazole in n-hexane. Then, a multi-set scaling damped Franck-Condon simulation was proposed and carried out for simulating the extremely solvent-enhanced absorbance and fluorescence, and here, the simulated spectra agreed well with the experimental ones. Five (four) representative solvent-enhanced normal modes corresponding to the combination of ring stretching and ring breathing vibrational motions were determined to be responsible for enhanced absorbance (fluorescence) in n-hexane solution. Furthermore, different scalings were applied to the ground and first-excited states, resulting in different enhancement of absorbance and fluorescence, and this analysis revealed atoms in the carbazole interacting with n-hexane solvent molecules and, hence, leading to different normal-mode vibrational vector patterns in the ground and first-excited states, respectively. Basically, the same conclusion was drawn from a simulation with HF-CIS and the three functionals (TD)B3LYP, (TD)B3LYP-35, and (TD)BHandHLYP. The present multi-set scaling damped Franck-Condon simulation scheme was demonstrated to successfully interpret extremely solvent-enhanced absorbance and fluorescence of carbazole in n-hexane-solvent.

Dynamic Stark-Induced Coherent π-Electron Rotations in a Chiral Aromatic Ring Molecule: Application to Phenylalanine.

We present the results of a theoretical study on dynamic Stark-induced coherent π-electron rotations in a chiral aromatic ring molecule. This is an extension of our previous papers, which have been published in Mineo , H. [ Phys. Chem. Chem. Phys. 2016 , 18 , 26786 - 26795 ] and Mineo , H. [ J. Phys. Chem. Lett. 2018 , 9 , 5521 - 5526 ]. In those papers, the time-dependent Schrödinger equation was solved under a restricted condition in which a degenerate excited state should be formed at the center of the two relevant excited states by dynamic Stark effects. The dynamic Stark-induced degenerate state (DSIDS) is essential to create unidirectional π-electron rotations. In the present theoretical treatment, the above restriction is relaxed and the DSIDS is set to be at any energy position between the two excited states. This indicates a wide applicability of the dynamic Stark effects to coherent control of photophysical properties in aromatic molecules, such as coherent ring currents and current-induced magnetic fluxes of low-symmetric aromatic molecules. Analytical expressions for the coherent π-electron angular momentum are derived within a three-electronic-state model by using the Laplace transform method. The validity of the developed theoretical procedure is demonstrated by carrying out simulations of the coherent angular momentum of l-phenylalanine. Effects of varying the DSIDS on the time-dependent coherent angular momentum and the populations in the three electronic states are examined, and the results are analyzed using approximate expressions for the time-dependent coherent angular momentum and the populations. Modulations in the time-dependent coherent angular momentum appear when the DSIDS is set at an energy position between the two excited states, while there are no beating modulations when the DSIDS is set at the center position. Such differences originate from whether interferences between the two dressed states take place or not.

First-principles study on sum-frequency generation spectroscopy of methanol adsorbed on TiO2(110) surface: Effects of substrate and molecular coverages.

In this work, starting from the general theory of sum-frequency generation (SFG), we proposed a computational strategy utilizing density functional theory with periodic boundary conditions to simulate the vibrational SFG of molecules/solid surface adsorption system. The method has been applied to the CH3OH/TiO2(110) system successfully. Compared with the isolated molecule model, our theoretical calculations showed that the TiO2 substrate can significantly alter the second-order susceptibilities of a methanol molecule which is directly related to the SFG intensity. In addition, the SFG spectra have obvious changes while the methanol coverage increases, especially for the OH vibration peaks. Our theoretical spectra agree reasonably well with experimental measurements at 1 ML coverage, and an interesting peak which is absent in the theoretical spectra is tentatively assigned to some CH3 stretch vibration of methanol adsorbed on the oxygen vacancy of TiO2.

Magnetism-tuning strategies for graphene oxide based on magnetic oligoacene oxide patches model.

Graphene oxide (GO) has wide application potential owing to its 2D structure and diverse modification sites for various targeted uses. The introduction of magnetism into GO structures has further advanced the controllability of the application of GO materials. Herein, the concept of modular design and modeling was applied to tune the magnetism of GO. To obtain desirable magnetic properties, diradical-structured GO patches were formed by the introduction of two functional groups to break the Kekule structure of the benzene ring. In these diradical GO patches, the energy of the triplet state was lower than those of the open-shell broken-symmetry singlet state and closed-shell singlet state. To create such multi-radical patches, a practical approach is to determine a substantial spatial separation of the α and ß spin densities in the molecule. Thus, systematic design strategies and tests were evaluated. The first strategy was extending the distance between the distribution center of the α and ß spin densities; the second was controlling the delocalization directions of the α and ß electrons; the third was controlling the delocalization extension of the α and ß electrons by oxidative modification, and finally introducing multi-radical structures into the molecular system and controlling the position of each radical. Herein, successful molecular models with a large magnetic coupling constant (∼3600 cm-1) were obtained. This study paves the way to explore ferromagnetic MGO guided by theoretical study, which may become reality soon.

Directed motion from particle size oscillations inside an asymmetric channel.

The motion of a spherical Brownian particle in an asymmetric periodic channel is considered. Under an external periodic stimulus, the particle switches between two states with different particle radius, every half-period. Using Brownian dynamics simulations, we show that the particle size oscillation, combined with the asymmetry of the channel, induces a drift along the channel axis, directed towards the steeper wall of the channel. The oscillation of the particle size is accompanied by a time variation of the space accessible to the particle and by an oscillation of its diffusion coefficient. The former underlies the drift inducing mechanism of purely entropic nature. The latter, combined with the former, leads to a significant amplification of the effect. The drift velocity vanishes when interconversion between the states occurs either very slow or very fast, having a maximum in between. The position and magnitude of the maximum are discussed by providing an analytical approach based on intuitively appealing assumptions.

Theoretical investigations of absorption and fluorescence spectra of protonated pyrene.

The equilibrium geometry and 75 vibrational normal-mode frequencies of the ground and first excited states of protonated pyrene isomers were calculated and characterized in the adiabatic representation by using the complete active space self-consistent field (CASSCF) method. Electronic absorption spectra of solid neon matrixes in the wavelength range 495-415 nm were determined by Maier et al. and they were analyzed using time-dependent density functional theory calculations (TDDFT). CASSCF calculations and absorption and emission spectra simulations by one-photon excitation equations were used to optimize the excited and ground state structures of protonated pyrene isomers. The absorption band was attributed to the S0 → S1 electronic transition in 1H-Py(+), and a band origin was used at 20580.96 cm(-1). The displaced harmonic oscillator approximation and Franck-Condon approximation were used to simulate the absorption spectrum of the (1) (1)A' ← X[combining tilde](1)A' transition of 1H-Py(+), and the main vibronic transitions were assigned for the first ππ* state. It shows that the vibronic structures were dominated by one of the eight active totally symmetric modes, with ν15 being the most crucial. This indicates that the electronic transition of the S1((1)A') state calculated in the adiabatic representation effectively includes a contribution from the adiabatic vibronic coupling through Franck-Condon factors perturbed by harmonic oscillators. The present method can adequately reproduce experimental absorption and fluorescence spectra of a gas phase.

Exploration of hydrogen bond networks and potential energy surfaces of methanol clusters using a two-stage clustering algorithm.

The potential energy surface (PES), structures and thermal properties of methanol clusters (MeOH)n with n = 8-15 were explored by replica-exchange molecular dynamics (REMD) simulations with an empirical model and refined using density functional theory (DFT) methods. For a given size, local minima structures were sampled from REMD trajectories and archived by a newly developed molecular database via a two-stage clustering algorithm (TSCA). Our TSCA utilizes both the topology of O-HO hydrogen bonding networks and the similarity of the shapes to filter out duplicates. The screened molecular database contains only distinct conformers sampled from REMD and their structures are further optimized by the two DFT methods with and without dispersion correction to examine the influence of dispersion on their structures and binding energies. Inspecting different O-HO networks, the binding energies of methanol clusters are highly degenerated. The degeneracy is more significant with the dispersion effect that introduces weaker but more complex C-HO bonds. Based on the structures we have searched, we were able to extract general trends and these datasets can serve as a starting point for further high-level ab initio calculations to reveal the true energy landscape of methanol clusters.

The generation of stationary π-electron rotations in chiral aromatic ring molecules possessing non-degenerate excited states.

The electron angular momentum is a fundamental quantity of high-symmetry aromatic ring molecules and finds many applications in chemistry such as molecular spectroscopy. The stationary angular momentum or unidirectional rotation of π electrons is generated by the excitation of a degenerated electronic excited state by a circularly-polarized photon. For low-symmetry aromatic ring molecules having non-degenerate states, such as chiral aromatic ring molecules, on the other hand, whether stationary angular momentum can be generated or not is uncertain and has not been clarified so far. We have found by both theoretical treatments and quantum optimal control (QOC) simulations that a stationary angular momentum can be generated even from a low-symmetry aromatic ring molecule. The generation mechanism can be explained in terms of the creation of a dressed-state, and the maximum angular momentum is generated by the dressed state with an equal contribution from the relevant two excited states in a simple three-electronic state model. The dressed state is formed by inducing selective nonresonant transitions between the ground and each excited state by two lasers with the same frequency but having different polarization directions. The selective excitation can be carried out by arranging each photon-polarization vector orthogonal to the electronic transition moment of the other transition. We have successfully analyzed the results of the QOC simulations of (P)-2,2'-biphenol of axial chirality in terms of the analytically determined optimal laser fields. The present findings may open up new types of chemical dynamics and spectroscopy by utilizing strong stationary ring currents and current-induced magnetic fields, which are created at a local site of large compounds such as biomolecules.

Induction of unidirectional π-electron rotations in low-symmetry aromatic ring molecules using two linearly polarized stationary lasers.

A new laser-control scenario of unidirectional π-electron rotations in a low-symmetry aromatic ring molecule having no degenerate excited states is proposed. This scenario is based on dynamic Stark shifts of two relevant excited states using two linearly polarized stationary lasers. Each laser is set to selectively interact with one of the two electronic states, the lower and higher excited states are shifted up and down with the same rate, respectively, and the two excited states become degenerate at their midpoint. One of the four control parameters of the two lasers, i.e. two frequencies and two intensities, determines the values of all the other parameters. The direction of π-electron rotations, clockwise or counter-clockwise rotation, depends on the sign of the relative phase of the two lasers at the initial time. An analytical expression for the time-dependent expectation value of the rotational angular momentum operator is derived using the rotating wave approximation (RWA). The control scenario depends on the initial condition of the electronic states. The control scenario with the ground state as the initial condition was applied to toluene molecules. The derived time-dependent angular momentum consists of a train of unidirectional angular momentum pulses. The validity of the RWA was checked by numerically solving the time-dependent Schrödinger equation. The simulation results suggest an experimental realization of the induction of unidirectional π-electron rotations in low-symmetry aromatic ring molecules without using any intricate quantum-optimal control procedure. This may open up an effective generation method of ring currents and current-induced magnetic fields in biomolecules such as amino acids having aromatic ring molecules for searching their interactions.

Intrinsic coordination for revealing local structural changes in protein folding-unfolding.

With a deformed object of a rigid rod inside, the local dislocations may be tracked relatively easily with respect to the internal rigid rod. We apply this concept on protein folding-unfolding to track the internal structural changes of an unfolded protein in solution. Proposed here is a protein internal coordination based on the major axis X of an ellipsoidal protein and the stable intrinsic transition dipole moment µ of the protein during unfolding. In this methodology, small-angle X-ray scattering (SAXS) is used to provide the protein global morphologies in the native and unfolded states. Furthermore, time-resolved fluorescence anisotropy (TRFA) provides the relative orientation between X and µ of Trp59 of the model protein cytochrome c. Hence observed in the protein unfolding with denaturants, acid, urea, or GuHCl, is the elongation of the native protein conformation along a reoriented protein major axis; accompanied are the different extents of relocations of the terminal α helices and loop structures of the protein in the corresponding unfolding.

Orientation hydrogen-bonding effect on vibronic spectra of isoquinoline in water solvent: Franck-Condon simulation and interpretation.

The excited-state orientation hydrogen-bonding dynamics, and vibronic spectra of isoquinoline (IQ) and its cationic form IQc in water have been investigated at the time-dependent density functional theory quantum chemistry level plus Franck-Condon simulation and interpretation. The excited-state orientation hydrogen bond strengthening has been found in IQ:H2O complex due to the charge redistribution upon excitation; this is interpreted by simulated 1:1 mixed absorption spectra of free IQ and IQ:H2O complex having best agreement with experimental results. Conversely, the orientation hydrogen bond in IQc:H2O complex would be strongly weakening in the S1 state and this is interpreted by simulated absorption spectra of free IQc having best agreement with experimental results. By performing Franck-Condon simulation, it reveals that several important vibrational normal modes with frequencies about 1250 cm-1 involving the wagging motion of the hydrogen atoms are very sensitive to the formation of the orientation hydrogen bond for the IQ/IQc:H2O complex and this is confirmed by damped Franck-Condon simulation with free IQ/IQc in water. However, the emission spectra of the IQ and IQc in water have been found differently. Upon the excitation, the simulated fluorescence of IQ in water is dominated by the IQ:H2O complex; thus hydrogen bond between IQ and H2O is much easier to form in the S1 state. While the weakened hydrogen bond in IQc:H2O complex is probably cleaved upon the laser pulse because the simulated emission spectrum of the free IQc is in better agreement with the experimental results.

Theory of Triplet Excitation Transfer in the Donor-Oxygen-Acceptor System: Application to Cytochrome b6f.

Theoretical consideration is presented of the triplet excitation dynamics in donor-acceptor systems in conditions where the transfer is mediated by an oxygen molecule. It is demonstrated that oxygen may be involved in both real and virtual intramolecular triplet-singlet conversions in the course of the process under consideration. Expressions describing a superexchange donor-acceptor coupling owing to a participation of the bridging twofold degenerate oxygen's virtual singlet state are derived and the transfer kinetics including the sequential (hopping) and coherent (distant) routes are analyzed. Applicability of this theoretical description to the pigment-protein complex cytochrome b6f, by considering the triplet excitation transfer from the chlorophyll a molecule to distant ß-carotene, is discussed.

Theoretical studies on the absorption spectra of cis-[Ru(4,4'-COO-2,2'-bpy)2(X)2](4-), (X = NCS, Cl) and panchromatic trans-terpyridyl Ru complexes including strong spin-orbit coupling.

In this paper, we theoretically and experimentally investigate the photophysical and chemical characteristics and absorption spectra of various ruthenium complexes in solution used as efficient dye-sensitized solar cells. The target molecules are two well-known complexes, cis-[Ru(4,4'-COO-2,2'-bpy)2(X)2](4-), (X = NCS, Cl) and trans-terpyridyl Ru. The experimental absorption spectra of these molecules, which show strong spin-orbit (SO) coupling, are simulated using first-order perturbation theory based on time-dependent density functional theory and quantum chemistry calculations. It turns out that the theory can simulate the experimental data very well, which indicates that SO coupling is very important and the mixing between singlet and triplet states is strong in these molecules because of the large SO coupling constant of the Ru atom. The exact absorption spectra can only be reproduced by including the perturbation by SO coupling. The physical and chemical differences between cis-[Ru(4,4'-COO-2,2'-bpy)2(X)2](4-), (X = NCS, Cl) and trans-terpyridyl Ru complexes are elucidated by natural bond orbital and natural transition orbital analyses. From these analyses, we have found that the two kinds of Ru complexes are quite different in terms of photoexcitation response and chemical bonding between the central Ru atom and the surrounding ligands.

Pressure-enhanced surface interactions between nano-TiO2 and ionic liquid mixtures probed by high pressure IR spectroscopy.

The pressure-dependent interactions between the ionic liquid mixture ([MPI][I1.5]) and nano-TiO2 surfaces have been studied up to 2.5 GPa. The results of infrared spectroscopic profiles of [MPI][I1.5] and [MPI][I1.5]-nano-TiO2 indicated that no appreciable changes in the C-H stretching bands with the addition of nano-TiO2 were observed under ambient pressure. As the pressure was elevated to 0.7 GPa, the C-H stretching absorption of [MPI][I1.5] underwent band-narrowing and red-shifts in frequency. In contrast to the results of [MPI][I1.5], the spectra of [MPI][I1.5]-nano-TiO2 do not show dramatic changes under high pressures. A possible explanation for this observation is the formation of certain pressure-enhanced C-H···nano-TiO2 interactions around the imidazolium C-H and alkyl C-H groups. As imidazolium C-H···I(-) is replaced by the weaker imidazolium C-H···polyiodide, the splitting of the imidazolium C-H stretching bands was observed. The experimental results indicate that both nano-TiO2 and polyiodides are capable of disturbing the self-assembly of ionic liquids. This study suggests the possibility to tune the efficiency of dye-sensitized solar cells via a high pressure method.

Intersystem Crossing Pathway in Quinoline-Pyrazole Isomerism: A Time-Dependent Density Functional Theory Study on Excited-State Intramolecular Proton Transfer.

The dynamics of the excited-state intramolecular proton-transfer (ESIPT) reaction of quinoline-pyrazole (QP) isomers, designated as QP-I and QP-II, has been investigated by means of time-dependent density functional theory (TDDFT). A lower barrier has been found in the potential energy curve for the lowest singlet excited state (S1) along the proton-transfer coordinate of QP-II compared with that of QP-I; however, this is at variance with a recent experimental report [J. Phys. Chem. A 2010, 114, 7886-7891], in which the authors proposed that the ESIPT reaction would only proceed in QP-I due to the absence of a PT emission for QP-II. Therefore, several deactivating pathways have been investigated to determine whether fluorescence quenching occurs in the PT form of QP-II (PT-II). The S1 state of PT-II has nπ* character, which is a well-known dark state. Moreover, the energy gap between the S1 and T2 states is only 0.29 eV, implying that an intersystem crossing (ISC) process would occur rapidly following the ESIPT reaction. Therefore, it is demonstrated that the ESIPT could successfully proceed in QP-II and that the PT emission would be quenched by the ISC process.

Contribution of thermal energy to initial ion production in matrix-assisted laser desorption/ionization observed with 2,4,6-trihydroxyacetophenone.

RATIONALE: Although several reaction models have been proposed in the literature to explain matrix-assisted laser desorption/ionization (MALDI), further study is still necessary to explore the important ionization pathways that occur under the high-temperature environment of MALDI. 2,4,6-Trihydroxyacetophenone (THAP) is an ideal compound for evaluating the contribution of thermal energy to an initial reaction with minimum side reactions. METHODS: Desorbed neutral THAP and ions were measured using a crossed-molecular beam machine and commercial MALDI-TOF instrument, respectively. A quantitative model incorporating an Arrhenius-type desorption rate derived from transition state theory was proposed. Reaction enthalpy was calculated using GAUSSIAN 03 software with dielectric effect. Additional evidence of thermal-induced proton disproportionation was given by the indirect ionization of THAP embedded in excess fullerene molecules excited by a 450 nm laser. RESULTS: The quantitative model predicted that proton disproportionation of THAP would be achieved by thermal energy converted from a commonly used single UV laser photon. The dielectric effect reduced the reaction Gibbs free energy considerably even when the dielectric constant was reduced under high-temperature MALDI conditions. With minimum fitting parameters, observations of pure THAP and THAP mixed with fullerene both agreed with predictions. CONCLUSIONS: Proton disproportionation of solid THAP was energetically favorable with a single UV laser photon. The quantitative model revealed an important initial ionization pathway induced by the abrupt heating of matrix crystals. In the matrix crystals, the dielectric effect reduced reaction Gibbs free energy under typical MALDI conditions. The result suggested that thermal energy plays an important role in the initial ionization reaction of THAP.

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.

Franck-Condon factors perturbed by damped harmonic oscillators: solvent enhanced X (1)Ag ↔ A(1)B1u absorption and fluorescence spectra of perylene.

Damped harmonic oscillators are utilized to calculate Franck-Condon factors within displaced harmonic oscillator approximation. This is practically done by scaling unperturbed Hessian matrix that represents local modes of force constants for molecule in gaseous phase, and then by diagonalizing perturbed Hessian matrix it results in direct modification of Huang-Rhys factors which represent normal modes of solute molecule perturbed by solvent environment. Scaling parameters are empirically introduced for simulating absorption and fluorescence spectra of an isolated solute molecule in solution. The present method is especially useful for simulating vibronic spectra of polycyclic aromatic hydrocarbon molecules in which hydrogen atom vibrations in solution can be scaled equally, namely the same scaling factor being applied to all hydrogen atoms in polycyclic aromatic hydrocarbons. The present method is demonstrated in simulating solvent enhanced X (1)Ag ↔ A(1)B1u absorption and fluorescence spectra of perylene (medium-sized polycyclic aromatic hydrocarbon) in benzene solution. It is found that one of six active normal modes v10 is actually responsible to the solvent enhancement of spectra observed in experiment. Simulations from all functionals (TD) B3LYP, (TD) B3LYP35, (TD) B3LYP50, and (TD) B3LYP100 draw the same conclusion. Hence, the present method is able to adequately reproduce experimental absorption and fluorescence spectra in both gas and solution phases.

Fluctuation-induced transport of two coupled particles: effect of the interparticle interaction.

We consider a system of two coupled particles fluctuating between two states, with different interparticle interaction potentials and particle friction coefficients. An external action drives the interstate transitions that induces reciprocating motion along the internal coordinate x (the interparticle distance). The system moves unidirectionally due to rectification of the internal motion by asymmetric friction fluctuations and thus operates as a dimeric motor that converts input energy into net movement. We focus on how the law of interaction between the particles affects the dimer transport and, in particular, the role of thermal noise in the motion inducing mechanism. It is argued that if the interaction potential behaves at large distances as x(α), depending on the value of the exponent α, the thermal noise plays a constructive (α > 2), neutral (α = 2), or destructive (α < 2) role. In the case of α = 1, corresponding piecewise linear potential profiles, an exact solution is obtained and discussed in detail.