Harmonic Potential Surfaces Research Articles

Vibrational line shapes of solvated molecules with a normal mode approach

We develop a theory of the vibrational absorption line shape of a solvated molecule. This approach is based on the instantaneous normal mode approximation, in which the fluid is taken to evolve on a harmonic potential surface whose curvature matches that of the true potential surface at the fluid’s initial configuration. We apply this method to the vibrational line shape of a harmonic diatomic molecule dissolved in an atomic solvent. The line shape is related to a configurationally averaged phonon Green’s function. A diagrammatic analysis of this Green’s function is shown to lead to a self-consistent approximation to the line shape. The only inputs to this calculation from other theory or simulation are the pair correlation functions for two solvent particles and for a solute atom and a solvent particle. The resulting spectra are compared with calculations for a similar model by Berne et al., based on the generalized Langevin equation [J. Chem. Phys. 93, 5084 (1990)].

Mean first-passage time and multiexponential relaxation of the activationless nonadiabatic electron transfer reaction.

We discuss the mean first-passage time of the activationless nonadiabatic electron transfer reaction by using the stochastic Liouville equation for the study of outer sphere electron transfer in polar solvents characterized by Debye dielectric relaxation. We obtain an approximate expression for the mean first-passage time which incorporates the width of the transition with an arbitrary initial condition far from equilibrium. We derive an analytical expression for the rate corresponding to harmonic potential surfaces in the overdamped regime. For Fokker-Planck operators of the Smoluchowski type, we introduce a method to solve all of the generalized moments by using the eigenfunction expansion method.

Generalized moment expansion for nonadiabatic transition reaction

We discuss the generalized moments of the nonadiabatic transition reaction by using the stochastic Liouville equation for the study of outer-sphere electron transfer in polar solvents characterized by Debye dielectric relaxation. We obtain an approximate expression for the generalized moments which incorporates the width of the transition with arbitrary initial condition far from equilibrium. For low barriers, we derive an analytical expression for the rate corresponding to harmonic potential surfaces in the overdamped regime. For Fokker-Planck operators of Smoluchowski type, we introduce a new method to solve all of the generalized moments by using the eigenfunction expansion method.

Optimal control of classical anharmonic molecules represented with piecewise harmonic potential surfaces: analytic solution and selective dissociation of triatomic systems

In this paper we apply optimal control theory to manipulate the dynamics of molecular systems with anharmonic potential surfaces modelled as a series of continuous piecewise harmonic sections within the classical approximation. In analogy to the simple harmonic case, an analytic solution is possible, yet bond dissociation is allowed. This methodology was applied to several model systems. It was found that the molecular objective of selective dissociation was achieved for a model linear triatomic system.

Theory of dynamic absorption spectroscopy of nonstationary states. 4. Application to 12-fs resonant impulsive Raman spectroscopy of bacteriorhodopsin

A time-dependent theory for femtosecond dynamic absorption spectroscopy is used to describe the creation and observation of molecular ground-state vibrational coherence through the resonance impulsive stimulated Raman mechanism. Model calculations show that the oscillatory absorption signal that arises from this ground-state coherence is maximized for a limited range of pulse lengths and that there is a complex relationship between the probe wavelength and the strength of the spectral oscillations. The generalized time-dependent linear susceptibility of the nonstationary system created by the impulsive pump pulse is defined and used to discuss the strong dependence of the measured signals on the properties of the probe pulse. Finally, calculations are presented to analyze the high-frequency oscillations ({approximately}20-fs period) recently observed in the transient absorption spectra of light-adapted bacteriorhodopsin (BR{sub 568}) following excitation with a 12-fs optical pulse. At the probe wavelengths used in this experiment, the contribution of stimulated emission is negligible at long times because of the extremely rapid excited-state isomerization; as a result, the spectral oscillations observed after this time are due to the impulsive excitation of coherent vibrations in the ground state. The transient response observed for BR{sub 568} is calculated using a 29-mode harmonic potential surface derived from amore » prior resonance Raman intensity analysis. Both the oscillatory signals and their dependence on the probe wavelength are satisfactorily reproduced. 68 refs., 11 figs.« less

Useful results for the determination of excited state geometries from resonance Raman spectra: Application to inorganic complexes

The time-dependent formulation of Raman scattering is used to derive simple expressions for fundamental and overtone intensities that depend on potential energy features in the Franck–Condon region and the homogeneous damping constant due to the bath modes. From the Raman excitation profiles, the dependence of the full width at half-maximum on the damping constant is calculated. The results are applied to the rich resonance Raman spectra and Raman excitation profiles of transition metal complexes, in particular, Cs3[Re2OCl10] and Cs4[W2OCl10], to determine the magnitude of the geometric changes occurring upon excitation of the molecule from the ground to the excited electronic state. For each compound, the multidimensional harmonic potential surface and damping constant derived for the excited electronic state are then used to simulate the observed Raman excitation profiles and resonance Raman spectrum.

Langevin modes of macromolecules: Applications to crambin and DNA hexamers

Langevin modes describe the behavior of atoms moving on a harmonic potential surface subject to viscous damping described by a classical Langevin equation. We present applications to the protein crambin and to the DNA duplex d(CpGpCpGpCpG)2 and its complex with ethidium. Our friction matrix is weighted according to surface area exposed to solvent, and results are reported for various values of the solvent viscosity and models for hydrodynamic interactions. Even for relatively small solvent friction (eta = 0.3 cp) a substantial number of modes are overdamped, and time correlation functions decay smoothly without the oscillations characteristic of gas-phase calculations. Perturbation theory starting from the gas-phase modes is accurate for many low-frequency modes (which are overdamped in the presence of solvent), but fails badly for higher modes. For correlation functions of interest to fluorescence depolarization or nmr relaxation, the plateau values are insensitive to solvent viscosity, but the relaxation times are not. The advantages and limitations of this analysis of macromolecular motions are discussed.

Characterization of ion trapping motion in the dual cell ion cyclotron resonance spectrometer: experimental and theoretical studies

Ions produced in one cell of a dual cell Fourier transform-ion cyclotron resonance mass spectrometer are partitioned by temporarily grounding the conductance limit separating mass two cells. Theoretical predictions for transfer behavior were produced by used of Runge—Kutta integration and an elliptical function method to calculate ion motion during the transfer period. Comparison between the experimental and theoretical results indicates that ion transfer between cells is occurring on a more harmonic potential surface than expected. These results suggest that only ions formed near the center of the source cell can be effectively transferred and/or detected.

Quantum-mechanical theory for 6 fs dynamic absorption spectroscopy and its application to nile blue

A time-dependent wavepacket theory for dynamic absorption spectroscopy is used to analyze the evolution of the transient absorption spectrum of nile blue following excitation with a 6 fs pulse. The femtosecond pump pulse creates nonstationary wavefunctions in both the ground and excited electronic states of the molecule. The absorption spectrum of these initial nonstationary states is expressed in terms of the Fourier transform of the time-dependent dipole matrix element between vibrational wavepackets moving on both the final and initial potential surfaces. The transient spectra observed for nile blue are successfully modeled using a one-dimensional system of displaced 590 cm −1 harmonic potential surfaces.

Theoretical analysis of resonance Raman spectra from the blue copper protein azurin

Using a time domain methodology, we have simulated low-temperature resonance Raman data from the blue copper protein azurin in an effort to determine an effective harmonic potential surface (relative to the ground-state geometry and normal coordinates) for the 600-nm transition of the blue copper active site. We find that utilization of overtone and combination intensities greatly facilitates the extraction of reliable parameter values. We are able in this manner to obtain accurate results for the linear displacements of the excited-state surface.

Diagrammatic theory of temperature-dependent resonance Raman scattering from polyatomic systems with general harmonic potential surfaces

A detailed derivation and discussion is given of our diagrammatic theory of temperature-dependent resonance Raman (RR) scattering and the optical absorption for multimode systems having general quadratic plus linear electron-vibrational coupling. By combining the time-correlator reformulation of RR scattering with suitably developed nonzero temperature many-body diagrammatic techniques, we obtain the RR excitation profiles and the absorption as one-dimensional Fourier transforms of analytic expressions involving just the model parameters and the temperature. The expressions are very convenient for explicit multimode model calculations. In addition, the theory brings out in a natural way the relation between RR profiles and the absorption, such that within well-defined special cases useful ‘‘transform’’ techniques can be developed for computing profiles directly from the observed temperature-dependent absorption. The many practical advantages of the theory for the analysis of experimental data have been demonstrated in earlier papers dealing with specific systems. In this paper we provide a comprehensive discussion of the theoretical details, which have not been given previously. The theory applies for any number of normal modes, and for arbitrary normal coordinate mixing, mode frequency shifts, and atomic equilibrium position shifts under electronic excitaiton. It involves products of phonon operators having both positive and negative time ordering, necessitating specialized combinatorial arguments. The use of an appropriate linked cluster expansion is shown to lead very naturally to a separation of the RR scattering into ‘‘orders,’’ which is the essential component producing the important general features listed above.Detailed derivations are given of the exact expressions for the first-order RR profiles in the most general model, and for first- , second- , and third-order profiles in the frequency-shift limit of no mode mixing. The latter formulas are recast into their absorption→profile ‘‘transform’’ versions, and these are simplified to more useful approximate forms for the practically important special case of small frequency shifts. Renormalizations of the linear electron-vibrational coupling parameters due both to mode mixing and to frequency shifts are also briefly discussed.

Classical dynamics of t r a n s-diimide: Intramolecular vibrational relaxation involving an active torsion

The classical dynamics of vibrationally excited trans-diimide (N2H2) were studied, focusing on the interaction of the torsion with the local NNH bends and NH stretches. Energy flow to and from the torsion is facilitated by nonlinear resonances between the NH stretch and a bend-torsion combination mode, as has been found for other types of vibrational motions. The significant frequency ratios for such combination resonances have been predicted by a simple, analytic model and verified using a local harmonic oscillator potential surface. Several of these resonances were observed for trans-diimide. Since the dominant couplings are primarily kinetic, these resonances should also be important in the torsional vibrational dynamics of other molecules.

Langevin modes of macromolecules

The standard gas phase normal mode analysis is generalized to condensed phases within the framework of the classical Langevin equation. The solution of this equation for a system of 3N atoms moving on a harmonic potential surface and subject to viscous damping described by a friction matrix (which is nondiagonal in the presence of hydrodynamic interactions) is reduced to the diagonalization of a 6N×6N real nonsymmetric matrix obtained from the mass-weighted force constant and friction matrices. Alternative formulations of the problem requiring either the diagonalization of a complex symmetric matrix or the solution to a generalized eigenvalue problem involving real symmetric matrices are also considered. The resulting eigenvalues, in general complex, are real when the motion is overdamped and occur in complex conjugate pairs in the underdamped case. It is shown that when the eigenvectors are normalized in a particular way, all quantities of interest (e.g., correlation functions) have simple spectral representations and can be expressed as summations over the eigenvalues and eigenvectors (‘‘Langevin modes’’) of the system. To treat situations in which the exact matrix solution is computationally prohibitive, a perturbation approach is developed which utilizes the gas phase normal mode results for the system. In zeroth-order each normal coordinate is a Langevin oscillator with a friction constant equal to a diagonal element of the normal mode-transformed friction matrix. This description is analytically tractable, can be improved perturbatively, and is exact in the limit that all atoms experience the same friction. The formalism developed in this paper provides a viable way of studying the influence of solvent on the dynamics of collective motions in macromolecules.

Three-dimensional analytical model for the photodissociation of symmetric triatomics. Absorption and fluorescence spectra of ozone

A full three-dimensional analytical Franck–Condon analysis is presented for dissociative spectroscopies of a symmetric triatomic system. Formulas derived for harmonic potential surfaces are applied to the theoretical calculation of ozone absorption spectrum in the Hartley continuum as well as to the determination of its fluorescence (or resonance Raman) spectrum when submitted to 266 nm wavelength excitation. The Hartley absorption line shape describing the 1 1B2←X 1A1 photodissociation is reproduced within very good agreement with experiment. The role of the so-called Duschinsky effect originating from the coupling of bending and stretching motions in the excited state is analyzed and comparisons are presented with previous more simple models neglecting this coupling. The influence of temperature is also taken into account through calculations dealing with vibrationally excited species. As for the photoemission fluorescence spectrum carrying a rich information content, the behavior is successfully predicted in its main lines, relative intensities of overtone progressions and of combination bands are fairly well reproduced.

Isotope effects in condensed phases, the benzene example. Influence of anharmonicity; harmonic and anharmonic potential surfaces and their isotope independence. Molar volume effects in isotopic benzenes

Application of the harmonic oscillator cell model (HOCM) for condensed phase isotope effects to the benzene/deuterobenzene system using temperature dependent force fields, which yield good calculated values for vapor pressure isotope effects (VPIE) and reasonable agreement with spectroscopically observed frequency shifts on condensation, gives calculated values of isotope effect on energies of vaporization which are not in agreement with experiment. Pseudoharmonic corrections are inadequate to restore agreement with experiment. The VPIE in benzene is dominated by the contributions of the CH/CD stretching vibrations and consistency is achieved by expanding the model to include anharmonic corrections for these frequencies. In this fashion a combination of thermodynamic measurements of free energies (by vapor pressure) and energies (from the temperature coefficient of the vapor pressure or from calorimetry) can be employed to yield shifts in anharmonic vibrational constants on condensation. In the second part of the paper the molar volume isotope effect (MVIE) and the effects of volume, pressure, and temperature on the internal energy and its isotope effects (as monitored by the expansivity, compressibility, heat capacity, and their derivatives) are interpreted. The low-lying lattice modes make important contributions to these properties.

Franck-Condon analysis of thermal and vibrational excitation effects on the ozone Hartley continuum

The Hartley ultraviolet continuum of vibrationally and rotationally excited ozone is studied by a Franck-Condon spectral synthesis. Harmonic potential surfaces and the Condon approximation are utilized. Good agreement with experiment is obtained for ozone spectra taken over a wide temperature range and for the spectrum of CO 2 laser-excited ozone. The accuracy of previous, semi-empirical spectra models is discussed in the light of the current work. A semi-empirical expression for the spectrum is developed which is accurate over a wide temperature range, and may be applied to other molecules as well. The usefulness of Franck-Condon sum rules in the analysis of electronic spectra is also demonstrated.

A simple sum rule in raman theory

It is shown that in the limit of large detuning energy and in the absence of coordinate dependence of the transition moment, the resonance Raman amplitude for a 0 → n transition on a harmonic potential surface is proportional to (δ 2 − + δ 2 +) −1 × ƒ ∞ −∞ exp (− q 2) · [∂ n Δ V( q)/ƒ q 11] d q< where Δ V( q) is the difference between the arbitrary excited-state surface and the initial harmonic potential. The resonant and non-resonant detuning energies are given by δ − = E − hv and δ + = E + hv, where v is the incident laser frequency and E is the minimum separation between the potential surfaces.

Polyatomic Raman scattering for general harmonic potentials

Lee and Heller’s time-dependent theory of resonance Raman scattering is reviewed. This theory is formally identical to the traditional Kramer–Heisenberg–Dirac (KHD) theory but, in its wave packet interpretation, the time-dependent theory has distinct calculational and conceptual advantages over the KHD sum-over-states method. For polyatomics with large Franck–Condon displacements and Duschinsky rotations, where typically the KHD sum is over 1010 states with complicated Franck–Condon factors, these advantages are most pronounced. Wave packet propagation on general harmonic potential surfaces (Franck–Condon displacement, frequency shifts, and Duschinsky rotation) is exact. Formulas for the propagated wave packet are given for various levels of harmonic sophistication. The role of symmetry in the wave packet dynamics is discussed and explicit formulas are derived for the overlap of the moving wave packet ‖φi(t)〉 with the final state of interest ‖φf〉. The half Fourier transform of this overlap gives the Raman amplitude α. The transform method of Tonks and Page, relating absorption and Raman excitation profiles, is shown to arise naturally in our approach. We show excitation profiles calculated by the time-dependent theory for multidimensional harmonic potential surfaces with and without Duschinsky rotation. For the no-Duschinsky cases, we compare our results with the profiles of Myers and Mathies and of Champion and Albrecht, which were calculated by a sum-over-states; we then discuss some discrepancies between the latter’s results and ours.

The numerical evaluations of the density of states weighted Franck–Condon factor in large molecules. Effects of frequency changes

The numerical integration method (NIM) is applied to calculations of the density of states weighted Franck–Condon (DWFC) factor which plays an important role in the theory of nonradiative transitions. The 200 point Legendre–Gauss quadrature formula is used to save computing time. Values of the integrand are given with the quadruple precision to avoid accumulated errors occuring in the numerical integration. The illustrative calculations are carried out for the T1→S0 intersystem crossing in benzene. If frequency changes of all vibrational modes are taken into account, the energy gap law applies very well, but it fails without frequency changes. This fact suggests that the ordinary saddle point method based on the harmonic displaced potential surface model is inaccurate. The DWFC factor is sensitive to frequency changes of a high frequency mode which plays an important role in nonradiative transition. When low and medium frequency modes are added, as a group, to high frequency ones, they give a significant effect upon the DWFC factor. If a frequency change of a low frequency mode is hypothetically reduced to an extremely low value, the DWFC factor rapidly increases. This relates to the proximity effect by Wassam and Lim in the nonradiative transition in N-heterocyclic aromatic hydrocarbons. Finally, Metz’s method is discussed.

Effects of anharmonicity on non-radiative transitions in polyatomic molecules

The effect of anharmonicity on the non-radiative transition in large molecules is examined within the Morse potential surface model. The vibrational wavefunctions are assumed to be the product of the harmonic and Morse oscillator wavefunctions. The method of factorization introduced by Gelbart et al. is used for the evaluation of a density weighted Franck-Condon factor. As an example, we choose the intersystem crossing 3 B 1u→1 A 1g in benzene. The numerical calculation shows that the anharmonicity causes an increase by a numerical factor ∼ 103 in the non-radiative transition rate. The electronic energy distribution over the vibrational modes in the final state is determined and compared with that obtained using the harmonic potential surface model.