Band-shape Function Research Articles

Theory of Nonradiative Energy Transfer between Two Optical Ions Using the Proper Adiabatic Approximation

Using adiabatic approximation of polyatomic molecules and methods of molecular quantum electrodynamics, we have derived the Förster-Dexter theory of energy transfer between two optical ions. Although we have assumed a single configurational coordinate, the theory can be easily formulated for multiple normal coordinate system. This approach leads smoothly to the energy transfer rates originally derived by Dexter1 for electric dipole-electric dipole interaction without any additional assumptions regarding the normalization of the wavefunctions in the energy space. Using the transfer rates thus derived, and the band shape functions, Wp , a new function F(T) has been proposed to study the temperature dependence of energy transfer rates due to the overlap of the emission and absorption line shapes of the donor and acceptor ions, respectively. This function has been studied for various scenarios of the relative alignment of the zero phonon lines of these ions in order to understand the participation of phonons in the energy transfer process. We have also compared this quantum mechanically derived F(T) function with that derived using semiclassical theory. The agreements are very good at high temperatures. D. L. Dexter, J. Chem. Phys. 21, 836(1953)

Theory of Nonradiative Energy Transfer between Two Optical Ions Using Proper Adiabatic Approximation

Using the adiabatic approximation of polyatomic molecules and methods of molecular quantum electrodynamics, we have derived the Förster-Dexter theory of energy transfer between two optical ions. Although we have assumed a single configurational coordinate, the theory can be easily formulated for a multiple normal coordinate system. This approach leads smoothly to the energy transfer rates originally derived by Dexter1 for electric dipole-electric dipole interaction without any additional assumptions regarding the normalization of the wavefunctions in the energy space. Using the transfer rates thus derived, and the band shape functions, W p , a new function F(T) has been proposed to study the temperature dependence of energy transfer rates due to the overlap of the emission and absorption lineshapes of the donor and acceptor ions, respectively. This function has been studied for various scenarios of the relative alignment of the zero phonon lines of these ions in order to understand the participation of phonons in the energy transfer process. We have also compared this quantum mechanically derived F(T) function with that derived using semiclassical theory. The agreements are very good at high temperatures.

Revisiting the Theory of Vibronic Transitions of Optical Ions in Solids

In order to improve the performance of phosphors in lighting and display devices, one needs to understand the temperature dependence of the radiative and nonradiative processes of optical ions in solids. In this presentation, we have revisited the theory of radiative transitions from an excited state of a polyatomic system within the framework of the proper adiabatic theory. First the transition rate for spontaneous emission from an excited state to the ground state has been derived rigorously using the methods of molecular quantum electrodynamics. It is assumed that this transition is associated with a loss of energy through the creation of p phonons in the system. It will be shown for the first time in a rigorous manner that only limited number of vibrational modes for which the Huang-Rhys parameter, s, is nonzero will participate. These modes will be identified as the active modes. Other modes will be completely passive during the relaxation process. It is speculated that these active modes are the local vibrational modes, and can be identified from the vibronic spectra of rare earth ions, particularly at low temperature. Based on the present formulation of the problem, the band shape function, W(p,T), will be derived. This band shape function will be used to interpret available vibronic spectra for some optical ions at low temperature.

Comparison between Theoretically and Experimentally Determined Electronic Properties: Applications to Two-Photon Singlet Oxygen Sensitizers

A series of oligo-phenylenevinylenes are investigated by spectroscopic measurements and calculations with the aim to rationalize the correlation between the electronic structure of the molecules and their efficiency as two-photon singlet oxygen sensitizers. The band-shape functions of selected two-photon absorption processes are analyzed and the corresponding spectra are deconvoluted. This analysis allows for a comparison between calculated and measured two-photon absorption cross sections where a reasonable agreement is found.

ChemInform Abstract: Experimental Determination of the S1 Torsional Potential of 9,9′- Bianthryl in 2-Methyl-butane by Simultaneous Franck-Condon and Band Shape Analysis of Temperature Dependent Optical Fluorescence Spectra.

Optical fluorescence spectra of anthracene (AN) and of 9,9’‐bianthryl (BA) have been measured in the apolar solvent 2‐methyl‐butane (2MB) at seven different temperatures in the range 123 K to 298 K. The AN spectra are strongly structured and consist of simple Franck–Condon (FC) progressions. The BA spectra on the other hand are broad and show unusual temperature dependencies. These effects are interpreted within a two state model with five vibrational and one torsional degree of freedom. A FC analysis is carried out using vibronic band shape functions which are convolutions of two functions. The first one is a Gaussian representing the spectral broadening due to librations and intermolecular interactions. The second one results from a semiclassical treatment of the intramolecular torsional mode (large amplitude motion, LAM). Optical transitions are assumed to occur from ensembles which are thermally equilibrated with respect to their nuclear degrees of freedom. The Boltzmann distribution with respect to the LAM potentials in combination with a change of the transition frequency as a function of the torsional angle φ is identified as the predominant broadening mechanism. The LAM potentials in the S0 and S1 state are represented by a three parameter cosine series. A nonlinear least‐squares fit of the experimental spectra leads to a single consistent set of model parameters valid for all temperatures. The S1 double minimum LAM potential of BA in 2MB is found to have minima at φeq=62° (and 118°) and a local barrier at φ=90° of 255 cm−1.

An Extended Fano–DeVoe Polarizability Theory Similar to the Bayley–Nielsen–Schellman Secular Matrix Method: CD Calculations of Polypeptides Having α-Helix, β -Sheet, and β -Turn Structures

Abstract We have expanded the previous Bayley–Nielsen–Schellman type polarizability theory so as to involve a new diagonal type of CAE (Condon, Alter, Eyring) interaction energy as {(the previous diagonal type of CAE interaction energy) − (the electrostatically corrected ground state energy)}, which corresponds to the term pointed out by Misra in his review article. For the CD and UV band shape functions which were previously derived in a different way, we have newly rederived the polymer CD and UV band shape functions by making use of the Green’s function method, starting from the well-known text book by Eyring et al. For a monomer polarizability tensor, the two most often used Lorentzian and Gaussian band approximations will be studied together with practically necessary relationships on the bandwidths. The theory has been applied to the CD and UV absorption calculations on polypeptides. Previously, unless one scaled the earlier diagonal type of CAE interaction energy, poor predictions were then obtained, especially for the β-sheet polypeptides as well as the small α-helix and β-turn tripeptides. Now, using the new diagonal type of CAE interaction energies, promising CD results have been obtained together with reasonable UV spectral shifts.

A Molecular Theory for Doubly Resonant IR−UV-vis Sum-Frequency Generation

The experiment for measuring doubly resonant infrared-visible (IR-vis) sum-frequency generation (SFG) has recently been developed by Shen and his co-worker and applied to Rhodamine 6G on silica surfaces. In this paper, based on the Born-Oppenheimer adiabatic approximation, a molecular theory for doubly resonant IR-vis SFG generation as a two-dimensional surface spectroscopy is presented. We shall show that this new nonlinear spectroscopy is closely related to IR and resonance Raman spectroscopy. In this preliminary theoretical derivation, the displaced harmonic potential energy surfaces model for the electronic ground and electronically excited states of the system are used to obtain the band shape functions for doubly resonant IR-vis SFG at a finite temperature. One of the unique and powerful abilities of this spectroscopy is that interference effects among IR-active modes can be observed. The calculated excitation profiles and IR spectra of the doubly resonant IR-UV SFG from Rhodamine 6G on fused silica are demonstrated. We shall show that this new nonlinear spectroscopy can provide experimentalists with access to microscopic properties of molecules on surfaces or at interfaces.

An operator approach to the construction of generating functions for the product of Laguerre polynomials: the thermal average bandshape function of a molecule

We introduce two different forms of mathematical identities, involving the product of Laguerre polynomials. These identities are a direct reflection of an operator identity. It is shown that these relations are useful in calculating the thermal average bandshape function of a molecular system. For a model displaced-oscillator system we derive how the displacement between two adiabatic potential surfaces affects the bandshape function.

A theoretical study of core excitation spectra of NO molecule

The N 1s and O 1s cores to antibonding transitions for the NO molecule have been investigated using a multiconfiguration self-consistent field approach. The core excitation spectra have been simulated using a Lorentzian band shape function with the full width at half maximum obtained from a time-dependent formulation. The calculations confirm the different split schemes for the N 1s and O 1s dipole-allowed core-excited states. Spectral characteristics related to nitrogen and oxygen excitation have been discussed in detail.

Simulated Infrared Emission Spectra of Highly Excited Polyatomic Molecules: A Detailed Model of the PAH‐UIR Hypothesis

A detailed description of the polycyclic aromatic hydrocarbon (PAH)/unidentified infrared band (UIR) mechanism is presented in which experimental spectral bandshape functions are used to simulate IR emission spectra for individual molecules. These spectra are additively superimposed to produce a conglomerate spectrum representative of a family of PAH molecules. Ab initio vibrational frequencies and intensities for nine PAHs (neutral and cationic) as large as ovalene are used in conjunction with measured bandshape and temperature-dependent redshift data to simulate the UIR bands. The calculated spectra of cations provide a closer match to the UIRs than do those of the neutrals. However, the PAH cations used in the simulations fail to reproduce the details of the UIR emission spectra. The discrepancies are potentially alleviated if both larger PAHs and a greater number of PAHs were included in the simulation.

Polarizability tensor theory by Green’s function matrix: Circular dichroism and ultraviolet absorption band shape calculations of cyclo(Gly-Pro-Gly-D-Ala-Pro) having β- and γ-turns

In a recent paper, we represented the partial polymer polarizability tensor equation in terms of partial monomer polarizability tensors and intermonomer unit dipole–dipole interaction tensors. Here, we represent the same equation in terms of a partial polymer Green’s function matrix equation being constructed from partial monomer Green’s matrix elements and intermonomer interaction energies. We also show that the circular dichroism (CD) and ultraviolet (UV) absorption band shape functions bear a strong resemblance to DeVoe’s scalar representation of the band shape functions, which we have rewritten, using our terminology, in terms of the three terms describing the dipole–dipole coupling, the electromagnetic coupling, and the intrinsic optical activity. The latter is capable of incorporating the one-electron mechanism. The present tensor theory offers five advantages. First, for describing physical phenomena, the tensor form may be more convenient than the scalar forms. Second, in both the partial polarizability tensor theory and DeVoe’s partial scalar polarizability theory, it is possible to place different transition dipoles within each monomer in different positions, whereas this is not possible in the usual Fano–DeVoe types of polarizability theories. Third, the intermonomer interaction energy can be represented readily either by the dipole approximation or by the monopole approximation. Fourth, various types of intermonomer interactions can be taken into account flexibly by making use of the Green’s matrix equation. Fifth, an efficient algorithm can be derived by reducing the matrix size by the factor 3×3 as compared with the partial polymer polarizability matrix tensor equation. By employing the dipole positions already known or discussed by other authors for the two nπ* and ππ* transitions within each amide monomer, the theory has been applied to the semiempirical calculations of the CD and UV absorption band shapes of a cyclic pentapeptide, cyclo (Gly-Pro-Gly-D-Ala-Pro), with comments on the adaptability of simplified parameterization.

Determination of integrated intensities of overlapped IR bands by curve-fitting, Fourier self-deconvolution and a combination of both methods

Curve-fitting, Fourier self-deconvolution, and a combination of these methods have been applied to determine integrated intensities of the bands arising from the C=O stretching mode of n-octanoic acid. When these bands are separated by more than an average band width, the most reliable results are supplied by curve-fitting. However, simple functions (Gauss or Lorentz) do not satisfactorily describe shapes of experimental bands, and more complex profiles should be used such as the sum or product of the Gauss and Lorentz functions or their convolution (Voigt profile). The applied Fourier self-deconvolution procedure assumes Lorentzian band shapes; therefore the determined intensities of the bands that deviate from this profile are always higher than the true values. An analysis combining the two methods mentioned above did not improve the results. The output bands from curve-fitting, with a high Lorentzian fraction, might have overestimated the band areas. Thus, the integration should be done over a limited wavenumber range. Our results clearly show that the band-shape function plays an important role during the determination of the integrated intensities using curve-fitting, Fourier self-deconvolution or a combination of both methods (referred to in this paper as the “combined method”).

Molecular theory of second-order sumdashfrequency generation

Recently the second-order sumdashfrequency generation (SFG) has become a powerful surface — spectroscopic technique. In this paper we present a molecular theory for SFG. To systematically treat this process, we have classified SFG into the resonance-resonance case, resonance-off-resonance case, off-resonance-resonance case and off-resonance-off-resonance case. Among these cases we may have IR-UV SFG and UV-UV SFG. Some of these SFG's are reported in this paper. In each SFG case, we have shown how to calculate the strength (or intensity) and band-shape function.

The non-Markovian master equation for stochastically perturbed systems: effect on spectral lineshape

Based on the system-reservoir model in which the intermediate states of the system are perturbed by stochastic interaction with the surroundings, we have generalized the master equation for the description of quantum dissipative processes and applied it to a three-level system to calculate transient optical response in pump-probe spectroscopy. We have shown that while the two damping processes act independently in the Markovian description, the interference of stochastic and system-reservoir interactions and the resulting non-Lorentzian structure of the band-shape function appear as the essential non-Markovian features in the dissipative dynamics.

Theoretical studies of IR-UV sum frequency generation applied to absorbed molecules

In this paper, we study the temperature effect and effect of the interaction between the adsorbed molecule and heat bath on the band-shape function of IR-UV sum frequency generation (SFG). Numerical calculations have been performed to demonstrate these effects. We have also compared the IR-visible SFG and FT-IR spectroscopy and considered the effects of inhomogeneities on the SFG and IR spectra.

Theoretical description of steady-state sum-frequency generation in molecular adsorbates.

In this paper, we present a density-matrix formalism for treating second-order sum-frequency generation (SFG) and difference-frequency generation (DFG). The theory can treat both steady-state and time-resolved SFG and DFG. However, only the steady-state SFG will be described in this paper. As a practical application, we show how the theory can be applied to study the infrared-visible SFG. The band-shape function of SFG is derived, which consists of both real and imaginary parts. The real part of the SFG band-shape function is related to the infrared spectral band-shape function and the imaginary part is related to the real part by Kramers-Kr\onig relations. The temperature effect on SFG is taken into account in our expression of the SFG band-shape function. We show that for a vibrational mode to be SFG active in infrared-visible SFG measurements, it has to be both infrared and Raman active.

Critical analysis of the band-shape function of a molecular system imbedded in a host crystal: Application to F centers

We present an investigation of the accuracy of the saddle-point method in application to band-shape functions and find that it gives excellent results for most commonly encountered situations. We also present a comparison with a new improved short-time approximation which shows promise in applications. We use this new method to examine the behavior of the band-shape function in a distorted oscillator model. As an application, we have used these methods to explain the absorption and emission spectra of an F-center defect (a prototype defect system) in four different host crystals KCl, KI, KBr, and NaCl.

Non-Markovian master equation for linear and nonlinear systems.

Based on the standard system-heat-bath model, we have formulated a theory of the non-Markovian master equation for linear and nonlinear systems for a correlation time that is short but finite. We have applied the theory to linear systems, such as harmonic oscillators and two-level atoms and calculated the band-shape functions for transient optical absorption in the pump-probe experiments. The significant interplay of nonlinearity and non-Markovian decay has been illustrated by another application of the theory to a Morse oscillator. The theory is valid for arbitrary nonlinearity and finite temperatures but within the weak-coupling limit and has been derived without ad hoc implementation of a phenomenological memory function.

Experimental determination of the S1 torsional potential of 9,9′-bianthryl in 2-methyl-butane by simultaneous Franck–Condon and band shape analysis of temperature dependent optical fluorescence spectra

Optical fluorescence spectra of anthracene (AN) and of 9,9′-bianthryl (BA) have been measured in the apolar solvent 2-methyl-butane (2MB) at seven different temperatures in the range 123 K to 298 K. The AN spectra are strongly structured and consist of simple Franck–Condon (FC) progressions. The BA spectra on the other hand are broad and show unusual temperature dependencies. These effects are interpreted within a two state model with five vibrational and one torsional degree of freedom. A FC analysis is carried out using vibronic band shape functions which are convolutions of two functions. The first one is a Gaussian representing the spectral broadening due to librations and intermolecular interactions. The second one results from a semiclassical treatment of the intramolecular torsional mode (large amplitude motion, LAM). Optical transitions are assumed to occur from ensembles which are thermally equilibrated with respect to their nuclear degrees of freedom. The Boltzmann distribution with respect to the LAM potentials in combination with a change of the transition frequency as a function of the torsional angle φ is identified as the predominant broadening mechanism. The LAM potentials in the S0 and S1 state are represented by a three parameter cosine series. A nonlinear least-squares fit of the experimental spectra leads to a single consistent set of model parameters valid for all temperatures. The S1 double minimum LAM potential of BA in 2MB is found to have minima at φeq=62° (and 118°) and a local barrier at φ=90° of 255 cm−1.

Non-Markovian effects on optical absorption.

In this paper, we shall report the investigation of the non-Markovian effects of the dynamical process on optical absorption. It will be shown that in steady-state experiments the non-Markovian effects cannot be probed. The non-Markovian effects on the transient pump-probe experiment are treated, and a couple of typical memory functions have been used to obtain the transient band-shape functions. Finally, probe-pulse duration effects on the absorption are considered.