Spectral lineshape function

For all of the cases considered earlier, a C(t) function is subjected to Fourier transformation to obtain a spectral lineshape function 1(G)), which then provides the essential ingredient for computing the net rate of photon absorption. In this Fourier transform process, the variable 0) is assumed to be the frequency of the electromagnetic field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to 1(G)) is ... 

As a final detail, we allow for broadening (e.g., Doppler broadening, which determines the lasing lineshape in He/Ne lasers) of the two-level transition. Assume that the spectral lineshape function is gf(v), normalized so that J g v)dv = 1. Then the negative absorption coefficient in Eq. 9.7 at the lineshape center frequency Vq should be multiplied by (vo), which will be inversely proportional to the lineshape fwhm Av since (v) is normalized the product givojAv will be about unity. This implies that the lasing criterion in Eq. 9.12 should be replaced by [1]... 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

If the matrices which appear in the calculations do not exceed the size of 2 x 2 one can analytically derive the lineshape function, and this tends to make the computation rather trivial. Such cases are represented by all A B exchange processes and some other two-site exchange systems which give first-order spectra, as well as by the mutual AB = BA exchange. The same is possible in calculations on spectral fragments of certain more complicated systems as reported in a previous review in this series by Sutherland. (53)... 

The DF spectra of wurtzite-structure ZnO within the VIS-to-VUV spectral region contain CP structures, which can be assigned to band-gap-related electronic band-to-band transitions Eq with a = A, B,C and to above-band-gap band-to-band transitions E13 with (3 = 1,..., 7. The F -related structures can be described by lineshape functions of the 3DMo-type (3.9 and 3.10), the CP structures with (3 = 3,4 by lineshape functions of the 2DMo-type (3.11), and the CP structures with (3=1,2,5,6,7 can be described by Lorentzian-damped harmonic oscillator functions (3.13). The CP structures Eq are supplemented by discrete (3.14) and continuum (3.16) excitonic contributions. Tables 3.9 and 3.10 summarize typical parameters of the CPs Eq and E, respectively, of ZnO [15]. 

Further observations from similar experiments are that the emission spectral lineshapes are not a function of the uranyl ion concentration in the zeolite. The lineshape does also not seem to be influenced by the relative acidity of the ion-exchanged zeolites. When the degree of hydration is changed from fully hydrated (stored in a dessicator under saturated NHi Cl aqueous soluttons) to vacuum dried at 1 x 10 Torr at room temperature, only the intensity of luminescence (and not the lineshape) changes. 

Computer simulation of molecular dynamics is concerned with solving numerically the simultaneous equations of motion for a few hundred atoms or molecules that interact via specified potentials. One thus obtains the coordinates and velocities of the ensemble as a function of time that describe the structure and correlations of the sample. If a model of the induced polarizabilities is adopted, the spectral lineshapes can be obtained, often with certain quantum corrections [425,426]. One primary concern is, of course, to account as accurately as possible for the pairwise interactions so that by carefully comparing the calculated with the measured band shapes, new information concerning the effects of irreducible contributions of inter-molecular potential and cluster polarizabilities can be identified eventually. Pioneering work has pointed out significant effects of irreducible long-range forces of the Axilrod-Teller triple-dipole type [10]. Very recently, on the basis of combined computer simulation and experimental CILS studies, claims have been made that irreducible three-body contributions are observable, for example, in dense krypton [221]. 

If the peptide is not oriented in the membrane but forms large immobilized aggregates, this is easily seen in the NMR spectrum. In this case, all different orientations of the peptide are present in the sample and a very broad lineshape is seen, as for a peptide powder. When this is the case, no orientation of the peptide can be determined. Looking at the NMR spectral lineshape is a useful method for investigating the aggregation behavior of peptides interacting with membranes, which may be intimately related to the function or malfunction of the peptide. 

A second group of techniques may be called lineshape analysis. Simple methods entail the measurements of linewidths or second moments as a function of temperature. More sophisticated methods involve the analysis or the model fitting of spectral lineshapes. A prominent method is ID lineshape analysis for deuterium-labeled polymers, which is sensitive to motions in the frequency range of lO -lO s (149). The 2D wideline separation NMR (WISE) experiment permits correlation of the high resolution spectrum with the wideline spectrum, which provides dipolar information (11,150). The linewidth is a function of the frequency of the polymer motion relative to the time scale of dipolar couplings. 

In this model, the symbols have the following meaning. ( )R(a3) ao(l)HD(a3) is the optical rotation produced by the vapour and depends on the number of absorption lengths ao and on the lineshape function D(o)) which takes the form of a Doppler-broadened dispersion curve for magnetic and electric field induced rotation ( )r will depend on the strength of the field and D(oa) on the direction and type of field (see table 2). The transmitted intensity 1 - Ij exp[-aoG(o))] where the lineshape function G(o)) for a single spectral component can usually be accurately described by a Doppler-broadened Lorentzian curve. Finally the terms B and C in equation (53) represent respectively the finite extinction ratio of the polarisers and a laser independent... 

As usual there is a reciprocal relationship between the time domain and the frequency domain. The more rapid is the damping (decay) of the signal (i.e., larger a and shorter lifetime t = 1/a), the wider the Lorentzian and dispersion lineshape functions become in the spectral domain (Figure 4). 

Where M2 is the second moment of the NMR lineshape, J the spectral density function, with (Dq the Larmor frequency, and (0i the frequency of the spin-locking field. The spectral density can be written in terms of the molecular correlation time, x, and the overall shape of the Tjp - temperature dispersion and the relatively shallow minima arc due to the correlation time distribution, although the location of the minimum is unaffected by this distribution. We have examined several models for the distribution, all of which give essentially the same results. One of the more simple is the Cole-Davidson function (75), which has also been applied to the analysis of dielectric relaxations. The relevant expression for the spectral density in this case is given by Equation 4. 

These two delta functions correspond to absorption and emission of radiation at frequency c o, respectively, with spectral lineshapes exhibiting zero full width at half maximum (fwhm). Such uninterrupted molecular rotation, in which the dipole correlation function (8.17) maintains perfect sinusoidal coherence for an indefinite period of time, produces no broadening in the lineshape function I co). 

More General Relaxation Functions and Spectral Lineshapes... 

If the off-diagonal matrix elements that describe the coherence between a ground state and an excited electronic state decay exponentially with time, the homogeneous absorption line should have a Lorentzian shape (Figs. 10.6 and 10.7 Eqs. (2.70) and (10.35)). More generally, as we discussed in Sect. 10.6, the spectral lineshape is the Fourier transform of the relaxation function ... 

Fig. 10.10 Absorption spectral lineshapes calculated as the Fourier transform of the Kubo relaxation function <)> (). (A) (indicated in arbitrary time units) is varied, while a is fixed at 1 reciprocal time unit (B) is fixed at 1 time unit and a is varied as indicated. To use the full Fourier transform (Eq. 10.70), < (r) is treated as an even function of time (Fig. 10.1 lA)...

To quantitate the spectral lineshape, we applied ( ) non-linear least squares methods based on the simplex algorithm. The intensity-intensity correlation function g (field-correlation function g (q,t) via... 

One way to improve the signal-to-noise ratio is through convolution of the spectrum with an appropriate function such as a boxcar, Lorentzian, or Gaussian function. The operation of spectral convolution has been presented in Section 2.3 Such operations tend to distort the spectrum, as the lineshape function is altered. The broader the convolution function, the greater is the distortion of the spectrum. The most common such convolution is the Savitzky-Golay smoothing algorithm [13]. 

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