Harmonic oscillator simplifying

Harmonic oscillators simplify the analysis. Normal coordinates and frequencies differ in principle from state to state and involve both state rotations and energy renormalization. In practice, unrotated ground-state coordinates are retained even when different frequencies are used in detailed fits [21,91,92]. Displaced harmonic oscillators yield analytical expressions [93,94] for Fp, when ground-state frequencies o>i are kept in excited states. Moreover, since o>i is typically smaller than electronic splittings, some of the sums in Eq. (30) can be evaluated by closure. Closure can readily be checked and holds unless an energy denominator becomes small. Closure leads to Franck-Condon averages [Pg.181]

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

To simplify the expression of the thermal average rate W, f given by Eq. (3.32), we shall assume that both potential energy surfaces of the two excited vibronic manifolds of the DA system consist of a collection of harmonic oscillators, that is,... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Let us now deal with the shape of the optical (absorption and emission) bands as a result of the previously discussed strong ion-lattice conpling. For this purpose, we consider a simplified two electronic energy states center, in which the initial i and final / states are both described by harmonic oscillators of the same frequency f and with minima at go and Q q respectively, as shown in Figure 5.12. 

A simplified model of the stretching frequency of 2 atoms and their connecting bond is offered by the harmonic oscillator of two masses connected by a spring (Herzberg 1950). 

The Hamiltonian (1) for quasi-one-dimensional two-electron quantum dots is simplified by neglecting the x and y degrees of freedom and by approximating the confining Gaussian potential by a harmonic-oscillator potential with >z... 

Figure 8. Spectra of CO calculated from the rigid rotor-harmonic oscillator approximation. The top spectrum is CO at 298 K. The bottom is CO at 20 K. This reduction in lines will be very important for simplifying the analysis in larger, more complicated species. 

The first usable results were obtained by Rice and Ramsperger" and Kassel,who were able to deduce from a simplified model of a molecule, consisting of a set of harmonic oscillators, that... 

A quantum-mechanical treatment has been given for the coherent excitation and detection of excited-state molecular vibrations by optical absorption of ultrashort excitation and probe pulses [66]. Here we present a simplified classical-mechanical treatment that is sufficient to explain the central experimental observations. The excited-state vibrations are described as damped harmonic oscillations [i.e., by Eq. (11) with no driving term but with initial condition Q(0) < 0.] We consider the effects of coherent vibrational oscillations in Si on the optical density OD i at a single wavelength k within the Sq -> Si absorption spectrum. Due to absorption from Sq to Si and stimulated emission from Si and Sq,... 

In order to extend the linearization scheme to non-adiabatic dynamics it is convenient to represent the role of the discrete electronic states in terms of operators that simplify the evolution of the quantum subsystem with out changing its effect on the classical bath. A way to do this was first suggested by Miller, McCurdy and Meyer [28,29[ and has more recently been revisited by Thoss and Stock [30, 31[. Their method, known as the mapping formalism, represents the electronic degrees of freedom and the transitions between different states in terms of positions and momenta of a set of fictitious harmonic oscillators. Formally the approach is exact, but approximations (e.g. semi-classical, linearized SC-IVR, etc.) must be made for its numerical implementation. 

To simplify our notation, we will suppress in what follows the polarization vector Cf, that is, the vector k will be taken to denote both wavevector and polarization. The time evolution of a mode k of frequency ct>k is determined by a harmonic oscillator Hamiltonian, and its quantum state—by the... 

Note that with the simplified potential (14.61) our problem becomes mathematically similarto that of a harmonic oscillator, albeit with a negative force constant. Because of its linear character we may anticipate that a linear transformation on the variables X and V can lead to a separation of variables. With this in mind we follow Kramers by making the ansatz that Eq. (14.62) may be satisfied by a function f of one linear combination of x and v, that is, we seek a solution of the form... 

The situation changes drastically if the field mode is allowed to interact with some detector placed inside the cavity. Following other findings [188,189] we demonstrate the effect in the framework of a simplified model, when a harmonic oscillator tuned to the frequency of the resonant mode is placed at the point of maximum of the amplitude mode function v /mn(x,y L ) in the 3D rectangular cavity. 

To simplify notation for these two terms let 2f0[G3(MP2)] s E0 and G3MP2 Enthalpy = //29g. The thermal correction to the enthalpy (TCH), converting energy at 0 K to enthalpy at 298, (H29% -E0 = -78.430772-(—78.4347736) = 0.0040016 h) is a composite of two classical statistical thermodynamic enthalpy changes for translation and rotation, and a quantum harmonic oscillator term for the vibrational energy. 

Now simplify the picture of the fluctuating dipole moment of each atom into a simple harmonic oscillator, with the force constant k, whose mass m is the mass of the moving cloud of electrons. The natural frequency of the oscillator in each of the atoms is then... 

The selection rule for the harmonic oscillator, Eq. (25.55), requires that An = 1. Under the influence of a light beam the harmonic oscillator makes transitions only to states immediately above and below its original state. The existence of selection rules simplifies the interpretation of spectra enormously. 

This formula simplifies considerably when all molecules are initially in the ground state and when det is a continuant, as it b when only transitions between nearest neighbor states occur (for example in the case of the simple harmonic oscillator model). We recall that A as defined in Eq. IV.6 has the property... 

Vibrational Motion in Polyatomic Molecules. Normal-mode analysis of vibrational motion in polyatomic molecules is the method of choice when there are several vibrational degrees of freedom. The actual vibrations of a polyatomic molecule are completely disordered, or aperiodic. However, these complicated vibrations can be simplified by expressing them as linear combinations of a set of vibrations (i.e., normal modes) in which all atoms move periodically in straight lines and in phase. In other words, all atoms pass through their equilibrium positions at the same time. Each normal mode can be modeled as a harmonic oscillator. The following rules are useful to determine the number of normal modes of vibration that a molecule possesses ... 

The Schiodinger equation is quite easy to solve nowadays with a desired accuracy for many systems. There are only a few ones for which the exact solutions are possible. These problems and solutions play an extremely important role in physics, since they represent a kind of beacon for our navigation in science, where as a rule we deal with complex systems. These may most often be approximated by those for which exact solutions exist. For example, a real diatomic molecule is difficult to describe in detail, and it certainly does not represent a harmonic oscillator. Nevertheless, the main properties of diatomics foUow from the simple harmonic oscillator model. When chemists or physicists must describe reality they always first try to simplify the problem, to make it similar to one of the simple problems described in the present chapter. Thus, from the beginning, we know the (idealized) solution. This is of prime importance when discussing the (usually complex) solution to a higher level of accuracy. If this higher-level description differs dramatically from that of the idealized one. most often this indicates that there is an error in our calculations, and no task is more urgent than to find and correct it. 

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