Independent harmonic oscillators

The extension of the quantum-mechanic interpretation of the vibrational motion of atoms to a crystal lattice is obtained by extrapolating the properties of the diatomic molecule. In this case there are 3 ( independent harmonic oscillators (9l is here the number of atoms in the primitive unit cell—e.g., fayalite has four... 

In the case of polyatomic molecules, the energy levels become quite numerous. Ideally, one can treat such a molecule as a series of diatomic, independent, harmonic oscillators and the above equation can be generalized ... 

One point which has not been addressed in the example of the time-independent harmonic oscillator is the non-perturbative treatment of the time dependence in the system Hamiltonians. Both the TL and the TNL non-Markovian theories employ auxiliary operators or density matrices, respectively, and can be applied in strongly driven systems [29,32]. This point will be shown to be very important in the examples for the molecular wires under the influence of strong laser fields. 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

Let us consider a particle of mass m, as described by its coordinate x and the conjugate momentum p, evolving in a potential V(x). The particle is coupled with a bath of independent harmonic oscillators of masses mn, described by the coordinates x and the conjugate momenta p (n = 1,..., 1V). The coupling between the particle and each one of the bath oscillators is taken as bilinear. The Hamiltonian of the global system constituted by the particle and the oscillators is... 

In order to evaluate the canonical partition function Q for a gas, we shall consider the system to be composed of an aggregate of essentially independent particles (molecules). As we shall see later, a crystal may be considered to a good approximation as an aggregate of independent harmonic oscillators. Each of these has its own microcanonical partition function ... 

The quantum-mechanical description of a polyatomic system may be extrapolated from the treatment of the diatomic molecule. Starting again with the harmonic oscillator, the energy levels for the entire system (E) can be given in terms of the characteristic frequencies (v,) and quantum numbers ( ,) of a series of independent harmonic oscillators ... 

A molecule with N atoms has a total of 37V degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 37V - 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N -5 vibrational modes. For example, the nonlinear molecule H2O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency Vj (expressed in cm ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency u . The normal modes of H2O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2O are... 

This linear problem is thus exactly soluble. On the practical level, however, one cannot carry out the diagonalization (4.11) for macroscopic systems without additional considerations, for example, by invoking the lattice periodicity as shown below. The important physical message at this point is that atomic motions in solids can be described, in the harmonic approximation, as motion of independent harmonic oscillators. It is important to note that even though we used a classical mechanics language above, what was actually done is to replace the interatomic potential by its expansion to quadratic order. Therefore, an identical independent harmonic oscillator picture holds also in the quantum regime. 

The Debye model discussed in Section 4.2,4 rests on three physical observations The fact that an atomic system characterized by small oscillations about the point of minimum energy can be described as a system of independent harmonic oscillators, the observation that the small frequency limit of the dispersion relation stems... 

We consider a classical equilibrium system of independent harmonic oscillators whose positions and velocities are denoted XjVy = iy, respectively. In fact, dealing with normal modes implies that we have gone through the linearization and diagonalization procedure described in Section 4.2.1. In this procedure it is convenient to work in mass-normalized coordinates, in particular when the problem involves different particle masses. This would lead to mass weighted position and... 

To summarize, we have found that the dielectric response of a polar medium can be described in terms of the Hamiltonian (16.88) that corresponds to a system of independent harmonic oscillators indexed by the spatial poison r. These oscillators are characterized by given equilibrium positions P (r) that depend on the electronic state Z. Therefore a change in electronic state corresponds to shifts in these equilibrium positions. 

When our problem is recast in this form it is clear that we are reconsidering it as one of matrix diagonalization. Our task is to find that change of variables built around linear combinations of the UiaS that results in a diagonal K. Each such linear combination of atomic displacements (the eigenvector) will be seen to act as an independent harmonic oscillator of a particular frequency, and will be denoted as normal coordinates . 

What we have learned is that our change to normal coordinates yields a series of independent harmonic oscillators. From the statistical mechanical viewpoint, this signifies that the statistical mechanics of the collective vibrations of the harmonic solid can be evaluated on a mode by mode basis using nothing more than the simple ideas associated with the one-dimensional oscillator that were reviewed in chap. 3. 

In light of our observations from above, namely that the vibrational contribution to the energy of the crystal may be written as a sum of independent harmonic oscillators, this result for the Helmholtz free energy may be immediately generalized. In particular, we note that once the vibrational density of states has been determined... 

As shown by the equations for x, y, and z, v0 is the frequency of the motion. It is seen that the particle may be described as carrying out independent harmonic oscillations along the x, y, and z axes, with different amplitudes x0, yo, and z0 and different phase angles i 8V, and 82, respectively. 

The last reference system we discuss is the lattice of interacting harmonic oscillators. In this system each atom is connected to its neighbors by a Hookean spring. By diagonalizing the quadratic form of the Hamiltonian, the system may be transformed into a collection of independent harmonic oscillators, for which the free energy is easily obtained. This reference system is the basis for lattice-dynamics treatments of the solid phase [67]. If D is the dynamical matrix for the harmonic system (such that element Dy- describes the force constant for atoms i and j), then the free energy is... 

Once again, in the absence of curvature coupling, this reduces to a sum of independent harmonic oscillator Hamiltonians. The wavefunction is written as a product of one-dimensional functions each expanded in a harmonic oscillator basis. 

The details of lattice vibrations will not be discussed here. But for the sake of discussion, the main results of one of the simpler models, namely, the Einstein solid, are given below without proof. By assuming the solid to consist of Avogadro s number jVav of independent harmonic oscillators, all oscillating with the same frequency Einstein showed that the thermal entropy per mole is given by... 

The separability of the Hamiltonian in the normal mode form implies that the dynamics is in some sense trivial. One must only consider the continuum limit of a collection of independent harmonic oscillators and a single parabolic barrier. As described in Sec. III.D, this simple dynamics leads to some important relations between the Hamiltonian approach and the more standard stochastic theories. Multidimensional generalization of the parabolic barrier case will be discussed briefly in Sec. VIII. 

The inclusion of molecular vibrations will effect the partition function which, in addition to the electronic contribution, will contain the vibration part [28]. Each molecule is assumed to behave as a set of (3 - 6) independent harmonic oscillators with the fundamental frequencies vf so that the vibration partition functions are... 

A general and convenient choice is to model the bath as a set of independent harmonic oscillators linearly coupled to the system. The Hamiltonian for a collection of oscillators of unit mass and frequencies w. is... 

The two baths are usually represented by sets of independent harmonic oscillators. For the molecular unit, we are mainly interested in 1-dimensional periodic chains made of N units with anharmonic force fields. System-bath coupling is either harmonic or nonlinear and is typically taken to be weak. We will assume that only the end atoms of the chain, 1 and Af, couple to the surfaces, L and R, respectively, and neglect direct interactions between the reservoirs. 

The molecular mode is coupled either linearly or nonlinearly to the L and R thermal baths, represented by sets of independent harmonic oscillators. 

In the harmonie approximation, the problem of small amplitude vibrations (discussed in Chapters 6 and 7) reduces to the 3N — 6 normal modes N is the number of atoms in the moleeule). Eaeh of the normal modes may be treated as an independent harmonic oscillator. A normal mode moves all the atoms with a certain frequency about their equilibrium positions... 

C.6. Small amplitude harmonic vibrations of a molecule (JV atoms) are described by 3iV—6 independent harmonic oscillators (normal modes). Each normal mode is characterized by an irreducible representatioiL A scheme of the vibrational energy levels of three normal modes corresponding to the irreducible representations Fj, T2, r3. The modes have different frequencies, so the interlevel separations are different for all of them (but equal for a given mode due to the harmonic potential). On the right side, all these levels are shown together. 

The following quantum mechanical hamiltonlan reproduces the same effects described above with a classical model. We consider a zeroth order hamiltonlan defined as the sum of a set of independent harmonic oscillators in a set of symmetrically related X-H bonds ... 

Upon Fourier-transforming h r) = Sj hq and restricting the functional to quadratic order we obtain the transformed Hamiltonian E hq = L lLq q hqf, which shows that the modes hq are independent harmonic oscillators. The equipartition theorem then implies that =k Tjl Kq, and thus fitting to... 

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