Harmonic Hamiltonian

We note that for the harmonic Hamiltonian in (5.25) the variance of the work approaches zero in the limit of an infinitely slow transformation, v — 0, r — oo, vt = const. However, as shown by Oberhofer et al. [13], this is not the case in general. As a consequence of adiabatic invariants of Hamiltonian dynamics, even infinitely slow transformations can result in a non-delta-like distribution of the work. Analytically solvable examples for that unexpected behavior are, for instance, harmonic Hamiltonians with time-dependent spring constants k = k t). 

Here, fe ) are the eigenkets of the fast mode harmonic Hamiltonian while (m)) are those of the harmonic slow mode, whereas k and (m) are the corresponding quantum number. These kets form two bases (w)) and fc ) that lead us to write, respectively,... 

The passage from representation // to /// does not affect the eigenstates of the slow mode harmonic Hamiltonian when the fast mode is in its ground state 0 >, but affects them when this mode has jumped on its first excited state 1 ). In this last situation there is... 

Again, the expansions of the corresponding eigenvectors on the basis in which the quantum harmonic Hamiltonian of the H-bond bridge is diagonal, are... 

Within the tensorial product basis, (in)) built up from the eigenvectors of the Hermitean harmonic Hamiltonians of the slow and fast modes, the eigenvectors are given by the expansion ... 

Harmonic Hamiltonians Describing the Slow and High Frequency Modes and the Thermal Bath... 

Next, observe that the trace trDav over the base spanned by the Hamiltonian reduces in the present situation to the product of traces tra and tr/, over the bases spanned by the H-bond bridge harmonic Hamiltonians of the two moieties. 

Unity operators involved in the theory of centrosymmetric cyclic dimers and built up, respectively, from the closeness relations on the eigenstates of the g and u symmetrized quantum harmonic Hamiltonians of the H-bond bridge. 

The /Ith eigenstates of the harmonic Hamiltonian of the H-bond bridge in quantum representations // and ///, Franck-Condon factors. 

Coherent state (of eigenvalue a0), fth eigenstate of the /lh harmonic Hamiltonian of the j th bending mode. 

Fast mode ftth eigenstates of the two moieties a and b harmonic Hamiltonians. Hydrogen-bond bridge nth eigenstates of the two moieties a and b harmonic Hamiltonians. 

Symmetric g and Antisymmetric u kth States of the Symmetrized High Frequency Mode Harmonic Hamiltonians. Symmetric g and antisymmetric u nth states of the symmetrized H-bond bridge harmonic Hamiltonians. 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a hannonic oscillator coupled linearly to a harmonic bath. Such model may represent a reasonable approximation to physical reality if the frequency of the oscillator under study, that is the mode that can be excited and monitored, is well embedded within the spectrum of bath modes. However, many processes ofinterest involve molecular vibrations whose frequencies are higherthan the solvent Debye frequency. In this case the linear coupling rate (13.35) vanishes, reflecting the fact that in a linear coupling model relaxation cannot take place in the absence of modes that can absorb the dissipated energy. The harmonic Hamiltonian... 

Until now, our treatment has been built in exactly the same terms that might have been used in work on normal modes of vibration in the latter part of the nineteenth century. However, it is incumbent upon us to revisit these same ideas within the quantum mechanical setting. The starting point of our analysis is the observation embodied in eqn (5.19), namely, that our harmonic Hamiltonian admits of a decomposition into a series of independent one-dimensional harmonic oscillators. We may build upon this observation by treating each and every such oscillator on the basis of the quantum mechanics discussed in chap. 3. In light of this observation, for example, we may write the total energy of the harmonic solid as... 

We have seen that as a consequence of the harmonic Hamiltonian that has been set up thus far, our oscillators decouple and in cases which attempt to capture the transport of energy via heating, there is no mechanism whereby energy may be communicated from one mode to the other. This shortcoming of the model may be amended by including coupling between the modes, which as we will show below, arises naturally if we go beyond the harmonic approximation. The simplest route to visualizing the physics of this problem is to assume that our harmonic model is supplemented by anharmonic terms, such as... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

When analytical solutions are not known and the approximate analytical methods give results of limited applicability, the numerical methods may be a solution. Let us first discuss a method based on the diagonalization of the second-harmonic Hamiltonian [48,49]. As we have already said, the two parts of the Hamiltonian Ho and Hi given by (55), commute with each other, so they are both constants of motion. The //0 determines the total energy stored in both modes, which is conserved by the interaction ///. This means that we can factor the quantum evolution operator... 

The derivation and application of normalised spherical harmonic Hamiltonians. J. C. Donini, B. R. Hollebone and A. B, P. Lever, Prog. Inorg. Chem., 1977,22, 226-307 (70. 

The equilibrium order determined within the mean-field theory is perturbed due to thermal fluctuations which give rise to collective excitations. Except in the close vicinity of the phase/structural transitions, the thermal fluctuations of the order parameter can be assumed small, and the free energy of the fluctuations can be considered a correction to the mean-field free energy. In such a case, the fluctuations of the liquid-crystaUine order are described consistently by a harmonic Hamiltonian of the form... 

It is convenient to first identily the constants of the motion in the harmonic Hamiltonian and use them as the new momenta. Consider then the momenta... 

With the force constants given by (3.21), the correlated gaussian wavefunction (3.18) is the exact solution of the harmonic Hamiltonian,... 

The harmonic Hamiltonian in Eq. (18) can be diagonalized exactly. First, one introduces Fourier-transformed displacement coordinates ... 

The anharmonic terms, i.e. the cubic and higher terms in the displacement expansion of the intermolecular potential and the rotational kinetic energy terms, which are neglected in the harmonic Hamiltonian, can be considered as perturbations. They affect the vibrational excitations of the crystal in two ways they shift the excitation frequencies and they lead to finite lifetimes of the excited states, which are visible as spectral line broadening. By means of anharmonic perturbation theory based on a Green s function approach [64, 65] it is possible to calculate the frequency shifts, as well as the line widths. 

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