Harmonic phonons

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

The scaling with the inverse of the square of the phonon frequency follows from harmonic phonon theory, such that low frequency modes have the larger amplitudes. The rotations of the octahedra will give a net reduction in volume ... 

The Fourier representation of the anharmonic propagator is obtained from Eqs. (73) and (72), by using Wick s theorem (69), together with the Fourier representation of the harmonic phonon propagators. The time integrations can then explicitly be performed, yielding the condition 2,- w, = 0 and a factor /3 at every interaction vertex. 

We have applied NESGET to study the charge conductivity of a molecular wire attached to two perfectly conducting leads. In the simplest approach the leads a and b are treated as two free electron reservoirs. Nuclear motions in the molecular region are described as harmonic phonons which interact with the surrounding electronic structure and the environment (secondary phonons) [26]. We first recast the general Hamiltonian, Eq. (1), in a single electron local basis and partition it as... 

We first want to recall some basic facts of the theory of elasticity. In the harmonic approximation the lattice energy of a crystal has contributions due to homogeneous strains and to harmonic phonons ... 

In the dielectric screening method the electron density response due to the motion of the ions around their equilibrium positions is calculated in first order perturbation theory. The potential energy of the crystal for an arbitrary configuration of the ions is expanded to second order in the ionic displacements from equilibrium. The expansion coefficients of the second order term form a matrix. The Fourier transform of this matrix is the dynamical matrix whose eigenvalues yield the phonon frequencies. The dynamical matrix has an ionic and electronic part. The electronic part can be expressed in terms of the electron density response matrix and of the ionic potential. This method has the advantage over the total energy difference m ethod that the phonon frequencies for any arbitrary wave vector can be calculated without additional difficulties. Furthermore in this method the acoustic sum rule is automatically satisfied as a consequence of the way the dynamical matrix is derived. However the dielectric screening method is limited to harmonic phonons. 

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