Normal Coordinates, Phonons

The generalization of the results derived for the diatomic chain (Sects. 2.1.4, 2.2.2-4) to the general three-dimensional crystal is trivial. It is only necessary to replace the scalar quantities q, u( ), x( ), 6(k1 ) by the corresponding three-dimensional vectors q, later 

In Appendix I it is shown that in terms of normal coordinates, the Hamiltonian (3.14) assumes the form (2.58) 

Furthermore, the normal coordinates satisfy the uncoupled harmonic oscillator equation 

As in the case of the diatomic chain, it is possible to introduce real normal coordinates a (j) (standing waves) or n(j) (running waves). They are defined by expressions analogous to (2.62,67), respectively. 

The classical mean energy of the system is given by the relations (2.73,74) 

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The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

According to the concept of the displacive-type ferroelectric phase transition [10], an increase in the dielectric constant corresponds directly to the softening of the IR-active transverse phonon. When the crystal can be regarded as an assembly of the vibrators of normal coordinates, the soft phonon... 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

The theory of multi-phonon electron transitions due to non-adiabatic interaction between the electron terms is expounded in this chapter. The participation of the large number of the vibration degrees of freedom in the transition is caused by the non-orthogonality of the oscillator wave function in the initial and final states. This non-orthogonality may be connected with the following factors the shift of the equilibrium positions of the oscillations, the change of the frequencies in the transition, and the change of the set of normal coordinates of the vibration system. 

The Raman scattering, developed as a Taylor series of the phonon normal coordinates, can be written as follows ... 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

The electron-phonon interaction can take on two different forms, depending on the type of phonons involved. The interaction with optical phonons is intramolecular and of the form %e 0 = X E, q(nh where X is the coupling strength and < , is the phonon normal coordinate at molecule i. The interaction with acoustical phonons is through the hopping integrals, which are modulated by intermolecular motion. The hopping between two molecules at the crystal positions R, and R is ti = t + (V ) (uj - uf) + , so... 

Here the displacement coordinate QA(t) is given a time dependence since we are adopting a classical description. As explained at length in Sect. 4.4.2, in a lattice the role of the molecular coordinate QA is taken by the corresponding resonance mode coordinate which is a superposition of the pure lattice normal coordinates qAj(= i = k jn) defined in (3.13). In terms of these and the electron-phonon coupling coefficients in (3.15)kAi(= k ), (5.1) can be rewritten as... 

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form... 

The model which has been most widely applied to the calculation of vibronic intensities of the Cs2NaLnCl6 systems is the vibronic coupling model of Faulkner and Richardson [67]. Prior to the introduction of this model, it was customary to analyse one-phonon vibronic transitions using Judd closure theory, Fig. 7d, [117] (see, for example, [156]) with the replacement of the Tfectromc (which is proportional to the above Q2) parameters by T bromc, which include the vibrational integral and the derivative of the CF with respect to the relevant normal coordinate. The selection rules for vibronic transitions under this scheme therefore parallel those for forced electric dipole transitions (e.g. A/ <6 and in particular when the initial or final state is /=0, then A/ =2, 4, 6). 

There is a formal similarity in the mathematics used to describe vibrational transitions pumped by a resonant radiation field [148] and vibrational transitions pumped by phonons in a crystal lattice. In the lowest-order approximations, the radiation field and the vibrational transition are coupled by a transition dipole matrix element that is a linear function of a coordinate. The transition dipole describes charge displacement that occurs during the transition. Some of the cubic anharmonic coupling terms described by Eq. (10) result in a similar coupling between vibrational transitions and a phonon coordinate. These generally have the form / vibVph, so that the energy of the vibration with normal coordinate /vib is linearly proportional to the phonon coordinate /ph. Thus either an incoherent photon field or an incoherent phonon field can result in incoherent... 

In diatomic VER, the frequency O is often mueh greater than so VER requires a high-order multiphonon proeess (see example C3.5.6.11. Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian E ( in equation tC3.5.21 in powers of normal coordinates [34, 63],... 

If a normal mode in a crystal, connected for example with a phonon or the photoionization of an impurity, gives rise to any change in the electric dipole moment p, then the dynamic dipole moment p = 9p/9 i is nonzero. Here qi is the normal coordinate, which characterizes the corresponding normal mode and can be derived from normal coordinate analysis based on classical physics [55], The value of p depends on the relative ionicity of the species and can be obtained only by quantum-chemical calculatious (see Ref. [61] and the literature therein). In general, the more polar the bond, the larger the p term. The matrix element of the dynamic dipole moment, (y p i), is called the transitional dipole moment (TDM) of the corresponding normal mode. 

The term pS represents the average number of modes involved in the jump. Following Ttyozawa (1967), Sq (the Huang-Rhys coupling parameter, Huang and Rhys 1950), which is directly linked to the amount by which the normal coordinates are displaced by the electron-phonon coupling, can be viewed also as the number of phonons emitted or absorbed in the transition. It is then possible to write pS=So, and eq. (135) can be written as... 

Starting from a set of harmonic force constants calculated in terms of internal coordinates, one can determine in this way the normal coordinates of the different vibrations (phonons) of an infinite chain as a function of (the columns of matrices L , ) as well as the corresponding phonon dispersion curves One should note that, in order to... 

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