Force constants from electronic wave functions

The time evolution of the electronic wave function can be obtained in the adiabatic or in the diabatic basis set. At each time step, one evaluates the transition probabilities between electronic states and decides whether to hop to another siu-face. When hopping occurs, nuclear velocities have to be adjusted to keep the total energy constant. After hopping, the forces are calculated from the potential of the newly populated electronic state. To decide whether or not to hop, a Monte Carlo technique is used Once the transition probability is obtained, a random number in the range (0,1) is generated and compared with the transition probability. If the munber is less than the probability, a hop occurs otherwise, the nuclear motion continues on the same surface as before. At the end of the simulation, one can analyze populations, distribution of nuclear geometries, reaction times, and other observables as an average over all the trajectories. 

Phonon dispersion curves w(q) can be measured by inelastic neutron scattering techniques this important method will be discussed in [1.35]. The curves determined by experiments are mainly of interest because they provide chance of testing various models of interatomic forces. The forces between the atoms are governed by the electronic structure of the atoms involved, and in simple cases it is possible to derive force constants from "first principles" which are based on the electronic wave functions [4.1-5]. With the exception of some qualitative remarks about metals, we shall not discuss these microscopic models here but consider only certain phenomenological models for the interatomic forces. A good model should contain only a few, but physically meaningful, parameters. 

It is possible to classify the corrections due to electron correlation (in other words the "correlation effects ) according to various criteria. A first distinction is between effects which are directly related to the amount of correlation taken into account in the wave function and effects which arise from the interdependence of correlation energy and molecular geometry. The latter were already mentioned in a different context at the end of the previous section they include changes in equilibrium distances and force constants. Usually, these molecular constants are considered to be insensitive to correlation, but such a statement is only true within limits. 

In particular, the phonon dispersion relations and polarization vectors can be calculated with reasonable accuracy using force-constant models [59] or the embedded atom method [60-62], In recent calculations of Fe-ph and X for surface states, wave functions obtained from the one-electron model potential [63, 64] have been used. For the description of the deformation potential, the screened electron-ion potential as determined by the static dielectric function and the bare pseudopotential is used, Vq z) = f dz e (z,2/ gy)qy), where (jy is the modulus of the phonon momentum wave vector parallel to the surface, and bare Fourier transform parallel to the surface of the bare electron-ion... 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

The PW basis set is universal, in the sense that it does not depend on the positions of the atoms in the unit cell, nor on their nature [458]. One does not have to construct a new basis set for every atom in the periodic table nor modify them in different materials, as is the case with locahzed atomic-hke functions and the basis can be made better (and more expensive) or worse (and cheaper) by varying a single parameter -the number of plane waves defined by the cutoff energy value. This characteristic is particularly valuable in the molecular-dynamics calculations, where nuclear positions are constantly changing. It is relatively easy to compute forces on atoms. Finally, plane-wave calculations do not suffer from the basis-set superposition error (BSSE) considered later. In practice, one must use a finite set of plane waves, and this in fact means that well-localized core electrons cannot be described in this manner. One must either augment the basis set with additional functions (as in linear combination of augmented plane waves scheme), or use pseudopotentials to describe the core states. Both AS and PW methods, developed in solid-state physics are used to solve Kohn-Sham equations. We refer the reader to recently published books for the detailed description of these methods [9-11]. 

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