Solutions of the Schrodinger equation

Before attempting to solve the Schrodinger equation with K(r) as given by Eq. (4.7), we consider solutions for two simple cases, namely K(r) = 0 and K(r) = Vq, a constant. 

If k is the wavevector of the incident wave in vacuum, then K is the magnitude of k after correction for the mean inner potential Uq of the crystal. The fact that K is different from k means that the crystal has a refractive index, and it is clear that the mean refractive index n must be related to Uq. Direct measurements show that n is of the order of 1 x 10 In the dynamical theory of electron diffraction, which we develop in this chapter, we formally make all the Fourier components Kg complex quantities, that is, we replace Kg by Vg + iV. The full physical significance of the procedure will become clear in due course, but for the moment it will be helpful to consider the consequences of making Kq complex. If Kq is complex, then the mean refractive index n must also be complex, and so we write n = n +in . 

A wave traveling through a medium of mean refractive index n without diffraction in the x direction can be written as 

It is clear from Eq. (4.16) that is a measure of the attenuation of the wave as it passes through the medium, and the linear absorption coefficient for intensity n is equal to 4xkn . We could equally well have written nk = K and made K complex, in which case the imaginary part K would 

We are now in a position to begin the development of the dynamical theory. K(r) is expressed as a Fourier series, 

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By learning the solutions of the Schrodinger equation for a few model systems, the student can better appreciate the treatment of the fundamental postulates of quantum mechanics as well as their relation to experimental measurement because the wavefunctions of the known model problems can be used to illustrate. 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

The Extended Hiickel method, for example, does not explicitly consider the effects of electron-electron repulsions but incorporates repulsions into a single-electron potential. This simplifies the solution of the Schrodinger equation and allows HyperChem to compute the potential energy as the sum of the energies for each electron. 

These atomic orbitals, called Slater Type Orbitals (STOs), are a simplification of exact solutions of the Schrodinger equation for the hydrogen atom (or any one-electron atom, such as Li" ). Hyper-Chem uses Slater atomic orbitals to construct semi-empirical molecular orbitals. The complete set of Slater atomic orbitals is called the basis set. Core orbitals are assumed to be chemically inactive and are not treated explicitly. Core orbitals and the atomic nucleus form the atomic core. 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

The cell in the lower right corner of the chart represents the exact solution of the Schrodinger equation, the limit toward which all approximate methods strive. Full Cl using an infinitely flexible basis set is the exact solution. ... 

W.Kohn and Rostoker, Solution of the Schrodinger equation in periodic lattices with an apphcation to metaUic hthium , Phys.Rev.94 1111 (1954). 

As an especially interesting case of -functions we consider the adiabatic solutions of the Schrodinger equation, i.e., the solutions of... 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The starting step of the present work is a specific analysis of the solution of the Schrodinger equation for atoms (section 1). The successive steps for the application of this analysis to molecules are presented in the section 2 (description of the optimised orbitals near of the nuclei), 3 (description of the orbitals outside the molecule), and 4 (numerical test in the case of H ). The study of other molecules will be presented elsewhere. 

In the asymptotic region, an electron approximately experiences a Z /f potential, where Z is the charge of the molecule-minus-one-electron ( Z = 1 in the case of a neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the ip orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrodinger equation for the hydrogen-like atom with atomic number Z. ... 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

The solution of the Schrodinger equation with Helec is the electronic wave function xTelec and the electronic energy Eelec. depends on the electron coordinates, while the nuclear coordinates enter only parametrically and do not explicitly appear in Pelec. The total energy Etot is then the sum of Eelec and the constant nuclear repulsion term, M M... 

We should also mention that basis sets which do not actually comply with the LCAO scheme are employed under certain circumstances in density functional calculations, i. e., plane waves. These are the solutions of the Schrodinger equation of a free particle and are simple exponential functions of the general form... 

Tfiese restrictions are in general the origin of the boundary conditions imposed on the solutions of the SchrOdinger equation, as illustrated in Chapter 5. 

We shall now concentrate on several cases where relations equations (18) and (19) simplify. The most favorable case is where ln<)>(f) is analytic in one halfplane, (say) in the lower half, so that In < ) (t) = 0. Then one obtains reciprocal relations between observable amplitude moduli and phases as in Eqs. (9) and (10), with the upper sign holding. Solutions of the Schrodinger equation are expected to be regular in the lower half of the complex t plane (which corresponds to positive temperatures), but singularities of In 4>(0 can still arise from zeros of < r(f). We turn now to the location of these zeros. 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the PES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schrodinger equation should, however, be kept in mind. 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function form Eq. (C.l) could be an exact solution of the Schrodinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

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