Schrodinger equation simplifying

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. 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Semi-empirical methods, such as AMI, MINDO/3 and PM3, implemented in programs like MOPAC, AMPAC, HyperChem, and Gaussian, use parameters derived from experimental data to simplify the computation. They solve an approximate form of the Schrodinger equation that depends on having appropriate parameters available for the type of chemical system under investigation. Different semi-emipirical methods are largely characterized by their differing parameter sets. 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

The atomic units system (au system) is a system of units meant to simplify the equations of molecular and atomic quantum mechanics. The units of the au system are combinations of the fundamental units of mass (mass of the electron), charge (charge of the electron), and of Planck s constant. By setting these three quantities to unity one gets simpler equations. Si in the usual SI system, Schrodinger equation takes the form ... 

At this point in the derivation, so as to simplify the notation, the subscript for a particular solution to the Schrodinger equation (2.1) and its associated energy will be dropped. Thus Eq. (2.7) can be rewritten as ... 

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. 

Because its base units directly underlie the quantum theory of electrons (i.e., the mass, charge, and angular momentum of the electron itself), the atomic units naturally simplify the fundamental Schrodinger equation for electronic interactions. (Indeed, with the choice me = e = h = 1, the Schrodinger equation reduces to pure numbers, and the solutions of this equation can be determined, once and for all, in a mathematical form that is independent of any subsequent re-measurement of e, me, and h in chosen practical units.) In contrast, textbooks commonly employ the Systeme International d Unites (SI), whose base units were originally chosen without reference to atomic phenomena ... 

Slater then proposed returning to one-electron Schrodinger equations (and therefore one-electron wave functions i and corresponding eigenvalues ,) but now using not the Hartree-Fock (non-local) potential in these equations but the simplified potential... 

One way to simplify the Schrodinger equation for molecular systems is to assume that the nuclei do not move. Of course, nuclei do move, but their motion is slow compared to the speed at which electrons move (the speed of light). This is called the Born-Oppenheimer approximation, and leads to an electronic Schrodinger equation. 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

To simplify the notation we have assumed that the light pulse has prepared the system in a single bound state. The probabilities for finding the system in states I J/ n) and 1 2(E,0)) at time t are aij(i) 2 and a2(t) E,/3) 2, respectively. Inserting (7.3) into the time-dependent Schrodinger equation with the full Hamiltonian which also includes the coupling W and utilizing (7.1) and (7.2) yields the coupled equations... 

Schrodinger equations for atoms and molecules use the the sum of the potential and kinetic energies of the electrons and nuclei in a structure as the basis of a description of the three dimensional arangements of electrons about the nucleus. Equations are normally obtained using the Born-Oppenheimer approximation, which considers the nucleus to be stationary with respect to the electrons. This approximation means that one need not consider the kinetic energy of the nuclei in a molecule, which considerably simplifies the calculations. Furthermore, the... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

So the simplified Schrodinger equation eq. (2.2.7) can be separated into two (familiar) equations, each involving only the coordinates of one electron ... 

A simplified model was necessary, because the exact calculation of all the orbitals for an atom with several shells of electrons is impossible there is no analytical solution for the Schrodinger equation for atoms with more than one electron. By concentrating on the outer electrons only, and by using the orbitals of these electrons to provide graphic images of electron density (i.e., of where electrons are most likely to be found), Pauling generated an intuitively... 

In terms of (14), imperfections appear as a modification of the local anisotropy K r) and lead to a nucleation-field and coercivity reduction [105, 110-112], The solution of the nucleation problem is simplified by the fact that Eq. (14) has the same structure as the single-particle Schrodinger equation, J i(r) and Hc being the respective micromagnetic equivalents of V(r) and E. Consider, for example, an imperfection in form of a cubic soft inclusion of volume l) in a hard matrix. The corresponding wave functions are particle-in-a-box states, and the nucleation field is [5]... 

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