Schrodinger model

The modern theory of the atom was initially introduced by Erwin Schrodinger in 1926. It is popularly known as the quantum-mechanical model. This model is based on some very complex mathematics, with which we will not concern ourselves here. The essence of the model is that electrons exist in principal (or main) energy levels, in energy sublevels within these principal levels, and in regions of space called orbitals within the sublevels. The electrons are also thought of as having a particular spin direction. 

The values fhaf n can take on (i.e., 1,2, 3,4,5, and so on) correspond to the first, second, third, fourth, fifth, and so on levels. The larger the number of the level is, the further that level is from the nucleus. Theoretically, the number of such levels is limitless. Practically, however, no atom that we currently know of has more than seven principal levels. Thus the configuration of electrons of any element in the periodic table can be described by using just seven principal levels. Since there are more than 100 elements in the periodic table, it follows that the average number of electrons that can be found in each level is relatively large. In addition, the number of electrons possible in each principal level varies with the level number. A maximum of only 2 electrons can be found in the n = 1 level, 8 in the n = 2 level, 18 in the n = 3 level, 32 in the 

FIGURE 4.7 A drawing depicting the first three energy levels around a nucleus. 

It is not sufficient to simply use the n value to describe the configuration of the electron structure around the nucleus since there are so many electrons. The electrons within each principal level are not equal in energy. In fact, the modern theory says that each is unique. To describe each one, we must explore the concepts of energy sublevels within the principal levels, orbitals within the sublevels, and electron spin direction. 

The different sublevels that exist are designated by the four letters s, p, d, and/ In other words, there are four possibilities for different sublevels within the principal energy levels for electrons. Not all sublevels are found in every principal level. For example, in the n = 1 level, there is only an s sublevel, since that is all that is required to contain the maximum of two electrons calculated using Equation 4.2, and in the n = 2 level, there are only s and p sub-levels. As the n value increases, more sublevels are required. This is because as n increases, more electrons are found, as is determined from calculations performed with Equation 4.2. Again, theoretically, the number of sublevels is unlimited however, all electrons in all known elements can be accounted for by only utilizing four sublevels—the four symbolized by the letters s, p, d, and/ We often refer to these sublevels as s, p, d, or/sublevels, to the orbitals as s, p, d, or /orbitals, and to the electrons they contain as the s, p, d, or J 

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The isotopic p —3 He2+ — e system has the same general properties as the originally studied p —4 He2+ — e system[36]. While the isotope effect on the energy levels within the Schrodinger model shows, as can be expected, a scaling which goes as the square root of the respective reduced masses. However, we note that the relativistic and QED corrections scale in a different fashion. This latter scaling is not yet obvious from the formal theory [36]. 

Suppose we compare the values of angular momentum in the Bohr and the Schrodinger models ... 

The peak of the radial probability distribution for the ground-state H atom appears at the same distance from the nucleus (0.529A, or 5.29x10 " m) as Bohr postulated for the closest orbit. Thus, at least for the ground state, the Schrodinger model predicts that the electron spends most of its time at the same distance that the Bohr model predicted it spent all of its time. The difference between most and all reflects the uncertainty of the electron s location in the Schrodinger model. 

Atomic structure, (a) In 1895, P. Lenard detected electrons outside a thin glass tube in which they were produced, and concluded that the atoms in the glass must have a very open structure. Explain these results in terms of the Rutherford atom, (b) Describe the fundamental difiEierences among the Dalton, Thomson, Rudierfbrd, and Schrodinger models of the atom. 

Two other models for the atom are important. One was proposed by a Danish scientist by the name of Neils Bohr in 1913. This model was called the solar system model because he proposed that the electrons orbit the nucleus much like the planets orbit the sun. The other was proposed by Austrian physicist Erwin Schrodinger in 1926. The Schrodinger model is called the quantum mechanical model and is the model that we use today to explain and predict atomic behavior. Each of these two models is explained more fully in the sections that follow. A summary of the history of the development of atomic theory is given in Table 4.1... 

The most famous experiment related to spin was the Stem-Gerlach experiment [13] in 1922. In that experiment, silver was vaporized in an oven and allowed to exit as a beam, which traveled between a long pair of magnet poles each machined to a knife edge. From chemistry, we know Ag has one outer electron and the Schrodinger model would describe the electron orbital occupancy as (li )(2i 2p ) 3s 3p 3d ) As 4d °)(5 ). As the Ag atoms traveled through the long path, the... 

One of the first models to describe electronic states in a periodic potential was the Kronig-Penney model [1]. This model is commonly used to illustrate the fundamental features of Bloch s theorem and solutions of the Schrodinger... 

This model considers the solution of wavefiinctions for a one-dimensional Schrodinger equation ... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master. With these thoughts in mind, the authors have organized this introductory section in a manner that first provides the student with a brief introduction to the two primary constructs of quantum mechanics, operators and wavefunctions that obey a Schrodinger equation, then demonstrates the application of these constructs to several chemically relevant model problems, and finally returns to examine in more detail the conceptual structure of quantum mechanics. 

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. 

Before moving deeper into understanding what quantum mechanics means, it is useful to learn how the wavefunctions E are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems. Knowing the solutions to these easy yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics because these model cases can be used as concrete examples. ... 

Having gained experience on the application of the Schrodinger equation to several of the more important model problems of chemistry, it is time to return to the issue of how the wavefunctions, operators, and energies relate to experimental reality. 

Macro Model Technical Manual Schrodinger, Portland OR (1999). 

HyperChem models the vibrations of a molecule as a set of N point masses (the nuclei of the atoms) with each vibrating about its equilibrium (optimized) position. The equilibrium positions are determined by solving the electronic Schrodinger equation. 

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. 

A theoretical model should be uniquely defined for any given configuration of nuclei and electrons. This means that specifying a molecular structure is all that is required to produce an approximate solution to the Schrodinger equation no other parameters are needed to specify the problem or its solution. 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs " " in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

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