Schrodinger equation for the hydrogen atom

Schrodinger s equation for a single electron and a nucleus with Z protons is an extension into three dimensions (x,y,z) of Equation 6.8, with the potential replaced by the Coulomb potential U(r) = Ze2/Ajt8or. This problem is exactly solvable, but it requires multivariate calculus and some very subtle mathematical manipulations which are beyond the scope of this book. 

We noted in Chapter 5 that only one component of the electron s angular momentum (typically chosen as the z-component, Sz) can be specified. A similar result holds for 

L only the single component Lz can be specified. There are (2/ + 1) levels of the azimuthal quantum number mi for each value of /. 

Note that even when mi has its maximum value (mi = /), L2Z L. Thus we never know the exact direction of the angular momentum vector. An atom does not really have any preferred direction in space, so all of the different mi levels have exactly the same energy such levels are called degenerate. In fact, for a hydrogen atom, states with different values of /, mi, and ms (but the same value of n) are degenerate as well. 

The orbital angular momentum quantum number / has other strange characteristics. Notice that / = 0 is allowed. However, for a classical orbit, circular or elliptical, we have L = mv R (Equation 5.19), which cannot be zero. The vector L points in the 

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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. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Suppose we get a little more sophisticated about our question. The more advanced student might respond that the periodic table can be explained in terms of the relationship between the quantum numbers which themselves emerge from the solutions to the Schrodinger equation for the hydrogen atom.5... 

As many textbooks correctly report, the number of electrons that can be accommodated into any electron shell coincides with the range of values for the three quantum numbers that characterize the solutions to the Schrodinger equation for the hydrogen atom and the fourth quantum number as first postulated by Pauli. 

There is another important reason for the existence of atomic units, namely, that quantum mechanical expressions, such as the Schrodinger equation, become simpler. When expressed in SI units, the Schrodinger equation for the hydrogen atom is... 

Calculations of the electronic structure of molecules, crystals and surfaces are often performed in atomic units. They are defined by setting the most important constants equal to unity h — eo — me — 1, where me is the electronic mass. The Coulomb law is written in electrostatic units V(r) = q/r, so that the time-independent Schrodinger equation for the hydrogen atom takes on the simple form ... 

To give the reader some further appreciation of the adiabatic correction, we next discuss the so-called Rydberg correction of the hydrogenic atom. The kinetic energy of the electron in the hydrogen atom is expressed by Equation 2.5, where mj is set equal to the electron mass, me. It is well known that the Schrodinger equation for the hydrogen atom separates into two equations, one of which deals with the motion of the center of mass of the system and the other with the motion of the electron... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Fock was then able to show that the momentum-space Schrodinger equation for the hydrogen atom has properly normalized solutions of the form... 

Slater. A function of the form x y" z" exp (-i r) where 1, m, n are integers (0,1, 2. . . ) and is a constant. Related to the exact solutions to the Schrodinger Equation for the hydrogen atom. Used as Basis Functions in Semi-Empirical Models. 

The explanation of the periodic system by quantum mechanics, for example, is only partial. The possible lengths of the various periods in the table follow deductively from the solution of the SchrOdinger equation for the hydrogen atom and the relationship between the four quantum numbers, which is also obtained deductively. However, the repetition of all but the first period length remains a source of debate (/). The repetition of all the other period lengths has not been deduced from first principles however (2). Stated more precisely, the empirical order in which the atomic orbitals are filled has not been deduced. If this were possible the explanation for the lengths of successive periods, including the repetitions, would follow trivially. 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

The electron distribution around an atom can be represented in several ways. Hydrogenlike functions based on solutions of the Schrodinger equation for the hydrogen atom, polynomial functions with adjustable parameters, Slater functions (Eq. 5.95), and Gaussian functions (Eq. 5.96) have all been used [34]. Of these, Slater and Gaussian functions are mathematically the simplest, and it is these that are currently used as the basis functions in molecular calculations. Slater functions are used in semiempirical calculations, like the extended Hiickel method (Section 4.4) and other semiempirical methods (Chapter 6). Modem molecular ab initio programs employ Gaussian functions. 

The Schrodinger equation for the hydrogen atom may therefore be written... 

The Schrodinger equation for the hydrogen atom is a second-order partial differential equation in three variables. A customary technique for solving this type of differential equation is by a procedure known as the separation of variables. In that way, a complicated... 

Some authors who use atomic units use the customary symbols for physical quantities to represent the numerical values of quantities in the form (physical quantity)/(atomic unit), so that all quantities appear as pure numbers. Thus, for example, the Schrodinger equation for the hydrogen atom is written in SI in the form... 

Although the detailed solution of the Schrodinger equation for the hydrogen atom is not appropriate in this text, we will illustrate some of the properties of wave mechanics and wave functions by using the wave equation to describe a very simple, hypothetical system commonly called the particle in a box, a situation in which a particle is trapped in a one-dimensional box that has infinitely high sides. It is important to recognize that this situation... 

When we solve the Schrodinger equation for the hydrogen atom, some of the solutions contain complex numbers (that is, they contain i = V-l). Because it is more convenient physically to deal with orbitals that contain only real numbers, the complex orbitals are usually combined (added and subtracted) to remove the complex portions. For example, the px and py orbitals shown in Table 12.1 are combinations of the complex orbitals that correspond to values of ml of +1 and —1. These orbitals are indicated with a brace in Table 12.1. The last four d orbitals listed are also obtained by combination of complex orbitals, as indicated by braces in Table 12.1. 

As we have seen, when we solve the Schrodinger equation for the hydrogen atom, we find many wave functions (orbitals) that satisfy it. Each of these orbitals is characterized by a set of quantum numbers that arise when the boundary conditions are applied. Now we will systematically describe these quantum numbers in terms of the values they can assume and their physical meanings. 

In contrast to the Schrodinger equation for the hydrogen atom, the Schrodinger equation for a polyelectronic atom cannot be solved exactly. For example, although the hydrogen and helium atoms are similar in many respects, the mathematical descriptions of these atoms are fundamentally... 

The angular wave functions Yn(0, (f>) for = 1, m = 1 and Yi i(0, (f>) for = 1, wr = -1 do not have a simple geometrical interpretation. However, their sum and their difference, which are also allowed solutions of the Schrodinger equation for the hydrogen atom, do have simple interpretations. Therefore, we form two new angular wave functions ... 

The difficulties in searching for viable options to address mathematics-related inadequacies increase considerably for the quantum chemistry course. The inadequacy of students familiarity with the mathematics required by that course is a rather common situation for quantum chemistry courses, also in other contexts. The course contents usually make provision for this, by including the development of familiarity with the needed mathematics (operators etc.) into the course. However, the characteristics of the UNIVEN context drastically reduce the viability of such option, because of the gap between students attained familiarity with mathematics, and what would be needed to cope with the mathematics for quantum chemistry. It is therefore opted to maximise the focus on the conceptual aspects and on the description of systems and behaviours, while only few mathematical procedures (e.g. the solution of the Schrodinger equation for the hydrogen atom) are presented, to provide at least some exposure to the ways of proceeding of quantum chemistry. 

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