One-electron atom, Schrodinger equation

Solving the One-Electron Atom Schrodinger Equation 115 TABLE 3.1 The angular wavefunctions ( ) of the one-electron atom. 

Solving the One-Electron Atom Schrodinger Equation 104 BIOSKETCH Peter Beiersdorfer 116 The One-Electron Atom Orbital Wavefunctions 121 Electric Dipole Interactions 135... 

This chapter begins with a description of the quantum picture of the chemical bond for the simplest possible molecule, Hj, which contains only one electron. The Schrodinger equation for Hj can be solved exactly, and we use its solutions to illustrate the general features of molecular orbitals (MOs), the one-electron wave functions that describe the electronic structure of molecules. Recall that we used the atomic orbitals (AOs) of the hydrogen atom to suggest approximate AOs for complex atoms. Similarly, we let the MOs for Hj guide us to develop approximations for the MOs of more complex molecules. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

The modern theory of chemical bonding begins with the article The Atom and the Molecule published by the American chemist G. N. Lewis in 1916 [1], In this article, which is still well worth reading, Lewis for the first time associates a single chemical bond with one pair of electrons held in common by the two atoms "After a brief review of Lewis model we turn to a quantum-mechanical description of the simplest of all molecules, viz. the hydrogen molecule ion H J. Since this molecule contains only one electron, the Schrodinger equation can be solved exactly once the distance between the nuclei has been fixed. We shall not write down these solutions since they require the use of a rather exotic coordinate system. Instead we shall show how approximate wavefunctions can be written as linear combinations of atomic orbitals of the two atoms. Finally we shall discuss so-called molecular orbital calculations on the simplest two-electron atom, viz. the hydrogen molecule. 

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. 

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... 

When the Schrodinger equation for a one-electron atom is solved mathematically, the restrictions on n and I emerge as quantization conditions that correlate with energy and the shape of the wave function. 

In the first approximation, energy levels of one-electron atoms (see Fig. 1.1) are described by the solutions of the Schrodinger equation for an electron in the field of an infinitely heavy Coulomb center with charge Z in terms of the proton charge ... 

The requirement that the basis functions should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei is easily satisfied by choosing hydrogen-like atom wave functions, t], the solutions to the Schrodinger equation for one-electron atoms for which exact solutions are available ... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

PROBLEM 3.1.3. The energy of a one-electron atom (nuclear charge Z e, electron charge — e, reduced mass of the electron-nucleus couple p) is obtained by solving the Schrodinger equation for the one-electron atom ... 

This means that the Schrodinger equation for the one-electron atom or ion can be recast as... 

TABLE 12.1 Solutions of the Schrodinger Wave Equation for a One-Electron Atom n ( m( Orbital Solution... 

A numerical method for solving the time-dependent Schrodinger equation for a one-electron atomic system in an intense, short-pulsed laser field is presented. An effective potential formalism is proposed and tested for representing the excitation of the valence electrons in rare gases. Results for ion production yields, photoelectron distributions and harmonic conversion are presented and compared to recent experimental results. 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

The time-independent electronic Schrodinger equation for a one-electron atom such as H or is... 

Solutions of Schrodinger s wave equation give the allowed energy levels and the corresponding wavefunctions. By analogy with the orbits of electrons in the classical planetary model (see Topic AT), wavefunctions for atoms are known as atomic orbitals. Exact solutions of Schrodinger s equation can be obtained only for one-electron atoms and ions, but the atomic orbitals that result from these solutions provide pictures of the behavior of electrons that can be extended to many-electron atoms and molecules (see Topics A3 and C4-C7). 

The second aim concerns a presentation of the theory of one-electron atoms starting from its relativisitic foundation, the Dirac equation. The nonrelativis-tic Pauli and Schrodinger theories are introduced as approximations of this equation. One of the major purpose, about these approximations, has been to display, on the one side, the enough good concordance between the Dirac and the Pauli-Schrodinger theories for the bound states of the electron furthermore, but to a weaker extent, for the states of the continuum close to the freedom energy and, on the other side, the considerable discordances for... 

Schrodinger constructed his wave equation in non-relativistic form, but he also brought Coulomb s law directly into the mix the fimctional form of the potential energy between the electron and the nucleus, between nuclei, or between electrons, is exactly Coulomb s law for two point particles. There was no guarantee it would work, except for the great success of the one-electron atomic system calculations. The kicker may have entered into the electron electron calculations. 

It should be mentioned, when this constmction is applied to the one-electron atomic case it reduces precisely to Schrodinger s equation for the one-electron atomic case at the limit as s approaches zero, i.e., at the minimum of the energy-e curve. 

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