Hydrogen atom Schrodinger equation

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

The Laguerre basis suits the symmetry of the problem, because its collocation points (think of nodes of the Laguerre functions) lie between 0 and oo, and the collocation grid spacing is nonuniform. Of course, we have prior intuition that this basis is better suited because the analytic solutions of the hydrogen atom Schrodinger equation are Laguerre polynomials. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

The Hydrogenic atom problem forms the basis of much of our thinking about atomic structure. To solve the corresponding Schrodinger equation requires separation of the r, 0, and (j) variables... 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

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. 

As I mentioned above, it is conventional in many engineering applications to seek to rewrite basic equations in dimensionless form. This also applies in quantum-mechanical applications. For example, consider the time-independent electronic Schrodinger equation for a hydrogen atom... 

So, let s get a bit more chemical and imagine the formation of an H2 molecule from two separated hydrogen atoms, Ha and Hb, initially an infinite distance apart. Electron 1 is associated with nucleus A, electron 2 with nucleus B, and the terms in the electronic Hamiltonian / ab, ba2 and are all negligible when the nuclei are at infinite separation. Thus the electronic Schrodinger equation becomes... 

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... 

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. 

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

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. 

To find the wavefunctions and energy levels of an electron in a hydrogen atom, we must solve the appropriate Schrodinger equation. To set up this equation, which resembles the equation in Eq. 9 but allows for motion in three dimensions, we use the expression for the potential energy of an electron of charge — e at a... 

Solving the Schrodinger equation for a particle with this potential energy is difficult, but Schrodinger himself achieved it in 1927. He found that the allowed energy levels for an electron in a hydrogen atom are... 

Unfortunately, the Schrodinger equation can be solved exactly only for one-electron systems such as the hydrogen atom. If it could be solved exactly for... 

In the case of hydrogen-1 ike atoms the Schrodinger equation can be written as (in atomic units) ... 

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. ... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonie oseillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

This, at first glance innocuous-looking functional FHK[p] is the holy grail of density functional theory. If it were known exactly we would have solved the Schrodinger equation, not approximately, but exactly. And, since it is a universal functional completely independent of the system at hand, it applies equally well to the hydrogen atom as to gigantic molecules such as, say, DNA FHK[p] contains the functional for the kinetic energy T[p] and that for the electron-electron interaction, Eee[p], The explicit form of both these functionals lies unfortunately completely in the dark. However, from the latter we can extract at least the classical Coulomb part J[p], since that is already well known (recall Section 2.3),... 

Other atoms with only a single electron (He+, Li2+, etc.) are known as hydrogen-like atoms. The Schrodinger equation for such a system is the same as that for the hydrogen atom... 

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