One-electron atom

In an atom that contains a single electron, the potential energy depends upon the distance between the electron and the nucleus as given by the Coulomb equation. The Hamiltonian thus takes the following form  

The wavefunctions are commonly referred to as orbitals and are characterised by three quantum numbers n, m and 1. The quantum numbers can adopt values as follows  

The angular part of the wavefunction is the product of a function of 6 and a fxmction of (p  

The solutions of the Schrodinger equation are either real or occur in degenerate pairs. These pairs are complex conjugates that can then be combined to give energetically equivalent real solutions. It is only when dealing with certain types of operator that it is necessary to retain a 

Finally, we should note that the solutions are all orthogonal to each other if the product of any pair of orbitals is integrated over all space, the result is zero unless the two orbitals are the same. Orthonormality is achieved by multiplying by an appropriate normalisation constant. 

The study of one-electron atoms in quantum chemistry is still of importance for two reasons at least. Firstly, exact analytic solutions are known in many cases, which, secondly, provide important reference data for other cases, where analytic closed-form solutions do not exist. And even in those cases where no closed-form solutions are available, approximate solutions of almost any desired accuracy can be obtained quite frequently. 

The eigenvalue problems, defined above for the radial functions through Eqs. (113) to (116), reduce to homogeneous problems in the case of one-electron atoms, since the various terms A (r) and j(r) are zero. Closed-form solutions are well known for the one-electron atom with a point-like nucleus, both in the non-relativistic and the relativistic framework, but do not exist for the large majority of finite nucleus models. A determination of the energy eigenvalue for a bound state i of the one-electron atom with a finite nucleus, is then possible in two ways, either by perturbation 

Prom first-order perturbation theory, to which we restrict ourselves here, we obtain for the absolute energy shift 

This distribution function is obtained straightforwardly from the known analytic expressions for the radial functions. Finally, the energy shift from first-order perturbation theory is obtained, for any finite nucleus model, in terms of a series expansion in X = 2ZR, where i is a model-specific radial nuclear size parameter, as 

2- /k — Z/cY (relativistic case). The function fi X) can be evaluated easily and accurately from the series expansion since usually X 0.1 a.u. holds. Explicit expressions for the required coefficients are given in [41, Sect. 3] for several finite nucleus models and for the (non-relativistic and relativistic) ground state of the one-electron atom. The resulting total energy shifts, for those models included in Sect. 4, are shown in Fig. 8 as functions of the nuclear charge number Z. 

The development of quantum chemistry, that is, the solution of the Schrodinger equation for molecules, is almost exclusively founded on the expansion of the molecular electronic wave function as a linear combination of atom-centered functions, or atomic orbitals—the LCAO approximation. These orbitals are usually built up out of some set of basis functions. The properties of the atomic functions at large and small distances from the nucleus determines to a large extent what characteristics the basis functions must have, and for this purpose it is sufficient to exanoine the properties of the hydro-genic solutions to the Schrodinger equation. If we are to do the same for relativistic quantum chemistry, we should first examine the properties of the atomic solutions to determine what kind of basis functions would be appropriate. 

However, the atomic solutions of the Dirac equation provide more than merely a guide to the choice of basis functions. The atoms in a molecule retain their atomic identities to a very large extent, and the modifications caused by the molecular field are quite small for most properties. In order to arrive at a satisfactory description of the relativistic effects in molecules, we must first of all be able to treat these effects at the atomic level. The insight gained into the effects of relativity on atomic structure is therefore a necessary and useful starting point for relativistic quantum chemistry. 

The last relationship follows from the fact that both the direct and exchange processes contribute to the amplitude for 

Scattering from alkali-metal atoms is understood as the three-body problem of two electrons interacting with an inert core. The electron—core potentials are frozen-core Hartree—Fock potentials with core polarisation being represented by a further potential (5.82). 

For electron scattering on lighter alkali-metals, spin asymmetry is due to the Pauli exclusion principle, not to relativistic effects. It tests the relationship between direct and exchange elements of the calculation. Since it is a ratio it is easier to measure accurately than the differential cross section, which varies over many orders of magnitude in the case of sodium. 

The theoretical treatment of asymmetry is rather generally tested for lithium by the energy-dependent measurements at three angles of Baum et al. (1986) for the ground (2s) state and Baum et al. (1989) for the first-excited (2p) state. 

The investigation of sodium as a critical test of the theoretical treatment of scattering is given a new dimension by the spin-dependent measurements of Kelley et al. (1992) in elastic and superelastic scattering experiments with polarised electrons on the polarised 3s and laser-excited 3p states. Not only have asymmetries been measured for these states, but spin-dependent observations of the magnetic substate parameter L have been made for the 3p state. 

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The Sclnodinger equation for a one-electron atom with nuclear charge Z is... 

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. 

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

Since s = j only, j is not a very useful quantum number for one-electron atoms, unless we are concerned with the fine detail of their spectra, but the analogous quantum number J, in polyelectronic atoms, is very important. 

The electronic energy of a free one-electron atom depends only on n ... 

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... 

Briefly, we write the atomie orbitals for a one-electron atom as... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

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. 

Notwithstanding, after hydrogen, helium is also the simplest naturally available atomic species, which, in contrast to one electron atoms, exhibits the additional electron-electron interaction, as a source of electronic correlations. Hence, helium is one of the simplest systems where electronic correlations can be studied. Direct manifestations of electronic correlations have been found, e.g., in doubly excited states of helium localized along highly asymmetric, though very stable, frozen planet configurations (FPC) (K. Richter et.al., 1990), or scarred by... 

For the example of a one-electron atom in the 2p state, Equation 9.9 leads to... 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

The possibility of adsorption on a virtual exciton was indicated by E. L. Nagayev (.14) on the simplest example of the adsorption of a one-electron atom. This problem is an example of the many-electron approach in chemisorption theory. Recently, V. L. Bonch-Bruevich and V. B. Glasko (16) have treated adsorption on metal surfaces by the many-electron method. 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

For an atom with many electrons, the first electron fills the lowest energy orbital, and the second electron fills the next lowest energy orbital, and so forth. For a one-electron atom or ion, the energy depends only on n, the principal quantum number but for a many-electron atom or ion, the value of I also plays a role in the energy. The order of atomic orbital energy is given by... 

Problem 8-15. A normalized hydrogenic Is orbital for a one-electron atom or ion of nuclear charge Z has the form ... 

The hydrogenic atom energy expression has no 1-dependence the 2s and 2p orbitals have exactly the same energy, as do the 3s, 3p, and 3d orbitals. This degree of degeneracy is only present in one-electron atoms and is the result of an additional symmetry (i.e., an additional operator that commutes with the Hamiltonian) that is not present once the atom contains two or more electrons. This additional symmetry is discussed on p. 77 of Atkins. 

In the Koopmans theorem Umit the photoemission of one-electron from an atom or a core in a solid is given by a single Une, positioned at the eigenvalue of the electron in the initial state. The intensity of this line depends on the cross-section for the event, which is determined by the one-electron atomic wavefunctions Wi ( j m)(-Eb) and Pfln(nM, m )(Ekin) (where the atomic quantum numbers are indicated as well as the eigenvalues En,i,m = Eb and E dn of the initial and final state) (the overlap integral of (13)... 

A final point about basis functions concerns the way in which their radial parts are represented mathematically. The AOs, obtained from solutions of the Schrbdin-ger equation for one-electron atoms, fall-off exponentially with distance. Unfoitu-nately, if exponentials are used as basis functions, computing the integrals that are required for obtaining electron repulsion energies between electrons is mathematically very cumbersome. Perhaps the most important software development in wave function based calculations came from the realization by Frank Boys that it would be much easier and faster to compute electron repulsion integrals if Gaussian-type functions, rather than exponential functions, were used to represent AOs. 

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 ... 

We see that due to the smallness of the fine structure constant a a one-electron atom is a loosely bound nonrelativistic system and all relativistic effects may be treated as perturbations. There are three characteristic scales... 

To evaluate both the coupling and the magnitude of the moments, we consider first an array of one-electron atoms in non-degenerate states, and let 0 denote an atomic orbital for an electron on atom (i). Then the electron on atom (i) can overlap on to neighbouring atoms (/), so we may write its wave function in the form... 

Antiferromagnetic insulators of the kind discussed here are sometimes called Mott insulators . It was thought until recently that a crystalline array of one-electron atoms at a sufficient distance apart would necessarily be of this type. In fact, however, many crystals in which each metal atom has a spin and that are... 

The present author (Mott 1949,1956,1961) first proposed that a crystalline array of one-electron atoms at the absolute zero of temperature should show a sharp transition from metallic to non-metallic behaviour as the distance between the atoms was varied. The method used, described in the Introduction, is now only of historical interest. Nearer to present ideas was the prediction (Knox 1963) that when a conduction and valence band in a semiconductor are caused to overlap by a change in composition or specific volume, a discontinuous change in the number of current carriers is to be expected a very small number of free electrons and holes is not possible, because they would form exdtons. 

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 ... 

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