One-electron systems

The reduced mass, f-i in this system is determined from the mass of an electron, m, and the atomic mass of the nucleus, mN. However, the mass of a proton and neutron is much more massive than an electron (1836.5 and 1838.7 times more massive respectively) hence, the reduced mass for the system can be taken as the mass of an electron. 

Since the radial and angular components are separable, the wavefunction will be a product of the angular function and a radial function, The 

To no surprise, substitution of Equation 8-3 into Equation 8-1 along with operation of on Y/ (6, ( )) and subsequent cancellation of Y (6, ( )) results in the same two-body radial Schroedinger equation as previously obtained for the vibration/rotation of diatomic molecules with a general expression for the potential V(r) (see Equation 6-10). 

The potential V(r) along the radial coordinate is Coulombic. The charge of the nucleus is +Ze where Z is the atomic number and e is the elementary charge (e = 1.602177 x 10 C). The charge of the electron is equal to -e. 

The term Co is the vacuum permittivity constant which in SI units is equal to 8.85419 X 10 T m . Substitution of Equation 8-5 into Equation 84 results in the following radial Schroedinger equation for a one-electron system. 

In hydrogen and hydrogen-like systems a single electron moves around the nucleus of charge Ze at a distance r in a central field, and its potential energy is given by 

If relativistic and quantum electrodynamic (QED) effects are considered, the far-reaching degeneracy for hydrogen-like systems is lifted. In Fig. 2.1 the simple energy-level diagram of hydrogen is shown. 

However, the i value specifies the wavefunctions and the states are characterized according to 

Besides hydrogen, hydrogen-like ions [2.23] and positronium (positron + electron) and muonium (proton H- muon) [2.24] have been much studied since they constitute good testing grounds for advanced theories. 

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In the case of 1,3-butadiene, RAMSES combines the two double bonds to form a single, delocalized r-electron system containing four electrons over all four atoms (Figure 2-50a). The same concept is applied to benzene. As shown in Figure 2-50b, the three double bonds of the Kekule representation form one electron system with six atoms and six electrons. 

Quantum mechanics (QM) is the correct mathematical description of the behavior of electrons and thus of chemistry. In theory, QM can predict any property of an individual atom or molecule exactly. In practice, the QM equations have only been solved exactly for one electron systems. A myriad collection of methods has been developed for approximating the solution for multiple electron systems. These approximations can be very useful, but this requires an amount of sophistication on the part of the researcher to know when each approximation is valid and how accurate the results are likely to be. A significant portion of this book addresses these questions. 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

Having stated the limitations (non-relativistic Hamilton operator and the Bom-Oppenheimer approximation), we are ready to consider the electronic Schrodinger equation. It can only be solved exactly for the Hj molecule, and similar one-electron systems. In the general case we have to rely on approximate (numerical) methods. By neglecting relativistic effects, we also have to introduce electron spin as an ad hoc quantum effect. Each electron has a spin quantum number of 1 /2. In the presence of an... 

It should be noted that several of the proposed functionals violate fundamental restrictions, such as predicting correlation energies for one-electron systems (for example P86 and PW91) or failing to have the exchange energy cancel the Coulomb self-repulsion (Section 3.3, eq. (3.32)). One of the more recent functionals which does not have these problems is due to Becke (B95), which has the form... 

Let us develop farther the simple example of a one-electron system... 

Wave mechanics is based on the fundamental principle that electrons behave as waves (e.g., they can be diffracted) and that consequently a wave equation can be written for them, in the same sense that light waves, soimd waves, and so on, can be described by wave equations. The equation that serves as a mathematical model for electrons is known as the Schrodinger equation, which for a one-electron system is... 

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... 

There is one more problem which is typical for approximate exchange-correlation functionals. Consider the simple case of a one electron system, such as the hydrogen atom. Clearly, the energy will only depend on the kinetic energy and the external potential due to the nucleus. With only one single electron there is absolutely no electron-electron interaction in such a system. This sounds so trivial that the reader might ask what the point is. But... 

This term does not exactly vanish for a one electron system since it contains the spurious interaction of the density with itself. Hence, for equation (6-32) to be correct, we must demand that J[p] exactly equals minus Exc[p] such that the wrong self-interaction is cancelled... 

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

The angular functions presented in Table 1 are derived from the wavefunc-tions for one-electron systems, e.g. the hydrogen atom. However, they can be... 

The occupancy of 0i is obviously the maximum possible in this one-electron system, so 0i is indeed a natural orbital. In terms of natural orbitals, the density operator takes the form... 

The preceding theorem falls well short of the Hohenberg-Kohn theorem because it is restricted to Coulombic external potentials. The theorem is not true for all external potentials. In fact, for any Coulombic system, there always exists a one-electron system, with external potential,... 

In view of the range of conclusions which will be derived from the approximate identification of EV2 with E0 it is appropriate to clarify what is involved. We mean here that EV2 = E6 + x the term x is omitted in the subsequent treatment because, as we will show below, it is both small and approximately constant under the relevant conditions. From the theory of polarography it follows that for reversible, one-electron systems... 

Cizek, J., and Paldus, J. (1977), An Algebraic Approach to Bound States of Simple One-Electron Systems, Inti. J. Quant. Chem. XII, 875. 

There was a problem with Bohr s model, however. It successfully explained only one-electron systems. That is, it worked fine for hydrogen and for ions with only one electron, such as He, Li, and Be. Bohr s model was unable, however, to explain the emission spectra produced by atoms with two or more electrons. Either Bohr s model was a coincidence, or it was an oversimplification in need of modification. Further investigation was in order. 

Finally, it should be noticed that the energy of the one-electron system FiJ increases less rapidly than the system of the states of the two-electron FI2. In consequence, for the larger values of w, the energy of the ionized system FI2 will be lower than the energy of some of the excited electronic states of FI2. 

Gauge-independent atomic orbitals. GlAOs are eigenfunctions of a one-electron system that have been perturbed by an external magnetic field. 

Although the r- and p-space representations of wavefunctions and density matrices are related by Fourier transformation, Eqs. (5.19) and (5.20) show that the densities are not so related. This is easily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The operations of Fourier transformation and taking the absolute value squared do not commute, and so the p-space density is not the Fourier transform of its r-space counterpart. In this section, we examine exactly what the Fourier transforms of these densities are. 

These considerations suggest that it might be useful to approximate molecular orbitals as sums of atomic orbitals. Thus for the hydrogen molecule ion hJ, a one-electron system, one could consider a wave function - based on the exact orbitals for the separated atoms - of the form, in... 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

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