Hydrogen-like

The ionization energy for hydrogen (or other hydrogen-like systems) can be found from the Rydberg equation... 

Millikan has shown that the overlap integral for hydrogen-like p orbitals in linear hydrocarbons is about 0.27 (Millikan, 1949). 

This equation is exaetly the same as the equation seen above for the radial motion of the eleetron in the hydrogen-like atoms exeept that the redueed mass p replaees the eleetron mass m and the potential V(r) is not the eoulomb potential. 

Plot the radial portions of the 4s, 4p, 4d, and 4f hydrogen like atomie wavefunetions. 

The hydrogen atom, consisting of a proton and only one electron, occupies a very important position in the development of quantum mechanics because the Schrddinger equation may be solved exactly for this system. This is true also for the hydrogen-like atomic ions He, Li, Be, etc., and simple one-electron molecular ions such as Hj. 

Table 1.1 Some wave functions for hydrogen and hydrogen-like atoms...

For the hydrogen atom, and for the hydrogen-like ions such as He, Li, ..., with a single electron in the field of a nucleus with charge +Ze, the hamiltonian (the quantum mechanical form of the energy) is given by... 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

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... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

Hydrodealkylation. Hydrodealkylation is a cracking reaction of an aromatic side chain in presence of hydrogen. Like hydrocracking, the... 

Different Types of Complete Sets. Importance of the Continuum in Using Hydrogen-like Orbitals... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

The discrete hydrogen-like orbitals (nlm) are given by the formula ... 

Comparing the two sets III.57 and III.55 for the same value of the effective charge rj, we find Is = Is, whereas all the other functions (nl) may be expanded in the functions (nl) and the associated continuum. In studying the function (2s), it has, e.g., been observed (Shull and Lowdin 1955,1958) that the discrete functions (nl) give a contribution of only 56.501 per cent, which implies that the remaining 43.499 per cent must come from the hydrogen-like continuum. The importance of the continuum part can hence hardly be enough emphasized. 

Unfortunately, the omission of the hydrogen-like continuum has in the literature led to several misleading conclusions and wrong results, and this question is still being discussed in connection with the correlation problem (Taylor and Parr 1952, Gerhauser and Mat sen 1955). 

The first excited orbitals are characterized by being localized mainly within the same region of space as Xv The function xni has (n—l 1) nodes but is otherwise not particularly hydrogen-like it may be expanded in the standard hydrogen-like functions only if a considerable contribution from the continuum is included. 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as... 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

A treatment of the hydrogen molecule by the Ritz method, applied to helium by Kellner (25), has been reported by S. C. Wang (Phys. Rev., 31, 579 (1928)). With this method the individual eigenfunctions p and

atom with atomic number Z differing from unity. The value found for Z is 1.166, and the... 

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