The hydrogen-like atom

The Schrodinger equation (6.12) for the relative motion of a two-particle system is applicable to the hydrogen-like atom, which consists of a nucleus of charge - -Ze and an electron of charge —e. The differential equation applies to H for Z = 1, He+ for Z = 2, Li + for Z = 3, and so forth. The potential energy V(r) of the interaction between the nucleus and the electron is a function of their separation distance r = r = (x + y + and is given by Coulomb s law (equation (5.76)), which in SI units is 

In this system, centimeter is the unit of length, erg is the unit of energy, and statcoulomb (also called the electrostatic unit or esu) is the unit of charge. In this book we accommodate both systems of units and write Coulomb s law in the form 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield 

If this expression is compared with equation (5.32), we see that 

Equation (6.12) cannot be solved analytically when expressed in the 

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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. 

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. 

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... 

The molal diamagnetic susceptibilities of rare gas atoms and a number of monatomic ions obtained by the use of equation (34) are given in Table IV. The values for the hydrogen-like atoms and ions are accurate, since here the screening constant is zero. It was found necessary to take into consideration in all cases except the neon (and helium) structure not only the outermost electron shell but also the next inner shell, whose contribution is for argon 5 per cent., for krypton 12 per cent., and for xenon 20 per cent, of the total. 

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. ... 

This assumption is the basis of the Bohr model for the hydrogen-like atom. When solved for m, this balancing equation is... 

Solving equation (6.21) for the energy E and replacing A by n, we obtain the quantized energy levels for the hydrogen-like atom... 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

Table 6.1. Radial functions R i for the hydrogen-like atom for n = to 6. The variable p is given by p = 2 Zrj na ...

There are also solutions to the radial differential equation (6.17) for positive values of the energy E, which correspond to the ionization of the hydrogen-like atom. In the limit r oo, equations (6.17) and (6.18) for positive E become... 

The energy levels of the hydrogen-like atom depend only on the principal quantum number n and are given by equation (6.48), with replaeed by ao, as... 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

Table 6.2. Real wave functions for the hydrogen-like atom. The parameter a j,...

The probability of finding the electron in the hydrogen-like atom, with the distance r from the nucleus between r and r + dr, with angle 6 between 6 and 6 + dO, and with the angle cp between tp and tp + dtp is... 

Figure 6.4 The radial functions for the hydrogen-like atom.

The expectation values of powers and inverse powers of r for any arbitrary state of the hydrogen-like atom are defined by... 

The theoretical results for the hydrogen-like atom may be related to experimentally measured spectra. Observed spectral lines arise from transitions of the atom from one electronic energy level to another. The frequency v of any given spectral line is given by the Planck relation... 

The hydrogen-like atomic energy levels are given in equation (6.48). If n and 2 are the principal quantum numbers of the energy levels E and E2, respectively, then the wave number of the spectral line is... 

If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator Hb for the hydrogen-like atom in a magnetic field B becomes... 

Consider a crude approximation to the ground state of the lithium atom in which the electron-electron repulsions are neglected. Construct the ground-state wave function in terms of the hydrogen-like atomic orbitals. 

The energy E of the hydrogen-like atom is related to X by equation (6.21). If we solve this equation for E and set X equal to n, we obtain... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

The Stark effect. The conditions l)v, 2)v just stated are rather complicated and may appear very restrictive at first sight. But in reality this is not so, although their rigorous verification is often rather tedious. We shall illustrate what these conditions mean in the case of the Stark effect of the hydrogen-like atom (Ex. 7, 12). Here we have... 

As the conclusion, we may safely say that the operator of the Stark effect of the hydrogen-like atom possesses just the correct number of pseudo-eigenvalues and no... 

The simple Bohr model of the hydrogen-like atom (one electron only) predicts that the X-ray energy or the transition energy, AE, is given as... 

Simple analytic formulas for a for the decoupled 2s-2p channels are derivable for any value of L and for any combination of the masses of the point charge, m0, and of the two particles, nit and m2, constituting the hydrogen-like atom [76] ... 

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