Hydrogen-atom-like

The wave functions ipnim can be plotted in several ways. For example, the function (nlm) = (100) or Is given by the formula 

We see that what the electron likes most is to sit on the nucleus. Indeed, if we chopped the space into tiny cubes, then computed the value of (Is) in each cube (the function is real, therefore, the modulus is irrelevant), and multiplied 

Since the Hamiltonian commutes with the square of the total angular momentum operator and with the operator of (cf. Chapter 2 and Appendix F on p. 955), then the functions iffnim are also the eigenfunctions of these two operators  

The wave functions (orbitals) with m 0 are difficult to draw, because they are complex. We may, however, plot the real part of nlm (Rc 4 nlm) taking the sum of lAnZm and 1-6. 2Re ipnim and the imaginary part of nlm (Im tjfnlm ) from 

please, that a linear combination of eigenfunctions is not necessarily an eigenfunction. It does, if the functions mixed, correspond to the same eigenvalue. This is why 2px and 2py are the eigenfunctions of the Hamiltonian, of the square of the angular momentum operator, but are not eigenfunctions of 7.  

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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. 

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

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... 

It is customary to express the empirical data concerning term values in the X-ray region by introducing an effective nuclear charge Zeff e in the place of the true nuclear charge Ze in an equation theoretically applicable only to a hydrogen-like atom. Often a screening constant S is used, defined by the equation... 

Stark effect of a hydrogen-like atom, using the Schrodinger wave mechanics. Their equation, obtained independently and by different methods, is... 

In Table IV are given values of the mole refraction of gaseous ions calculated from equations (24) and (29) with the use of the values found above for SE and ASe. Values for hydrogen-like atoms and ions are also included these are, of course, accurate, since no screening constant is needed. Table IV is... 

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. 

According to the discussion in Section II, the quantity TT" represents the electron density about the nucleus in a hydrogen-like atom. The electron... 

Added February 10, 1927.—J. H. Van Vleck in Proc. Nat. Acad. America, vol. 12, p. 662 (December, 1926), has discussed the mole refraction and the diamagnetic susceptibility of hydrogen-like atoms with the use of the wave mechanics, obtaining results identical with our equations (24) and (34). He also considered the effect of the relativity corrections (which is equivalent to the effect of a central field) and concluded that equation (24), derived by the use of parabolic instead of spherical co-ordinates, is not invalidated.]... 

Now many physical properties depend mainly on the behaviour of the electron in the outer part of its orbit. As an example we may mention the mole refraction or polarizability of an atom, which arises from deformation of the orbit in an external field. This deformation is greatest where the ratio of external field strength to atomic field strength is greatest that is, in the outer part of the orbit. Let us consider such a property which for hydrogen-like atoms is found to vary with nrZ t. Then a screening constant for this property would be such that... 

Added in proof Professor J. H. Van Vleck has pointed out to us that the functions were used in the treatment of dispersion by hydrogen-like atoms by B. Podolsky, Proc. Nat. Acad. Sci. 14, 253 (1928). 

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

The main aspect of the eq.(17) is that the orbital energy e occurs only in the coefficients Uk with k > 2. Therefore we obtain here the same results as the one obtained in the case of hydrogen-like atoms ( 1.1 and 1.2) ... 

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... 

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