One-Electron Atom Quantum Numbers

The theory of one-electron atom seems to be the simplest one but it is also the most important one. 

An one-electron atom contains two particles, that is, a positively charged nucleus and the negatively charged electron. The two particles are bound together by the Coulomb attraction force. In fact, both particles move around a common center of masses. The ratio of the proton mass to the electron mass mp/mg is 1835.55. Thus, the massive nucleus may by considered as almost completely stationary. 

The time-independent Schrodinger equation for one-electron atom is expressed as 

As before, we require that an eigenftmction xp x,y,z) and its derivatives xl) x, y, z) must be finite, single valued and continuous. Such functions are called well-behaved.  

There are three unknown independent variables in (3.2), namely x, y, z. One should separate the variables in order to split the partial differential equation into a set of three ordinary differential equations, each involving only one coordinate. However, the separation of variables cannot be carried out when rectangular coordinates are employed because the Coulomb potential energy (3.1) could not be represented as a product of functions each depending on one variable only. 

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Within the hydrogen atom, the lower the value or n, the more stable will be the orbital. For the hydrogen atom, the energy depends only upon n for atoms with more than one electron the quantum number / is important as wel1. 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

Multiple line core spectra are produced also if the atom has an open valence shell, provided that the crystalline environment has not wiped out the J, L, S, M quantization of that shell the core vacancy is variously coupled to the open shell to yield a set of final states. For example, if the open shell has the one-electron orbital quantum numbers n and l and total spin S, a core s vacancy will be observed in two final states having spins (S +1/2) and (S — 1/2), with the latter spin state lying higher in energy. According to Condon-Slater-Racah theory, the energy separation is... 

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. 

Schrodinger s equation required the use of quantum numbers to describe each electron within an atom corresponding to the orbital size, shape, and orientation in space. Later it was found that one needed a quantum number associated with the electron spin. 

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... 

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)... 

For example, for a normal He atom with two electrons having quantum numbers n — 1, /== 0, m — 0, one electron shall be in s = + state and the other in s— — state. On the other hand, for an electronically excited He atom, since now the two electrons reside in two different energy states, i.e. n values differ the two electrons may have the Same spin values. 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

According to this principle, the quantum numbers, n, 1, mp and ms, can never be identical for two electrons in an atom. This means that at least one of the quantum numbers must be different. For example, even if two electrons have identical values for n, 1 and m, (as a result of being in the same orbital), their magnetic spin quantum numbers must be different. That is, these electrons are said to have opposing spins. In fact, we have already mentioned that each electron may be described by a set of the four quantum numbers ... 

The integer n in Eq. (7-3), called the principal quantum number, determines the energy levels in a one-electron atom or ion and largely determines the average distance of the electron from the nucleus. A complete description of the H atom requires two additional quantum numbers ... 

For atoms with more than one electron, we must take account of a fourth quantum number, ms, the electron spin quantum number, which has only two values, ms = 1/2. An electron has a magnetic moment which can be rationalized by imagining that electrons spin about an... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

Spin-orbit coupling arises naturally in Dirac theory, which is a fully relativistic one-particle theory for spin j systems.11 In one-electron atoms, spin s and orbital angular momentum l of the electron are not separately conserved they are coupled and only the resulting total electronic angular momentum j is a good quantum number. 

Before we delve further into the properties of the nucleus, let us momentarily shift our attention back to one of the electrons zooming around the nucleus. Just like photons, electrons exhibit both wave and particle properties. Each electron wave in an atom is characterized by four quantum numbers. The first three of these numbers can be taken as the electron s address and describe the energy, shape, and orientation of the volume the electron occupies in the atom. This volume is called an orbital. The fourth quantum number is the electron spin quantum number s, which can assume only two values, or - f. (Why J was selected rather than, say, 1 will be described a little later.) The Pauli exclusion principle tells us that no two electrons in an atom can have exactly the same set of four quantum numbers. Therefore, if two electrons occupy the same orbital (and thus possess the same first three quantum numbers), they must have different spin quantum numbers. Therefore, no orbital can possess more than two electrons, and then only if their spins are paired (opposite). 

The concept of electron spin was developed by Samuel Goudsmit and George Uhlenbeck in 1925 while they were graduate students at the University of Leyden in the Netherlands. They found that a fourth quantum number (in addition to n, t, and me) was necessary to account for the details of the emission spectra of atoms. The new quantum number adopted to describe this phenomenon, called the electron spin quantum number (ms), can have only one of two values, + and — j. 

In the SAE calculations we solve an equation identical to Eqs. (6) and (10) except that the 1/r term is replaced by the proper radial potential corresponding to the particular /-component of the wave function. This does not alter the structure of the coupled equations, and so the propagation is essentially the same as for a hydrogen atom. Note that if we wish to consider the excitation of one of the p-electrons with quantum number m different from zero, the aj coefficients defined in Eqs. (10-12) become... 

It is clear that what fundamentally distinguishes the study of a polyelec-tronic atom from a monoelectronic one is the Pauli principle and the repulsion between electrons. The Pauli principle is directly related to the spin of the electron, whose quantum number is 1/2 the electron is called a fermion particle. Let us first consider this question in more detail. For such a purpose, we will ignore the electron repulsion, for the moment. This means that each... 

When a one-electron atom or ion undergoes a transition from a state characterized by quantum number W to a state lower in energy with quantum number Wf (m > Wf), light is emitted to carry off the energy hv lost by the atom. By conservation of energy, E, = Ef + hv thus,... 

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