Book alonso & finn volume 3.capitulate 4

section 4.1 Introduction

All atoms – except hydrogen and certain ions of the light elements – contain several electrons. Therefore it may seen that the one-electron problem studied in the previous chapter is of little practical value. However, an understanding of the properties of one-electron atoms is very helpful in analyzing many-electron atoms.

The first difficulty with the many-electron atom is the impossibility of describing the motion of each individual electron because, in addition to considering the electron interaction of each electron with the nucleus, we must consider the mutual interactions among the electrons. Thus the potential energy of the whole atom is:equation 4.1

The last summation provides a coupling int the motion of the electrons, and hence we cannot consider each electron as moving independently of the others. Any modification in the motion of one electron must, of necessity, affect the motion of all the other electrons. Therefore we cannot speak of the individual energy of each electron but only of the energy of the whole atom (or ion). For the same reason, we do not speak of a wave function for each electron, but only a wave function for the complete atom.

The problem of many electron cannot be solved exactly; therefore certain approximations are required. We shall illustrate these approximations by discussing the helium atom.

See picture 4.1

4.2 The Helium Atom

Of all the many-electrons atoms, the simplest are those with tow electrons, such as the negative hydrogen ion H- (Z = 1), the helium atom He (Z = 2), the singly ionized lithium atom Li+ (Z = 3), and so on. The potential energy of the electron in this case is: Picture 4-1 and equation 4-2

The first two terms correspond to the attraction between the nucleus and each of the electron and the last term to the repulsion between the two electrons. Even for this relatively simple problem of only two electrons, it is impossible to solve Schrodinger's equation exactly; it is necessary to make several approximations. Since the mathematical discussion of heliumlike atoms is beyond the scope of this book, we shall limit ourselves to a physical description, followed, in example 4.2, by a brief explanation of how the energy is a calculated.

As a first approximation, we may ignore the last term (or the electron-electron interaction term) in eq.(4.2). This is equivalent to assuming that each electron moves independently of the other. Thus we may call this approximation the independent-particle model. Therefore the motion of each electron can be described by hydrogenlike wave functions of the type (x), specified by the orbital quantum numbers n, l, ml. The energy of the electronic motion in the ground state, n=1 (if we set z=2 in eq. 3.5) would be Ehe= 2 x (-54,4eV) = -108,8eV. The experimental value, however, is Ehe= -78,98eV. The reason for this is that we neglected the repulsion of the two electrons, which tends to raise the energy of the atom.We can improve our approximation by considering the perturbation of the electronic motion caused by the mutual interaction of the two electrons. A plausible improvement is to consider that each

electron moves not only in the central field of the nucleus but also in a average central field produced by the other electron. Thus the apparent net effect of each electron on the motion of the other is to screen the charge of the nucleus by a certain amount. The energy of the atom in the ground state may then be written asE = 2(Z - S)2Eh where Eh = -13,6eV (the energy for hydrogenlike motion) and S is the screening constant, which for the ground state of helium must have the value of 0,32 to agree with the observed value of Ehe . That is, the screening effect of each electron on the other is equivalent to about one-third of the electronic charge.

As an aid, let us label the electrons 1 and 2. Since in our first approximation we assumed that the electrons moved independently, the probability of finding electron 1 at a given position and at the same time finding electron 2 at another given position is the product of the probability distribution for each electron, because the two events are uncorrelated; that is, Patom=P(1)P(2). Therefore we conclude that in the independent-particle model the wave function of the atom should be the product of the wave functions for each electron. If we designate the orbital quantum numbers n, l, ml, of electron 1 by a and the quantum numbers of electron 2 by b, we must then write: Eq 4.3resulting in a probability distributionEq. 4.4

Due to the motion of the electrons, the average central field that one electron produces on the other deviates from the 1/r coulomb field produced by the nucleus. This obviously requires a slight modification of the wave functions “qsi”1 and “qsi”2, which are no longer identical to the hydrogenlike wave functions. The charge affects the radial part Rnl but not the part Ylm1 of the wave function because the resultant force on each electron is still a central force. Using appropriate mathematical techniques, we can optimize the electron wave function, and so obtain the energy levels if the atom with relatively good accuracy.However, even if the functions appearing in eq. 4.3 are so optimized, this expression for the wave function of the atom cannot be correct. Wave function 4.3 says that electron 1 is in state a and electron 2 is in state b. But the wave function 4.5 corresponding to electron 2 in state a and electron 1 in state b, must represent a state of the same energy as the wave function of Eq. 4.3 and should describe the state of the atom just as well Eq. 4.3; The fact that the wave functions given by Eqs. 4.3 and 4.5 correspond to the same energy is called exchange degeneracy. Now, electrons are identical and indistinguishable, and the most we can say is that the wave function “qsi”atom be such that |”qsi”|square (whith gives the probability distribution of both electrons) be symmetric with respect to the two electrons, in such a way that both electrons play the same role. Neither the atomic wave function given by eq. 4.3 nor the given by eq. 4.5 meets this requirement. But we may obtain an adequate atomic wave function ( incorporating the fact that the electrons are indistinguishable) by making appropriate linear combinations of eqs. 4.3 and 4.5, which happen to be the functions.Eq 4.6