Many-electron Systems
 
 
All atoms - except hydrogen and some ions of light elements - consist of many electrons. It may seem unimportant to know features of one-electron systems which were discussed in the last chapter. However, one-electron system characteristics are very important to understand many-electron system "functioning". 
 
The first difficulty to describe many-electron system is the impossibility to describe each particular electron motion since one must also pay attention to interactions between electrons and electron-nucleus interaction. And therefore the potential energy of the whole atom Ep is as follows:
 
Ep  = all electronsS − Ze²/4peori  + all pairsS e²/4peorij
(V.1)
 
The last sum corresponds to binding between electrons and that's why we can't talk about particular movement of an electron independently of one another. Each change in electron motion should influence on the motion of all other electrons. That's why we can talk about the total energy of atom (or ion) rather than particular energy of an electron. According to this reason we're talking about the whole atom (or ion) wavefunction rather than the wavefunction of an individual electron. 
 
Many-electron problem can't be solved accurately; therefore there are some approximations need to be applied. We will discuss these approximations on the Helium atom example. 
 
Figure 1: Helium-like atom or ion.
 
The two electrons atoms are the simplest ones among many-electron atomic systems. For instance, negative hydrogen ion H− (Z = 1), Helium atom He (Z = 2) and once ionized Li+ ion (Z = 3) and so on. The potential energy of electrons is as follows in this case (figure 1):
 
Ep  =  − Ze²/4peor1 − Ze²/4peor2 + e²/4peor12
(V.2)
 
The first two terms correspond to the attraction between nucleus and each of the electrons and the last term corresponds to the interaction between two electrons. Even for this extremely simple problem it's impossible to find precise solution of Schroedinger equation: it's necessary to do some approximations. Since the mathematical discussion of helium-like atoms exceeds the bounds of this lecture we will be limited by quantitative description. 
 
We can neglect the last term (or the elctron-electron interaction) in (V.2) in the first approximation. It corresponds to the assumption that each electron moves independently from all other electrons. This is so-called "Independent particle model". Then we can describe electron motion by using hydrogen-like wavefunctions ynlml = Rnl(r) Ylml(θ, φ) which are characterized by quantum numbers n, l, ml.  The energy of atom can be obtained from formula E = − RZ²/n² (R: Rydberg constant = 13,6 eV) for the sum of particular electrons. That's why the electronic motion energy would be then for Helium in the ground state n = 1 (Z = 2), EHe = 2 · (−54,4 eV) = − 108,8 eV (two electrons). The experimental value is another one: EHe = − 78,98 eV. And so one can clearly see that our first approximation gives energy lying too low. The reason is our neglect of interaction between both electrons. It will give the increasing of atomic energy. 
 
We can improve our approximation by taking electron interaction into account. Each electron is situated not only in the nuclear central field but also in average central symmetrical field made by other electron. So the total effect of each electron on the motion of another electron consists in the covering of nuclear field. The atomic energy in the ground state is then as follows:
 
E  =  2(Z − δ)² EH
 
where EH = − 13,6 eV (the energy of hydrogen-like motion) and δ is the shielding constant which is equal to 0,32 for ground state of He in order to match energy value with experimental measurements. 
 
We would like to assign electrons as 1 and 2. Since we have assumed electrons move independently of one another the detection probability of electron 1 in one coordinate and electron 2 in other coordinate simultaneously is equal to the product of probability distribution of individual electrons since both events doesn't depend of each other: PAtom = P(1) P(2). We can draw a conclusion here that the wavefunction in the independent particles model should be a product of individual electron wavefunctions (this is a direct consequence of Schroedinger equation for the system of independent particles). If we assign the quantum numbers (n, l, ml) of  electron 1 as a and quantum numbers of electron 2 as b then we will have:
 
yAtom  = ya(1) yb(2)
(V.3)
 
That finally gives the probability distribution:
 
|yAtom|²  =  |ya(1) yb(2)|²  =  |ya(1)|² |yb(2)|²
(V.4)
 
Due to the electron movement the average central field differs from 1/r-Coulomb field which is produced by nucleus. It apparently demands light change of wavefunctions ya(1) and yb(2) that is not similar to hydrogen-like wavefunctions. The change concerns the radial part Rnl rather than angle part Ylml of the wavefunction since force acting upon electrons is central force. Applying corresponding mathematical methods one can optimize the electron wavefunction and obtain the atomic energy states with quite good accuracy. 
 
Even if we optimize the wavefunctions in (V.3) nevertheless this expression couldn't be correct for the atomic wavefunction. Wavefunction (V.3) shows us that electron 1 is in state a and electron 2 is in state b. But the wavefunction 
 
yAtom  = ya(2) yb(1)
(V.5)
 
corresponds to electron 2 in state a and electron 1 in state b and it must describe the same energy state as the wavefunction in (V.3). The fact that expressions (V.3) and (V.5) corresponds to wavefunctions of the same energy is called exchange degeneracy. 
 
Now electrons are identical and undistinguishable and we can only talk concerning atom that one electron is situated in state a and another one is in state b. It demands that wavefunction yAtom and correspondingly |yAtom|² (which corresponds to the probability distribution of both electrons) is symmetric relative to both electrons namely both of them play the same role. Neither wavefunction (V.3) nor (V.5) will comply with this demand. Nevertheless we obtain appropriate wavefunction (which includes the mixture of electrons); expressions (V.3) and (V.5) give rise to new above-mentioned appropriate wavefunction:
 
yAtom  =  ya(1) yb(2)  ±  ya(2) yb(1)
(V.6)
 
In both cases (±) the expression for |yAtom|² is symmetrical relative to both electrons.
 
We will call the wavefunction in expression (V.6) as the orbital wavefunctions in the future since they describe the space or orbital behavior of electrons in atom neglecting spin. The orbital wavefunction having positive sign is as follows:
 
yS(1, 2)  =  ya(1) yb(2)  +  ya(2) yb(1)
(V.7)
 
which is symmetrical for two electrons and stays the same when exchanging by electrons; i.e. yS(1, 2)  =  yS(2, 1). On the other side there is another orbital wavefunction: 
 
yA(1, 2)  =  ya(1) yb(2)  −  ya(2) yb(1)
(4.8)
 
this wavefunction is antisymmetrical for two electrons and changes its sign when exchanging by electrons; i.e. yA(1, 2)  =  − yA(2, 1).
 
This symmetry behavior can be demonstrated by the following important feature: the atomic energy which corresponds to yS doesn't match with the energy corresponding to yA. When the two electrons 1 and 2 are very close to each other then both expressions are almost identical and therefore we have very small or zero yA. That's why the antisymmetric wavefunction yA describes the state in which electrons don't come close to each other and have quite small average repulsion energy. On the other hand, the symmetric wavefunction yS doesn't map out the opportunity that electrons could be very close to each other at definite moments and therefore the average repulsion energy of yS state is higher than that of  yA state. Here we have come to the following fact:
 
Helium-like atoms can be situated in two different states which have different energies and different orbital atomic wavefunctions yS  and yA having the same orbital quantum numbers a and b of both electrons in the independent particle model.
 
To put this another way, atom having two electrons possesses two different stationary states and energy levels. One of these levels are described by using symmetrical orbital wavefunctions and another one - by using antisymmetrical orbital wavefunctions. This purely quantum mechanical effect follows from undistinguished characteristic of an electron.
 
The only exception of the above-mentioned statement is the case when both sets of orbital quantum numbers are identical; i.e. a = b. It means that                       yA = ya(1) ya(2) − ya(2) ya(1) = 0. If both electrons have the same set of orbital quantum numbers then there will be only symmetrical state possible. 
 
 
Figure 2: Spin states of two electron system.
 
 
We have considered only wavefunctions which describe the space distribution of electrons, so far. The complete description of atomic state demands to take electron spin into account. Each electron has spin ½. Spin of one electron can be directed parallel or antiparallel relative to spin of another electron; it gives the total spin (S = 1) or zero        (S = 0) (capital letters are used to assign the quantum numbers of total orbital kinetic moments and small letters are used to mark corresponding values of individual particles).  Spin states with    S = 0 are called Singlets which are made up in a way shown on figure 2 (to the left). If however S = 1 the resulting spin-vector can have three space orientations that are as follows,  MS = +1, 0 and -1 correspondingly (Figure 2, to the right). That's why spin-states having S = 1 and three spin-wavefunctions belong to Triplets.
 
One can prove that the total spin-wavefunction for singlet-states (S = 0) is antisymmetrical wavefunction, while the total spin-wavefunction of triplet-states (S = 1) is symmetrical wavefunction. These spin-wavefunctions are assigned as χ+, c- for particular electrons:
 
χA   =   1/Ö2 [χ+(1) c-(2) − χ+(2) c-(1)] MS  =  0
χS  = ì 
í 
î 		 χ+(1) χ+(2) MS  =  +1
1/Ö2 [χ+(1) c-(2) + χ+(2) c-(1)] MS  =  0
c-(1) c-(2) MS  =  −1
 
The value MS = ms1 + ms2 of the component Sz of the total spin is given for particular wavefunctions. The faktor 1/Ö2 is just for normalization. Resuming one can have:
 
Singlet-state (S = 0): antisymmetric spin-wavefunction χA
Triplet-state (S = 1): symmetric spin-wavefunction χS
 
We obtain the total atomic wavefunction by combining orbital wavefunctions yS or yA and spin-wavefunctions χS or χA; i.e.
 
ytotal  =  (orbital wavefunction) . (spin-wavefunction)
 
The ytotal symmetry depends obviously on both factors symmetry.  Since there are two types of orbital wavefunctions and two types of spin-wavefunctions there are totally four possible combinations. Now the helium energy levels study gives the following: the symmetric orbital wavefunction yS describes singlets (S = 0) and and therefore it corresponds to antisymmetric spin-wavefunctions χA, whereas the antisymmetric orbital wavefunctions yA having triplets states (S = 1) correspond to symmetric spin-wavefunctions χS. It's obviously that only the following states are possible:
 
ytotal  =  (symmetric orbital wavefunction) . (antisymmetric spin-wavefunction)  =  yS χA         Singlets
oder
ytotal  =  (antisymmetric orbital wavefunction) . (symmetric spin-wavefunction)  =  yA χS          Triplets
 
ytotal is antisymmetric wavefunction in any case since it's the product of antisymmetric and symmetric factors.  This statement can be written down in the following more general form for either electron number: 
 
The total wavefunction of electron system should be antisymmetric one.