|
The Schrödinger equation for all electrons of an atom or molecule cannot be solved. We can derive, however, some very good approximations using the variation principle. The problem with the Schrödinger equation for more than one electron is the repulsion between the electrons. Without these interactions we could write the solution as a product of functions that each depend on just one electron; i.e.,
| Y(t1,t2,...,tN,) = y1(t1)y2(t2) ... yN(tN). |
|
In this expression tn stand for all coordinates (spatial and spin) of electron n. The functions yn are all functions of the coordinates of just one electron. They are called one-electron wave functions or orbitals .
The product form above can only be a solution of the Schrödinger equation if there is no interaction between the electrons. As the real Hamiltonian does have interactions between the electrons, the product form cannot be a solution. We can, however, assume that it is a good approximation for the solution. What we have to do then is to use the variation principle, and find the orbitals yn that give a lowest value for the expectation value of the Hamiltonian. This is a very lengthy and tricky procedure, which we will skip. The result of this procedure is not so difficult to understand. The best orbitals are solutions of the so-called Fock equation .
Here F is the Fock operator, and e(n) is an energy that is called the orbital energy .
The Fock equation looks a lot like the Schrödinger equation. The latter has solutions that are all-electron wave functions and their corresponding energies. The former has solutions that are one-electron wave functions and their corresponding energies. The Fock operator also looks a lot like the Hamiltonian. It contains three terms, which we can identify as a term corresponding to the kinetic energy of one electron, a term for the interaction of one electron with the nuclei, and and a term for the interaction of one electron with all other electrons. We have already seen the first two terms in the Schrödinger equation. The kinetic energy term is
The term corresponding to the interaction with the nuclei is
| - |
1
4pe0
|
|
å
a
|
|
Za e2
|r-Ra|
|
. |
|
The summation is over all nuclei; Za is the atom number of nucleus a and Ra is its position.
The term corresponding to the interaction with the other electrons is a bit more complicated. It can be written as
| + |
1
4pe0
|
|
ó
õ |
dr¢ |
ern(r¢)
|r-r¢|
|
. |
|
In this expression rn is defined as
| rn(r) = e |
å
m ¹ n
|
|ym(r)|2. |
|
Because the summation explicitly excludes orbital yn, the term is different for each orbital. The quantity rn is called a charge cloud or a charge density. The term in the Fock operator describes the Coulomb interaction of one electron with a cloud of charge that is formed by the other electrons. This cloud depends on the probabilities |ym|2 of finding an electron in ym at some place.
|
|
