4. Beyond exact solutions
Unfortunately, exact solutions of the Schrödinger equation are not possible for systems containing more than one electron.
4.1 The Hamiltonian for the helium atom
Earlier we saw that the full Hamiltonian for the H atom, expressed in spherical polar coordinates, is
This may be written for a hydrogenic atom with nuclear charge Z as:
How must this be adapted for the He atom? Consider the
following schematic diagram of the He atom:
The Hamiltonian for this system is:
where is the Laplacian for each electron i.
represents the kinetic energy of the electrons.
is the nucleus-electron attraction and
is the Coulomb repulsion between the two electrons.
In order to proceed we make the assumption that the repulsion between the two electrons can be mathematically ignored and somehow absorbed into an effective potential (i.e. we ignore the term).
We are thus assuming that the electrons move independently of one another. We can therefore assign each electron its own hydrogenic wavefunction Ψ1s (1) and Ψ 1s (2) and the wavefunction for He can be written:
Ψ (1,2) =Ψ1s (1)Ψ1s (2)
This is called the one-electron or orbital approximation.
4.2 Electron spin and the Pauli exclusion principle
An electron has an intrinsic property which could be visualised as it spinning on its own axis. In our discussion of the particle on a ring, we saw that the z component of the orbital angular momentum is restricted to be Lz =±ml
By analogy, the component of the spin angular momentum in the z direction is confined to be sz = ±ms.
The spin quantum number ms can take one of only two values: ±1/2; we usually call these spin-up and spin-down or α and β .
The Pauli exclusion principle may be stated:
No two electrons in an atom may have the same set of the four quantum numbers n, l, ml and ms.
There is, however, a more fundamental form. of this principle. Consider the effect on Ψ(1, 2) of interchanging the two electrons, i.e.: Ψ(1,2) →Ψ (2,1) . Electrons are indistinguishable and hence this process cannot affect the physical properties of the system. The probability distribution Ψ *Ψ must remain unchanged. For this to be true: either Ψ(1,2) =Ψ (2,1) or Ψ(1,2) = -Ψ(2,1)
It turns out that the second condition is the correct restriction for electrons, and the more fundamental form of the Pauli exclusion principle is: the total wavefunction of a system must change sign when any two electrons are interchanged. We say that the total wavefunction must be antisymmetric.
4.3 The ground state of the helium atom
Let us write α for a spin-up electron and β for a spin-down electron. Then the spin possibilities for the two electrons 1 and 2 in the He atom are:
α(1)α(2) β(1)β(2) α(1)β(2) α(2)β(1)
The first two cases (i.e. α(1)α(2) and β(1)β(2)) are OK, but there is a problem with α(1)β(2) and α(2)β(1) . Electrons are indistinguishable, so we cannot say for sure that electron 1 has α spin and electron 2 has β spin, or vice versa. When electrons have opposite spins, there must always be equal probabilities of α(1)β(2) and
α(2)β(1)which we can ensure by taking linear combinations of these spin arrangements:
α(1)β(2)+α(2)β(1) and α(1)β(2)-α(2)β(1) .
We saw earlier that the wavefunction of the He atom could be written Ψ (1, 2) =Ψ1s (1)Ψ1s (2) . We now combine this spatial wavefunction with the spin wavefunctions:
Ψ (1, 2) =Ψ1s(1)Ψ1s(2)α(1)α(2)
Ψ (1, 2) =Ψ1s(1)Ψ1s(2)β(1)β(2)
Ψ (1, 2) =Ψ1s(1)Ψ1s(2) (α(1)β(2) + α(2)β(1))
Ψ (1, 2) =Ψ1s(1)Ψ1s(2) (α(1)β(2) -α(2)β(1))
Which of these satisfies the Pauli exclusion principle? The spatial wavefunction is symmetric with respect to the interchange of the two electrons, and hence the spin wavefunction must be antisymmetric. Only the fourth spin function is antisymmetric, and hence the ground state wavefunction of He is:
when we include normalisation.
If we have two electrons in the 1s orbital of He, their n, l and ml values are all the same: 1, 0, 0. Thus ms must be different for the two electrons, as is the case for Ψ(1, 2) .
Ψ (1, 2)may also be written in determinantal form.
This is known as a Slater determinant. Each term in the determinant has a hydrogenic spatial orbital multiplied by a spin function, and is known as a spin-orbital.
4.4 Excited states of the helium atom
The ground state of He has two electrons in the 1s orbital, i.e. the electronic configuration 1s2. Excited states of He can be generated by promoting an electron from the 1s orbital to the 2s to yield the configuration 1s12s1. What states does this configuration give rise to?
We saw earlier that the spin wavefunctions α(1)β(2) and α(2)β(1)are not valid, due to the indistinguishability of electron 1 and electron 2. The same argument applies to the excited spatial wavefunction:
Ψspatial (1s, 2s ) =Ψ1s (1)Ψ2s (2)
Satisfactory spatial wavefunctions (including normalisation) may be obtained through linear combinations:
The first of these is symmetric with respect to the interchange of electron 1 and electron 2, and hence must be combined with the antisymmetric spin wavefunction to satisfy the Pauli principle:
This state is a singlet state (there is only one contributing wavefunction).
The second spatial wavefunction is antisymmetric, and hence can be combined with any of the three symmetric spin wavefunctions to satisfy the Pauli principle:
This state is a triplet state (there are three contributing wavefunctions).
The Moodle document “The first excited state of the Helium atom, and the energetic effect of electron exchange” shows that the energy of the singlet state is E singlet = E1s + E2s + J + K and the triplet state is E triplet = E1s + E2s + J - K .
Hence there is a difference in energy between the singlet and triplet states of 2K, where the exchange integral Note how this differs from the Coulomb intergral
The exchange integral K > 0, and hence the triplet state is the more stable (c.f. Hund’s rule of maximum multiplicity).