and therefore have the same analytical form as for the hydrogenic one-electron atom.
Allowable solutions again only exist for
where Zeff = Z - nl.
Zeff is the effective nuclear charge and nl is the shielding constant. This gives rise to the shell model for multi-electron atoms.
Atoms with two valence electrons
Includes He and Group II elements (e.g., Be, Mg, Ca, etc.). Valence electrons are indistinguishable, i.e., not physically possible to assign unique positions simultaneously.
This means that multi-electron wave functions must have exchange symmetry:
which will be satisfied if
That is, exchanging labels of pair of electrons has no effect on wave function.
The “+” sign applies if the particles are bosons. These are said to be symmetric with respect to particle exchange. The “-” sign applies to fermions, which are anti-symmetric with respect to particle exchange.
As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be anti-symmetric with respect to particle exchange.
Helium wave functions
He atom consists of a nucleus with Z = 2 and two electrons.
Must now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin wave functions:
As electrons are indistinguishable => must be anti-symmetric. See table for allowed symmetries of spatial and spin wave functions.
Helium wave functions: spatial
State of atom is specified by configuration of two electrons. In ground state, both electrons are is 1s shell, so we have a 1s2 configuration.
In excited state, one or both electrons will be in higher shell (e.g., 1s12s1). Configuration must therefore be written in terms of particle #1 in a state defined by four quantum numbers(called ). State of particle #2 called .
Total wave function for a excited atom can therefore be written:
But, this does not take into account that electrons are indistinguishable. The following is therefore equally valid:
Because both these are solutions of Schrödinger equation, linear combination also a solutions:
where is a normalisation factor.
Helium wave functions: spin
There are two electrons => S = s1+ s2 = 0 or 1. S = 0 states are called singlets because they only have one ms value. S = 1 states are called triplets as ms = +1, 0, -1.
There are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry.
Only one possible anitsymmetric spin eigenfunction:
There are three possible symmetric spin eigenfunctions:
Helium wave functions: spin
Table gives spin wave functions for a two-electron system. The arrows indicate whether the spin of the individual electrons is up or down (i.e. +1/2 or -1/2).
The + sign in the symmetry column applies if the wave function is symmetric with respect to particle exchange, while the - sign indicates that the wave function is anti-symmetric.
The Sz value is indicated by the quantum number for ms, which is obtained by adding the ms values of the two electrons together.
Helium wave functions
Singlet and triplet states therefore have different spatial wave functions.
Surprising as spin and spatial wavefunctions are basically independent of each other.
This has a strong effect on the energies of the allowed states.
Singles and triplet states
Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin angular momentum (S), where
is the sum of the spin angular momenta of the two electrons.
The magnutude of the total spin and its z-component are quantised:
where ms = -s, … +s and s = 0, 1.
If s1 = +1/2 and s2 = -1/2 => s = 0.
Therefore ms = 0 (singlet state)
If s1 = +1/2 and s2 = +1/2 => s = 1.
Therefore ms = -1, 0, +1 (triplet states)
Angular momenta of electrons are described by l1, l2, s1, s2.
As Z<30 for He, use LS or Russel Saunders coupling.
Consider ground state configuration of He: 1s2
Orbital angular momentum: l1=l2 = 0 => L = l1 + l2 = 0
Gives rise to an S term.
Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1
Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet)
J = L + S, …, |L-S| => J = 1, 0.
Therefore there are two states: 11S0 and 13S1 (also using n = 1)
But are they both allowed quantum mechanically?
Must consider Pauli Exclusion principle: “In a multi-electron atom, there can never be more that one electron in the same quantum state”;or equivalently, “No two electrons can have the same set of quantum numbers”.
Exchange splitting is part of gross structure of He - not a small effect. The value of 2J is ~0.8 eV.
Exchange energy is sometimes written in the form
which shows explicitly that the change of energy is related to the relative alignment of the electron spins. If aligned = > energy goes up.
Orthohelium states are lower in energy than the parahelium states. Explanation for this is:
Parallel spins make the spin part of the wavefunction symmetric.
Total wavefunction for electrons must be antisymmetric since electrons are fermions.
This forces space part of wavefunction to be antisymmetric.
Antisymmetric space wavefunction implies a larger average distance between electrons than a symmetric function. Results as square of antisymmetric function must go to zero at the origin => probability for small separations of the two electrons is smaller than for a symmetric space wavefunction.
If electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy.