2006-7 Quantum Theory Slides Lecture 8

Quantum theory and atomic spectroscopy Lecture 8 Many electron atoms

Quantum mechanics is brilliant! • Solve the Schrödinger equation for hydrogen in spherical co-ordinates with a fixed nucleus • Not only do we get the wavefunctions, but we also get the energies of the orbitals too! • So what about the rest of the periodic table….. But it doesn’t really handle SPIN….

Today’s question is…… • Can we solve the Schrödinger equation for many electron atoms, find their wavefunctions and energies and explain the periodic table? Well, no!

The Schrödinger equation for helium • This is a many particle system • If we assume that the nucleus is stationary, we find the Schrödinger equation is 2 2 h h 2 2 Hˆ = − 2 ∇ el1 − 2 ∇ el 2 + V (r1 , r2 , r12 ) 8π me 8π me

2q 2 2q 2 q2 V(r1 , r2 , r12 ) = − − + 4πε 0 r1 4πε 0 r2 4πε 0 r12

Electron correlations • Electrons are not truly independent of one another • This is known as electron correlation and is the result of the repulsive interaction between the electrons • Even this simple 2 electron problem cannot be solved analytically

So introduce APPROXIMATIONS

The orbital approximation • We use the hydrogen wavefunctions as the basis for the manyelectron wavefunction e.g 1s2 • The increased nuclear charge means more “compact” orbitals • But what about the electron correlation?

Ψ (r1 , r2 ,.....rN ) = ϕ(r1 )ϕ(r2 ).....ϕ(rN )

Effective nuclear charge • We introduce simple concepts to deal with the electron-correlation effects and explain relative energies • Shielding effects • Penetration of electron wavefunctions

The two electron atom • Helium is a true two-electron atom, but others can be treated in this way e.g. calcium • In its ground state it has a filled orbital, but in its excited states this is usually no longer true. • What will these wavefunctions be like?

Degenerate wavefunctions • Look at two electrons in just two possible orbitals: 1s and 2s (a two level atom again!) • Remember using quantum mechanics we have four different total energies, where classically we would expect three • Where do we get the extra level?

Superposition Two electrons in two levels leads to four distinct configurations Quantum – four different total energy states (SUPERPOSITION of c2 + c3)

These wavefunctions are indistinguishable! • The simplest two-electron wavefunctions to form would be just products of the orbitals • These two degenerate wavefunctions are for all intents and purposes identical • To see this, just consider ionisation of the helium- which electron has been ionised? • Also, they are not orthogonal ψ 1s (1)ψ 2s (2) • Known as product states

ψ 2s (1)ψ 1s (2 )

Superposition of wavefunctions • The true wavefunction is thus a superposition of the two orbital product wavefunctions ψ 1s (1)ψ 2s (2 )

ψ 2s (1)ψ 1s (2 )

a (ψ 1s (1)ψ 2 s (2 )) + b(ψ 2 s (1)ψ 1s (2 )) • This clearly shows that orbitals really are only an approximation to the real two electron wavefunction

Looks familiar…. • This is very similar to the free particle, where we had two possible solutions

aψ 1s (1)ψ 2s (2 ) + bψ 2s (1)ψ 1s (2 )

ae

ikx

+ be

− ikx

• The adding of wavefunctions in this way is called SUPERPOSITION • This time, however, we have a BOUNDARY CONDITION….

Symmetric and anti-symmetric wavefunctions • Ensure superposition produces orthogonal wavefunctions • There are only two solutions found in this case Do these have the same energy? Let’s look at the wavefunctions…

Ψ+ (1,2 ) = a [ψ 1s (1)ψ 2s (2 ) + ψ 2s (1)ψ 1s (2 )]

Ψ− (1,2 ) = a [ψ 1s (1)ψ 2s (2) − ψ 2s (1)ψ 1s (2 )]

Seeing electron correlation • The two (radial) wavefunctions are very similar, except at very short distances from the nucleus • Notice there is an (anti)-correlation between the electrons

Fermi-heaps and Fermi-holes • The anti-symmetric wavefunction has an decreased probability of finding both electrons near the nucleus. This is a Fermi-hole. • By contrast, the symmetric wavefunction has a Fermi-heap • Both are a result of the superposition of the 1s2s product wavefunctions (configurations)

Energy effect of electron correlation • The increased magnitude of the two-electron wavefunction at the nucleus decreases the stability of the symmetric wavefunction because of the electronelectron repulsion!

• Key here is that the electron repulsion does not “create” the Fermi-hole, but it does effect the final energy of the wavefunctions!

The helium atom again • We seem to have two sets of energy levels • They correspond to different values of total spin • Orthohelium energy levels much lower! • Now, we’ll draw this a slightly different way…

Spatial wavefunctions This time, notice that all singlet terms are symmetric spatial wavefunctions, and the antisymmetric triplet!

• This is due to a fundamental symmetry of quantum objects…. but we’ll leave it till next year!!!

Answers to the earlier questions.. • We cannot solve a Schrödinger equation for many electron atoms analytically • Instead, we introduce some approximations and use numerical techniques to help us • Electron correlations can be revealed in many-electron wavefunctions, such as Fermi heaps and holes • Superposition of product wavefunctions is imposed by the indistinguishability of electrons

To summarise • Quantum mechanics was a marriage between experimental observation and theoretical insights • The key seems to be people willing to think the unthinkable!