1. PY3P05 Lectures 10-11: Multi-electron atoms oSchrödinger equation for oTwo-electron atoms. oMulti-electron atoms. oHelium-like atoms. oSinglet and triplet states. oExchange energy.

2. PY3P05 System of non-interacting particles oWhat is probability of simultaneously finding a particle 1 at (x 1,y 1,z 1 ), particle 2 at (x 2,y 2,z 2 ), etc. => need joint probability distribution. oN-particle system is therefore a function of 3N coordinates:  (x 1,y 1,z 1 ; x 2,y 2,z 2 ; … x N,y N,z N ) oMust solve(1) oFirst consider two particles which do not interact with one another, but move in potentials V 1 and V 2. The Hamiltonian is (2) oThe electron-nucleus potential for helium is oThe eigenfunctions of H 1 and H 2 can be written as the product: x y z e1e1 e2e2 r1r1 r2r2

3. PY3P05 System of non-interacting particles oUsing this and Eqns. 1 and 2, oThat is, where E = E 1 +E 2. oThe product wavefunction is an eigenfunction of the complete Hamiltonian H, corresponding to an eigenvalue E which is the sum of the energy eigenvalues of the two separate particles. oFor N-particles, oEigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus oCan be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction.

4. PY3P05 Application to helium oAssuming each electron in helium is non-interacting, can assume each can be treated independently with hydrogenic energy levels: oTotal energy of two-electron system in ground state (n(1) = n(2) = 1) is therefore oFor first excited state, n(1) = 1, n(2) = 2 => E =-68 eV. eV -50 -60 -70 -80 -90 -100 -110 Energy (eV) Neglecting electron-electron interaction Observed

5. PY3P05 System of interacting particles oFor He-like atoms can extend to include electron-electron interaction: oThe final term represents electron-electron repulsion at a distance r 12. oOr for N-electrons, the Hamiltonian is: and the corresponding Schrödinger equation is again of the form where x y z e1e1 e2e2 r 12 r1r1 r2r2 and

6. PY3P05 Wave function for system of interacting particles oThe solutions to the equation can again be written in the form oThe radial wave functions are solutions to and therefore have the same analytical form as for the hydrogenic one-electron atom. oAllowable solutions again only exist for where Z eff = Z -  nl. oZ eff is the effective nuclear charge and  nl is the shielding constant. This gives rise to the shell model for multi-electron atoms.

7. PY3P05 Atoms with two valence electrons oIncludes 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. oThis means that multi-electron wave functions must have exchange symmetry: which will be satisfied if oThat is, exchanging labels of pair of electrons has no effect on wave function. oThe “+” 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. oAs electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be anti- symmetric with respect to particle exchange.

8. PY3P05 Helium wave functions oHe atom consists of a nucleus with Z = 2 and two electrons. oMust now include electron spins. Two-electron wave function is therefore written as a product spatial and a spin wave functions: oAs electrons are indistinguishable =>  must be anti- symmetric. See table for allowed symmetries of spatial and spin wave functions. x y z e1e1 e2e2 r1r1 r2r2 Z=2 r 12

9. PY3P05 Helium wave functions:  spatial oState of atom is specified by configuration of two electrons. In ground state, both electrons are is 1s shell, so we have a 1s 2 configuration. oIn excited state, one or both electrons will be in higher shell (e.g., 1s 1 2s 1 ). Configuration must therefore be written in terms of particle #1 in a state defined by four quantum numbers (called  ). State of particle #2 called . oTotal wave function for a excited atom can therefore be written: oBut, this does not take into account that electrons are indistinguishable. The following is therefore equally valid: oBecause both these are solutions of Schrödinger equation, linear combination also a solutions: where is a normalisation factor. Symmetric Asymmetric

10. PY3P05 Helium wave functions:  spin oThere are two electrons => S = s 1 + s 2 = 0 or 1. S = 0 states are called singlets because they only have one m s value. S = 1 states are called triplets as m s = +1, 0, -1. oThere are four possible ways to combine the spins of the two electrons so that the total wave function has exchange symmetry. oOnly one possible anitsymmetric spin eigenfunction: oThere are three possible symmetric spin eigenfunctions: singlet triplet

11. PY3P05 Helium wave functions:  spin oTable 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). oThe + 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. oThe S z value is indicated by the quantum number for m s, which is obtained by adding the m s values of the two electrons together.

12. PY3P05 Helium wave functions oSinglet and triplet states therefore have different spatial wave functions. oSurprising as spin and spatial wavefunctions are basically independent of each other. oThis has a strong effect on the energies of the allowed states. S m s  spin  spatial 0 +1 1 0 Singlet Triplet

13. PY3P05 Singles and triplet states oPhysical 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. oThe magnutude of the total spin and its z-component are quantised: where m s = -s, … +s and s = 0, 1. oIf s 1 = +1/2 and s 2 = -1/2 => s = 0. oTherefore m s = 0 (singlet state) oIf s 1 = +1/2 and s 2 = +1/2 => s = 1. oTherefore m s = -1, 0, +1 (triplet states) z s = 1 +1 0 triplet state s 1 =1/2 s 2 =1/2 msms s 1 =1/2 s 2 =-1/2 s = 0, m s = 0 singlet state

14. PY3P05 Helium terms oAngular momenta of electrons are described by l 1, l 2, s 1, s 2. oAs Z<30 for He, use LS or Russel Saunders coupling. oConsider ground state configuration of He: 1s 2 oOrbital angular momentum: l 1 =l 2 = 0 => L = l 1 + l 2 = 0 oGives rise to an S term. oSpin angular momentum: s 1 = s 2 = 1/2 => S = 0 or 1 oMultiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet) oJ = L + S, …, |L-S| => J = 1, 0. oTherefore there are two states: 1 1 S 0 and 1 3 S 1 (also using n = 1) oBut are they both allowed quantum mechanically?

15. PY3P05 Helium terms oMust 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”. oConsider the 1 1 S 0 state: L = 0, S = 0, J = 0 on 1 = 1, l 1 = 0, m l1 = 0, s 1 = 1/2, m s1 = +1/2 on 2 = 1, l 2 = 0, m l2 = 0, s 2 = 1/2, m s2 = -1/2 o1 1 S 0 is therefore allowed by Pauli principle as m s quantum numbers differ. oNow consider the 1 3 S 1 state: L = 0, S = 1, J = 1 on 1 = 1, l 1 = 0, m l1 = 0, s 1 = 1/2, m s1 = +1/2 on 2 = 1, l 2 = 0, m l2 = 0, s 2 = 1/2, m s2 = +1/2 o1 1 S 1 is therefore disallowed by Pauli principle as m s quantum numbers are the same.

16. PY3P05 Helium terms oFirst excited state of He: 1s 1 2p 1 oOrbital angular momentum: l 1 = 0, l 2 = 1 => L = 1 oGives rise to an P term. oSpin angular momentum: s 1 = s 2 = 1/2 => S = 0 or 1 oMultiplicity (2S+1) is therefore 2(0) + 1 = 1 or 2(1) + 1 = 3 oFor L = 1, S = 1 => J = L + S, …, |L-S| => J = 2, 1, 0 oProduces 3 P 3,2,1 oTherefore have, n 1 = 1, l 1 = 0, s 1 = 1/2 and n 2 = 2, l 2 = 1, s 1 = 1/2 oFor L = 1, S = 0 => J = 1 oTerm is therefore 1 P 1 oAllowed from consideration of Pauli principle No violation of Pauli principle => 3 P 3,2,1 are allowed terms

17. PY3P05 Helium terms oNow consider excitation of both electrons from ground state to first excited state: gives a 2p 2 configuration. oOrbital angular momentum: l 1 = l 2 =1 => L = 2, 1, 0 oProduces S, P and D terms oSpin angular momentum: s 1 = s 2 = 1/2 = > S =1, 0 and multiplicity is 3 or 1 o * Violate Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P) LSJTerm 000 1S01S0 101 1P11P1 202 1D21D2 011 *3 S 1 112, 1, 0 3 P 2,1,0 213, 2, 1 *3 D 3,2,1

18. PY3P05 Helium Grötrian diagram oSinglet states result when S = 0. oParahelium. oTriplet states result when S = 1 oOrthohelium.

19. PY3P05 Exchange energy oNeed to explain why triplet states are lower in energy that singlet states. Consider oThe expectation value of the Hamiltonian is oThe energy can be split into three parts, E = E 1 + E 2 + E 12 where oThe expectation value of the first two terms of the Hamiltonian is just where E R = 13.6 eV is called the Rydberg energy.

20. PY3P05 Exchange energy oThe third term is the electron-electron Coulomb repulsion energy: oEvaluating this integral gives E 12 = D   J  where the + sign is for singlets and the - sign for triplets and D  is the direct Coulomb energy and J  is the exchange Coulomb energy: oThe resulting energy is E 12 ~ 2.5 E R. Note that in the exchange integral, we integrate the expectation value of 1/r 12 with each electron in a different shell. See McMurry, Chapter 13.

21. PY3P05 Exchange energy oThe total energy is therefore where the + sign applies to singlet states (S = 0) and the -sign to triplets (S = 1). oEnergies of the singlet and triplet states differ by 2J . Splitting of spin states is direct consequence of exchange symmetry. oWe now have, E 1 + E 2 = -8E R and E 12 = 2.5E R => E = -5.5E R = -74.8 eV oCompares to measure value of ground state energy, 78.98 eV. oNote: oExchange splitting is part of gross structure of He - not a small effect. The value of 2J  is ~0.8 eV. oExchange 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.

22. PY3P05 Helium terms oOrthohelium states are lower in energy than the parahelium states. Explanation for this is: 1.Parallel spins make the spin part of the wavefunction symmetric. 2.Total wavefunction for electrons must be antisymmetric since electrons are fermions. 3.This forces space part of wavefunction to be antisymmetric. 4.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. 5.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.