Atoms, Energy Levels and SpectroscopyEnergy Levels in Atoms (13.2, 13.3, 13.4, 13.5)Coupling of Angular Momenta (13.7, 13.8)Term Symbols and Selection Rules (13.9)Russell-Saunders Coupling (13.9)

ExamplesHydrogen and Alkali Metal AtomsHeliumSelection rules and other polyelectronic atoms

Supplementary NotesCoupling of Angular MomentaHyperfine SplittingZeeman InteractionStark Effect

Atomic Spectroscopy ofPolyelectronic Atoms

Atomic SpectroscopyElectronic spectroscopy is the study of absorption andemission transitions between electronic states in an atom ora molecule.Atomic spectroscopy is concerned with the electronictransitions in atoms, and is quite simple compared toelectronic molecular spectroscopy. This is because atomsonly have translational and electronic degrees of freedom,and sometime influenced by nuclear spin (molecules alsohave vibrations and rotations).Recall (Lecture 5) that for hydrogen and hydrogen-likeions (i.e., He+, Li2+, Be3+, etc.) with a single electron and anucleus with charge +Ze, the hamiltonian is:

, ' &£ 2

2µL2

&Ze 2

4B,0r

µ = (memp)/(me + mp) reduced masse = 1.6021 × 10-19 C charge on an electronr = distance between +e and -e,0 = 8.854 × 10-12 C2 s2 kg-1 m-3 permitivity of free spaceFor a polyelectronic atom, the hamiltonian becomes:

, ' &£ 2

2mej

iL2

i & ji

Ze 2

4B,0ri

% ji<j

e 2

4B,0rij

summed over i electrons. The new term describescoulombic repulsions between pairs of electrons which aredistance rij apart (second term is coulombic attractionbetween nucleus and electrons)

SCF MethodThe electron-electron repulsion term creates a many-bodyproblem, in which the hamiltonian cannot be broken downinto a sum of contributions from each electron - the result isthat the Schroedinger equation cannot be solved exactly.

Methods of approximation have been devised, includingthat of Hartree, who replaces the electron repulsions witha sum of potential energy functions from separate electrons.This is known as the self-consistent field (SCF) method,under which the Schroedinger equation can be solved.

, • &£ 2

2mej

iL2

i & ji

Ze 2

4B,0ri

% ji

V(ri)

The electron repulsions remove degeneracy of orbitals withdifferent orbital angular momenta (like 3s, 3p, 3d, whichare degenerate in the H atom) - see next page

The atomic orbital (AO)energies vary with the principalquantum number n = 1, 2, 3, ...and the orbital angularmomentum, R = 0, 1, 2, ... for s, p,d, ... orbitals.

Plots of the SCF Hartree-Fockradial distribution functions areshown for sodium - electrondensity is grouped into shells, asanticipated by early chemists!

Electron ConfigurationsThe value of the orbital energy increases with the nuclearcharge of the atom (the energy to remove an electron fromthe 1s orbital, the ionization energy, is 13.6 eV for He and870.4 eV for Ne).A comparison of the AO energy levels in a hydrogen-likeatom/ion and a polyelectronic atom/ion are shown below:

The aufbau or building-up principle: electrons are “fed”into the orbitals in order of increasing energy until theelectrons are used up - this gives the ground configurationof the atom.The Pauli-exclusion principle: no two electrons may havethe same set of quantum numbers: n, R, mR, ms. Since mR canhave 2R + 1 possible values and ms = ±½, each orbitalcharacterized by n and R can have 2(2R + 1) electrons.

Configurations & StatesOrbital: particular values of n and RShell: all orbitals with same value of nFor n = 1, 2, 3, 4, ... shells are labelled K, L, M, N, ...

A configuration describes the manner in which theelectrons are distributed among the orbitals, but a givenconfiguration may give rise to more than one state - thenature of the states depend on how the electrons couplewith one another, resulting in states of different energies.

Trends in the periodic table:alkali metals: outer ns1 configuration, monovalentalkaline earth metals: outer ns2 configuration, divalentnoble gases: outer np6 configuration, filled orbital sub-shellresults in their characteristic chemical inertnessfirst transition series: characterized by filling of the 3dorbital (3d and 4s are similar in energy, but their separationchanges along the series).Examples:Cu has a KLM4s1 or KL3d104s1 ground configuration, thecompletely filled 3d10 orbital has an “innate stability”Cr has a KL3s23p63d54s1 ground configuration, in somecases half-filled d-orbitals confer stability onto a particularconfigurationlanthanides: filling of the 4f orbital is characteristic of thelanthanides, but since 4f and 5d are of similar energy, insome cases one electron will go into the 5d orbital

Angular Momenta & Magnetic MomentsAn electron in an atom has two possible sources of angularmomenta: orbital angular momentum and spin angularmomentum. The orbital angular momentum vector for asingle electron is given by:

[R(R % 1)]1/2£ ' R(£

where R = 0, 1, 2, ... (the quantity [A(A+1)]1/2 happens somuch in these discussions that we denote it as A*)The spin angular momentum vector for a single electron is:

[s(s % 1)]1/2£ ' s (£

where s = ½. For an electron with both types of angularmomentum, there is a quantum number j, assigned to thetotal angular momentum, which is a vector quantity:

[j(j % 1)]1/2£ ' j (£

For one-electron atoms, where s = ½, j is not very useful(unless considering fine structure), but the quantum numberJ (for polyelectronic atoms) is very important.If the nucleus has a non-zero spin quantum number I, theremay be a coupling from nuclear spin angular momentum:

[I(I % 1)]1/2£ ' I (£Since the nucleus is so large compared to the electron, thisangular momentum is considerably smaller, so cannormally be neglected - however, if looking at finestructure, this additional angular momentum results inhyperfine splitting in atomic spectra.

j ' R % s, R % s & 1, ..., |R & s|

A charge -e circulating in an orbit is equivalent to currentflowing through a circular wire. The latter causes amagnetic field perpendicular to the plane of the loop:

Angular Momenta & Magnetic Moments, 2

while the former results in a magnetic moment.R

µR

µs

s

R

µR

µs

s

The magnetic moment µR from orbital angular momentumis opposite direction to the orbital angular momentumvector, R. The classical picture of the electron spinningabout its own axis gives a magnetic moment µs in theopposite direction to the spin angular momentum vector s.µR and µs may be parallel or antiparallel, and can beregarded as tiny classical bar magnets that interact

Singlet & Triplet StatesHe: 1s12s1 excited configuration (ground: 1s2), so electronsdo not have to be paired since they are in different orbitals. According to Hund’s rules, the state of the atom with spinsparallel lies lower in energy than the state in which they arepaired (though both states are allowed).Parallel and antiparallel spins differ in overall angularmomentum. When they are paired, zero net spin, called thesinglet state:

F&(1,2) ' 2&1/2 ["(1)$(2) & "(2)$(1)]

The spin angular momenta of two parallel spins addtogether for non-zero total spin, this is the triplet state:

"(1)"(2)F%(1,2) ' 2&1/2 ["(1)$(2) % "(2)$(1)]

$(1)$(2)

In the singlet state, theanti-parallel pair ofspins are precisely anti-parallel, and theresultant vector is zero.

In the triplet state,the parallel pair ofspins are notprecisely parallel;rather, the anglebetween the spinangular momentumvectors is alwaysconstant, so all threearrangements havethe same total spinangular momentum.

Coupling of Angular MomentaThe interaction of µR and µs (like small classical barmagnets) is referred to as coupling of angular momenta. The larger the magnetic moments involved, the greater thesize or strength of the coupling.The coupling of two vectors a and b produces a resultantvector c. If these vectors represent angular momenta, thena and b undergo precession around c. The rate ofprecession (or precession frequency) increases as thestrength of the coupling increases.

Normally, c precesses about some arbitrary direction inspace, unless in the presence of an electric or magneticfield (Stark and Zeeman effects, respectively), in whichcase space quantization may be observed.

The spin of one electron can interact with:(a) spins of other electrons(b) its own orbital motion (ls or jj- coupling, oftenimportant only for few states of heavy atoms)(c) orbital motion of other electrons (very small)

Russell-Saunders CouplingIf we make the approximation that (i) the coupling betweenthe spin of an electron and its orbital momentum can beneglected (no jj-coupling), but that (ii) coupling betweenorbital momenta is significant and (iii) coupling betweenspin momenta is weak but appreciable, then this is theopposite extreme to jj-coupling known as the Russell-Saunders coupling approximation, and provides a usefulway of describing the states of most atoms.Non-equivalent electrons are those which have differentvalues of n or R: e.g., 3p13d1 or 3p14p1 configurations havenon-equivalent electrons, but 2p2 electrons are equivalent.Consider the coupling of orbital angular momenta of twonon-equivalent electrons, which is known as RR coupling.For example, the He atom with excited configuration2p13d1, where the 2p and 3d electrons are labelled 1 and 2so that R1 = 1 and R2 = 2, which have vectors withmagnitudes of 21/2£ and 61/2£, respectively. These vectorscouple to give a resultant L of magnitude:

[L(L % 1)]1/2£ ' L (£

The values of the total orbital angular momentum numberL are limited, since the relative orientations of R1 and R2 arelimited to the Clebsch-Gordan series:

L ' R1 % R2, R1 % R2 & 1, ..., |R1 & R2|

In this example, L = 3, 2 or 1 and magnitude of L is 121/2£,61/2£ or 21/2£ (see next page).

One can calculate the possible values of the magnitude of Land then construct vector diagrams showing this coupling:

RR Coupling

The terms of the atoms are labelled as S, P, D, F, G, ... forL = 0, 1, 2, 3, 4, ... (capital letters are for multielectronatoms). So the 2p13d1 configuration gives rise to P, D andF terms (Figure a). If a third electron is present, thencoupling in a third vector to any of the L in the figure abovewill give terms arising from three non-equivalent electrons.For a filled sub-shell (e.g., 2p6 or 3d10), L = 0.Space quantization of the total orbital angular momentumproduces 2L + 1 components, ML = L, L-1, ..., -L. In a filledsub-shell, 3i(mR)i = 0 (sum over all electrons in sub-shell). Since ML = 3i(mR)i, it follows that L = 0.

Coupling between spin momenta is known as ss coupling. The coupling of s vectors is treated the same way as RRcoupling. However, since s = ½, the vector for eachelectron always has a magnitude of 31/2£. Two s vectorstake up orientations w.r.t. one another such that

ss Coupling

[S(S % 1)]1/2£ ' S (£

where S is the total spin quantum number, restricted to:S ' s1 % s2, s1 % s2 & 1, ..., |s1 & s2|

For two electrons, S = 0 or S = 1 (Figure b), withmagnitudes 0 and 21/2£.The labels for the terms indicate the value of S by having a2S + 1 pre-superscript on the S, P, D, F, ... labels. Thisvalue of 2S + 1 is called the multiplicity, and is the numberof values that MS can take:

MS ' S, S & 1, ..., &SFor two electrons, S = 0 or 1, and the multiplicity is 1 or 3,giving terms called singlet or triplet, respectively. For afilled orbital, just like L = 0, S = 0 for the same reasons.For C and Si excited configurations:

C 1s22s22p13d1

Si 1s22s22p63s23p13d1

both possess P, D and F terms and considering multiplicityboth possess 1P, 3P, 1D, 3D, 1F and 3F terms. Noble gases,which have all occupied orbitals filled, have only 1S termsin their ground state configurations.

Total Angular MomentumSome terms arising from configurations of non-equivalentand equivalen electrons are shown below:

Coupling between resultant orbital angular and spinmomenta is known as LS coupling (spin-orbit interaction). This interaction results from the positive Ze charge on thenucleus and is proportional to Z4.The total angular momentum implies orbital plus electronspin, and has the symbol J = L + S (if nuclear spin isincluded, it has the symbol F = L+S+I). The magnitude is

J ' L % S, L % S & 1, ..., |L & S|

[J(J % 1)]1/2£ ' J (£

and J is restricted to

If L > S, then J has 2S + 1 values. If L < S, J has 2L + 1values.

Term SymbolsFigure c illustrates 3 ways of coupling L and S, e.g., in a 3Dterm with L = 2, S = 1 and J = 3, 2 or 1. The value of J isattached as a post-script, so that the three components of3D are 3D3, 3D2 and 3D1.

For the previously discussed excited configurations of Cand Si, we now have:

1P1, 3P0, 3P1, 3P2, 1D2, 3D1, 3D2,, 3D3, 1F3, 3F2, 3F3, 3F4

So the term symbol has the following structure:

3multiplicity,2S + 1

total angularmomentum,J = L + S, ..., *L -S*

orbital angular momentumL = 0, 1, 2, 3, ... 6 S, P, D, F, ...

Configuration: gross approximation on filling orbitals. All configurations give rise to at least one term or state:So, the configuration 1s12s1 of He has two states 1S0 and 3S0

Term is used to describe what arises from an approximatetreatment of electron configuration, and state is used todescribe something experimentally. For instance, the1s22s22p13d1 configuration of C has a 3P term, which whenspin-orbit coupling is taken into account splits into 3P1, 3P2and 3P3 states. Since spin-orbit coupling is excluded intheory, there is no real experimental observation of the 3Pterm: confusing - since term and state are often mixed up!!***Nuclear spins can further split states (so bad name in the first place) often called components of states

Term Symbols, 2An easy way to remember how term symbols work isshown in the flow chart below.

1. The letter S, P, D, F, G, H, I, ...indicates the total orbital angularmomentum quantum numberL = 0, 1, 2, 3, 4, 5, 6, ...

2. The left superscript gives themultiplicity of the term (2S+1)

3. The right superscript on the termsymbol is the value of the totalangular momentum quantumnumber J

Configurations can be written toindicate the absence of an electron:e.g., F, [He]2p5 / [Ne]2p-1

To derive a term symbol:1. Write the configurations, ignore closed inner shells.2. Couple the orbital angular momenta to find L.3. Couple the spin angular momenta to find S.4. Couple L and S to find J.5. Express the term as 2S+1{L}J, where {L} is the

appropriate letter.***Try example 13.7, Atkins 7th ed. p. 402.

Spin-Orbit CouplingSo again considering the picture of spin-orbit coupling,which is the interaction between spin and orbital magneticmoments:

(a) When the magnetic momentsare parallel, they are alignedunfavourably, and this is the high jor high energy arrangement: j = R + s

(b) When the magnetic momentsare antiparallel, they are alignedfavourably, and this is the low j orlow energy arrangement:j = R - s

The dependence of the spin orbit coupling on j is expressedin terms of the spin-orbit coupling constant, A (usually awavenumber), and energies of levels with quantumnumbers of s, R and j are adjusted by

ER,s,j ' ½hcA[j(j % 1) & R(R % 1) & s(s % 1)]

Spin orbit coupling splitting the 2Pterm of the 2p1 configuration/If A is +ve, the component withsmallest value of J is lowest inenergy and the multiplet is normal. If A is -ve, the multiplet is inverted.

Atoms with ground configurations having orbitals whichare half filled always have an S ground state (e.g., N(2p3),Mn(3d5), Eu(4f7))

Spin-Orbit Coupling, 2

For excited states, there are no general rules for normal orinverted multiplets.One further addition to symbolism is that sometimes asuperscript “o” is used. For example, the ground state of B:

2P o1/2

This symbol means that the arithmetic sum 3iRi for allelectrons in the atom is an odd number (1 in this case). Ifthere is no subscript, the total sum is even.

The fine structure resultingfrom spin-orbit coupling inelectronically excited Naresults from two separatetransitions: one from a j = 3/2level and one from a j = 1/2level. The yellow line from asodium discharge lamp, at 589nm, is actually a doublet withtwo lines at 589.76 and 589.16nm (spin orbit coupling effectsthe energies of the split 3Pstates by about 17 cm-1)*** Try example 13.5,

Atkins p. 399

Quantum No. Values

(mR)1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 -1

(mR)2 1 0 0 0 0 -1 -1 -1 -1 0 -1 -1 -1 -1 -1

(ms)1 ½ ½ ½ -½ -½ ½ ½ -½ -½ ½ ½ ½ -½ -½ ½

(ms)2 -½ ½ -½ ½ -½ ½ -½ ½ -½ -½ ½ -½ ½ -½ -½

ML = 3i (mR)i 2 1 1 1 1 0 0 0 0 0 -1 -1 -1 -1 -2

MS = 3i (ms)i 0 1 0 0 -1 1 0 0 -1 0 1 0 0 -1 0

Rearrange pairs of values of ML and MS

ML 2 1 0 -1 -2 1 0 -1 1 0 -1 1 0 -1 0

MS 0 0 0 0 0 1 1 1 0 0 0 -1 -1 -1 01D 3P 1S

Equivalent ElectronsThe Russell-Saunders approximation for two (or more)equivalent electrons is a bit more lengthy to apply. Theelectrons have the same values of n and R, for example, inthe ground configuration for carbon:

C 1s22s22p2

For filled orbitals, L = 0, S = 0. For the 2p electrons, n = 2and l = 1. In order not to violate the Pauli-exclusionprinciple, 2 electrons must always have different mR and ms

Label one of the p electrons 1, R1 = 1, (mR)1 = +1, 0, -1 ands1 = ½ and (ms)1 = +½, -½. Same for electron 2.

(mR)1 and (ms)1 cannot simultaneously have the same valuesas for (mR)2 and (ms)2 due to Pauli exclusion principle.

There are 15 possible combinations summarized below:

- Highest value of ML = 2, highest value of L as well, andthere is a D term

- ML = 2 has only MS = 0, so 1D term - accounts for 5 ofthe combinations listed in the table.

- The next highest value of ML = 1, with MS = +1, 0, -1, soit is a 3P term, associated with 9 combinations.

- This leaves ML = 0 and MS = 0, comprising a 1S term.

Equivalent Electrons, 2

Of the 1S, 3S, 1P, 3P, 1D, 3D terms which arise for 2equivalent p electrons, only the 1S, 3P and 1D are allowed(Pauli-exclusion forbids the others).

For 3 equivalent electrons, similar treatment can be applied(somewhat more lengthy).For 4 equivalent electrons, there is a useful rule that says avacancy in a sub-shell behaves like an electron. So for theground configurations of C and O:

C 1s22s22p2

O 1s22s22p4

both have the same terms: 1S, 3P and 1DThe excited configurations of C and Ne also have the sameallowed terms:

C 1s22s22p13d1

Ne 1s22s22p53d1

Terms: 1,3P, 1,3D, 1,3F

Hund’s Rules(1) Of the terms arising from equivalent electrons, those

with the highest multiplicity lie lowest in energy(2) Of these, the lowest is that with the highest values of LFor the ground configurations of both C and O:

C 1s22s22p2

O 1s22s22p4

the 3P term is the lowest in energy.The ground configuration of Ti is KL3s23p63d24s2, and ofterms arising from the d2 configuration, 3F is lowestThe splitting of a term by the spin-orbit interaction isproportional to J:

EJ & EJ& 1 ' AJ

where EJ is the smallest energy corresponding to J. In thiscase a multiplet results.If A is +ve, the component with the smallest value of J islowest in energy and the multiplet is said to be normal.If A is -ve, the multiplet is inverted(3) Normal multiplets arise from equivalent electrons

when a partially filled orbital is less than half full.(4) Inverted multiplets arise from equivalent electrons

when a partially filled orbital is more than half full.For Ti, 3F is split by spin-orbit coupling into a normalmultiplet, and the ground state is 3F2. The lowest energyterm of C, 3P, splits into a normal multiplet, the groundstate is 3P2; for O, there is an inverted multiplet withground state 3P0

jj CouplingRussell-Saunders coupling fails when spin-orbit coupling islarge (happens for heavy atoms). The individual spin andorbital angular momenta are coupled into individual jvalues, which are combined into a grand total, J: this isknown as jj coupling.For example, p2 configuration, j are 3/2 and 1/2 for eachelectron. If s and R are coupled strongly, each electron isconsidered a particle with j = 3/2 or 1/2. Then:

j1 '3

2and j2 '

3

2J ' 3, 2, 1, 0

j1 '3

2and j2 '

1

2J ' 2, 1

j1 '1

2and j2 '

3

2J ' 2, 1

j1 '1

2and j2 '

1

2J ' 1, 0

For heavy atoms, it is best to use thequantum numbers above to discussenergies, but the terms derived fromRS coupling can still be used aslabels.

The correlation diagram on the leftshows how energies of the atomicstates change with increasing spin-orbit coupling. So for low spin-orbitcoupling (RS-coupling) and highspin-orbit coupling (jj-coupling)schemes, the RS labels are used.

Key Concepts1. Electronic spectroscopy is the study of absorption

and emission transitions between electronic states inan atom or a molecule. Atomic spectroscopy isconcerned with the electronic transitions in atoms.

2. The Hartree-Fock SCF method is used toapproximate the potential energy resulting from theinteraction of electrons in a multi-electron atom.

3. The aufbau principle and Pauli exclusion principleare used to construct the ground configuration for allatoms in the periodic table.

4. There are two sources of angular momentum withinan atom orbital angular momentum and spinangular momentum. The total angular momentum isthe sum of these other contributions.

5. The interaction of magnetic moments from spin andorbital angular momenta is known as spin-orbitcoupling, and is responsible for the splitting of energy levels and fine structure in atomic spectra. This can be accurately described by the Russell-Saunders coupling approximation.

6. Term symbols are used to describe differentelectronic atomic states.

7. Hund’s rules are used to determine the relativeenergy levels of electronic atomic states.

Hydrogen & One Electron IonsThe hydrogen atom and one-electron ions are the easiest totreat, since there are no electron-electron repulsions. However, there are a series of degeneracies that are absentin all other polyelectronic atoms which feature prominentlyin the atomic spectra of hydrogen and one-electron ions.The alkali metal ions are the next simplest case, as they allhave a single electron in the outer ns orbital, where n = 2,3, 4, 5, 6 for Li, Na, K, Rb and Cs.If the electrons in these systems only changed orbitals (i.e.,values of n), the spectra would resemble the simple atomicspectra for hydrogen. Whereas the H atom only shows theBalmer series in the visible region of the spectrum, thereare at least three different series for alkali metal atoms:

The series are shown above for Li: (i) principal series:observed via absorption through column of vapour; (ii)sharp & (iii) diffuse series: so-named because of theirappearance; & (iv) fundamental series: sometimes observed

Alkali Metal AtomsThe Grotrian diagramis an energy leveldiagram used todescribe atomic spectra. To the right is theGrotrian diagram for theground configuration(1s22s1) of Li.

The lowest energy levelon the diagram, labelled2s, corresponds to theground configuration. Higher levels, e.g., 4p,correspond to excitedconfigurations, e.g.,1s24p1

There are large separations between energy levels withdifferent values of R (e.g., 3s, 3p, 3d) for all atoms excepthydrogen. The selection rules for promotion of an electronto excited orbital (and return to lower energy orbital) are:

(a) )n is unrestricted(b) )R = ±1

These rules lead to the sharp, principal, diffuse andfundamental series, in which the promoted electron is inan s, p, d and f orbital, respectively.

Alkali Metal Atoms, 2Some excited configurations and states of Li (involvingpromotion of the valence electron only) are shown below:Configuration States Configuration States1s22s1 2S1/2 1s2nd1 (n=3, 4, ...) 2D3/2, 2D5/21s2ns1 (n=3, 4, ...) 2S1/2 1s2nf1 (n=4, 5, ...) 2F5/2, 2F7/21s2np1 (n=2, 3, ...) 2P1/2, 2P3/2

Spin-orbit coupling splits apart the two components of the2P, 2D, 2F, ... terms, and the splitting descreases withincreasing R and n, but increases with increasing atomicnumber (for Li, Grotrian diagram spectrum is not observed,but is easily observed for heavier alkali metal atoms).2P1/2 and 2P3/2 states result from the promotion of the 3svalence electron to any np orbital with n > 2 (states areoften labelled with n: n2P1/2 and n2P3/2)

2P3/2

2P1/2

2S1/2

spectrum<

The splitting of the 32P1/2and 32P3/2 states of sodiumis 17.2 cm-1, which reducesto 5.6, 2.5, 1.3 cm-1 for n =4, 5 and 6, respectively.

The splitting of the 32D1/2and 32D3/2 states of sodiumis only 0.1 cm-1, asincreasing R serves todecrease the splitting.

Pictured above is the “simpled doublet” of atomic sodium

There is a fine structure selection rule:)J = 0, ±1 except J = 0 : J = 0

This means that the principal series will consist of pairs of2P1/2-2S1/2 and 2P3/2-2S1/2 transitions (N-M notation, N higherstate and M lower state, as for vibrational spectroscopy),which are known as simple doublets. All sharp seriesmembers show up as simple doublets.

Alkali Metal Atoms, 3

The 32P1/2 and 32P3/2 excited states involved in the sodiumD line are the lowest excited energy states of the atom. Discharge in the vapour (where collisional deactivation ofexcited states readily occurs) results in many of the atomsbeing in these state before emission of radiation. The Dline is prominent, given sodium lamps their characteristicyellow colouration.

2D5/22D3/2

spectrum<

In the diffuse series,compound doublets areoften observed, but thesplitting between the closelyspaced 2D3/2 and 2D5/2 statesmay be too close to beresolved (why it is called acompound doublet insteadof a triplet).

2P3/22P1/2

Atomic HydrogenUnique case for development of quantum mechanics:(1) Schroedinger equation is exactly soluble(2) Orbital energies (at this level of approximation) are

indepedent of Rsince the electron moves in a coulombic field free from theeffects of electron repulsions.

Dirac included the effect of relativity in quantummechanical treatments, and predicted the splitting of the n= 2 level into two components 0.365 cm-1 apart (n = 2, R = 1or 0). Since s = 1/2, j can be 3/2 or 1/2 for R = 1 and 1/2 forR = 0. One component of the n = 2 level has j = 3/2, R = 1,other is doubly degenerate with j = 1/2, R = 0, 1.In 1947, Lamb & Rutherford observed the 22P3/2-22S1/2transition using µwave techniques (off by 0.0354 cm-1 ofDirac’s prediction). The shift of the 22P3/2 level is the Lambshift (quantum electro-dyanmics or QED is the modifiedDirac theory). The 12S1/2 state is shifted but not split, nuclearspin (s = 1/2) splits it into two components 0.0475 cm-1 apart.

Hydrogen and AstronomyInterpreting spectra of atomic hydrogen is of paramountimportance in the study of stars, which are largelycomposed of atomic hydrogen.The interior temperature of a star is ca. 106 K, whereas theexterior (photosphere) is ca. 103 K. The observedabsorption spectrum has the interior as a continuum sourceand the photosphere as the absorber. Thus, the absorptionspectrum of a star shows the Lyman (n = 1) and Balmer (n= 2) series (the ratio of n=2:n=1 populations is only 2.9 ×10-5 at 103 K, but the high concentration of H atoms andlong absorption path length through the photosphere makesit possible to observe these series quite easily.

In the first five series of theatomic H spectrum there are aninfinite number of energy levels,and there are also an infinitenumber of series. As nOincreases, levels become closerso that series with high nO valuesstart showing up in theradiofrequency region.Scanning of the interstellarmedium with radiotelescopes hasresulted in the observations ofthe nO = 90, 104, 109, 126, 156,158, 159 and 166 series

The quantity of H atoms in stars is determine by measuringthe F = 1-0 hyperfine transition

Helium and Alkaline Earth MetalsThe emission spectrum of discharge in helium gas in thevisible and UV regions appears like the spectrum of twoalkali metals. There are two sets of lines which convergeslowly to high energy which can be divided into one groupof single lines & one group of double lines - no transitionsare observed between the two sets of levels.In 1925, when electron spin was taken into account, itbecame apparent that the two groups arise from singlet andtriplet forms of helium.For hydrogen and alkali metal atoms, there is only oneelectron with an unpaired spin (all states are doublet states)If spin-orbit coupling is small (it is in He), the totalelectronic wavefunction Re can be factorized into an orbitalpart and a spin part:

Re = ReoRe

s

The spin part is derived by labelling electrons 1 and 2, andwriting the four functions: "(1)"(2), $(1)$(2), "(1)$(2) and$(1)"(2). The latter two are neither symmetric norantisymmetric w.r.t. electron exchange, so we write thelinear combinations:

2&1/2 ["(1)$(2) & "(2)$(1)]

"(1)"(2)2&1/2 ["(1)$(2) % "(2)$(1)]

$(1)$(2)

1 antisymmetricfunction (singlet)

3 symmetricfunctions (triplet)

For the orbital part of the wavefunction, electrons in twodifferent atomic orbitals, Pa and Pb, are considered. Thereare two ways of placing the electrons into these orbitals,giving the wavefunctions Pa(1)Pb(2) and Pa(2)Pb(1) - butsince these are not symmetric to exchange, we must use:

Helium and Alkaline Earth Metals, 2

Roe ' 2&1/2 [Pa(1)Pb(2) % Pa(2)Pb(1)]

Roe ' 2&1/2 [Pa(1)Pb(2) & Pa(2)Pb(1)]

The most general statement of the Pauli exclusion principlefor electrons is that the total wavefunction must beantisymmetric w.r.t. electron exchangeSo for He, the singlet spin wavefunction can only combinewith the symmetric orbital wavefunctionRe ' 2&1 [Pa(1)Pb(2) % Pa(2)Pb(1)]["(1)$(2) & $(1)"(2)]

and for the triple spin wavefunctions:Re ' 2&1 [Pa(1)Pb(2) & Pa(2)Pb(1)]["(1)"(2)]or 2&1/2 [Pa(1)Pb(2) & Pa(2)Pb(1)][$(1)$(2)]or 2&1/2 [Pa(1)Pb(2) & Pa(2)Pb(1)]["(1)$(2) % $(1)"(2)]

The ground configuration, 1s2, has an orbital wavefunction

Roe ' Pa(1)Pb(2)

which is symmetric to electron exchange. Thisconfiguration has only a singlet term, whereas an excitedconfiguration has singlet & triplet terms, where the latter islower in energy.

The Grotrian diagram for He is shown below, whichderives from the selection rules:

)R = ±1, for promoted electron, and )s = 0

Helium and Alkaline Earth Metals, 3

Atoms which get into the lowest triplet state, 23S1, do noteasily revert to the 11S0 state: forbidden by both orbital andspin selection rules - so the lowest triplet state ismetastable, and in a typical discharge has a lifetime on theorder of 1 ms!The first excited singlet state, 21S0, is also metastable sincethe transition to ground state is forbidden by the )Rselection rule, but the transition is not spin forbidden so itis not as long lived as the 23S1 state.

All triplet terms, except 3S, are split into three components(in the case of the 3P term, for instance, L = 1 and S = 1, soJ can be 2, 1, 0 (Clebsch-Gordan series).

Helium and Alkaline Earth Metals, 4

The fine structure of a 3P-3S transition of an alkaline earthmetal is shown below. The )J selection rule results in asimple triplet (small separation of 23P1 and 23P2 in heliumaccounts for early description of the low resolutionspectrum of triplet helium consisting of “doublets”)

The 3D-3P transition, shown (b), has six components. Inmedium resolution, the 6 lines appear as a triplet(compound triplet), since the multiplet splitting decreasesrapidly with increasing L.

Spectra of Polyelectronic AtomsThe emission spectra of most other atoms are quitecomplex and no obvious series are observed. The Russell-Saunders coupling approximation (or jj coupling, heavyatoms) can be used to derive the states which arise fromany configuration:1. )L = 0, ±1 except L = 0 : L = 0

This general rule applies to the promotion of any numberof electrons and involves the total angular momentumquantum number, L

2. even : even, odd : odd, even : oddEven and odd refer to the arithmetic sum 3iRi over allelectrons (called the Laporte rule). This rule forbidstransitions between states arising from the sameconfiguration. For instance, the 1P - 1D transition isforbidden for the 1s22s22p13d1 configuration of carbon,though it is allowed by the )L and )S selection rules. Similarly, any transitions arising from the 1s22s22p13d1

configuration, with 3iRi = 3 and the 1s22s23d13f1

configuration, with 3iRi = 5, are also forbidden. This isalso consistent with )R = ±1, when only one electron ispromoted from the ground configuration.

3. )J = 0, ±1, except J = 0 : J = 0Same for all atoms.

4. )S = 0Applies to atoms with small nuclear charge (see nextpage)

In atoms with large nuclear charge the factorization of Reno longer applies and states are no longer accuratelydescribed as singlet, doublet, triplet, etc.For example, Hg, KLMN5s25p65d106s2 ground configurationis similar to an alkaline earth metal. Promotion of anelectron from the 6s to the 6p orbital results in 61P1, 63P0,63P1 and 63P2 states, with three components of 63P widelysplit by spin-orbit coupling. This interaction also breaksdown the spin selection rule so much that the 63P1-63S0transition at 253.652 nm is one of the strongest in the Hgemission spectrum.

Spectra of Polyelectronic Atoms, 2

Worked Example:From the ground electron configuration of Zr derive theground state (values of L, S and J). Then derive the statesarising from the configuration: KLM4s24p64d15s25f1

1. The ground electron configuration of Zr is:KLM4s24p64d25s2

- all filled orbitals contribute zero to all angular momenta- only the 2 equivalent 4d electrons need to be consideredAccording to the table (tot. ang. mom, earlier), terms are:

1S, 3P, 1D, 3F and 1G

2. Hund’s rule: 3F is the lowest energy term, since it has thehighest spin multiplicity (2S+1=3) and the highest valueof L (3 for an F term). For the F term:

J = L + S, L + S - 1, *L - S* = 4, 3, 2

Thus giving states: 3F4, 3F3, 3F2

Spectra of Polyelectronic Atoms, 3Since the unfilled orbital, 4d, is less than half full, the 3Fmultiplet is normal and the component with the lowestvalue of J (i.e., 3F2) is lowest in energy. Ground state is 3F2

3. In the excited electron configuration there are twoelectrons in partially filled orbitals (4d and 5f, electrons#1 and #2).For the coupling of orbital angular momenta, R1 = 2, R2 =3, and L = R1 + R2, R1 + R2 - 1, ..., *R1 - R2* = 5, 4, 3, 2, 1giving H, G, F, D and P terms.

For the coupling of spin angular momenta, s1 = ½, s2 =½, and S = s1 + s2, s1 + s2 - 1, ..., *s1 - s2* = 1, 0giving multiplicities (2S + 1) of 3 and 1.

4. The total number of terms:1H, 1G, 1F, 1D, 1P, 3H, 3G, 3F , 3D, 3P

5. For each of the triplet terms, there are three states. Forexample, for the 3H term, L = 5 and S = 1, soJ = L + S, L + S - 1, ..., *L - S* = 6, 5, 4So the total number of states is:1H5, 1G4, 1F3, 1D2, 1P1, 3H6, 3H5, 3H4, 3G5, 3G4, 3G3,3F4, 3F3, 3F2, 3D3, 3D2, 3D1, 3P2, 3P1, 3P0

So it is no wonder the atomic spectra of polyelectronicatoms are so complex - remember that the Laporte ruleforbids transitions between states arising from the sameconfiguration!

Key Concepts1. The atomic spectra of alkali metal ions are relatively

simple to interpret, as they all have a single electronin the outer ns orbital, where n = 2, 3, 4, 5, 6 for Li,Na, K, Rb and Cs. This gives rise to the (i) principalseries: observed via absorption through column ofvapour; (ii) sharp & (iii) diffuse series: so-namedbecause of their appearance; & (iv) fundamentalseries: sometimes observed.

2. The atomic spectra of hydrogen exhibit fascinatingfine structure due to effects of Einstein’s specialrelativity (included by Dirac) and quantum electro-dynamics (measured by Lamb & Rutherford).

3. The atomic spectra of helium gas (or alkaline earthmetals) in the visible and UV regions appears like thespectrum of two alkali metals. There are two sets oflines which converge slowly to high energy which canbe divided into one group of single lines & one groupof double lines - no transitions are observed betweenthe two sets of levels. When electron spin is takeninto account, it became apparent that the two groupsarise from singlet and triplet forms of helium.

4. The emission spectra of most other atoms are quitecomplex and no obvious series are observed. TheRussell-Saunders coupling approximation (or jjcoupling, heavy atoms) can be used to derive thestates which arise from any configuration.