Lecture notes on Hund’s rules

In the non-relativistic approximation the electrons in the atom move under the influ-ence of the Coulomb field of the nucleus and that of the other electrons. The Hamiltonianof an Z-electron atom, given by

H = − h2

2m

Z∑

i=1

∇2i −

Ze2

4πε0

Z∑

i=1

1

|~ri| +e2

4πε0

1

2

∑

i6=j

1

|~ri − ~rj| , (1)

is independent of the spin of the electrons. Strictly speaking, on should deal with a many-electron wave function and not single electron wave functions. An approximate descriptionwhich works extremely well involves defining single electron states; these are defined interms of the solutions of the one-particle Schrodinger equation with an effective, centrallysymmetric field. The central field, however, is not the same for all electrons. Moreover,the the wave function of one of the electrons determines the force on the others and viceversa. So the fields have to be determined self-consistently.

Our discussion is based on the above picture. Since the effective field is centrallysymmetric each state of the electron is characterized by the same quantum numbers asthe electron in the hydrogen atom, n, `, and m. These one-electron states are calledorbitals. In contrast to the hydrogen atom the energies of the orbitals depend on the valueof `, in general. For example, the 2p state has higher energy than the 2s state, roughlyspeaking, due to screening effects discussed in class. The set of 4` + 2 orbitals is calleda shell. So for a system of electrons we can specify the total orbital angular momentum~L ≡ ∑Z

i=1~i and ~S =

∑Zi=1 ~si since they are both separately conserved. The degeneracy

of a state given L and S is clearly (2L+1)(2S +1). When one includes relativistic effects

such as spin-orbit coupling are included ~L and ~S are no longer separately conserved butthe total angular momentum ~J = ~L + ~S is conserved. So the states of the system arespecified by {L, S, J,mJ}. We will ignore spin-orbit coupling first.

Using the same notation as for the hydrogen atom the electrons occupy successivelythe states diagonally from the first to third quadrants as shown in class:

1s2s 2p3s 3p 3d4s 4p 4d 4f5s 5p 5d 5f · · ·6s 6p 6d · · ·

(2)

The periodic table of Mendeleev is obtained by filling these levels taking into account theeffect of degeneracies: for a given n and ` we have a degeneracy of 2(2`+1) states and thisis referred to as a shell. The rule shown fails occasionally, for example, Gd, Gadoliniumwith Z = 64.

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Consider Silicon, Si, with 14 electrons; these occupy the 1s2, 2s2, 2p6, 3s2, 3p2 states.This is referred to as a configuration. Except for the two electrons in the 2p state theother levels are completely filled and are characterized by zero total orbital and total spinangular momentum. Verify this!

Now the 2p state is 6-fold degenerate and the two electrons can occupy these one-electron states in 6 × 5/2 = 15 ways. These 15 states do not have the same energy.Which are the lowest states? Hund’s rules addresses the issue of which of the allowedstates(terms) for a given configuration has the lowest energy: for the case of Si, the twoelectrons in the outermost shell both are labeled by the quantum numbers ` = 1, s = 1/2;what is the lowest energy state?. The rules apply to most atoms except the heaviestatoms (see later).

Hund’s rules which are empirical state (the first and second) that the term structurewith the maximum possible S and the largest possible L for the given S compatible withthe Pauli exclusion Principle has the lowest energy.

There is no simple proof of this rule; it has been verified experimentally and by nu-merical calculations. They can be justified crudely as follows. In the case of two electronsthe possible values of S are 0 or 1. The S = 0 state has an antisymmetric spin wavefunction and consequently a symmetric spatial part of the wave function. Remember thatidentical fermions must be described by an antisymmetric (total) wave function. Similarlythe S = 1 state is symmetric in spin space and antisymmetric in coordinate space. In thespatially antisymmetric state (ψ(~r1, ~r2) = −ψ(~r2, ~r1) and ψ(~r, ~r) = 0 and therefore,) theprobability of finding the electrons close to each other is small and the repulsive electro-static energy is reduced. In addition, when the electrons are farther apart they screen thenucleus less increasing the attractive energy. Thus the lowest energy state has the maxi-mum possible S. This argument can be generalized to more number of electrons since thestate with the highest spin is maximally symmetric and the corresponding spatial wavefunction has to be totally antisymmetric.

Next it is reasonable that the electrons are farthest apart when the total orbital angu-lar momentum is the largest. One way to see this within a classical picture for the statewith maximum mL = L. Clearly for a system of point masses rotating about the z-axisthe larger the angular momentum along z the farther apart they are. Now remember thatfor a given L all the states with different mL values have the same energy and so ourargument for mL = L should be OK. This justifies the L part of the rule.

So for the 2p2 electrons clearly the lowest energy state has S = 1. What is thevalue of L? Given `1 = 1 and `2 = 1 the total angular momentum assumes values`1 + `2, `1 + `2 − 1, ......|`1 − `2 which in this case yields L = 2, 1, 0. The trick is touse mS and mL to guide our thinking remembering that the specific valuesdo not matter by spherical symmetry. Since S = 1 choose mS = 1, i.e, ms1 = 1/2and ms2 = 1/2. Now both m`1 and m`2 cannot be the maximum value of 1 by Pauli

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exclusion principle. Be sure of this! The highest allowed value of mL corresponds tom`1 = 1 and m`2 = 0 or vice versa. Therefore, the maximum allowed value of L cor-responds to 1. Thus the lowest energy state of Silicon has L = S = 1. Thus there are(2L + 1)(2S + 1) = 3× 3 = 9 states with the same energy. What are these states? Theyare labeled by ML and MS. ML = 1, 0,−1 since L = 1 and similarly for S.

Given L = 1 and S = 1 the allowed values of the total angular momentum areJ = 2, 1, 0. Now since the shell is less than half-filled, the ground state corresponds tothe lowest value of J . (See below!)

The trick of using the maximum possible values of ML and MS is a useful way ofdeducing the maximum L and S. So if we have three spins in the 3d orbital we have ` = 2for each of the electrons. First we find the maximum S by finding the maximum MS

and then find the maximum allowed L consistent with the Pauli principle by consideringdifferent m` values.

Table 1: Three 3d electrons

Quantum number 1 2 3 Totalms ↑ ↑ ↑ MS = 3/2ms 2 1 0 ML = 3

At the risk of confusing you this trick of focusing on the maximum MS and ML doesnot mean that all the spins cannot point down. Of course they can; when we say S = 3/2,MS can be ±3/2, ±1/2 and there are 4 possible states. This is a quick way finding themaximum allowed S and L.

So we have decided which values of L and S are allowed, and in the jargon of spec-troscopy we know the LS multiplet. Finally, we include the effect of spin-orbit cou-pling (fine structure arising from relativistic corrections) perturbatively. The degeneracy((2L + 1)(2S + 1)) is partially lifted and the energy eigenstates are given in terms of|L, S, J,mJ〉 rather than |L, S, mL,mS〉. The different mJ values are degenerate at thislevel.

Hund’s third rule (which applies for atoms or ions with a single unfilled shell) statesthat if the unfilled shell is not more than half-filled the lowest value of J has the lowestenergy while if it is more than half-filled the largest value of J has the lowest energy.

So for Si the ground state corresponds to J = 0. the spectroscopic notation is arcaneand gives the L value in terms of a letter of the alphabet: S,P,D,F,G,H,I, .... for L =0, 1, 2, 3, 4, 5, 6, 7... respectively. The S value is specified by giving the spin degeneracy,2S + 1, as a left superscript and the J value as a right subscript. For Si the spectroscopicnotation is

3P0 .

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Look at Praseodymium, Pr or triply ionized Neodymium. These have three 4f elec-trons. You can verify that S = 3/2, L = 6, and J = 9/2. Thus the term structureis

4I9/2

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