Chapter 3:Many Electron Atoms

Question 1

In the central field approximation how do the wavefunctions associated with each singleelectron orbital differ from those in a one-electron atom?

Answer 1

As in a one-electron atom, in the central field approximation the spatial part of the total wavefunction can be separated into radial and angular functions.However, in the central field approximation the radial functions, Fni,i (ri), differ from those in a one-electron atom because of the deviation of the potential from a pure Coulomb potential at short range, while the angular functions, Yi,mi (θi , φi), are the same.

Question 2

To what types of atoms, or electronic configurations, is the central field approximation best applied (give some examples)?

Answer 2

Atoms or electronic configurations with one isolated outer electron. Examples: the alkali metal (Group I) atoms which all have one electron in an outer ns orbital, the single charged alkaline earth (Group II) atomic ions which have similar electronic configurations, or any other atom with one electron in a highly excited orbital, e.g., the 1s 52d configuration in helium.

Question 3

For a particular value of n how do the quantum defects of an alkali metal atom change with increasing values of l?

Answer 3

For a particular value of n as l increases the quantum defects reduce because the centrifugal barrier reduces the penetration of the outer electron into the region close to the non-hydrogenic ion core.

Question 4

What is meant by the term effective principal quantum number?

Answer 4

(n − ∆nl), where n is the principal quantum number and ∆nl is the quantum defect, is often referred to as the effective principal quantum number.

Question 5

State the Pauli principle as it applies to a quantum system containing many indistinguishable spin-1/2 particles.

Answer 5

Spin 1/2 particles are Fermions. The Pauli principle states that the total wavefunction for a system containing many indistinguishable Fermions must be antisymmetric with respect to exchange of any two of the particles.

Question 6

Why in general can the Schr¨odinger equation for a many-electron atom not be separated into single-electron components?

Answer 6

The Coulomb interactions between all of the individual electrons in a many electron atom mean that the instantaneous energy of each individual electron is dependent on the instantaneous position of all of the other electrons. Because of this the Schr¨odinger equation for a many-electron atom cannot be separated into single electron components.

Question 7

Hamiltonianfor multi electron atom

Answer 7

Question 8

For N electron atom, V_rep has ___ terms

Answer 8

N(N-1)/2

Question 9

List all possible spin wavefunctions for a system containing two electrons, giving the total spin quantum number, the projection of the total spin onto the z-axis, the spin multiplicity, and the symmetry with respect to exchange of the two electrons for each.

Answer 9

Question 10

The above two-electron spin wavefunctions can describe the electrons in the helium
atom. What other systems could these spin wavefunctions also be used to represent
(give one or two examples)?

Answer 10

The two electrons in the Li+ ion, or two protons in a single quantised motional
state of a magnetic trap.

Question 11

An excited electronic conguration of the helium atom has the form 4s2. Give an ex-
pression for the total wavefunction of the helium atom in this electronic conguration.

Answer 11

Question 12

What causes the energy dierence between the singlet and triplet spin states of the 1s 2s
electronic conguration in helium?

Answer 12

The exchange interaction.

Question 13

In a many electron atom, what is the maximum number of electrons that can be con-
tained within an ng sub-shell?

Answer 13

An ng orbital has l = 4. Maximum number of electrons is 2(2l + 1) = 18.

Question 14

What do the quantum numbers L and S represent?

Answer 14

L is the total electron orbital angular momentum quantum number and S is
the total electron spin quantum number.

Question 15

What are the electron spin multiplicities of the following terms:
¹S, ²F, and ⁵P

Answer 15

Question 16

In electronic congurations with equivalent electrons how does the Pauli principle aect
the combinations of S and L of the allowed terms?

Answer 16

The Pauli principle requires that the total wavefunction of a system containing
many indistinguishable Fermions (electrons) must be antisymmetric with respect to the
exchange of any two of the particles. For electronic congurations with equivalent
electrons the only allowed combinations of L and S are therefore ones for which the
symmetry of the total spin wavefunction is opposite to that of the spatial wavefunction.
If the total spin wavefunction is symmetric, i.e., S = 1, the spatial wavefunction must
be antisymmetric, i.e., L = 1; 3; 5; : : : If the total spin wavefunction is antisymmetric,
i.e., S = 0, the spatial wavefunction must be symmetric, i.e., L = 0; 2; 4; : : :.

Question 17

What is the origin of the spin-orbit interaction in atoms?

Answer 17

The spin-orbit interaction is the interaction of the magnetic dipole moment
(magnetic eld) associated with the electron spin in an atom with the magnetic dipole
moment associated with the electron orbital angular momentum.

Question 18

How does the spin-orbit interaction a ect the 1 ²S term in the H atom?

How does the spin-orbit interaction a ect the 5 ²D term in the H atom?

Answer 18

In the 1 2S term in the H atom, S = 1=2 and L = 0. The spin magnetic
moment is therefore non-zero but the because the orbital angular momentum quantum
number is zero the orbital magnetic moment is zero and there is no spin-orbit splitting.
In the 5 2D term in the H atom, S = 1=2 and L = 2. The spin magnetic moment and
the orbital magnetic moment are therefore both non-zero and the resulting spin-orbit
interaction will split this term into levels with total angular momentum J = 3=2 and
J = 5=2 which will be separated in energy by the spin-orbit splitting.

Question 19

What are the values of the electron spin g-factor, ge, and the orbital g-factor, gL?

Answer 19

The electron spin g-factor is ge = 2.002319304362 ‘ 2. The orbital g-factor is gL = 1.

Question 20

How does the spin-orbit energy splitting in an atom depend on the nuclear charge Z? How does the spin-orbit energy splitting in an atom depend on the principal quantum number n?

Answer 20

The spin-orbit energy splitting depends on Z ⁴ , and on 1/n³

Question 21

Which of the following terms exhibit spin-orbit energy splittings of their associated levels? ¹P, ³D, ²F, and ⁵S Give the symbols for the resulting levels in complete spectroscopic notation.

Answer 21

Question 22

The Land´e interval rule gives the energy difference between levels with total angular momentum quantum numbers J −1 and J. Give an expression for this energy difference in SI units in terms of the spin-orbit coupling constant.

Answer 22

Question 23

In terms of the total electron angular momentum in a many-electron atom, what is the distinction between LS- and jj-coupling?

Answer 23

Question 24

For what kinds of atoms does the jj-coupling scheme give a more appropriate description of the electronic structure than the LS-coupling scheme?

Answer 24

The jj-coupling scheme gives a more appropriate description of the electronic structure of atoms with highly charged nuclei (high Z atoms) than the LS-coupling scheme.

Question 25

How is the parity of an atomic term calculated, and how is this indicated in the spectroscopic notation for the term symbol?

Answer 25

Question 26

What are the selection rules for electric dipole transitions in one-electron atoms?

Answer 26

Question 27

What are the selection rules for electric dipole transitions in multi-electron atoms?

Answer 27

Question 28

To what type of atom is the central field approximation applicable

Answer 28

many-electron atoms with a single (excited) outer electron

Question 29

Within the central-field approximation, the eigenenergies of a single active electron can be expressed as

Answer 29

Question 30

How do the values of the quantum defect differ with n and l

Answer 30

Question 31

State the Pauli Principle

Answer 31

Question 32

Spin of an individual electron

Answer 32

1/2

Question 33

What does M_s represent

Answer 33

projection of the total spin onto the z-axis

Question 34

Practice Exchange Interaction

Question 35

How many electrons in a shell

How many electrons in a sub-shell or orbital

Answer 35

2n²

4l+2

Question 36

Complete Term Symbol:

Answer 36

^2S+1L_J^Parity

Question 37

How to determine parity of the Spacial wavefunction

Answer 37

If (−1)^L = +1, the function is symmetric while if (−1)L = −1 the function is antisymmetric.

Question 38

State and explain Hund’s 1st rule

Answer 38

For a given electronic configuration the term with the highest spin multiplicity, i.e., largest value of S, is lowest in energy.

This rule is a consequence of the effect of spin-spin interactions on the energy of a term. Terms with high spin multiplicity are those for which the electron spins are aligned parallel to each other. As discussed above in the case of the exchange interaction (Section 3.5), electrons with parallel spins tend to avoid each other and cannot be located in the same place. Because of this they are spatially separated from each other and the Coulomb repulsion between them is reduced. As a result they are more tightly bound (lower in energy).

Question 39

State and explain Hund’s 2nd rule

Answer 39

For a particular value of S in a given electronic configuration, the term with the highest total orbital angular momentum, i.e., largest value of L, is lowest in energy.

When the total orbital angular momentum of a term is largest, the orbital angular momentum vectors of the individual electrons in the configuration must be aligned parallel to each other, and in a classical description the electrons must be orbiting in the same direction about the nucleus. When orbiting in the same direction as each other, two electrons encounter each other less often than when orbiting in opposite directions. They are therefore on average further apart from each other reducing the Coulomb repulsion between them and lowering the term energy

Question 40

State and Explain Hund’s 3rd rule

Answer 40

3a. Normal case: For a particular term of a given electronic configuration, if the outer subshell of the configuration is less than half full the level energies increase with increasing values of J. Therefore the level with the smallest value of J is lowest in energy.
3b. Inverted case: For a particular term of a given electronic configuration, if the outer subshell of the configuration is more than half full the level energies decrease with increasing values of J and the level with the largest value of J is lowest in energy.
3c. For a particular term of a given electronic configuration, if the outer subshell is half full there is no multiplet energy splitting.

Question 41

Ground level of 1s² 2s² 2p²

Answer 41

³P₀

Question 42

Ground level of 1s² 2s² 2p¹

Answer 42

2P⁰ _1/2

Question 43

What is the total atomic nuclear spin vector

Answer 43

The combination of the individual spins of the Protons and Neutrons

Question 44

How to calculate quantum number F

Answer 44

|J − I |, . . . , |J + I | in steps of 1.

Question 45

As an example of the hyperfine structure of an atomic energy level, consider the 1 ²S_1/2 ground state of the H atom. This has

Answer 45

S = 1/2 J = 1/2, and because the proton has a spin of 1/2, I = 1/2. Therefore the possible values of F are F = 0, 1

F=J-L, J+L