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Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect CHAPTER 9 Beyond Hydrogen Atom Beyond Hydrogen Atom What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior. - J. Bronowski Dimitri Mendeleev
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Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Dec 16, 2015

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Page 1: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Multi-electron AtomPauli Exclusion PrincipleAtomic Structure and the Periodic

TableTotal Angular MomentumAnomalous Zeeman Effect

CHAPTER 9Beyond Hydrogen AtomBeyond Hydrogen Atom

What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior.

- J. Bronowski

Dimitri Mendeleev

Page 2: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Independent Particle Approximation

The potential of an electron depends on its distance from the nucleus as well as the its distance from other electron. The independent Particle approximation simply assumes that the potential for each particle is the same as that of a single electron orbiting the nucleus like that in the Hydrogen atom.

Page 3: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Multi-electron atoms

When more than one electron is involved, the potential and the wave function are functions of more than one position:

1 2( , ,..., )NV V r r r

1 2( , ,..., , )Nr r r t

Solving the Schrodinger Equation in this case can be very hard. But we can approximate the solution as the product of single-particle wave functions:

1 2 1 1 2 2( , ,..., , ) ( , ) ( , ) ( , )N N Nr r r t r t r t r t

And it turns out that, for electrons (and other spin ½ particles), all the i’s must be different. This is the Pauli Exclusion Principle.

Page 4: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Pauli Exclusion Principle

To understand atomic spectroscopic data, Pauli proposed his exclusion principle:

No two electrons in an atom may have the same set of quantum numbers (n, ℓ, mℓ, ms).

It applies to all particles of half-integer spin, which are called fermions, and particles in the nucleus are fermions.

The periodic table can be understood by two rules:

The electrons in an atom tend to occupy the lowest energy levels available to them.

The Pauli exclusion principle.

Page 5: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Symmetry of the Wave function

Two electron Schrodinger equation

The probability density does not change if we interchange the particles

The total wave function has two possibilities

symmetric

Antisymmetric

2 22 21 1 2 2 1 2 1 2 1 22 2

( , ) ( , ) ( , ) ( , )2 2

r r r r V r r E r rm m

2

1 2( , )dP

r rdV

2 2

1 2 2 1( , ) ( , )r r r r

1 2 2 1( , ) ( , )s sr r r r

1 2 2 1( , ) ( , )A Ar r r r

Page 6: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Symmetry of the Wave function

If a and b are the four quantum numbers of the two electrons then we can construct linear combinations

Symmetric

Antisymmetric

For two electrons with symmetric wave function the two particles can exist simultaneously with a=b, however the anitsymmetric wave function is zero if a=b, the electrons cannot be in the same state.

So according to quantum mechanics the anitsymmetric wave function is consistent with the Pauli's exclusion principle.

1 2 1 2 1 2( , ) ( ) ( ) ( ) ( )s a b b ar r C r r r r

1 2 1 2 1 2( , ) ( ) ( ) ( ) ( )A a b b ar r C r r r r

Page 7: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Example

9.1 An atom with four electrons is in an excited state. One electron is in energy level 1.04 eV, one electron is in energy level 4.16 eV and two are in energy level 9.36 eV. When the atom returns to its ground state what would the possible energies of the photons.

Page 8: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Atomic Structure

Hydrogen: (n, ℓ, mℓ, ms) = (1, 0, 0, ±½) in ground state.

In the absence of a magnetic field, the state ms = ½ is degenerate with the ms = −½ state.

Helium: (1, 0, 0, ½) for the first electron.

(1, 0, 0, −½) for the second electron.

Electrons have anti-aligned (ms = +½ and ms = −½) spins.

The principle quantum number also has letter codes.

n = 1 2 3 4...

Letter = K L M N…

n = shells (eg: K shell, L shell, etc.)

nℓ = subshells (e.g.: 1s, 2p, 3d)

Total number of electrons in any shell is

Electrons for H and He atoms are in the K shell.H: 1s2 He: 1s1 or 1s1

2

0

2 (2 1) 2(1 3 5 ... 1) 2n

nl

N l n n

Page 9: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Atomic StructureHow many electrons may be in each subshell?

Recall: ℓ = 0 1 2 3 4 5 …

letter = s p d f g h …

ℓ = 0, (s state) can have two electrons.

ℓ = 1, (p state) can have six electrons, and so on.

The lower ℓ values have more elliptical orbits than the higher ℓ values.

Electrons with higher ℓ values are more shielded from the nuclear charge.

Electrons lie higher in energy than those with lower ℓ values.

4s fills before 3d.

Total

For each mℓ: two values of ms 2

For each ℓ: (2ℓ + 1) values of mℓ 2(2ℓ + 1)

Page 10: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Examples

9.2 write down all possible sets of quantum numbers for an electron in a 4f and 2p subshels.

9.3 Suppose the outer electron in a potassium atom is in a state with l=2. Compute the magnitude of L. What are the possible values of j and the possible magnitudes of J?

9.4 write down the electron configuration of Carbon.

9.5 what element has this ground state electron configuration 1s22s22p63s22p2

9.6 which of the following atoms would you expect to have its ground state split by the spin orbit interaction: Li, B, Na

Page 11: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

The Periodic

Table

Page 12: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Groups and Periods

Groups:

Vertical columns.

Same number of electrons in an ℓ orbit.

Can form similar chemical bonds.

Periods:

Horizontal rows.

Correspond to filling of the subshells.

Page 13: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

The Periodic Table

Inert Gases:

Last group of the periodic table

Closed p subshell except helium

Zero net spin and large ionization energy

Their atoms interact weakly with each other

Fig a) Closed shell at 2, 10, 18, 36 and 54

Fig b) Increase in atomic radius

Page 14: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

The Periodic TableHalogens:

Need one more electron to fill outermost subshell

Form strong ionic bonds with the alkalis

More stable configurations occur as the p subshell is filled

Transition Metals:

Three rows of elements in which the 3d, 4d, and 5d are being filled

Properties primarily determined by the s electrons, rather than by the d subshell being filled

Have d-shell electrons with unpaired spins

As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

Page 15: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

The Periodic Table

Lanthanides (rare earths):

Have the outside 6s2 subshell completed

As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves

The large orbital angular momentum contributes to the large ferromagnetic effects

Actinides:

Inner subshells are being filled while the 7s2 subshell is complete

Difficult to obtain chemical data because they are all radioactive

Have longer half-lives

Page 16: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Coupling of Angular MomentaCorrections to Energy Levels

Orbital – Orbital Coupling, spin-spin coupling and spin orbit coupling, jj coupling

Orbital-Orbital Coupling

Important because coulomb repulsion between electrons means the electron probability density is dependent on L. Atoms with low Z have the largest effect on energies

For 2 electron system:

States with larger value of l have lower energy

1 2

1 2 1 2 1 2

2 2

allowed values of , 1,...

( 1)

L L L

l l l l l l l

L l l

Page 17: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Coupling of Angular MomentaSpin-Spin Coupling

Magnetic moment between electrons is weak, but for atoms with low Z values, the Pauli exclusion principle is important.

For 2 electron system

If spins are parallel the energy levels are further apart then when spins are antiparallel. Total wave function is antisymmetric so if the sping part is symmetric, the space part must be antisymmetric and thus Pauli principle requires the energy level to be further apart. One of the energy level is lower than normal

1 2

1 2

2 2

1

2

total intrinsics angular momentum quantum number s

be zero or one.

( 1)

s s

S S S

S s s

Page 18: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Coupling of Angular MomentaSpin-Orbit Coupling

Much smaller effect on energy levels if Z is not very large. Often called LS coupling or Russell-Saunders coupling.

allowed values of j , 1, ,...

J L S

l s l s l s

Page 19: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Coupling of Angular MomentaJ-J Coupling

For large Z values the spin-orbit coupling is much stronger than the spin-spin and orbital-orbital coupling. In this case we first find the spin-orbit coupling correction and do the other two next.

1 1 1

2 2 2

1 2

1 2

and are the two total angular momentum quantum numbers.

J L S

J L S

j j

J J J

Page 20: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Many-Electron Atoms

Hund’s rules:

The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle.

Insofar as rule 1 is not violated, L should also be maximized.

For atoms having subshells less than half full, J should be minimized.

For a two-electron atom

There are LS coupling and jj coupling to combine four angular momenta J.

Page 21: Multi-electron Atom Pauli Exclusion Principle Atomic Structure and the Periodic Table Total Angular Momentum Anomalous Zeeman Effect Beyond Hydrogen Atom.

Examples

9.7 Consider a system of two electron each with l=1 and s=1/2. a) what are possible values of the quantum number L for the total angular momentum? b) what are the possible values of the quantum number S for the total spin? c) find the possible quantum numbers j for the combination J=L+S. d) what are the possible quantum number j1 and j2 for the total angular momentum of each particle? e) find the possible values of j from the combination j1 and j2