Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 21 Many-Electrons Atom

Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 21 Chapter 21 Many-Electrons Atom

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms

ObjectivesObjectives

• Using of Variational Method• Introduce Hartree-Fock Self-Consistent

Field Method



OutlineOutline

1. Helium: The Smallest Many-Electron Atom

2. Introducing Electron Spin3. Wave Functions Must Reflect the

Indistinguishability of Electrons4. Using the Variational Method to Solve

the Schrödinger Equation5. The Hartree-Fock Self- Consistent Field

Method6. Understanding Trends in the Periodic

Table from Hartree-Fock Calculations



21.1 Helium: The Smallest Many-Electron Atom21.1 Helium: The Smallest Many-Electron Atom

• Orbital approximation is to express a many-electron eigenfunction in terms of individual electron orbitals.

• Each of the Фn(r) is associated with a one-electron orbital energy εn.

• The orbital approximation allows a many-electron wave function to be written as a product of one-electron wave functions.

nnn rrrrrr .....,....,, 221121



21.2 Introducing Electron Spin21.2 Introducing Electron Spin

• Electron spin plays an important part in formulating the Schrödinger equation for many-electron atoms.

• The spin operators follow the commutation rules and have the following properties:

1**

2ˆ ,

2ˆ

12

1

21ˆ

12

1

21ˆ

222

222

dd

hhms

hhms

hsshs

hsshs

szsz



21.3 Wave Functions Must Reflect the 21.3 Wave Functions Must Reflect the Indistinguishability of Indistinguishability of Electrons Electrons

• There are 2 types of wave functions with respect to the interchange of the two electrons:

1. Symmetric wave function 2. Antisymmetric wave function

Postulate 6Wave functions describing a many-electron

system must change sign (be antisymmetric) under

the exchange of any two electrons.




• Postulate 6 is known as the Pauli exclusion principle.

• It states that different product wave functions of the type must be combined such that the resulting wave function changes sign when any two electrons are interchanged.

• Wave function is zero if all quantum numbers of any two electrons are the same.

• A configuration specifies the values of n and l for each electron.



Example 21.1Example 21.1

Consider the determinant

Evaluate the determinant by expanding it in the cofactors of the first row.

b. Show that the value of the related determinant

in which the first two rows are identical, is zero.723

124

124

723

124

513




c. Show that exchanging the first two rows changes the sign of the value of the determinant.



SolutionSolution

a. The value of a 2x2 determinant

We reduce a higher order determinant to a 2 x 2 determinant by expanding it in the cofactors of a row or column (see the Math Supplement). Any rowor column can be used for this reduction, and all will yield the same result.

bcaddc

ba



SolutionSolution

a. The value of a 2x2 determinant

We reduce a higher order determinant to a 2 x 2 determinant by expanding it in the cofactors of a row or column (see the Math Supplement). Any rowor column can be used for this reduction, and all will yield the same result.

bcaddc

ba



SolutionSolution

For the given determinant,

3361152121074

23

1311

73

5312

72

5114

723

513

124

c.

068132822144

23

2411

73

1412

72

1214

723

124

124

b.

368532812143

23

2415

73

1411

72

1213

723

124

513

312111

312111

312111




• Pauli exclusion principle requires that each orbital have a maximum occupancy of two electrons.



21.4 Using the Variational Method to Solve the 21.4 Using the Variational Method to Solve the Schrödinger Schrödinger Equation Equation

• Hartree-Fock self-consistent field method combined with the variational method is used to approximate eigenfunctions and eigenvalues of total energy for the many-electron atom.

• Variational theorem states that the energy is always greater than or equal to the true energy.

0*

ˆ*E

d

dHE



21.5 The Hartree-Fock Self-Consistent Field 21.5 The Hartree-Fock Self-Consistent Field MethodMethod

• Hartree-Fock method allows the best one-electron orbitals and the corresponding orbital energies to be calculated.

• The one-electron Schrödinger equations have the form

where = central field approximation

• The energy calculated with the variational method is greater than the true energy.

nirrrm

hiiiiii

effii ,....,1 ,

22

2

ieffi r




The effective nuclear charge seen by a 2s electron in Li is 1.28. We might expect this number to be 1.0 rather than 1.28. Why is larger than 1? Similarly, explain the effective nuclear charge seen by a 2s electron in carbon.



SolutionSolution

The effective nuclear charge seen by a 2s electron in Li will be only 1.0 if all the charge associated with the 1s electrons is located between the nucleus and the2s shell. As Figure 20.10 shows, a significant fraction of the charge is located farther from the nucleus than the 2s shell, and some of the charge is quite closeto the nucleus. Therefore, the effective nuclear charge seen by the 2s electrons is reduced by a number smaller than 2.



SolutionSolution

On the basis of the argument presented forLi, we expect the shielding by the 1s electrons in carbon to be incomplete and we might expect the effective nuclear charge felt by the 2s electrons in carbon to be more than 4. However, carbon has four electrons in the n=2 shell, and although shielding by electrons in the same shell is less effective than shielding by electrons in inner shells, the total effect of all four n=2 electrons reduces the effective nuclear charge felt by the 2s electrons to 3.22.



21.6 Understanding Trends in the Periodic Table from 21.6 Understanding Trends in the Periodic Table from Hartree-Hartree- Fock Calculations Fock Calculations

• Main results of Hartree-Fock calculations for atoms:

1. Orbital energy depends on both n and l.2. Electrons in a many-electron atom are

shielded from the full nuclear charge.3. Ground-state configuration for an atom

results from a balance between orbital energies and electron-electron repulsion.



21.6 Understanding Trends in the Periodic Table from 21.6 Understanding Trends in the Periodic Table from Hartree-Hartree- Fock Calculations Fock Calculations

• 2 parameters calculated using the Hartree-Fock method:

1. Covalent atomic radius 2. Degree to which atoms will accept or donate

electrons to other atoms in a reaction

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Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 21 Many-Electrons Atom