Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid
Chapter 21 Chapter 21 Many-Electrons Atom
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
ObjectivesObjectives
• Using of Variational Method• Introduce Hartree-Fock Self-Consistent
Field Method
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
21.3 Wave Functions Must Reflect the 21.3 Wave Functions Must Reflect the Indistinguishability of Indistinguishability of Electrons 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.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
Example 21.1Example 21.1
c. Show that exchanging the first two rows changes the sign of the value of the determinant.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
21.3 Wave Functions Must Reflect the 21.3 Wave Functions Must Reflect the Indistinguishability of Indistinguishability of Electrons Electrons
• Pauli exclusion principle requires that each orbital have a maximum occupancy of two electrons.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
Example 21.3Example 21.3
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.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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.
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Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 21: Many-Electron Atoms
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