Physical Chemistry 2nd Edition

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Transcript Physical Chemistry 2nd Edition

Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid

Chapter 21

Many-Electrons Atom

Objectives

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

Outline

Helium: The Smallest Many-Electron Atom Introducing Electron Spin Wave Functions Must Reflect the Indistinguishability of Electrons Using the Variational Method to Solve the Schr ö dinger Equation The Hartree-Fock Self- Consistent Field Method Understanding Trends in the Periodic Table from Hartree-Fock Calculations

21.1 Helium: The Smallest Many-Electron Atom • • • Orbital approximation is to express a many electron eigenfunction in terms of individual electron orbitals.





1 ,

2 ,....,

r n



  1

   

1  2 2 .....



 

Each of the

Ф

(r)

is associated with a one electron orbital energy

ε n

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

21.2 Introducing Electron Spin • • Electron spin plays an important part in formulating the Schr ö dinger equation for many electron atoms.

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

ˆ 2  

s s

 

2 2   1 2

ˆ 2  

   

2 2   1 2 1     1    

   

m s h

 * 

    

2  ,

 * 

  1 

m s h

  

2 

21.3 Wave Functions Must Reflect the Indistinguishability of Electrons •

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

Symmetric wave function Antisymmetric wave function Postulate 6

Wave functions describing a many-electron system must change sign (be antisymmetric) under the exchange of any two electrons .

21.3 Wave Functions Must Reflect the Indistinguishability of 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 for each electron.

n

and

l

Example 21.1

Consider the determinant 3 4 1  2 5 1 3 2 7 Evaluate the determinant by expanding it in the cofactors of the first row.

b. Show that the value of the related determinant 4  2 1 4  2 1 3 2 7 in which the first two rows are identical, is zero.

Example 21.1

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

Solution a. The value of a 2x2 determinant

a b





bc c d

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 row or column can be used for this reduction, and all will yield the same result.

Solution a. The value of a 2x2 determinant

a b





bc c d

Solution For the given determinant, 3 4 1  2 5 1  3   1  1  2 2 1 7  1   1  2 4 3 1 7 3 2 7  3   14  2   28  3   8  6    5   1  3  3 4 3  2 2 b.

4 4  2  2 1 1  4   1  1  2 2 1 7    1  2 4 3 3 2 7  4   14  2   28  3   8  6   0 1 7  1   1  3 4 3 c.

4  2 1 3 1 5  4   1  1 1 2 5 7  2   1  2 3 3 5 7  1   1  3 3 3 3 2 7  4  7  10   21  15   6  3   3 1 2

 2 2

21.3 Wave Functions Must Reflect the Indistinguishability of Electrons •

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

21.4 Using the Variational Method to Solve the Schr ö dinger 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.



 

  * * 

 

 

21.5 The Hartree-Fock Self-Consistent Field Method

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

The one-electron Schr ö dinger equations have the form   

2 2



2  

i eff

   

  

i i

 



 

 1 ,....,

• where = central field approximation

 

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

Example 21.3

The effective nuclear charge seen by a 2

s

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 2

s

electron in carbon.



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

Solution On the basis of the argument presented for Li, we expect the shielding by the 1 s electrons in carbon to be incomplete and we might expect the effective nuclear charge felt by the 2 s 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 2 s electrons to 3.22.

21.6 Understanding Trends in the Periodic Table from Hartree Fock Calculations • 1.

Main results of Hartree-Fock calculations for atoms: Orbital energy depends on both n and l .

Electrons in a many-electron atom are shielded from the full nuclear charge.

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 Hartree Fock Calculations • 1.

2 parameters calculated using the Hartree-Fock method: Covalent atomic radius Degree to which atoms will accept or donate electrons to other atoms in a reaction