Chem 373- Lecture 19: Many Electron Atoms

8/3/2019 Chem 373- Lecture 19: Many Electron Atoms

1/23

Lecture 19: Many Electron Atoms

The material in this lecture covers the following in Atkins.The structure of many-electron atoms

13.4 The orbital approximation

(a) The helium atom

(b) The Pauli principle(c) penetration and shielding

(d) the building-up principle

(e) The configuration of ions

(f) Ionization energies and electron affinities

13.5 Self-consistent field method

Lecture on-line

Many electron atoms (PDF Format)Many electron atoms (PowerPoint)

Handout for this lecture


2/23

The Helium atom and Helium like ions

Or

The two electron atom

We shall consider an atom made up of two electronsand a nucleus of charge Z

X

Z

Y

ee12

r2

r1

r12

Z

Electronic Structure of Many -Electron Atoms He atom


3/23

We might also describe the two electrons in sphericalcoordinates

X

Z

Y

ee12

r2

r1

r12

Z

1

1 2

2

Electron #1 : (r1,1,1)Electron #2 : (r2,

2,2)

The Hamiltonian of the system is given by

H =

h2 2

12 412

m

Ze

re o+

h2

22

2

22 4m

Ze

re o+

e2

124or



4/23

Here

ho(1) =

h2

12 2

12 4m

Ze

re ois the Hamiltonian of a single electron in the field of a

nucleus of charge Z

X

Z

Y

e1

r1

Z

1

Same for

ho(2) =

h2

22

2

22 4m

Ze

re o

X

Z

Y

e2

r2

Z

2

2



5/23

The term e2

124orrepresents repulsion between

two electrons at the distance r 12 .

Y

Z

Y

ee12

r2

r1

r12

Z



6/23

Electronic structure of many - electron atoms

For a many electron atoms wehave the exact Hamiltonian

H

m

Ze

r

e

rei

o i j

n e

o iji j

n

i

n=

+

>=

[ ]h

22

2

1 2 4 4

H h Z ie

r

Here

h Z im

Ze

r

j

n e

o iji j

n

i

n

ei

o i

= +

=

>=( , )

( , )

4

2 4

1

22

2

h

Electron electronrepulsion

between

electron i andelectron j


7/23

We can approximately add this potential to the attractive potential

due to the nucleus as V (r) to modify h asio( )i

Electronic Structure of Many -Electron Atoms Shielding

h'( , ) ( )Z im

Ze

rV r

ei

o ii i=

+

h2 2 2

2 4i = 1,2,4,....

Repulsion fromother electrons

H H h Z ii

N

i = =' ' ( , )1


8/23

Electronic Structure of Many -Electron Atoms

For the approximate Hamiltonian

H' = h'

and the corresponding Schrdinger equation is

H' (1,2,3,4..,n) = E (1,2,3,4..,n)

i ( , )Z i

We shall now show that the the many - electronwavefunction (1,2,3,..) that is a solution to theabove Schrdinger equation can be written as

(1,2,3,..,n) = 1 2 3 4 n ( ) ( ) ( ). ( ).... ( )r r r r rr r r r rn1 2 2 4

Orbitalapproximation

Where is a solution to :

or each i = 1,2, ..n, and

i i i i i i i

e i o i i i

r r r

f

Z i m

Ze

r V r

h

h'

i'( ) ( ) ( )

( , ) ( )

=

=

+

h2

22

2 4


9/23


We have(1, 2, 3, ..,n) = { 1 2 3 4 nH' h'i

i

( )} ( ) ( ) ( ). ( ).... ( )i r r r r rn r r r r r1 2 2 4

= h'

ii

( ) ( ) ( ) ( ). ( ).... ( )i r r r r rn 1 2 3 4 n

r r r r r

1 2 2 4

= i

i nr r r r r 1 2 3 4 n( ) ( ) ( ). ( ).... ( )r r r r r1 2 2 4

= { } ( ) ( ) ( ). ( ).... ( ) i nr r r r r

i1 2 3 4 nr r r r r1 2 2 4

ThusH' (1, 2, 3,..,n) = E (1, 2, 3,..,n)

E = { i

i=1

n }

Orbitalapproximation


10/23

The

i im

Ze

rV r r ri

ei

o ii i i i i i i

solutions to the equations

h'i

( ) ( ) ( ) ( ) ( )

=

+

=

h2

22

2 4


+ + ++

+ =

h h2 2

2

2

22

2

4

1

2

{ ) {( )

}

( )

R (r) R (r)R (r)

R (r) R (r)

nl nlnl

nl nl

r r r

Ze

r

l l

mr

V r E

o

i i

can

r Y R r

However ri lm nl i

i

again be written on the form

i ( ) ( . ) ( ), ( )

=now R is a solution to :nl

Shielding


11/23

The

r Y R r

where R r

i limi i i nili i

nl i

final solution for a many - electron atom with nuclearcharge Z can to a good approximation be written as :

has the same analytical form as forthe hydrogenic one - electron atom with atom charge Z.

i ( ) ( . ) ( )

( )

r=


Only throughout Z is replaced by Z - = Znlnl

Shielding constant

Effective nuclear chargeWith

Z enl Z

the energy

Z,n

=

2 4

22h


12/23

n=1

n=2n=3

n=4


1. in orbitals with

quantum number n = n * willbe shielded from the nucleus byelectrons of lower n - quantum

number that are closer to the nuclei.

Thus

increases with n

and Z (the effective nuclear charge)will decrease with n

nl

nl

Electrons

Z = Z -nl nl


13/23


E Z e

nn

nli

o

( )2 42 2 2 232

h

Orbitals with th same n - quantum number have different energy

if l is different in many - electron atoms but the same energy inhydrogenic atoms

3s 3p 3d

E Z en

no

= 2 4

2 2 2 232

h

3s 3p 3d

For a given n

The shielding increases with l

and

the effective nuclear charge Z decreaseswith l.

nl

nl


14/23


Z = Zeff nl' = Z nlZ

Shielding


15/23


He atom

Each orbital (different n, l,m) can only contain

two electrons .The electrons must have different

spins ( (i)/ (i) )

nlm

To construct the wavefunction we use thePauli exclusion principle :

1s(1)

r / a

Z=2

Z= 1.68

Z=1.0

Thus :

(1, 2) = 1s(1) (1)1s(2) (2)

E s s= + 1 1

1 32sHe = . ; Z' = 2 - .32

= -

(Z

32nl

' 2

2

) e

no

4

2 2 2h

Building upprinciple


16/23


Li atom

Each orbital (different n, l,m) can only contain

two electrons .The electrons must have different

spins ( (i)/ (i) )

nlm

According to the Pauli exclusion principle the

K - shell (1s) is full and the next electron must

go into 2p or 2s orbitals of L - shell

2


In many - electron shells orbitals of same n (shell) butdifferent l do not have the same energy.The energy increases with l as the orbitals penetrates

less and less to the nucleiThus within a subshell : s < p < d < f


17/23

2s (Z= 3.0)

2s(Z=1.7)

1s (Z=3)

1s (Z=2.68)

Electronic Structure of Many -Electron AtomsThus the electron configuration

og Li is 1s22

1 2 3 1 1 1 1 1 1 2 1 1

1 2 3 1 1 1 1 1 1 2 1 1

s

s s s

or s s s

( , , ) ( ) ( ) ( ) ( ) ( ) ( )

( , , ) ( ) ( ) ( ) ( ) ( ) ( )

=

=

the radial functions for 1s and 2s

of Li have exponents of 2.68 and

1.7, respectively, rather than 3

For Be we have 1s22 2s

( , , , ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 3 4 1 1 1 1 1 1

2 1 1 2 1 1

=

s s

s s



18/23


px py pz

B : 1s22 22 1s px

C : 1s22 2 22 1 1s p px ypx py pz

N: 1s22 2 2 22 1 1 1s p p px y z px py pz

Hunds rule :

Electrons occupy different orbitals of sub - shellbefore double occupation

Parallel spin prefered


and

configurationof atoms


19/23


O : 1s22 2 2 22 2 1 1s p p px y zpx py pz

F : 1s22 2 2 22 2 2 1s p p px y zpx py pz

Ne : 1s22 2 2 22 2 2 2s p p px y zpx py pz


and

configurationof atoms


20/23


Occupation of the sub - shellsfollow the order

1s < 2s < 2p < 3s < 3p < 4s< 3d < 4p < 5s < 4d< 5p < 6s

with some exceptions


21/23


Ionization potentials

Z,n =

Z e

nl Z

2 4

22h


22/23


Electron affinities

Z,n =

Z enl Z2 422h


23/23

You should unders d the

concept of a shielding cons t

.

tan

tanYou should also know that

depends on a particular

element as well as the n and l

quantum number ( ) and

that the effective charge

is given by

Zn,leff

Z n l

Z n lZ

, ,

, ,=

Be

R Yl m

aware that the orbitals ina many - electron atom can bewritten as

n,l,m(r, , )=R , )n,l

Z

( ) (

,

Be able to construct the

electron configuration foran atom using the Pauliexclusion principle andHundsbuild - up principle

Realize that orbitalswith the samen - quantumnumber but differentl - quatum number have differentenergies in many electron atoms

What you should know from this lecture

Understand the concept ofionization potentials andelectron affinities and how

they are related to shieldingand the effective nuclearcharge

Documents

Chem 373- Lecture 19: Many Electron Atoms