of 23
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
Electronic Structure of Many -Electron Atoms He atom
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
Electronic Structure of Many -Electron Atoms He atom
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
5/23
The term e2
124orrepresents repulsion between
two electrons at the distance r 12 .
Y
Z
Y
ee12
r2
r1
r12
Z
Electronic Structure of Many -Electron Atoms He atom
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
9/23
Electronic Structure of Many -Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
Electronic Structure of Many -Electron Atoms
+ + ++
+ =
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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=
Electronic Structure of Many -Electron Atoms Shielding
Only throughout Z is replaced by Z - = Znlnl
Shielding constant
Effective nuclear chargeWith
Z enl Z
the energy
Z,n
=
2 4
22h
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
12/23
n=1
n=2n=3
n=4
Electronic Structure of Many -Electron Atoms Shielding
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
13/23
Electronic Structure of Many -Electron Atoms Shielding
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
14/23
Electronic Structure of Many -Electron Atoms
Z = Zeff nl' = Z nlZ
Shielding
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
15/23
Electronic Structure of Many -Electron Atoms
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
16/23
Electronic Structure of Many -Electron Atoms
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
Building upprinciple
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
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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
Building upprinciple
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
18/23
Electronic Structure of Many -Electron Atoms
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
Building upprinciple
and
configurationof atoms
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
19/23
Electronic Structure of Many -Electron Atoms
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
Building upprinciple
and
configurationof atoms
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
20/23
Electronic Structure of Many -Electron Atoms
Occupation of the sub - shellsfollow the order
1s < 2s < 2p < 3s < 3p < 4s< 3d < 4p < 5s < 4d< 5p < 6s
with some exceptions
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
21/23
Electronic Structure of Many -Electron Atoms
Ionization potentials
Z,n =
Z e
nl Z
2 4
22h
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
22/23
Electronic Structure of Many -Electron Atoms
Electron affinities
Z,n =
Z enl Z2 422h
8/3/2019 Chem 373- Lecture 19: Many Electron Atoms
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