Lecture 20
Helium and heavier atoms
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Helium and heavier atoms



We use the exact solutions of hydrogenic
Schrödinger equation or orbitals to construct
an approximate wave function of a manyelectron atom, the helium and heavier atoms.
Unlike the hydrogenic atom, the discussion
here is approximate and some rules
introduced can have exceptions.
Spins and antisymmetry of fermion wave
functions start to play a critical role.
Helium and heavier atoms


The Schrödinger equation for hydrogenic
atoms can be solved exactly, analytically.
Those for many-electron atoms and
molecules cannot be solved analytically.
The wave function is a coupled function of
many variables:
(
Y r1,r2 ,… ,rn
)
Coordinates of electron 1
The orbital approximation

We introduce the following approximation
(the orbital approximation):
(
)
( ) ( )
Y r1,r2 ,… ,rn » j1 r1 j 2 r2

Hydrogenic orbital
j n (rn )
For the helium atom, this amounts to
 r1, r2   1 r1 2 r2 
The orbital approximation

The approximation is equivalent to neglecting
interaction between electrons 1 and 2,
Hˆ = -
2
2m
Ñ 2
1
Ze
2
4pe 0r1
-
2
2m
Ñ 2
2
Ze
2
4pe 0r2
+
e
2
4pe 0r12
Hydrogenic electron 1 Hydrogenic electron 2 Interaction

… so that,
Hˆ » -
2
2m
Ñ 2
1
Hˆ 1
Z
-
2
4pe 0r1 2m
Ñ 2
2
Hˆ 2
Z
4pe 0 r2
= Hˆ 1 + Hˆ 2
The orbital approximation
H »-
2
2m
Ñ 2
1
Z
4pe 0 r1
-
2
Ñ 2
2
2m
Z
4pe 0 r2
= H1 + H 2
( )
( ) exact
hydrogenic
H j ( r ) = E j ( r ) problem
( H + H )j ( r )j (r ) =
( H j ( r ) ) j ( r ) + ( H j ( r ) )j ( r ) =
( E + E )j (r )j (r ) Eigenfunction
H1j1 r1 = E1j1 r1
2
2
1
1 1
2
2
1
2
1
2
1
2
2
1
2
1
1
2
2
2
2
2
2
2
2
1
1
The orbital approximation

We construct a helium wave function as the
product of hydrogenic orbitals with Z = 2.
 r1, r2   1 r1 2 r2 

Issue #1: an electron is fermion and fermions’
wave function must be antisymmetric with
respect to interchange (the above isn’t):
Y

(
,rn , ,rm ,
) = -Y (
,rm , ,rn ,
)
Issue #2: each electron must be either spin α
or β (the above neglects spins).
Spins

Let us first append spin factors
  r1 , r2   1  r1  2  r2   (1) (2)
  r1 , r2   1  r1  2  r2   (1)  (2)
  r1 , r2   1  r1  2  r2   (1)  (2)
  r1 , r2   1  r1  2  r2   (1) (2)

None of these is antisymmetric yet
Y
(
,rn , ,rm ,
) = -Y (
,rm , ,rn ,
)
(Anti)symmetrization

Symmetrization:
Sym.
g+ ( x, y ) = f ( x, y ) + f ( y, x )
g+ ( y, x ) = f ( y, x ) + f ( x, y ) = g+ ( x, y )

Antisymmetrization:
Antisym.
g- ( x, y ) = f ( x, y ) - f ( y, x )
g- ( y, x ) = f ( y, x ) - f ( x, y ) = -g- ( x, y )
Antisymmetric function
Antisym.
Sym. Antisym.
h- ( x, y ) = g+ ( x, y ) g- ( x, y )
{
}
h- ( y, x ) = g+ ( y, x ) g- ( y, x ) = g+ ( x, y ) -g- ( x, y ) = -h- ( x, y )
Helium wave functions
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 )a (1)a (2) Already symmetric and cannot
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) b (1)b (2) be made antisymmetric
Neither sym.
or antisym.
Antisym.
Sym.
{
}
Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2)
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) a (1)a (2)
1
2
1
{
Antisym.
{
Sym.
1
2
2
1
2
2
1
}
Sym.
}
Antisym.
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) {a (1)b (2) + b (1)a (2)}
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 )a (1)b (2) Neither sym.
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) b (1)a (2) or antisym.
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) + j1 ( r2 )j 2 ( r1 ) {a (1)b (2) - b (1)a (2)}
Triplet states

These three have the same spatial shape –
the same probability density and energy –
triply degenerate (triplet states)
Antisym.
Sym.
{
}
Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2)
Y ( r ,r ) µ {j ( r )j ( r ) - j ( r )j ( r )} {a (1)b (2) + b (1)a (2)}
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) - j1 ( r2 )j 2 ( r1 ) a (1)a (2)
1
2
1
1
2
2
1
2
2
1
1
2
1
1
2
2
1
2
2
1
φ1 and φ2 cannot have the same spatial
form (otherwise this part becomes zero).
Electrons 1 and 2 cannot be in the same
orbital or same spatial position in triplet
states (cf. Pauli exclusion principle)
Singlet state

There is another state which is nondegenerate (singlet state):
{
Sym.
}
Antisym.
Y ( r1 ,r2 ) µ j1 ( r1 )j 2 ( r2 ) + j1 ( r2 )j 2 ( r1 ) {a (1)b (2) - b (1)a (2)}
Opposite spins
φ1 and φ2 can have the same spatial
form because the anti-symmetry is
ensured by the spin part. Electrons 1
and 2 can be found at the same
spatial position.
Energy ordering

For the helium atom, depicting α- and β-spin
electrons by upward and downward arrows,
we can specify its electron configurations.
2s
2s
2s
1s
1s
1s
Triplet states
Singlet state
A
Singlet state
B
The orbital approximation
Ze
Ze
e
2
H =Ñ Ñ2 +
2m
4pe 0 r1 2m
4pe 0 r2 4pe 0 r12
2
2
2
2
2
2
1
Hydrogenic electron 1 Hydrogenic electron 2
Interaction
2s
2s
1s
1s
2s
Triplet states
159856 cm-1
Singlet A
166277 cm-1 1s
0 cm-1 Singlet B
Beyond helium …



A many-electron atom’s groundstate configuration can be obtained
by filling two electrons (α and β
spin) in each of the corresponding
hydrogenic orbitals from below.
When a shell (K, L, M, etc.) is
completely filled, the atom
becomes a closed shell – a
chemically inert species like rare
gas species.
Electrons partially filling the
outermost shell are chemically
active valence electrons.
Shielding



In a hydrogenic atom (with
only one electron), s, p, d
orbitals in the same shell
are degenerate.
However, for more than
one electrons, this will no
longer be true.
Nuclear charge is partially
shielded by other
electrons making the outer
orbitals energies higher.
Shielding

Electrons in outer, more
diffuse orbitals experience
Coulomb potential of nuclear
charge less than Z because
inner electrons shield it.
Zeff  Z  
Effective nuclear charge
Shielding


The s functions have greater probability
density near the nucleus than p or d in the
same shell and experience less shielding.
Consequently, the energy ordering in a shell
is
Lower
40
energy
s< p<d < f <
3p
30
3d
20
3s
10
0
5
10
15
20
Aufbau principle

This explains the well-known building-up
(aufbau) principle of atomic configuration
based on the order (exceptions exist).
1s < 2s < 2 p < 3s < 3p < 4s < 3d < 4 p < 5s < 4d < 5p < 6s <
6s 6p 6d 6f 6g 6g
5s 5p 5d 5f 5g
4s 4p 4d 4f
3s 3p 3d
2s 2p
1s
Hund’s rule

An atom in its ground state adopts a
configuration with the greatest number of
unpaired electrons (exceptions exist) – why?
2p
2s
1s
Oxygen
Hund’s rule

Spin correlation or Pauli exclusion rule
explains Hund’s rule.
{
}
Y ( r1 ,r2 ) µ j 2 p ( r1 )j 2 p ( r2 ) - j 2 p ( r2 )j 2 p ( r1 ) a (1)a (2)
y
z
y
z
2p
Y ( r1 ,r2 ) µ j 2 p ( r1 )j 2 p ( r2 ) {a (1)b (2) - b (1)a (2)}
y
y
Two electrons can be in the same
spatial orbitals and the same position
Spatial part is antisymmetric and the two electron
cannot occupy the same spatial orbitals or the same
position – energetically more favorable
Summary



We have learned the orbital approximation,
an approximate wave function of a manyelectron atom that is an antisymmetric
product of hydrogenic orbitals.
We have learned how the (anti)symmetry of
spin part affects the spatial part and hence
energies and the singlet & triplet helium
atom and explains Hund’s rule.
Shielding explains the aufbau principle.