Download Lectures 10-11: Multi-electron atoms System of non

System of non-interacting particles
Lectures 10-11: Multi-electron atoms
o
What is probability of simultaneously finding a particle 1 at (x1,y1,z1), particle 2 at (x2,y2,z2),
etc. => need joint probability distribution.
o
N-particle system is therefore a function of 3N coordinates:
!(x1,y1,z1; x2,y2,z2; … xN,yN,zN)
o
Must solve
o
First consider two particles which do not interact with one
another, but move !
in potentials V1 and V2. The Hamiltonian is
(2)
Hˆ = Hˆ 1 + Hˆ 2
$ h2 2
' $ h2 2
'
= &"
#1 + V1 ( rˆ1 )) + &"
# 2 + V2 ( rˆ2 ))
% 2m1
( % 2m2
(
o Schrödinger equation for
o Two-electron atoms.
o Multi-electron atoms.
o Helium-like atoms.
o Singlet and triplet states.
o Exchange energy.
Hˆ "( rˆ1, rˆ2 ,..., rˆN ) = E"( rˆ1, rˆ2 ,..., rˆN ) (1)
2e 2
4 #$0 ri
r1
x
i = 1,2
o
The eigenfunctions of H1 and H2 can be written as the product:
!
!
!
e1
y
The electron-nucleus potential for helium is
Vi = "
e2
r2
o
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z
"( rˆ1, rˆ2 ) = #1 ( rˆ1 )# 2 ( rˆ2 )
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!
System of non-interacting particles
o
Using this and Eqns. 1 and 2,
o
That is, Hˆ "( rˆ1, rˆ2 ) = E"(rˆ1, rˆ2 ) where E = E1+E2.
o
Application to helium
o
Hˆ "( rˆ1, rˆ2 ) = ( Hˆ 1 + Hˆ 2 )#1 ( rˆ1 )# 2 ( rˆ2 )
= (E1 + E 2 )#1 ( rˆ1 )# 2 ( rˆ2 )
Assuming each electron in helium is non-interacting, can assume each can be treated
independently with hydrogenic energy levels:
13.6Z 2
n2
Total energy of two-electron system in ground
state (n(1) = n(2) = 1)is therefore
!
The product wavefunction is an eigenfunction of the complete Hamiltonian H, corresponding
!to an eigenvalue E which is the sum of the energy eigenvalues of the two separate particles.
o
For N-particles,
o
Eigenvalues of each particle’s Hamiltonian determine possible energies. Total energy is thus
o
!
"( rˆ1, rˆ2 ,...rˆN ) = #1 ( rˆ1 )# 2 ( rˆ2 )L# N ( rˆN )
N
!
E = " Ei
i=1
o
Observed
En = "
Can be used as a first approximation to two interacting particles. Can then use perturbation
theory to include interaction.
!
o
E = E1 (1) + E1 (2)
# 1
1 &
= "13.6Z 2 %
+
(
$ n(1) 2 n(2) 2 '
#1 1&
= "13.6(2) 2 % 2 + 2 (
$1 1 '
= "109 eV
-50
-60
Energy
(eV)
For first excited state, n(1) = 1, n(2) = 2 => E =-68 eV.
-70
-80
-90
-100
-110
!
Neglecting electron-electron interaction
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System of interacting particles
o
For He-like atoms can extend to include electron-electron
interaction:
& "h 2 2
Ze 2 ) & "h 2 2
Ze 2 )
e2
Hˆ = (
#1 "
#2 "
++(
++
4 $%0 r1 * ' 2m
4 $%0 r2 * 4 $%0 r12
' 2m
o
o
z
e2
The final term represents electron-electron repulsion at a
distance r12.
e1
r12
r2
!
o
Wave function for system of interacting particles
The solutions to the equation
can again be written in the form " i ( rˆi ,# i , $ i ) = Rn i li ( rˆi )%li m i (# i , $ i )
r1
o
d 2 Rn i li
! are solutions to
The radial wave functions
y
Or for N-electrons, the Hamiltonian is:
' #h 2
Ze 2
e2 *
Hˆ i" i ( rˆi ) = ))
$ 2i #
+
," i ( rˆi )
4 %&0 ri 4 %&0 rij ,+
( 2m
= E i" i ( rˆi )
dri 2
!
+
2 dRn i li 2µ $
e2 '
+ 2 &E +
)Rn l = 0
r dri
h %
4 "#0 ri ( i i
and therefore have the same analytical form as for the hydrogenic one-electron atom.
x
N
Hˆ = " Hˆ i
o
i=1
' #h 2 2
Ze 2 * N e 2
= ")
$i #
,+"
2m
4
%&0 ri + i> j 4 %&0 rij
i=1 (
! for
Allowable solutions again only exist
N
2
En = "
where Zeff = Z - !nl.
and the corresponding Schrödinger equation is again of the
form
Hˆ "( rˆ1, rˆ2 ,..., rˆN ) = E"( rˆ1, rˆ2 ,..., rˆN )
!
N
where
"( rˆ1, rˆ2 ,..., rˆN ) = #1 ( rˆ1 )# 2 ( rˆ2 )L# N ( rˆN ) and E = " E i
o
Z eff µe 4
(4 #$0 ) 2 2h 2 n 2
Zeff is the effective nuclear charge and !nl is the shielding constant. This gives rise to the shell
model for multi-electron !
atoms.
i=1
!
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!
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!
Shell model
o
Shell model
Electrons in orbitals with large principal quantum
numbers (n) will be shielded from the nucleus by innershell electrons.
Zeff = Z - !nl.
o
!nl increases with n => Zeff decreases with n.
o
!nl increases with l => Zeff decreases with l.
o
In hydrogenic one-electron model, the energy levels
of a given n are degenerate in l:
En = "
o
Z 2µe 4
(4 #$0 ) 2 2h 2 n 2
3s
3p
3d
3s
3p
3d
Not the case in multi-electron atoms. Orbitals with the
same
! n quantum number have different energies for
differing values of l.
2
En = "
n=1
n=2
n=3
n=4
o
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Z eff µe 4
(4 #$0 ) 2 2h 2 n 2
As Zeff = Z - !nl is a function of n and l, the l
degeneracy
is broken by modified potential.
!
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Shell model
Shell model
o
Wave functions of electrons with different l are found to have different amount of penetration
into the region occupied by the 1s electrons.
o
This penetration of the shielding 1s electrons exposes them to more of the influence of the
nucleus and causes them to be more tightly bound, lowering their associated energy states.
o
In the case of Li, the 2s electron shows more penetration inside the first Bohr
radius and is therefore lower than the 2p.
o
In the case of Na with two filled shells, the 3s electron penetrates the inner
shielding shells more than the 3p and is significantly lower in energy.
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Atoms with two valence electrons
Helium wave functions
o
Includes He and Group II elements (e.g., Be, Mg, Ca, etc.). Valence electrons are
indistinguishable, i.e., not physically possible to assign unique positions simultaneously.
o
This means that multi-electron wave functions must have exchange symmetry:
o
He atom consists of a nucleus with Z = 2 and two electrons.
o
Must now include electron spins. Two-electron wave
function is therefore written as a product spatial and a spin
wave functions:
| "( rˆ1, rˆ2 ,..., rˆK , rˆL ,..., rˆN ) |2 =| "(rˆ1, rˆ2 ,..., rˆL , rˆK ,..., rˆN ) |2
which will be satisfied if
z
r2
" = # spatial ( rˆ1, rˆ2 )# spin
"( rˆ1, rˆ2 ,..., rˆK , rˆL ,..., rˆN ) = ±"( rˆ1, rˆ2 ,..., rˆL , rˆK ,..., rˆN )
o
!
o
That is, exchanging labels of pair of electrons has no effect on wave function.
o
The “+” sign applies if the particles are bosons. These are said to be symmetric with respect to
particle exchange. The “-” sign applies to fermions, which are anti-symmetric with respect to
particle exchange.
o
As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be antisymmetric with respect to particle exchange.
!
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As electrons are indistinguishable => ! must be antisymmetric. See table for allowed symmetries of spatial and
spin wave
! functions.
r12
e2
y
e1
r1
Z=2
x
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Helium wave functions: "spatial
Helium wave functions: "spin
o
State of atom is specified by configuration of two electrons. In ground state, both electrons are
is 1s shell, so we have a 1s2 configuration.
o
There are two electrons => S = s1+ s2 = 0 or 1. S = 0 states are called singlets because they
only have one ms value. S = 1 states are called triplets as ms = +1, 0, -1.
o
In excited state, one or both electrons will be in higher shell (e.g., 1s12s1). Configuration must
therefore be written in terms of particle #1 in a state defined by four quantum numbers (called
#). State of particle #2 called $.
o
There are four possible ways to combine the spins of the two electrons so that the total wave
function has exchange symmetry.
o
Total wave function for a excited atom can therefore be written: " = #$ ( rˆ1 )# % ( rˆ2 )
o
Only one possible anitsymmetric spin eigenfunction:
o
But, this does not take into account that electrons are indistinguishable. The following is
therefore equally valid:
" = # $ ( rˆ1 )#% ( rˆ2 )
!
o
Because both these are solutions of Schrödinger equation, linear combination also a solutions:
!
where
!
1/ 2 is
1
"S =
(#$ ( rˆ1 )# % ( rˆ2 ) + # % ( rˆ1 )#$ ( rˆ2 ))
2
1
"A =
(#$ ( rˆ1 )# % ( rˆ2 ) & # % ( rˆ1 )#$ ( rˆ2 ))
2
1
[(+1/2,"1/2) " ("1/2,+1/2)]
2
o
There are three possible symmetric spin eigenfunctions:
!
o
o
(+1/2,+1/2)
Symmetric
1
[(+1/2,"1/2) + ("1/2,+1/2)]
2
("1/2,"1/2)
Asymmetric
triplet
a normalisation factor.
!
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!
Helium wave functions: "spin
o
singlet
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Helium wave functions
Table gives spin wave functions for a twoelectron system. The arrows indicate whether
the spin of the individual electrons is up or
down (i.e. +1/2 or -1/2).
o
Singlet and triplet states therefore have different spatial wave functions.
o
Surprising as spin and spatial wavefunctions are basically independent of each other.
o
This has a strong effect on the energies of the allowed states.
The + sign in the symmetry column applies if
the wave function is symmetric with respect to
particle exchange, while the - sign indicates that
the wave function is anti-symmetric.
The Sz value is indicated by the quantum
number for ms, which is obtained by adding the
ms values of the two electrons together.
S
ms
%spin
Singlet
0
0
1/ 2("1#2 $#1"2 )
Triplet
1
+1
0
!
-1
1/ 2("1#2 + #1"2 )
#1#2 !
!
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"1"2
%spatial
1
("# ( rˆ1 )" $ ( rˆ2 ) + " $ ( rˆ1 )"# ( rˆ2 ))
2
1
("# ( rˆ1 )" $ ( rˆ2 ) % " $ ( rˆ1 )"# ( rˆ2 ))
2
!
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Singles and triplet states
o
Helium terms
Physical interpretation of singlet and triplet states can be obtained by evaluating the total spin
angular momentum (S), where
Sˆ = Sˆ1 + Sˆ 2
is the sum of the spin angular momenta of the two electrons.
o
The magnutude of the total spin and its z-component are quantised: S = s(s + 1)h
!
Sz = m sh
where ms = -s, … +s and s = 0, 1.
o
o
If s1 = +1/2 and s2 = -1/2 => s = 0.
o Therefore ms = 0
(singlet state)
If s1 = +1/2 and s2 = +1/2 => s = 1.
o Therefore ms = -1, 0, +1 (triplet states)
singlet
state
s1=1/2
!
s1=1/2
s2=1/2
s2=-1/2
ms
+1
0
o
Angular momenta of electrons are described by l1, l2, s1, s2.
o
As Z<30 for He, use LS or Russel Saunders coupling.
o
Consider ground state configuration of He: 1s2
o
Orbital angular momentum: l1=l2 = 0 => L = l1 + l2 = 0
o Gives rise to an S term.
o
Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1
o Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet)
o
J = L + S, …, |L-S| => J = 1, 0.
o Therefore there are two states: 11S0 and 13S1
z
triplet
state
s=1
-1
o
(also using n = 1)
But are they both allowed quantum mechanically?
s = 0, ms = 0
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Helium terms
o
o
Helium terms
Must consider Pauli Exclusion principle: “In a multi-electron atom, there can never be more
that one electron in the same”; or equivalently, “No two electrons can have the same set of
quantum numbers”.
o
First excited state of He: 1s12p1
o
Orbital angular momentum: l1= 0, l2 = 1 => L = 1
o Gives rise to an P term.
o
Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1
o Multiplicity (2S+1) is therefore 2(0) + 1 = 1 or 2(1) + 1 = 3
o
For L = 1, S = 1 => J = L + S, …, |L-S| => J = 2, 1, 0
o Produces 3P3,2,1
Consider the 11S0 state: L = 0, S = 0, J = 0
o n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2
o n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = -1/2
o
11S0 is therefore allowed by Pauli principle as ms quantum numbers differ.
o
Now consider the 11S1 state: L = 0, S = 1, J = 1
o
o n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2
o n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = +1/2
Therefore have, n1 = 1, l1 = 0, s1 = 1/2
and
n2 = 2, l2 = 1, s1 = 1/2
o
For L = 1, S = 0 => J = 1
o Term is therefore 1P1
o Allowed from consideration of Pauli principle
o
11S1 is therefore disallowed by Pauli principle as ms quantum numbers are the same.
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No violation of Pauli principle
=> 3P3,2,1 are allowed terms
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Helium terms
Helium Grötrian diagram
o
Now consider excitation of both electrons from ground state to first excited state: gives a 2p2
configuration.
o
Orbital angular momentum: l1 = l2=1 => L = 2, 1, 0
o Produces S, P and D terms
o
o
o
Singlet states result when S = 0.
o Parahelium.
o
Triplet states result when S = 1
o Orthohelium.
Spin angular momentum: s1 = s2= 1/2 = > S =1, 0 and multiplicity is 3 or 1
*Violate
L
S
J
Term
0
0
0
1S
0
1
0
1
1P
1
2
0
2
1D
0
1
1
*3S
1
1
2, 1, 0
3P
2
1
3, 2, 1
*3D
2
1
2,1,0
3,2,1
Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P)
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Exchange energy
o
Exchange energy
o
Need to explain why triplet states are lower in energy that singlet states. Consider
The third term is the electron-electron Coulomb repulsion energy:
& "h 2 2
Ze 2 ) & "h 2 2
Ze 2 )
e2
Hˆ = (
#1 "
#2 "
++(
++
4 $%0 r1 * ' 2m
4 $%0 r2 * 4 $%0 r12
' 2m
= Hˆ + Hˆ + Hˆ
1
o
o
2
The expectation value of the Hamiltonian is
E =
## "
*
spatial
Hˆ " spatial d 3 rˆ1d 3 rˆ2
o
!
The energy can be split into three parts,
E!
= E1 + E2 + E3
where
*
E i = ## " spatial
Hˆ i" spatial d 3 rˆ1d 3 rˆ2
## "
*
spatial
Hˆ 12" spatial d 3 rˆ1d 3 rˆ2
E = E1 + E 2
#1
1&
= "4 E R % 2 + 2 (
$ n1 n 2 '
o
where ER = 13.6 eV is called the Rydberg energy.
!
*
spatial
=
## "
*
spatial
Evaluating this integral gives
Hˆ 12" spatial d 3 rˆ1d 3 rˆ2
e2
" spatial d 3 rˆ1d 3 rˆ2
4 $%0 r12
E12 = D#$ ± J#$
!
where the + sign is for singlets and the - sign for triplets and D#$ is the direct Coulomb energy
and J#$ is the exchange Coulomb energy:
The expectation value of the first two terms of the Hamiltonian is just
!
## "
12
E12 =
o
E12 =
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D"# =
e2
4 $%0
'' &
J"# =
e2
4 $%0
'' &
( rˆ1 )& #* ( rˆ2 )
1
&" ( rˆ1 )& # ( rˆ2 )d 3 rˆ1d 3 rˆ2
r12
( rˆ1 )& #* ( rˆ2 )
1
& # ( rˆ1 )&" ( rˆ2 )d 3 rˆ1d 3 rˆ2
r12
*
"
*
"
The resulting energy is E12 ~ 2.5 ER. Note that in the exchange integral, we integrate the
expectation value!of 1/r12 with each electron in a different shell. See McMurry, Chapter 13.
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Exchange energy
o
The total energy is therefore
Helium terms
#1
1&
E = "4 E R % 2 + 2 ( + D)* ± J)*
$ n1 n 2 '
o
Orthohelium states are lower in energy than the parahelium states. Explanation for this is:
where the + sign applies to singlet states (S = 0) and the -sign to triplets (S = 1).
1. Parallel spins make the spin part of the wavefunction symmetric.
o
Energies of the singlet and!triplet states differ by 2J#$. Splitting of spin states is direct
consequence of exchange symmetry.
2. Total wavefunction for electrons must be antisymmetric since electrons are fermions.
o
We now have,
3. This forces space part of wavefunction to be antisymmetric.
o
Compares to measure value of ground state energy, 78.98 eV.
o
Note:
o Exchange splitting is part of gross structure of He - not a small effect. The value of 2J#$
is ~0.8 eV.
E1 + E2 = -8ER
and
E12 = 2.5ER => E = -5.5ER = -74.8 eV
o Exchange energy is sometimes written in the form "E exchange = #2J$% Sˆ1 & Sˆ 2
which shows explicitly that the change of energy is related to the relative alignment of
the electron spins. If aligned = > energy goes up.
!
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4. Antisymmetric space wavefunction implies a larger average distance between electrons
than a symmetric function. Results as square of antisymmetric function must go to zero
at the origin => probability for small separations of the two electrons is smaller than for a
symmetric space wavefunction.
5. If electrons are on the average further apart, then there will be less shielding of the
nucleus by the ground state electron, and the excited state electron will therefore be more
exposed to the nucleus. This implies that it will be more tightly bound and of lower
energy.
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