Solution of the Schrodinger Equation for multi electron atomic systems cannot be done with great
precision.
It is because of the repulsion energy terms of the
potential energy of such systems cannot be
handled mathematically accurately, analytically.
Many Electron Atoms
Chapter 21
Approximate (numerical) methods however handle
such systems numerically.
Such numerical methods has allowed to consider the
complications brought about in multi-electron systems;
such as indistinguishability of electrons, electron spin,
interaction between orbital & spin magnetic moments.
Time-independent Schrödinger Equation for H atom:
∇=
Wavefunction
Separation of variables
KE
PE
∇ el2 =
∂
∂
∂
+ +
∂x ∂y ∂z
and ∇ 2 =
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
KE of the electron
∂2
∂2
∂2
+ 2+ 2
2
∂x ∂y
∂z
H atom
PE
∇ el2
KE
H atom: PE
r
PE
PE term is spherically
symmetric; 1/r
r
H atom
He Atom – two electron system
Orbital Approximation:
∇ el2 ,1 + ∇el2 ,2
As the number of electrons the ri’s increase and so
are the number of rij’s.
Vel ,1 + Vel ,2
= r12
V1,2
SE for He atom:
He atom
V1,2
Insoluble analytically.
They are not spherically symmetrical and appears
in the repulsive potential energy terms between
electrons. Such repulsive terms with rij’s cannot be
handled mathematically in an analytical manner, only
numerically.
Therefore a wave function relating all coordinates of
n-electrons as a single function would be impossible.
el 2
SE
⇒
H atom
r
el 1 - distribution
If there is a way to treat all the electrons as
independent, with no rij appearing between electrons,
the solution of SE would yield a wave functions for
each electron.
r2
r2
r1
Now, the effect of electron 1 (el1) on el2 would be a
repulsion that minimize the attractive force on the el 2
by the nucleus. The repulsion would be spherically
symmetric!.
He atom
In the orbital approximation, the many (n) electron
wave function is expressed as a product of functions
where each function depend only on the coordinates
of one electron.
The functional form of each one electron orbitals φi (ri)
are similar to the H atom orbitals ψ nlml(r,θ,φ); like 1s,
2s, 2p etc., with associated energies.
Place the second electron in the spherically symmetric
potential field due to nucleus and the spherically symmetric
electron distribution from el 1.
H like 1s
of He atom
φi (ri) = ‘orbital’
This approach makes solution of an n-electron wave
function from SE, to solving n, one-electron SE
equations.
Solution of which would yield n, φ’s (atomic orbitals)
H like 1s of
n electron
atom
1 Z 
 
π  a0 
1 ζ 
 
π  a0 
3/ 2
−
e
3/ 2
−
e
Zr
a0
ζ<Z
ζr
a0
ζ = effective nuclear charge (< Z) felt by the electron.
el 2
Z
ζ<Z
el 1 distribution
to use for Veff calculation.
The effect of el 1 on el 2 is like smearing the electrons
e1 uniformly around the atom. It makes the effective
nuclear charge < nuclear charge, Z. It would be the
same for el 1 due to el 2.
el 1
el 2 distribution
to use for Veff calculation.
Finding ζ is an iterative process iteration done until
it is self consistent.
Electron Spin
z
In multi-electron systems the electron spin plays
an important role in formulating the SE.
s
Electron spin with has two intrinsic states, two
z-components of spin momentum, as shown below,
1
1
+ ℏ or - ℏ .
2
2
1
sz = ± ℏ .
2
The intrinsic electron spin is a vector (angular
momentum) s. with q.n. ms = +1/2 and ms = -1/2
and the respective spin wavefunctions α and β
(α & β are orthogonal).
sz
sz
s
One electron wave function (a wave function
associated with one electron) will then have
a set of four quantum numbers.
n, l , ml and ms i.e.ψ n ,l ,ml ,ms ( r ,θ , φ ).
ms = +
1
2
ms = −
1
2
Spin function is not dependent on r , θ or φ .
and is intrinsic.
orthogonality
Spin variable
1sα
So the wave function of which the space part is;
1sβ
spatial
1s wave functions
redefined – spin orbital.
spin
energy
The wave function describing an electron in an atomic
orbital has two parts, space and spin parts
e.g, 1s(1)α(1), 1s(1)β(1); an electron ‘labeled’ 1 in
1s space orbital with α spin and electron ‘labeled’ 1 in
1s space orbital with β spin.
must enjoin with its spin part to form the
total wave function.
For He a two electron atom ψ(r1, r2)= ;
1sα
1sβ
1s*
1s
The indistinguishability of electrons makes
identifying electrons in an atomic system with labels
like 1, 2, .. etc meaningless;
ψ (1, 2) = φ1 (1)α (1)φ2 ( 2)β ( 2)
ψ ( 2,1) = φ1 ( 2)α ( 2)φ2 (1)β (1) symmetric
interchange of elecrtrons
1sα
1sα
and also if ψ is expressed as
ψ ( 2,1) = −φ1 ( 2)α ( 2)φ2 (1)β (1) antisymmetric
is acceptable because ψ 2 would be the same
1s*
1s
in both cases.
Postulate 6 deals with the indistinguishable
character of the electrons in a multi-electron system
after Pauli.
φ1s (1)α (1)φ1s ( 2)β ( 2)
Postulate 6 requires the wavefunction be
anti-symmetric (changes sign) with respect to
exchange of any two ‘electron labels’
φ1s ( 2)α ( 2)φ1s (1)β (1)
Linear combination of above - a better representation.
A [φ1s (1)α (1)φ1s ( 2)β ( 2) + φ1s ( 2)α ( 2)φ1s (1)β (1)]
But Not anti-symmetric.
Anti-symmetric!!
Slater Determinant (total wave function – ground state)
φ (1)α (1)
ψ (1, 2) = A 1s
φ1s ( 2)α ( 2)
φ1s (1)β(1)
φ1s (2)β(2)
1 φ1s (1)α (1)
2 φ1s ( 2)α ( 2)
φ1s (1)β(1)
φ1s (2)β(2)
ψ (1, 2) =
total wave function
Anti symmetric character
is in engrained in the spin part
of the total wave function.
Each column same spin orbital
Each row same electron label
n = # electrons
1s12s1
m = n/2 (n even) or =(n/2 +1/2) (n odd)
(total wave function – excited states He)
3Li
(ground state)
If the three electrons are placed in the same orbital
the total wave function would be;
φ1s (1)α (1)
ψ (1, 2, 3) =
φ1s ( 2)α ( 2)
3
φ1s (3)α (3)
1
1
2
3
4
φ1s (1)β(1)
φ1s (2)β(2)
φ1s (3)β(3)
φ1s (1)α (1)
φ1s (2)α (2)
φ1s (3)α (3)
1 ψ (1, 2) = A
φ1s (1)α (1)
φ1s ( 2)α ( 2)
φ2 s (1)β(1)
φ2 s (2)β(2)
2 ψ (1, 2) = A
φ1s (1)α (1)
φ1s ( 2)α ( 2)
φ2 s (1)α (1)
φ2 s (2)α ( 2)
ψ (1, 2, 3) = 0 No wave function at all !!
3 ψ (1, 2) = A
φ1s (1)β (1)
φ1s ( 2)β ( 2)
φ2 s (1)β(1)
φ2 s (2)β(2)
4 ψ (1, 2) = A
φ1s (1) β (1)
φ1s ( 2)β ( 2)
φ2 s (1)α (1)
φ2 s (2)α (2)
i.e. three electrons (or more) cannot occupy the
same space orbital, max number of electrons in a
space orbital is two; Pauli Exclusion Principle.
The space part of the wave function is the hard to
determine in any multi electron system.
Approximate Wave Functions and the Variation
Method
They are H like functions but are not the same.
For many situations it is not practical to obtain a
wave function by the solution of the wave equation
that describes the system.
H like functions on different atoms are not easy to
integrate as well.
Yet, it is possible to perform calculations using the
Variation Theorem.
Variation Theorem
Energy associated with a wave function ψ 0 (true,
exact wave function) is calculated using;
E0 =
∫ψ Hψ dτ
∫ψ ψ dτ
*
0
0
*
0
0
For any other (trial) wave function ψ which is an
approximation (a function constructed to emulate
the state ψ 0) the calculated energy would be;
E=
∫ψ
∫ψ
*
trial
Hψ trial dτ
ψ trial dτ
*
trial
≥ E0
Improving the constructed trial function ψ trial would
improve the calculated E and in it’s limit will be equal
to E0.
The trial functions usually contain parameters and
finding the best parameters is the objective of the
variation method.
Variation theorem provides the basis for the variation
method, which states any trial function other than the
true function would give an energy E that is higher
than the true energy E0.
A trial function will be constructed with adjustable
parameters so that it could be varied to improve the
‘trial function’.
The best parameterized parameters (optimal values)
would give the lowest (minimum) energy
corresponding to the trial function, closer to E0.
In the case of atoms the functions of the form of the
H atom solutions is a good starting point.
Set α = 0 and calculate;
E1 = 0.125
h2
↔ψ 0
ma 2
E=
∫ψ
∫ψ
*
trial
Hψ trial dτ
ψ trial dτ
*
trial
Illustration: Particle in a Box
Fact:
H =−
ℏ2 d 2
2m dx 2
E1 =
h2
h2
=
0
.
125
8ma 2
ma 2
An acceptable trial wave function for n=1 state is;
Changing α would change the energy estimated.
Variation Method
The best value for α gives the lowest possible energy
(minimizes energy) for the trial function. Variation
method allows us to find the ‘optimized value for the
parameter’ - parameterization.
a. Find the expression for energy in terms of the
parameter (s)
b. Partial differentiation w.r.t each parameter
and set the resulting expression to zero.
c. Solve for parameters.
Illustration: Particle in a Box - continuation
Differentiating E w. r. t. α and setting it to zero gives
α = -5.74 and -0.345; latter gives the minimum
energy.
E1 = 0.127
h2
ma 2
E1 = 0.125
h2
↔ψ 0
ma 2
Completing the integration gives an expression for E.
Self-consistent field theory
α=0
An important unsolved problem in QM - how to deal
with indistinguishable, interacting particles.
The key to success of the
Variation Method is the
selection of a trial function.
The basic problem is that if particles interact, that
interaction must represented in the Hamiltonian.
α= -0.345
So until we know where the particles are, we can’t
write down the Hamiltonian, and until we know the
Hamiltonian, we can’t tell where the particles are.
Hartree-Fock theory
Hartree: To solve the Schrödinger equation for an
electron moving in the potential of the nucleus and
all the other electrons, start with a guess for the
trial electron charge density.
Solve n/2 one-particle Schrödinger equations to
obtain n electron wave functions.
Then construct the potential for each wave function
from that of the nucleus and that of all the other
electrons, symmetrize it, and solve the n/2
Schrödinger equations again.
Fock: Improved on Hartree’s method by using the
properly anti-symmetrised wavefunction (Slater
determinant) instead of simple one-electron
wavefunctions.
This accounts for the exchange interaction.
The methodology is used (HF-SCF) for molecules.
This is a variational method, so wherever we refer to
wavefunctions, we assume that they are expanded in
some appropriate basis set.
Written as an algorithm.
HF-SCF Method
el 2
el 1 distribution
to use for Veff calculation.
Guess ψ
Improved
old ψ
improved ψ
el 1
el 2 distribution
to use for Veff calculation.
ψ !!
Solving SE for many electron systems.
Use orbital approximation and Pauli exclusion principle.
Every electron feels the spatially averaged electron
charge distribution by the rest of the (n-1) electrons.
Antisymmetric wave function (as Slater determinant)
No rij terms, so spherically symmetric potential exist;
angular part of the orbitals are same as that of H atom.
Every φj is a modified H atom orbital.
Radial part φj(r) varies from case to case and the
variation method allows us to find the optimum
parameters of the radial part of wave function.
The functions for φj(r)
Each φj(r) of the determinant is constructed as a
linear combination of some well behaved (easily
mathematically handled ) basis functions fi(r).
e.g. acceptable φ(r) for 1s of H
1s
Variational parameters
m=3
Variational parameter
suitable
Less suitable
best fit 1s
sum of 3 Gaussians
e.g. acceptable φj(r) for 2p.
2p
Variational parameters
f i ( r ) = N i r exp( −ζ i r / a0 )
m=4
Hartree Fock radial functions can be used to obtain
the radial probability for multi-electron atoms.
ni = # electrons in the shell.
P(r)
In multi-electron atoms
the energy of an orbital
depends on (n + l),
for same (n + l) values,
the higher n value takes
precedence, unlike in
H atom (n only)