Last hour: Electron Spin
• Triplet electrons “avoid each other”, the WF of the system goes to zero if the two
electrons approach each other. Consequence: less Coulomb repulsion in
triplet states than in singlet states of the same electron configuration.
• In multielectron systems, the WFs represented by Slater determinants:
– N electrons  N x N determinant
– Each column contains one of the possible spin-orbitals (e.g. 1s)
– Each row contains these spin-orbitals with the “index” of one electron,
ensuring antisymmetry w.r. to exchange of two electrons.
1
– Normalization factor:
N!
Hartree Self Consistent Field Method
e1
2+
e2
actual situation: electrons with correlated motion
Basic considerations about multielectron atoms:
2 N
2
N

Ze
Hˆ  
j 


2m j 1
j 1 40 r j
Hˆ 0

N
e2
 4 r
j 1 k  j
0 jk
Hˆ '
• Perturbation theory problematic, because H’ is a large perturbation
 either we need high orders of P.T., or the result will be very inaccurate.
• Variational theory better suited. In principle, however, we have to change the
functional form of the trial w.f., not only parameters
 linear variation functions.
• To have a physical picture of the result, we choose the trial w.f.’s to be products of
1-e--w.f.’s. This is not accurate, as H’ is not separable, so the “true” w.f. cannot be
written as a product of n 1-e--w.f.’s !
Hartree Self Consistent Field Method
e1
2+
e2
Start e.g. with H functions
Sort electrons into orbitals, two for each orbital (Pauli Exclusion Principle!)
Result: starting “population” of orbitals with electrons
Hartree Self Consistent Field (SCF) Approach:
Choose a trial wave function 0 with
N
 0  s1 ( r1 ,1 ,1 ) s2 ( r2 , 2 , 2 )sN ( rN , N , N )   s j ( rj , j , j )
j 1
where each sj is a product of a normalized function of rj and a
spherical harmonic:
sj(rj,j,j) = h(rj)·Yℓm(j,j)
Assume that we can treat each electron as moving through an
average charge distribution caused by the other electrons.
Focus on potential energy term for electron j:
sj
2
2
Ze
V j (rj ,  j ,  j )  e 2  
d j 
40 rj
k  j 40 r jk
 e-e-repulsion averaged over e--cloud
Hartree Self Consistent Field Method
e1
2+
approximation: assume that all electrons except one are “smeared out”
only treat average field from electron population of orbitals
solve radial Schrödinger equation to get new shape of orbital
Hartree Self Consistent Field Method
2+
e2
repeat this process, “focusing” once on each electron
result: “new & improved” radial functions for each electron
Central field approximation:
Assume that the actual electric field only depends on r:
2

 d  d
j
V j (r j ) 
0
0
2
j
sin  j V j (r j , j ,  j )

 d j  d j sin  j
0
1

4
2

 d  d
j
0
j
sin  j V j (r j , j ,  j )
0
0
Note:
• 1st approximation: WF=product of one-e- fcts (only true for separable
Hamiltonian)
• 2nd approximation: treat each electron in the average field of all other
electrons (clearly an approximation!)
• 3rd approximation: assume central fields (also clearly an approximation!)
• although sj looks like a H-like w.f., the field is not a simple Coulomb
potential, so h(rj) is not a H-like radial function!
• maintains idea of orbitals for each electron
How to get the wave functions:

Recall that, in principle, we have to change functional form  very impractical for
systematic automated minimization procedure

hj = k ajk k
Expand radial functions in full basis set:
where k are basis functions. Minimize energy by variation of the aj

Slater-type radial functions (STOs) are sometimes used as basis functions:
n  12
 2 
 a 
0

(2n)!
normalization
r
n 1
e
r
a0
 leading term in Laguerre pol.

 = (Z-s)/n is called “orbital exponent”. Determines “size” of radial function

The parameter s is a “screening constant” that reduces the effective nuclear charge.

Another possibility (very common): Gaussian functions mimicking STO’s

Usually, several STO’s or other orbitals needed for each radial function to get the
correct number of nodes, etc.
Hartree Self Consistent Field Method
e1
2+
Calculate total energy
Repeat previous step with “new & improved” functions until
no significant changes from one iteration to the next.
Strategy: Iterative procedure
1. Choose “reasonable” starting wave functions: H-like w.f.’s
2. “Fill” the orbitals paying attention to the Pauli Exclusion Principle, never putting
more than two electrons in the same orbital.
3. Use w.f.’s for electrons to approximate Vj(rj)  makes H^ separable into Hj and Hkj
4. Use Vj to solve the radial Schrödinger equation
 2

 j  V j (r j ) h j (r j )  E j h j (r j )

 2m

5. for improved w.f.’s hj, determine energies of orbitals Ej
6. Return to (3) with improved w.f.’s hj. Repeat until “no significant” change in sj, and Ej
after each new iteration (“self-consistent”).
Note:
Ej represents energy of electron j in the j-th orbital.
Each Ej contains the repulsion with each other electron, e.g., electron k. To avoid
“double-counting” of repulsion energies, the total energy is given by
N
E  Ej  e
j 1
s



N
2
j 1 k  j
2
j
sk d j d k
2
rjk
•Radial w.f.’s of Hartree-orbitals are not H-like radial functions, but the solutions are
still labeled by quantum numbers n, ℓ.
•Angular w.f.’s are the same Yℓm as in H-atom (by design!)
•Set of orbitals with the same n  shell
•Set of orbitals with the same ℓ  subshell
•Filled shells and subshells yield a spherically symmetric probability density (Unsöld’s
Theorem)
Hartree-Fock SCF Method
• Need to include spin explicitly  use Slater determinants as total wave function
instead of just the product
n
 s (r ,
j
j
j
, j )
j 1
• HF-SCF (“configuration”) yields approximate wave function for n-e- atom
HF-SCF ground state energy:
Recall ground state from degenerate perturbation theory for He:
where
and
e2
Ja  a |
40 r12
K ab  a |
e2
40 r12
|a
(Coulomb integral)
|b
(exchange integral)
Ej  Ej
( 0)
 J a  K ab
The effective Hamiltonian of electron j for the HF-SCF method can be written as
( 0)
Fˆ j  Hˆ j   Jˆk ( j )   Kˆ jk
k j
k j
where Jk and Kjk are the operators similar to the Coulomb and exchange integrals, F is
called the Fock operator.
(details see e.g. Szabo and Ostlund “Modern Quantum Chemistry”)
In practice:
•For N electrons, use M spatial orbitals as basis functions, yields 2M spin-orbitals
(M up, M down), 2M ≥ N
•N lowest spin-orbitals are called “occupied”
•2M – N remaining spin-orbitals are called “virtual” or “unoccupied”
•larger and more complete basis set  lower energies
•converges to “Hartree-Fock limit”
Summary:
• Despite many approximations in HF-SCF, the w.f.’s and energies obtained are
reasonably good.
Example:
He(1s2) 
EHF = -77.9 eV
while
Eexact = -79.0 eV
Note that EHF > Eexact (linear variation method!)
• HF-SCF gives us the basis for the buildup principle that determines the ordering of
low energy configurations for multi-electron atoms
• Concept of  as product of 1-e- atomic orbitals (AO’s) is approximate, but our only
chance to discuss multi-electron systems in a simple way.
• The energies of the 1-e- AO’s correspond approximately to the negative of the
ionization energies of these electron, i.e., j is about the energy necessary to
remove an electron from the orbital j (Koopmans’ Theorem).
Comparison of Koopmans’ Theorem with exp.
Ionization Energies
from McQuarrie & Simon: Physical Chemistry