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Notes 7
Slater Determinant
Overall wavefunctions that satisfy the Puli Principle can be obtained through the Stater
determinant.
Remember that since electrons are fermions, another way of expressing the wavefunction
of ground state Helium:
(1,2) = 1s(r1)1s(r2)- spin(1,2) = 1s(r1)1s(r2)-spin(1,2)
= 1/21 / 2 1s(r1)1s(r2)(1)(2)-(1)(2)]
the terms in the determinant can be simplified by introducing the concept of a spinorbital,
which is a joint space-spin orbital
1sα (1) = 1s(r1)(1)
1s(r2)(1) = 1sβ (1)
Then the ground state can be expressed from the following determinant
If the labels of the electrons 1 and 2 are interchanged then the rows are interchanged.
It is generally a property of determinants that when the rows are interchanged, then it
results in a change in the sign of the determinant. This reflects the overall antisymmetry.
Now suppose the electrons have the same spin and occupy the same orbitals. The Slater
determinant for this would be:
Note that a property of determinants is that determinants with two identical columns have
the value 0. So this state does not exist.
Note that when two electrons are in an antisymmetrical spatial state, then the electrons
avoid each other, and their spin state must be symmetrical and S=1 overall.
So parallel spins tend to avoid one another, and is called spin correlation. Spin
correlation is only an indirect consequence of spin working through the Pauli principle.
If the spins of the electrons are parallel, then the Pauli principle requires them to have an
antisymmetrical spatial wavefunction. This is a purely Coulombic effect due to the fact
that the electrons must stay away from one another.
Many Electron Atoms
A crude description of the ground state of the helium atom is 1s2 with both electrons in
hydrogenic 1s orbitals with Z=2.
Could still write as orbital approximation
For Neon:
 = (1/10!)½ det | 1sα (1) 1sβ (2) 2sα (3) 2sβ (4) …2pβ (10) |
Electronic Structure of a multielectron atom consists of a series of shells labeled by n
Each shell consists of n subshells with a common value of l, there are 2l+1 orbitals in a
subshell.
In hydrogen all subshells of a given shell are degenerate, but the presence of electronelectron repulsion will change that.
Improvement requires taking into account the repulsion between the electrons.
Penetration and Shielding
The explanation of the order of the subshells is based on the central field approximation.
The complicated electron-electron repulsion terms given by
V = e2 / (40r1i) = jo  1/rij
can be simulated by using an effective nuclear charge Zeff e.
using:
V =  jo / r1
jo = e2 /(4o)
where -e is an effective charge that repels the charge -e of the electron of interest.
Zeff = Z - 


 is the nuclear screening constant. (See the table on page
Justification for this large approximation is:
The effective reduction of the nuclear charge due to the presence of other electrons in an
atom is called the shielding.
It magnitude is determined by the amount that the electron penetrates to the core regions
of the atom and close to the nucleus. Really  should be a function of the r but we can
get by using an average value of .
What provides us with information on where the electron is most likely to be in a certain
orbital?
This shifts the energy of the orbitals relative to one another.
The rationalization of the observed configurations is expressed in terms of the buildingup or Aufbau principle.
Two features of Aufbau should be remembered.
1) Electrons will occupy separate degenerate orbitals before "pairing-up".
2) When they do this it is with parallel spin. (Hunds Rule) Traced to spin correlation.
Can rationalize size, ionization energy, and electronegativity from periodic table.
The orbital approximation is extremely primitive. Need to have available a much better
set of approximate atomic orbitals. These then can help with modeling of atoms an
models with more sophisticated approximations
Use Slater Type Orbitals
Constructed by
1. orbital with n, l, ml belonging to a nucleus of an atom of atomic # Z is written
2. The effective principal quantum number, neff, is related to true n by
3. The effective nuclear charge, Zeff, is taken from the Table on page
Two words of CAUTION!
1. STOs with different values of n but the same values of l and ml are not orthogonal.
2. Also ns-orbitals with n>1 have zerto amplitude at the nucleus.
Example:
One can use a short hand for the Slater Determinant by writing its diagonal elements.
(1,2,3, … N) = (1/N!) det |a(1)b(2) … z(N) |
Often need to evaluate matrix elements between the Slater Determinants of either a oneelectron (electric dipole moment) or two-electron (electron-electron repulsion energy).
Slater developed rules for evaluating the diagonal matrix element of one and two-electron
operators and Condon generalized his results to off-diagonal elements. These are known
as the Condon-Slater Rules.
Condon Slater fules
one and two-electron operators
1 = i (ri)
2 = ½ (ri,rj)
The ground-state one-electron excited state, and two-electron excited state wavevunctions
are labeled as:
, mp , mrp q ,
the subscripts denote the spinorbital from which the electron has been promoted, and the
superscripts denote the spinorbital to which it has been promoted.
NO need to consider promotion of more than one electron for a one-electron operator and
of more than two electron for a two-electron operator for all such matrix-elements vanish
due to the orthogonality of the spin orbitals.
The non-vanishing one-electron integrals are:
The non-vanishing two-electron integrals are:
We'll need these.
Of course the best Atomic Orbitals are found by Numerical Solution of
Schrodinger's Eqn.
Use the Self Consistent Field Method originally made by Hartree and improved by Fock
and Slater to include the effects of Electron Exchange. CALLED Hartree-Fock Orbitals.
OVERALL IDEA of Hartree technique is that any one electron moves in a potential
which is a spherical average of the potential due to all the other electrons and the nucleus,
wand this is expressed as a single charge centered on the nucleus. (This is the central
field approximation; but it is not assumed that the charge has a fixed value.)
The Hartree-Fock procedure adds to this nucleus-centered potential a non-classical
exchange term that cannot be identified with a central point charge. Then the
Schrodinger eqn., which is essentially converted into a set of Hartree-Fock eqns., is
integrated numerically for that electron, and the effective potential in which it moves,
taking into account, for instance, the fact that the total charge inside the sphere is defined
by the potion of the electron varies as the distance of the electron from the nucleus varies.
This assumes that the wavefunctions of all the other electrons are already known sot thet
the effective potential can be calculated.
This of course is not in general tru, so the calculation starts out from some approximate
form of the wavefunctions such as approximating them by STOs. The Hartree-Fock eqn.
for the electron is then solved, and the procedure is repeated for all the electrons in the
atom.
At the end of the first round, one has a set of improved wavefunctions for all of the
electrons. These improved wavefunctions are then used to calculate the effective
potential, and the cycle is repeated until an improved set of wavefunctions does not vary
significantly from the wavefunctions at the start of that cycle.
The Hartree-Fock Eqns. are tricky to derive
Hamiltonian is:
The Coulomb operator is Jr is given by
The Exchange-operator, Kr, is given by
Koopman's Theory
Hartee-Fock SCF's (HF-SCF) are not the most refined orbitals that can be obtained. The
true wavefucntion for an atom, depends explicitly on the separations of the electrons, not
just their distances from the nucleus.
The incorporation of the separations rij explicitly into the wavefunction is the know as
Electron Correlation.
Also Relativistic effects could be taken into account and prove to be important for heavy
atoms.
Restricted HF-SCF
Here it is assumed that the spatial component of the spin orbitals are identical for each
member of a pair of electrons for closed-shell states of atoms (assume # of electrons is Ne,
so there are ½ Ne spataial arbitals of the form m(ri).
The Restricted HF (RHF) wavefunction is then given by
o = 1/(Ne!)½ det |aα (1)aβ (2)bα (3) … zβ (3)|
For electrons in Open Shells (Two ways)
Restricted Open-shell Formalism
All electrons except those occupying open-shell orbitals are forced to occupy doubly
occupied spatial orbitals.
For Li this would give RHF of o =
The problem with this is that although the 1s electron has an exchante interaction with
the 2s electron , the 1s electron cannot, which makes the Variational ground state
energy inaccurate.
Unrestricted Open Shell HF (UHF)
Here the electrons are not constrained to the same spatial wavefunctions and for example
in the case of Li the two 1s electrons are not constrained to be in the same spatial
wavefunction.
So o =
This gives a lower variational energy than the open-shell RHF case. Here however the
total spin angular momentum is not well defined. S2 is not well defined.
In practice the S2 is computed and compared with the true value of S(S+1)ħ2 for the
ground state. If there is minimal discrepancy then the UHF method is taken to have
given a reasonable molecular wavefunction. UHF wavefunctions are often used anyway
as a first approximation. May use projection operator on UHF wavefunctions to improve
them and get more accurate S2 values.
Density Functional Theory
Entirely different approach to the calculation of the electronic structures of atoms was
developed by Thomas and Fermi.
Idea is that the kinetic energy and potential energy of the electrons could be directly
related to the electron density, (r), rather than the wavefunction of the electrons.
Rationalization of this idea using the particle in a box model leads to:
The kinetic energy T
T[] = C ∫ (r)5 / 3 dr
called a functional (a single value depends on the global
property of the function (r))
The potential energy due to electron nuclear attraction
VeN[-jo Z ∫ (r)/r dr
where jo = e2 /(4o)
The potential energy due to electron-electron repulsion
Vee[] = ½ jo ∫ (r1) (r2) / ( |r1 - r2| ) dr1 dr2
there the classical Coulombic interaction between a charge -ep(r1)dr1 in the volume
element dr1 at r1 and the charge -e(r2)dr2 in the volume element dr2 at r2. The ½
prevents double counting.
So the TOTAL ENERGY for the atom is
ETF[] = T[] + VeN[Vee[]
It is supposed that the Variation Principle applies here, and that the optimum electron
density corresponds to minimum of the energy functional. There is a constraint which
says that he total number of electrons must be constant as  is varied.
∫ (r) dr - Ne = 0
(A)
including this constraint means:
δ{ETF[] -  ∫ (r) dr } = 0
and thus for a minimum energy
δ ETF[] /δ = 
5/3 C(r)2 / 3 - ξ(r)
(B)
ξ(r) is the electrostatic potential (a potential energy)
ξ(r) = jo Z/r - jo ∫ (r1) / (|r-r1|) dr1
So find  from (B) and insure the constraint (A) is satisfied.
For neutral atoms, this is simpler because at large distances from the nucleus, both (r)
and ξ(r) are zero, so there  = 0 by (B)
This leads to
(r) = (3/(5C))3 / 2 ξ(r)3 / 2 = 23 / 2 /(32 ) ξ(r)
(C)
For an atom the electron density and electrostatic potential have spherical symmetry so
that ξ(r) is function of only r.
ξ(r) = Z (r)/r
Putting this in (C) with change of variable x = r and  = 1.1295 Z1 / 3
See Fig
Lead to E[ = -0.7687 Z7 / 3
DFT Dirac Fermi Thomas Method
One problem above is that there is nothing to account for the effect of electron exchange.
To fix this Vee[p] needs to be written as a sum of two terms: classical and exchange
contributions
Vee[] = J[] - ½ K[]
J[] is the classical part =
expressed in the "one-point" electron density (r) =
and K[] is the exchange part =
This requires the "two-point" density (r,r') =
This lead to C(r)2 / 3 - 2/3 Cex(r)1 / 3 - ξ(r)
We will a more modern version of this.
Many electron atoms - Term Symbols
See fig
Russel Saunders Coupling
j-j Coupling
Selection Rules for Many Electron Atoms
J = 0, 1 J=0 -> J=0 is not allowed
L = 0, +/-1 L=0 -> L=0 is not allowed
l = +/- 1
S = 0
Conservation of Angular momentum
Conservation of Angular momentum of the actual electron excited by photon
Rule regarding S reflects the fact that the electric component of the electromagnetic
field has no effect on the spin moment of the electron
Hunds Rules and Racah Parameters
1. The term with the maximum multiplicity lies lowest in energy.
2. For terms with the same multiplicity, the term with the highest value of L has the
lowest energy.
3. For atoms with less than half-filled shells, the level with the lowest value of J lies
lowest in energy, and if more than half-filled then the one with the highest J lies lowest in
energy.
Atoms in External Fields
Zeeman Effect
Normal
Anomalous
Stark Effect