we've seen that Hartree's method provides a good approximation to the
energies and configuration of electrons in atoms
however we try to extend these ideas to molecules we run into some challenges
the central field approximation is key to the success of our trees method
treating the net force on each electron due to the nucleus and the other
electrons has always directed towards the center of the atom
greatly simplifies the problem 0
in a central feel the angular behaviour the way function is the same as for the
already solved cases the hydrogen atom
all that remains is to solve an equation the single
radio variable are and as in the case of hydrogen
each orbital is characterized by a set of quantum numbers
this allows us to enforce the exclusion principle by simply assigning different
quantum numbers to each electron
however the central field approximation is not generally applica both molecules
all the net attractive force on electron do the polls two or more nuclei
will not be spherically symmetric about some central point and the reports were
forced to do the other electrons
even treated in average sense is also unlikely to have spare kal symmetry
service we will not be able to reduce shriners equation to a single
radio variable are this leaves us with a computationally much more challenging
problem
0 moreover in a non central feel the angular momentum up
individual electrons will not be conserved without angular momentum
quantum numbers
we cannot enforce the exclusion principle by simply assigning different
quantum numbers to different electrons
this is a fundamental issue we need to resolve before we can move forward
so far we stated the Pauli exclusion principle along the lines up
no two electrons can be in the same quantum state when the posh was a
quantum mechanics is that the state have a quantum system
is specified by its way function so rigorously
the exclusion principle should be a statement about the wavefunction
this label the spin of an electron by a disc recorded
ass that has two values up or down in for a two-electron system
the way function would be a function other heat variables the XY and z.
position coordinates
plus the spin courted breaching the two electrons
simplify the notation less represent the four corners of an electron
by a single boldface corn attacks and our way function is
cya x1 x2 and this gives the probability amplitude for finding electron one
with Gordon 61 an electron to with coordinates x two
similar configuration where electron one which we color orange
has scored its a and electron to which we
color green has coordinates be the probability in this configuration
specified by the magnitude squared
up sigh %uh a be if instead it was electron one that had corn its be
telecom to the head according to a and the probability factor would be
magnitude squared up sigh be
K
mathematically these could be different state to the system with different
probabilities
and if instead of electrons we were keeping track of macroscopic objects
like billiard balls
planets it would definitely be different distinguishable States
the state have been as being closer to the Sun than Earth is
very different from that earth being closer to the Sun and Venus
electrons has elementary particles
are indistinguishable we can't physically label their paint them
and given the uncertainty principle there's no way to talk about the
distinct path electron one
purses the PAP electron to inside a molecule
the most that a measurement of a two-electron system could tell us even
in principle
is that there is an electron with court and say
and an electron with court and speak
so the probabilities must be equal the behavior a physical system cannot depend
on how we arbitrarily assigned subscript
indistinguishable particles in a mathematical model 0
to elect one-way function must have the property that its magnitude squared it's
not change its according to the electrons are exchanged
says this is equivalent simply relabeling indistinguishable particles
now what does this imply about the way function itself
if you squared equals C squared what we say about you
NV one possibility is that you equals be
certainly a side x1 x2
equals sire x2 x1 the probabilities will be equal
away function with this property is said to be symmetric with respect to the
exchange accords
but another possibility is that you equals minus p
after all two squared equals four but so does -2 scored
so we could have cya x1 x2
equals minus side affects 2x1 0
away function with this property is said to be antisymmetric with respect to the
exchange accordance
in nature we find two types of elementary particles the beaux-arts
such as the proton have some magic wave functions
and do not obey the exclusion principle the fermions
such as the electron have antisymmetric way functions
and do a baby exclusion principle the exclusion principle follows from this
antisymmetric
for
sample suppose both sets of court x1 and x2
are equal to axe the two electrons have the same spin
an art same location wave function and asymmetry then requires that
science axe axe equals its negative
at but there's only one number that he calls its negative
and thats 0 there is zero probability that two electrons with the same spinner
at the same place
which is particular case have two electrons in the same quantum state
a row and an electron wave function must be antisymmetric with respect to the
exchange accordance
any pair of electrons this is a rigorous statement at the exclusion principle
and forms an additional posh love quantum mechanics but how do we enforce
this condition
elegant solution is given by the Slater determine
due to chance later in heart res method
we represent the wave function as a two-electron system byproduct to one
electron wave functions
a so-called Hartree product slater pointed out that this form a
wavefunction
is not intrinsically and its metric has it is not rigorously satisfy the
exclusion principle
but we can make it an isometric
by subtracting the product with the cornice exchange
no matter what the one-electron way functions are
this to electron wave function is explicitly anti-semitic
it is equivalent to so-called determine it up a matrix
in which the columns correspond to different one electron wave functions
and the rows correspond to different electron cornett
normalizing factor in this case one over the squared
to is usually included let's see how this works by considering the example
two electrons with the same span
occupying hydrogen one ass into s orbitals
we take the 1s and 2s
way functions conform the Slater determine it to electron wave function
consider the situation where one electron is that radius
a but arbitrarily take us to be electron to
and replace are two by a this gives us a way function for the other electron
a function at the single variable or one 0
1r1 equals hey this way function is zero
here will plot this way function for different guys obey which represent by
Red Dot
when the second electron is far from the nucleus
the first electron essentially occupies the 1s orbital
but as the second electron moves towards the nucleus
way function for the first electron changes dramatically
source to always be 0 at the location of the second electron
apply the way function by the radius and square the result
we get the probability density for electron want to be in a particular
radius
which is more physical significance
NC how electron to squeeze as electron ones probability distribution
was to always keep the two-electron separate
three electrons there are three different pairs of electrons we can swap
to have to be careful about our bookkeeping bus represent the Hartree
product shown as 12
3 we can swap electrons 2&3
to get the permutation one 3 to friend symmetry
a single pair swapped term should have a negative sign we can also swap electrons
wanted to
to get the permutation to 13
or we could swap electrons 13 to get the permutation
3 to one 0
we can get two more permutations by swapping a second pair of electrons
it in the 213 permutation we swap electrons 13
we get the permutation 231 0
per to swaps we have a minus minus sign which is a plus like
finally in the permutation 321
we can swap electrons 2&1 to get the permit Asian 31
to
later determined by adding the two positive permutations
231 and 312
followed by subtracting the three negative permutations
132 213 and 321
0 we end up with a total of six terms
three positive and three negative at formally we
right this is the terminate up a three by three metrics which the calls
correspond to different one electron wave functions or orbitals
and the rows correspond to different electron court
normalizing factor in this case is one over the square
sex
turning to look unwieldy for and electrons
there are and factorial terms in this later determined
because there are and factorial permutations are the numbers 1 through
an
in factorial is an times and -1
10gen -2 down to one 0
the water molecule has only ten electrons get 10 factorial is more than
three million
fortunately we can always take our single electron orbitals to be a
so-called
or for normal set in which the projection have any orbital onto another
0
and projection have an orbital onto itself as one in this case the vast
majority
terms in calculations slater determine its r0
we end up with tractable expressions
create develop his method using intuitive concept slater provided a
theoretical foundation for the hearty matted
indeed if one starts with the assumption that the an electron wave function has a
heart re product
and one electron wave functions the Hartleys method can be shown to be
optimal
at the same time slater pointed out that the Hartree product way function
does not have the an asymmetry required by the exclusion principle
what was needed was a way to extend the ideas behind our trees method
two-way function represented by a Slater determine