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Multi Electron Wavefunctions
----------------------------
Pauli Exclusion Principle:
No two electrons in an atom can have the same
four quantum numbers n, l, m_{l}, m_{s} (i.e. occupy
the same quantum state simultaneously). This
is an example of a general principle which applies
not only to electrons but also to other particles
of half-integer spin (fermions). It does not apply
to particles of integer spin (bosons). For two
electrons residing in the same orbital, n, l, and
m_{l}, m_{s} must have opposite spins.
The nature of the Pauli exclusion principle can be
illustrated by supposing that electrons 1 and 2 are
in states a and b respectively. The wavefunction
for the two electron system would be:
χ(a,b) = χ_{1}(a)χ_{2}(b) - The HARTREE PRODUCT
where χ is the product of the spatial (ψ,φ) and
spin (α,β) wavefuntions, ψα or φβ. χ is referred
to as a SPIN-ORBITAL.
But this wavefunction is unacceptable because in
quantum mechanics the electrons are identical and
indistinguishable. For this to be true, the wave
function needs to be antisymmetric. An antisymmetric
wave function can be mathematically described as
follows:
χ(a,b) = -χ(b,a)
This restriction can be overcome by taking a linear
combination of both products. The wavefunction for
the state in which both states "a" and "b" are
occupied by the particles can be written as:
Bosons: χ = χ_{1}(a)χ_{2}(b) + χ_{1}(b)χ_{2}(a)
Fermions: χ = χ_{1}(a)χ_{2}(b) - χ_{1}(b)χ_{2}(a)
The minus sign in the above relationship forces the
wavefunction to vanish identically if both states
are "a" or "b", implying that it is impossible for
both Fermions to occupy the same state.
The latter can be written as a 2 x 2 SLATER
DETERMINANT as follows:
= (1/√2)|χ_{1}(a) χ_{2}(a)|
|χ_{1}(b) χ_{2}(b)|
Interchanging any pairs of rows (equivalent to
interchanging electrons) changes the sign of the
determinant. Moreover, in accordance with the PE
principle, the determinant goes to zero if any
two wave functions are the same (2 rows in the
determinant are equal).
Expanding the above, we get
χ(a,b) = (1/√2)|ψ_{1}(a)α_{1}(a) ψ_{2}(a)β_{1}(a)|
|ψ_{1}(b)α_{2}(b) ψ_{2}(b)β_{2}(b)|
= (1/√2){ψ_{1}(a)α_{1}(a)ψ_{2}(b)β_{2}(b)
- ψ_{1}(b)α_{2}(b)ψ_{2}(a)β_{1}(a)}
= ψ_{1}(a)ψ_{2}(b)(1/√2)[α_{1}(a)β_{2}(b) - α_{2}(b)β_{1}(a)]}
If ψ_{1}(a)ψ_{2}(b) is symmetric then:
(1/√2){α_{1}(a)β_{2}(b) - α_{2}(b)β_{1}(a)} must be antisymmetric.
This represents a SINGLET state.
If ψ_{1}(a)ψ_{2}(b) is antisymmetric then the spin
wavefuntions must be symmetric. There are 3 ways
that this can happen:
α_{1}(a)α_{2}(b)
(1/√2){α_{1}(a)β_{2}(b) + α_{2}(b)β_{1}(a)}
β_{1}(a)β_{2}(b)
These represent a TRIPLET state.