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Multi Electron Wavefunctions
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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 and spin wavefuntions ψα or ψβ (again,
the Hartree product). 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.