From the above we see that we supply a number to a function to get a number,
and we supply a function to a functional to get a number. By analogy, we supply a
functional to a “2-functional” to get a number. I leave a specific example as an
exercise for the reader.
Chapter 7, Harder Questions, Answers
Q3
Why is it that the HF Slater determinant is an inexact representation of the
wavefunction, but the DFT determinant for a system of noninteracting electrons
is exact for this particular wavefunction?
The HF (Hartree–Fock) Slater determinant is an inexact representation of the
wavefunction because even with an infinitely big basis set it would not account
fully for electron correlation (it does account exactly for “Pauli repulsion” since if
two electrons had the same spatial and spin coordinates the determinant would
vanish). This is shown by the fact that electron correlation can in principle be
handled fully by expressing the wavefunction as a linear combination of the HF
determinant plus determinants representing all possible promotions of electrons
into virtual orbitals: full configuration interaction. Physically, this mathematical
construction permits the electrons maximum freedom in avoiding one another.
The DFT determinant for a system of noninteracting electrons is exact for this
particular wavefunction (i.e. for the wavefunction of the hypothetical noninteract-
ing electrons) because since the electrons are noninteracting there is no need to
allow them to avoid one another by promotion into virtual orbitals.
For an account of DFT that is at once reasonably detailed, clear and concise see
Cramer [1].
Reference
- Cramer CJ (2004) Essentials of computational chemistry. Wiley, Chichester, England, chapter 8
Chapter 7, Harder Questions, Answers
Q4
Why do we expect the “unknown” term in the energy equation (Exc[r0], in
Eq. 7.21) to be small?
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