CHEM6085: Density Functional Theory

Lecture 6

Kohn-Sham DFT

C.-K. Skylaris

CHEM6085 Density Functional Theory 1

 Replacing one problem with another
Shortcomings of orbital-free DFT
• Pure (or “orbital-free”) DFT attempts to compute the energy of interacting electrons,
as a functional of the density
• While this brute force approach is in principle correct, in practice it is not very
accurate
• This is due to the lack of accurate approximations for the kinetic energy functional

The Kohn-Sham suggestion

• Kohn and Sham proposed an alternative approach in order to overcome this problem
[ W. Kohn and L. J. Sham. “Self-consistent equations including exchange and
correlation effects”, Phys. Rev. 140, A1133 (1965) ]

CHEM6085 Density Functional Theory 2

 Replacing one problem with another
The Kohn-Sham suggestion
• Proposed to work instead with a system of non-interacting electrons
• In general, the wavefunction of the non-interacting electrons is different from that of
the interacting ones and so is the density
• However in the Kohn-Sham approach, a fictitious non-interacting system is
constructed, in such a way, that its density is the same as that of the interacting
electrons
• The challenge is then shifted from finding the universal H-K functional to finding the
fictitious system of non-interacting electrons which has the same density as the “real”
one with the interacting electrons
• This allows to set up more accurate DFT calculations as the expression for the kinetic
energy of non-interacting electrons is known (so no need for approximate kinetic
energy functionals of the density)
• In order to see how it works we first need to examine the Schrödinger equation for
non-interacting electrons

CHEM6085 Density Functional Theory 3

 Interacting electrons

Exact nucleus of real molecule

Hamiltonian
operator for all electron
electrons
together

Construct and solve Schrödinger equation to

obtain exact wavefunction for interacting electrons

Exact electron
density

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 Non-interacting electrons

Hamiltonian operator nucleus of real molecule

for one electron
moving in an
electron
“average” potential
caused by the rest of
the electrons

Construct and solve Schrödinger equation for one electron. Combine many
one-electron wavefunctions (molecular orbitals) to obtain approximate
wavefunction for all electrons

Approximate
electron
density

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 Kohn-Sham non-interacting electrons

Hamiltonian operator nucleus of “fictitious” molecule

for one electron
moving in an “average”
electron
potential caused by a
special fictitious system
of electrons

Construct and solve Schrödinger equation for one electron. Combine one-
electron wavefunctions (molecular orbitals) to obtain approximate
wavefunction for all electrons

Exact electron
density

CHEM6085 Density Functional Theory 6

 These are not any non-interacting electrons... These are Kohn-Sham electrons!

• Interacting electrons • Non-Interacting electrons • Kohn-Sham non-

interacting electrons
• Exact Hamiltonian for • Approximate Hamiltonian • Approximate Hamiltonian
real molecule for real molecule for fictitious molecule

Density of interacting Density of non-interacting Density of fictitious system of

electrons (=exact density) electrons (=approximate non-interacting electrons (= exact
density) density of interacting electrons)

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 Interacting and non-interacting electrons

• We have seen that the exact electronic Hamiltonian is

• This describes a system of interacting electrons (as the 1/|ri-rj| can not be
broken into a sum of terms containing only ri and only rj)
• A much simpler problem is that of non-interacting electrons: This involves
replacing the direct interaction by some one-electron operator (potential)
Vav that describes the “average” effect of the interaction

Why Vav can be no better than

an “average” interaction?

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 Hamiltonian for non-interacting electrons
• In the case of the non-interacting electrons the Hamiltonian operator is
simply a sum of one-electron operators

• So the entire Hamiltonian is just a sum of one-electron Hamiltonians: one

for each electron

CHEM6085 Density Functional Theory 9

 Wavefunction for non-interacting electrons

•The Hamiltonian is a sum of one-electron Hamiltonians

• We can write (and solve) a Schrödinger equation for each one-electron

Hamiltonian separately

• From this we obtain molecular orbitals and their energies

CHEM6085 Density Functional Theory 10

 Molecular orbitals
• A wavefunction for a single electron is called a molecular orbital (MO)
• MO with spatial and spin coordinates are called spin orbitals and are
products of a spatial orbital and a spin function, e.g.

Spin orbital Spatial orbital Spin function

• As our one-electron Hamiltonians depend only on spatial coordinates, the

spin orbitals are also acceptable eigenfunctions

CHEM6085 Density Functional Theory 11

 Wavefunction for (many) non-interacting electrons

• Let’s see what happens for two non-interacting electrons

• In a similar way, we can show that the (exact) solution to the

Schrödinger equation for Nel non-interacting electrons is

• The total energy eigenvalue is simply the sum of the spin orbital energies

CHEM6085 Density Functional Theory 12

 Hartree products

• So the exact wavefunction of the independent electrons is simply the

product the spin-orbital for each electron

• Such a wavefunction is called a Hartree product

• A Hartree product is a mathematically acceptable solution to the

Schrödinger equation for independent electrons

• However it is not physically acceptable because it does not satisfy some of

the conditions we have set for wavefunctions (which ones?)

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 Slater determinants
You can verify that the following wavefunction for two electrons is
antisymmetric:

It can also be written as a determinant:

• This can be generalised to any number of electrons

• A wavefunction of this form is called a Slater determinant
• It obeys antisymmetry by construction (determinants change sign when
two rows or columns are interchanged)
• It also respects the indistinguishability of the electrons

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 Slater determinants
Slater determinant for N electrons:

Normalisation
constant

Shorthand representation, as a ket:

Spin orbitals in Slater determinants are always orthonormal, i.e.

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 The Pauli principle
• A determinant is zero if two or more of its rows or columns are identical
• Therefore, a given spin orbital can be included only once in a Slater
determinant, as otherwise the determinant would become zero
everywhere in space and such a wavefunction is not acceptable

Examples of not
Allowed cases allowed cases

•The Pauli principle follows naturally as a spatial orbital can result in at

most two spin orbitals (one with up and one with down spin functions)
• In other words, a spatial orbital can be occupied by no more than two
electrons of opposite spins

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 Electronic density of a Slater determinant

• We have seen that the electronic density of any wavefunction is defined

by the following expression

• When the wavefunction is a Slater determinant, the above expression

simplifies to (assuming an even number of electrons, doubly occupying
the spatial orbitals)

•Therefore, the electronic density of a Slater determinant wavefunction

is simply equal to the sum of the squares of the occupied orbitals

CHEM6085 Density Functional Theory 17

 Orbital-free vs Kohn-Sham DFT

Orbital-free approach Kohn-Sham approach

“ Find approximation for “ Find a Veff(r)=VKS(r) such that

FHK[n] “ nKS(r) = n0(r) “

Compute density n0(r) by Solve for the wavefunction

minimising (Slater determinant) of the
E[n]=FHK[n]+Eext[n] non-interacting electrons (i.e.
find the molecular orbitals)

Compute energy E[n0]

Build the density n0(r) from
the molecular orbitals

Compute energy E[n0]

CHEM6085 Density Functional Theory 18

 The Kohn-Sham trick

• In pure (orbital-free) DFT the energy is given by the functional

and the biggest obstacle is the lack of an accurate expressions for the kinetic
energy functional

• With the Kohn-Sham DFT approach we can re-write the energy as

• Where Ekin,KS[n] is the energy of the (non-interacting) Kohn-Sham electrons

• We have an exact expression for Ekin,KS[n], albeit in terms of molecular
orbitals, rather than the density
• The difference (Ekin[n]-Ekin,KS[n]) is smaller than the errors in the
approximations for Ekin[n] in pure (orbital-free) DFT

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 Kohn-Sham energy expression
• The total energy functional is now written as the sum of a functional of
the molecular orbitals for the kinetic energy part and functionals of the
density for the rest of the energy components

• Approximations are sought for E’xc[n], which even though it now contains
also contributions from the kinetic energy, it is still just called “Exchange-
correlation functional” and from now on we will be writing it as Exc[n]

CHEM6085 Density Functional Theory 20

 Homework

1) Write down an expression for the energy of a system of non-interacting electrons (e.g. in the Kohn-Sham approach)
in terms of molecular orbitals

2) Compare the above expression with the Kohn-Sham expression for the energy of the interacting system. In what do
they differ? Can you write the one in terms of the other?

3) Write down an explicit (i.e. one you can use for computations) approximate expression for the energy of interacting
electrons in the Kohn-Sham approach, using the Thomas-Fermi-Dirac orbital-free DFT model for inspiration. In
actual DFT calculations this approach is called exchange-only LDA (local density approximation).

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 5-minute quiz
Name : Date :
Surname:

1) Which operator prevents the full electronic Hamiltonian from being an “non-interacting electron”
Hamiltonian?

2) What is special about the system of non-interacting electrons in the Kohn-Sham DFT approach as compared
to any other collection of non-interacting electrons?

3) Write down the expression used for the kinetic energy in Kohn-Sham calculations. How is this different from
the exact kinetic energy functional in orbital-free DFT?

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