This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
2. Many Particle Problem
Now, our Hamiltonian operator is given by: 𝐻 = 𝑇 + 𝑉
Where, 𝑇 is the Kinetic energy operator and 𝑉 is given by,
𝑉 =
𝑞 𝑖 𝑞 𝑗
𝑟 𝑖−𝑟 𝑗
𝐻ψ 𝑟𝑖, 𝑅𝐼 = 𝐸ψ 𝑟𝑖, 𝑅𝐼
e e
- -
+ +
2
3. Born-Oppenheimer Approximation
According to this approximation, the nucleus is large and slow as compared to electrons which
are small and fast. Thus we can separate out our general wavefunction into a products of electron
wavefunction and nuclei wavefunction,
ψ 𝑟𝑖, 𝑅𝐼 = ψ 𝑒 𝑟𝑖 ∗ ψ 𝑁 𝑅𝐼
So now we first solve for ground state of electrons by considering fixed nuclei centres.
𝐻ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 = 𝐸ψ 𝑟1, 𝑟2, … , 𝑟 𝑁
Where,
𝐻 =
−ℏ2
2𝑚 𝑒
𝑖
𝑁 𝑒
∇𝑖
2
+ 𝑖
𝑁 𝑒
𝑉𝑒𝑥𝑡 𝑟𝑖 + 𝑖=1
𝑁 𝑒
𝑗>1 𝑈 𝑟𝑖, 𝑟𝑗
e + e e e
_ _ _ _
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4. From wavefunctions to electron densities
We now define the electron density in a region as,
𝑛 𝑟 = ψ∗ 𝑟1, 𝑟2, … , 𝑟 𝑁 ψ 𝑟1, 𝑟2, … , 𝑟 𝑁
Where n(r) is the electron density. Since you just need 3 coordinates to define density of a charge
configuration, clearly our problem now reduces from 3N dimensions to 3 dimensions.
The jth electron is treated as a point charge in the field of all other electrons. This reduces our
many electron problem to single electron problem.
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5. From wavefunctions to electron densities
Again we can make simplifications,
ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 = ψ 𝑟1 ∗ ψ 𝑟2 ∗ ψ 𝑟3 ∗ ⋯ ∗ ψ 𝑟 𝑁
This is nothing but the Hartree Product. So now we can define electron density in terms of single
electron wavefunctions:
𝑛 𝑟 = 2 𝑖 ψ∗
𝑟 ψ 𝑟
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6. Hohenberg and Kohn
Theorem 1: The ground state energy E is a unique functional of the electron density.
𝐸 = 𝐸 𝑛 𝑟
The external potential corresponds to a unique ground state electron density.
- A given ground state electron density corresponds to a unique external potential.
- In particular, there is a one to one correspondence between the external potential and the ground
state electron density.
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𝑉𝑒𝑥𝑡 𝑟 ψ 𝐺 𝑟1, 𝑟2, …
n𝐺 𝑟
7. Hohenberg and Kohn
Theorem 2: The electron density that minimizes the energy of the overall functional is the true
ground state electron density.
𝐸 𝑛 𝑟 > 𝐸0 𝑛0 𝑟
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𝐸 𝑛 𝑟
𝑛 𝑟𝑛0 𝑟
𝐸0
8. The Energy Functional
𝐸 {ψ𝑖} = 𝐸 𝑘𝑛𝑜𝑤𝑛 {ψ𝑖} + 𝐸 𝑋𝐶 {ψ𝑖}
Where
𝐸 𝑘𝑛𝑜𝑤𝑛 {ψ𝑖} =
−ℏ
𝑚 𝑒
𝑖 ψ𝑖
∗
∇2ψ𝑖 𝑑3 𝑟 + 𝑉 𝑟 𝑛 𝑟 𝑑3 𝑟 +
𝑒2
2
𝑛 𝑟 𝑛 𝑟′
𝑟−𝑟′ 𝑑3 𝑟 𝑑3 𝑟′ + 𝐸𝑖𝑜𝑛
And
𝐸 𝑋𝐶 {ψ𝑖} : Exchange-Correlational Functional which includes all the quantum-mechanical terms
and this is what needs to be approximated
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e + e e e + +
_ _ _ _
9. Kohn-Sham Approach
Solve a set of single electron wavefunctions that only depend on 3 spatial variables, ψ 𝑒(𝑟)
−ℏ2
2𝑚 𝑒
∇2 + V r + 𝑉𝐻 𝑟 + 𝑉𝑋𝐶 𝑟 ψ𝑖 𝑟 = ϵ𝑖ψ𝑖 𝑟
In terms of Energy Functional, we can write the Kohn-Sham equation as:
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Exchange Correlation potentiale
+ e
-
-
n(r)
𝐸 𝑛 = 𝑇𝑠 𝑛 + 𝑉 𝑛 + 𝑊𝐻 𝑛 + 𝐸 𝑋𝐶 𝑛
10. Self Consistency Scheme
Step 1: Guess initial electron density n(r)
Step 2: Solve Kohn-Sham equation with n(r)
and obtain ψ𝑖 𝑟
Step 3: Calculate electron density based on
single electron wavefunction
Step 4: Compare; if n1(r) == n2(r) then stop
else substitute n1(r) = n2(r) and continue with
step 2 again
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−ℏ2
2𝑚 𝑒
∇2
+ V r + 𝑉𝐻 𝑟 + 𝑉𝑋𝐶 𝑟 ψ𝑖 𝑟 = ϵ𝑖ψ𝑖 𝑟
𝑛 𝑟 = 2
𝑖
ψ∗
𝑟 ψ 𝑟
n1(r)
n2(r)
11. Kohn-Sham Approach with LDA
The exchange-correlation functional is clearly the key to success of DFT. One of the great
appealing aspects of DFT is that even relatively simple approximations to VXC can give quite
accurate results. The local density approximation (LDA) is by far the simplest and known to be
the most widely used functional.
𝐸 𝑋𝐶
𝐿𝐷𝐴
= n 𝑟 ϵ 𝑋𝐶
𝑢𝑛𝑖𝑓
n 𝑑𝑟
Where ϵ 𝑋𝐶
𝑢𝑛𝑖𝑓
n is the exchange correlation energy per particle of infinite uniform electron gas
with density n. Thus, in LDA, the exchange correlation energy per particle of an inhomogeneous at
spatial point r of density n(r) is approximated as the exchange-correlation energy per particle of the
uniform electron gas of the same density.
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12. Kohn-Sham Approach with LDA
We can write,
𝜖 𝑋𝐶
𝑢𝑛𝑖𝑓
𝑛 = ϵ 𝑋
𝑢𝑛𝑖𝑓
𝑛 + ϵ 𝐶
𝑢𝑛𝑖𝑓
𝑛
ϵ 𝐶
𝑢𝑛𝑖𝑓
𝑛 cannot be calculated analytically. This quantity has been obtained numerically using
Quantum Monte-Carlo calculations and fitted to a parameterized function of n.
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−
3
4
3
π
1 3
𝑛1 3 LDA Exchange functional
13. Ionic Ground State
Forces on atoms can be easily calculated once the electronic ground state is obtained.
By moving along the ionic forces (steepest descent), the ionic ground state can be calculated. We
can then displace ion from ionic ground state and calculate the forces on all other ions.
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𝐹𝑙 = −
ⅆ𝐸
ⅆ𝑟𝑙
= − 𝜓𝑖
𝜕 𝐻
𝜕𝑟𝑙
𝜓𝑖
15. References
•Fundamentals and applications of density functional theory,
https://www.youtube.com/watch?v=SXvhDLCycxc&t=1166s
•Local-density approximation (LDA),
https://www.youtube.com/watch?v=GApI1I9AQMA&t=242s
•Gritsenko, O. V., P. R. T. Schipper, and E. J. Baerends. "Exchange and correlation energy in density
functional theory: Comparison of accurate density functional theory quantities with traditional Hartree–Fock
based ones and generalized gradient approximations for the molecules Li 2, N 2, F 2." The Journal of
chemical physics 107, no. 13 (1997): 5007-5015.
•Wikipedia.
•An Introduction to Density Functional Theory, N. M. Harrison,
https://www.ch.ic.ac.uk/harrison/Teaching/DFT_NATO.pdf
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