Density functional theory (DFT) provides an alternative approach to calculate properties of molecules by working with electron density rather than wave functions. DFT relies on two theorems linking the ground state energy and electron density. Approximations must be made for the exchange-correlation functional, with popular approximations including LDA, GGA, and hybrid functionals. DFT calculations can determine properties like molecular geometries, energies, vibrational frequencies, and more using software packages. While computationally efficient, DFT has limitations such as its reliance on approximate exchange-correlation functionals.

1. • Density FunctionalTheory
• Exchange And CorrelationFunctional
• Its Application
Presentedby: Saibalendu Sarkar
Poulomi Nandi
SuryaPratapSingh
Presentedto: DR. BHABANIS. MALLIK
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2. DENSITY FUNCTIONAL THEORY
2

3. INTRODUCTION
In the period 1995-2000, density functional theory (DFT) showed
a meteoric rise to popularity in quantum-chemistry
calculations:
“a substantial majority of the [quantum chemistry] papers
published today are based on applications of the density
functional theory” [K. Raghavachari, Theor. Chem. Acc., 103,
361 (2000)].
In DFT, one does not attempt to calculate the molecular wave
function. Instead, one works with the electron probability
density, ρ(x,y,z).
Its advantages include less demanding computational effort, less
computer time, and in some cases better agreement with the
experimental values than is obtained from Hartree-Fock
procedures.
3

4. The schrodinger wave equation for many particle system is given as:
Ĥᴪ({ri},{RI}) = E ᴪ({ri},{RI})
Ĥ = T̂ + V̂coulomb
4
MANY PARTICLE SYSTEM PROBLEM

5. Born-Oppenheimer Approx.
According to this approximation :
1. Mass of Nuclei is very greater than mass of
electrons.
2. Thus Nuclei are slow and electrons are fast.
3. So, we can decouple the dynamics of Nuclei
and electrons.
4. In other words we can write the Schrodinger
wave equation by separating the electronic
and Nuclei terms.
5. THIS MAKES LIFE MUCH MORE EASIER. 5

6. Born-Oppenheimer Approx.
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7. PROBLEMS WITH THE DIMENSIONS
Let us take a simple molecule H2O :-
Total number of electrons in the molecule = 8+2 =
10 electrons
Three spatial coordinates per electron.
So, Schrodinger equation becomes a 30
Dimensional Problem.
For ‘N’ numbers of electrons there will be ‘3N’
numbers of Dimensions.
WE HAVE TO THINK OF A BETTER WAY!!!!
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8. 8

9. DENSITY FUNCTIONAL THEORY
-FROM WAVE FUNCTION TO ELECTRON DENSITY
ELECTRON DENSITY :
n(r) =ᴪ*(r1,r2,....,rN)ᴪ(r1,r2,....,rN)
Z
ELECTRON DENSITY
X
Y
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10. DENSITY FUNCTIONAL THEORY
-FROM WAVE FUNCTION TO ELECTRON DENSITY
Now, let the “i”-th electron is treated as
point charge in the field of all other
electrons. This simplifies the “many
electron problem” to “many one-electron
problem” and is written as Hartree
product of N electrons:
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11. DENSITY FUNCTIONAL THEORY
-FROM WAVE FUNCTION TO ELECTRON DENSITY
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12. DENSITY FUNCTIONAL THEORY
-FROM WAVE FUNCTION TO ELECTRON DENSITY
• Hohenberg and Kohn- at the heart of DFT(1964)
THEOREM 1: The ground state energy E is a
unique functional of electron density:
E = Eo[ρo(r)]
where ρ (r) represents the density function
which itself is a function of position (r).
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13. DENSITY FUNCTIONAL THEORY
-FROM WAVE FUNCTION TO ELECTRON DENSITY
THEOREM 2: The electron density that
minimizes the energy of the overall functional
is the true ground state electron density:
E[ρ(r)]>Eo[ρo(r)]
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14. THE ENERGY FUNCTIONAL
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15. EXCHANGE-CORRELATION
FUNCTIONAL, Exc[n(r)]
• Includes all quantum mechanical terms.
• Not known-needs to be approximated.
• It is a smaller fraction of the total energy.
• Improved results are obtained by relating
exchange correlation with the first derivative
of density.
• Both exchange-correlation are long and short
distance.
• Long distance exchange-correlation is static
and short distance is dynamic. 15

16. THE KOHN-SHAM SCHEME
Solve a set of single-electron wave functions
that only depends on three spatial variables:
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17. HOW TO SOLVE???????
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18. SELF-CONSISTENCY SCHEME
1.Assume a trial density function, ρ(r)
COMPARE
3.Calculate the electron density 2.Solve Kohn-sham eq. With ρ (r).
based on the single electron Obtain single electron wave functions.
wave functions
ρ (r) = ∑ᴪ*(r)ᴪ(r)
A. If different, then begin the process from step 2.
4.
B. If identical, true ground state density is obtained.
trial
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Calculated

19. APPROX. METHOD AND SOFTWARE
The approximation methods used are
 LDA (Local Density Approximation)
 LSDA (Local Spin Density Approximation)
 GGA (Generalized Gradient Approximation)
 The Xa Method(Hartree–Fock–Slater method)
 Meta-GGA Functionals
 Hybrid GGA
The software which is most popular in calculating DFT is the
VASP (Vienna Ab initio Simulation Package)
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20. Density Functional Theorem :
Approximations of Exchange
and Correlation
Functional
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21. H{H-F} =T(K.E) + V(e-e) + V(N-e) + E(Exchange)
H{K-S} =[T(K.E,Ks) +V(e-e,ks) +V(N-e)]+{T(K.E) -T(K.E,Ks)}+
{V(e-e) - V(e-e,Ks)} + Ex
=[T(K.E,Ks) + V(e-e,ks) + V(N-e)]+ [∆T + ∆V + Ex]
=T(K.E,Ks) + V(e-e,ks) + V(N-e) + Vxc
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22. The Local-Density Approximation
(LDA)
• Exc depends upon electron density (ρ) at each point in space i.e
local value of ρ.
• Homogeneous (or uniform) electron gas (HEG/UEG) model :
Jellium
•If ρ varies extremely slowly with position then Exc is accurately
given by,
εx = exchange-correlation energy per electron of a homogeneous electron
gas of electron density ρ.
For spin unpolarised system
22

23. Separation of Exchange and
Correlation part
Ex = exchange-energy functional
Ec = correlation-energy functional
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24. The Functionals Ex and Ec
•
•Only limiting expressions for Ec(ρ) are known exactly, leading
to numerous different approx. for Ec(ρ) .
•Ec(ρ) can be determined numerically by Monte Carlo simulations
• Ex(ρ) takes a simple analytical form for HEG model
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25. Calculation of Vxc
• Therefore, functional derivative of Exc
25

26. The Local-Spin-Density
Approximation(LSDA)
• For open-shell molecules and molecular geometries near
dissociation it gives better result.
• LSDA allows electrons with opposite spins to have different spatial KS
orbitals
26

27. Generalization Of Density-
functional Theory
• E.g CH3 and O2 having N(↑) ≠ N(↓) , ρ (↑) ≠ ρ(↓) spin-
DFT will give different orbitals for electrons with different
spins.
• Species with all electrons paired and molecular geometries
in the region of the equilibrium geometry, we can expect
that ρ (↑) = ρ(↓) and LSDA → LDA
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28. Generalized Gradient
Approximation(GGA)
• Why do we need gradient corrected functional?
Ex(BECK)
• Becke’s exchange functional is (commonly used Ex [GGA])
= Ex (LSDA)+ ∆ Ex (BECK)
• Commonly used GGA Ec[GGA] include the Lee–Yang–Parr
(LYP) functional
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29. Meta-GGA Functionals
• Exchange correlation density functional depend on KS orbital
through the kinetic-energy density i.e on the laplacian of the
orbitals.
• Kohn–Sham kinetic-energy density for the spin-a electrons
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30. Hybrid GGA
• It incorporates a portion of exact exchange from H-F
theory with exchange and correlation from other
sources (ab initio or empirical)
• Exc(hybrid) = Ex(exact) + Ex(GGA) + Ec(GGA)
30

31. Density Functional Theory (DFT)
: Applications
Employing DFT for Computation of Various
Properties
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32. Computer Simulation and Modeling: A
Principal Tool in Computational Chemistry
Process of building a computer model, and the
interplay between experiment, simulation, and
theory.
https://en.wikipedia.org/wiki/Computer_simulation#/media/File:Molecula
r_simulation_process.svg
A computer model involves
the expression of the system
that is to be studied in terms
of some Quantum-
Mechanical equations and
algorithms. By contrast,
computer simulation
involves the running of a
computer based program
that contains these
equations.
Simulation, therefore, is the
process of running a model.
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33. Systematic Development of
Density Functional Theory (DFT)
Calculations
Hohenburg, Kohn, and Sham’s work was published in 1964 and
1965.
 Soon thereafter, KS DFT using LSDA approximation was applied by
physicists to study crystalline solids.
 Around 1980, LSDA DFT molecular calculations achieved good
results for molecular geometries but failed to give accurate
dissociation energies.
 In mid-1980s, Becke’s work gave accurate dissociation energies.
 In 1988, analytic gradients were implemented in DFT which gave
good result for equilibrium geometries.
 In 1993, Gaussian was employed for DFT calculations.
 In the mid-1990s, a huge growth was experienced in molecular
calculations based on DFT.
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34. What Molecular Properties can be
Calculated using Density Functional Theory
(DFT)Calculations ?
The result of a DFT calculation is an energy value, and It involves
using the probability density functions of the electrons in a
molecule to determine other properties of the molecule.
 Molecular Geometries : Bond Angles, Bond Lengths, Dihedral Angles,
Transition-State Structures( KIE); etc.
 Energy Changes : Atomization Energies, Heats of Formation, and
Heats of reaction; Energy changes related with Isodemic Reactions and
Bond-Separation Energies, Isomerization Energies, Energy Difference
between Conformers, Rotational Barriers, Activation Energies; etc.
 Other Properties : Dipole Moments, Vibrational Frequencies,
Entropies, Gibbs Energy of Solvation, NMR Shielding Constants; etc.
 Hydrogen Bonding
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35. How to Do Our Density Functional
(DFT) Calculations ?
 Involves the use of a computer software.
The choice of software depends on its own quirks,
learning curves, availability of documentation,
open/closed-source, and whether the originator of the
software has been banned for life from using it (amongst
other people).
 Most Generally used Quantum Mechanical codes are
Abinit, Quantum Espresso (QE), VASP and Gaussian.
 A list of Quantum Chemistry and Solid State Physics
software with language, basis sets, and methods is as
follows.
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36. 36

37. https://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_software
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38. A Comparison Of Different
Computational Methods
1. Molecular Geometry : Mean Absolute Errors (MAE) in bond lengths (A⁰)
where; A : G2 subset (32 molecules containing only first-row atoms), B : : (108
molecules including first- and second-row atoms), and C : : (40 molecules containing third-row
atoms Ga-Kr.
Level of Theory A B C
MO Models
HF 0.022 0.021
LSDA Functional
SVWN 0.017 0.016
GGA and MGGA
BLYP 0.014 0.021 0.048
Hybrid Functionals
B3LYP 0.004 0.008 0.030
B3PW91 0.008 0.011 0.020
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39. 2. Energetics :
 Atomization Energies : MAE and maximum errors (kcal/mol)
where; A = G2 subset (32 molecules containing only first-row atoms) , B = G2 set
(55 molecules including first- and second-row atoms), and C = 108 molecules
including first- and second-row atoms.
Level of Theory A B C
MO Models
HF/6-31G 85.9 80.5(184.5) 150.6
LSDA Functional
SVWN/6-31G 35.7 36.4 (84.0)
GGA and MGGA Functional
BLYP/6-31G 5.6 5.3 (18.8)
Hybrid Functionals
B3LYP/6-31G 5.2 (31.5)
B3PW91/6-311++G 4.8
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40.  Metal- Ligand Binding Energies : MAE (kcal/mol)
where ; Complex MX+ ; M = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and X = ligand.
 Enthalpy of Acivation and Forward Reactions : MAE (kcal/mol)
Level of Theory H CH3 CH2
MO Model
PCI 80 2 2 4
Density Functionals
SWVN 12
B3LYP 5 6 4
Level of Theory Activation Reaction
MO Model
HF/ 6-31G 18.7 3.8
Density Functionals
B3LYP 1.7 4.1 40

41.  IPs and EAs : MAE (eV)
3. Dipole Moment : D
Level of Theory IPs/G2 EA/G2
MO Model
MP2 0.1 0.1
LSDA Functionals
SVWN/ 6-311 + G 0.7 0.7
GGA and MGGA Functionals
BLYP/ 6-311 + G 0.19 0.11
Hybrid Functionals
B3LYP / aug-cc-pVDZ 0.2 0.1
B3PW91 / 6-311 + G 0.14 0.10
Molecule HF BLYP B3LYP Experiment
NH3 1.62 1.48 1.52 1.47
H2O 1.98 1.80 1.86 1.85
HF 1.92 1.75 1.80 1.83
SO2 1.99 1.57 1.67 1.63 41

42. Limitations Of Density Functional
Theory (DFT)
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43.  The Hohenberg-Kohn-Sham theory is basically a ground state theory. LR-TDDFT
can only be applied to certain kind of excited states.
 Since approximate functionals are used, KS-DFT can give an energy below the
true ground state energy; e.g.- a B3LYP/cc-pVTZ geometry optimization of H2O
give an energy of -76.460 hartrees, compared with the true nonrelativistic
energy of -76.438 hartrees.
 Many of the currently used Exc functionals fail for van-der-Waals molecules;
e.g.- the BLYP, B3LYP, and BPW91 functional predict no binding in He2 and Ne2.
However, the PBE functional work fairly well .
Towards Systematic Improvability
 Iterative methods have been devised that take a very accurate ground-sate
molecular energy density found from a high-level calculation (e.g- CI) and use it
to calculate vs for the corresponding reference system and vs can be used to
find vxc and in turn, Exc can be calculated.
 DFT has been applied to give the quantitative definitions of such chemical
concepts as electronegativity, hardness and softness; etc.
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44. REFERENCES
• IRA N. LEVINE
• DONALD A. MCQUARRIE
• JULIO DE PAULA AND PETER ATKINS
• CHRISTOPHER J CRAMER
• WIKIPEDIA
• YOUTUBE VIDEOES
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