This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
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Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
(If visualization is slow, please try downloading the file.)
Part 1 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
(If visualization is slow, please try downloading the file.)
Part 1 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
Field energy correction with discrete chargesSergio Prats
This document introduces a correction in the classical electromagnetic field energy density when there are discrete charges instead of a whole density of charge based in the fact that the EM field induces by a particle does not affect itself and does not contribute to the potential in the hamiltonian.
A deeper analysis is done on how to deal with the radiated field energy, the reaction force and its analogy with quan Vacuum
A model is proposed to show that the electron spin may not be purely intrinsic but the result of a loop of current with two different components interacting between them
Describe the Schroedinger wavefunctions and energies of electrons in an atom leading to the 3 quantum numbers. These can be also observed in the line spectra of atoms.
This Presentation "Energy band theory of solids" will help you to Clarify your doubts and Enrich your Knowledge. Kindly use this presentation as a Reference and utilize this presentation
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
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
_ _ _ _
3
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.
4
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
5
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.
6
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
7
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
8
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:
9
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
10
−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.
11
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.
12
−
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.
13
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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