![Hohenberg-kohn Theorem 1
▪ The Ground state energy E is the unique functional of
Electron Density
E = E 。 [po(r)]
▪ where p (r) represents the density function which itself is a
function of position (r).](https://image.slidesharecdn.com/densityfunctionaltheoryoverview-241215172219-46ba724c/85/Density-Functional-Theory-DFT-Overview-pptx-21-320.jpg)
![▪ The electron density that minimizes the energy of the
overall functional is the true ground state electron density:
E[p(r)]>E.[p.(r)]
Hohenberg-kohn Theorem 2](https://image.slidesharecdn.com/densityfunctionaltheoryoverview-241215172219-46ba724c/85/Density-Functional-Theory-DFT-Overview-pptx-22-320.jpg)
Density Functional Theory (DFT) Overview.pptx
- 1. DENSITY FUNCTIONAL THEORY MOMNA QAYYUM Department of Chemistry University of Management and Technology, Lahore
- 2. CONTENTS ▪ DFT ▪ Why DFT ▪ H 2O Molecule ▪ Electron Density ▪ Many Particle System ▪ Born-Oppenheimer Approximations ▪ Hohenburg Kohn Theorem ▪ References
- 3. Density Functional Theory Density functional theory (DFT) is a quantum-mechanical atomistic simulation method to compute a wide variety of properties of almost any kind of atomic system: molecules, crystals, surfaces, and even electronic devices when combined with non-equilibrium Green's functions (NEGF).
- 5. Why DFT Schrodinger Equation can be solved easily for one electron system. For multiple electronic system, it is very hard to solve the Schrodinger equation. ▪ So we introduce some Approximations, one of them is DFT ▪ Reduce electron-electron interactions.
- 6. H20 Molecule
- 7. H₂O Molecule Explained by DFT ▪ Density Functional Theory (DFT): ▪ Models the behavior of electrons in the H₂O molecule using electron density, not wave functions. ▪ Efficiently predict molecular geometry, bond strengths, and electronic properties. ▪ Electron Density Distribution: ▪ DFT calculates the 3D electron density, showing how electrons are distributed around the oxygen and hydrogen atoms. ▪ Highlights the polar nature of the H₂O molecule, with a higher electron density near oxygen.
- 8. H₂O Geometry by DFT ▪ Bond Angle: ~104.5° (due to lone pair repulsion on oxygen). ▪ Bond Lengths: ~0.96 Å for O–H bonds (close to experimental values). ▪ Potential Energy Surface (PES): DFT maps the energy changes as the bond lengths and angles vary, providing insights into molecular stability.
- 10. Electron Density ▪ Electron density significantly speeds up the calculation. ▪ Many body electronic wavefunction is a function of 3N variables the electron density is only a function of x, y, z only three variables. ▪ The Hohenburg-Kohn theorem asserts that the density of any system determines all ground-state properties of the system. ▪ In this case the total ground state energy of a many-electron system is a functional of the density.
- 11. Many Particle System ▪ Find the ground state for a collection of atoms by solving the Schrodinger equation ▪ Η Ψ({ri} {RI})= Ε Ψ ( {ri} {RI}) ▪ So here we have a bunch of nuclei & bunch of electron which makes a complicated equation to solve.
- 12. Born-Oppenheimer Approximations ▪ According to this approximation: ▪ Mass of nuclei is very greater than mass of electrons N>>>>e ▪ Thus nuclei are slow and electrons are fast. ▪ In other words we can write the Schrodinger wave equation by separating the electronic and nuclei terms.
- 13. Key Concepts of Born-Oppenheimer Approximations 1. Separation of Motion: ▪ Assumes that nuclei are much heavier than electrons, so their motion can be treated separately. ▪ Electrons adjust instantly to the movement of nuclei. 2. Simplifies the Schrödinger Equation: ▪ Treats the electronic and nuclear wavefunctions independently, drastically reducing computational complexity. 3. Energy Surface: ▪ The nuclei move on a potential energy surface defined by the electronic energy.
- 14. Why is This Important ▪ Simplifies Molecular Calculations: Reduces a complex multi- particle system into more manageable parts. ▪ Basis for Quantum Chemistry: Enables understanding of chemical bonding and molecular dynamics. ▪ Accurate Models: Provides realistic approximations for molecular structure and spectroscopy.
- 15. Practical Applications ▪ Molecular Vibrations: Explains vibrational spectra by treating nuclei separately. ▪ Chemical Reactions: Helps map potential energy surfaces to understand reaction pathways. ▪ Spectroscopy: Provides a framework for interpreting electronic, vibrational, and rotational spectra. ▪ Molecular Dynamics Simulations: Simplifies calculations for simulating large systems. ▪ Quantum Chemistry Software: Foundational principle behind tools like Gaussian and VASP.
- 16. Hohenberg-kohn Theorem ▪ The Ground state energy E is the unique functional of Electron Density This foundational concept in Density Functional Theory (DFT) has two key parts: 1. The Ground State Density Uniquely Defines All Properties ▪ The total energy and all properties of a quantum system can be determined entirely from the electron density, not the many-electron wave function. ▪ This simplifies complex quantum calculations by reducing dimensions. 2. Existence of a Universal Energy Functional ▪ There exists a mathematical function that relates the energy of the system to the electron density. ▪ The ground-state energy corresponds to the minimum of this energy functional.
- 17. Breaking Down the Hohenberg-kohn Theorem 1. Why Focus on Electron Density? ▪ Electron density tells us where electrons are most likely to be in a molecule or material. ▪ It’s much simpler than the many-body wave function but still contains all the essential information. 2. Implications of the Theorem ▪ The energy and behavior of a system can be calculated without solving the complex Schrödinger equation directly. ▪ This reduces the problem's complexity from many-electron interactions to just density distribution.
- 18. Key Concepts of Hohenberg-kohn Theorem ▪ The ground-state electron density uniquely determines all properties of a quantum system (e.g., energy, reactivity). ▪ Reduces complexity: Depends on 3 spatial variables, not the 3N variables of the wavefunction (N = number of electrons). ▪ A mathematical function relates the energy of the system to the electron density. ▪ The ground-state energy corresponds to the minimum of this functional. ▪ The system is in its ground state (lowest energy). ▪ The relationship between energy and density is universal for all systems.
- 19. Why is This Important ▪ Simplifies Quantum Calculations: Reduces complexity from multi-electron wavefunctions to a 3-variable electron density. ▪ Foundation of DFT: Forms the basis for Density Functional Theory, a key tool in computational chemistry and physics. ▪ Reduces Computational Cost: Enables accurate predictions for large systems with minimal computational effort. ▪ Broad Applicability: Widely used in material science, catalysis, energy storage, and drug discovery. ▪ Revolutionized Modern Chemistry: Transformed the study and design of molecules and materials with efficient modeling techniques.
- 20. Practical Applications ▪ Material design: Predicts stability, conductivity, and magnetic properties of materials. ▪ Drug Discovery: Models molecular interactions with biological targets. ▪ Catalysis: Optimizes catalysts for faster and greener reactions. ▪ Battery Development: Designs more efficient energy storage materials. ▪ Environmental Science: Models pollutant behavior and chemical reactions in the atmosphere.
- 21. Hohenberg-kohn Theorem 1 ▪ The Ground state energy E is the unique functional of Electron Density E = E 。 [po(r)] ▪ where p (r) represents the density function which itself is a function of position (r).
- 22. ▪ The electron density that minimizes the energy of the overall functional is the true ground state electron density: E[p(r)]>E.[p.(r)] Hohenberg-kohn Theorem 2
- 23. From Wave function to Electron Density ▪ So, we can separate the wave function for electron and nuclei Ψ({ri} {RI})=ΨN {RI} Ψe {ri} ▪ DFT explains the electronic density. ▪ That reduce to N dimensional to 3 spatial dimension ▪ So electron density is only 3 dimensional
- 25. References ▪ Roienko, O., Lukin, V., Oliinyk, V., Djurović, I., & Simeunović, M. (2022). An Overview of the Adaptive Robust DFT and its Applications. Technological Innovation in Engineering Research, 4, 68-89. ▪ Chakraborty, D., & Chattaraj, P. K. (2021). Conceptual density functional theory based electronic structure principles. Chemical Science, 12(18), 6264-6279. ▪ Fiechter, M. R., & Richardson, J. O. (2024). Understanding the cavity Born–Oppenheimer approximation. The Journal of Chemical Physics, 160(18). ▪ Liebert, J., & Schilling, C. (2023). An exact one-particle theory of bosonic excitations: from a generalized Hohenberg–Kohn theorem to convexified N-representability. New Journal of Physics, 25(1), 013009.
- 26. THANK YOU