The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
This presentation is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
(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.
The all-electron GW method based on WIEN2k: Implementation and applications.ABDERRAHMANE REGGAD
The all-electron GW method based on WIEN2k:
Implementation and applications.
Ricardo I. G´omez-Abal
Fritz-Haber-Institut of the Max-Planck-Society
Faradayweg 4-6, D-14195, Berlin, Germany
15th. WIEN2k-Workshop
March, 29th. 2008
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 is the introduction to Density Functional Theory, an essential computational approach used by Physicist and Quantum Chemist to study Solid State matter.
(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.
The all-electron GW method based on WIEN2k: Implementation and applications.ABDERRAHMANE REGGAD
The all-electron GW method based on WIEN2k:
Implementation and applications.
Ricardo I. G´omez-Abal
Fritz-Haber-Institut of the Max-Planck-Society
Faradayweg 4-6, D-14195, Berlin, Germany
15th. WIEN2k-Workshop
March, 29th. 2008
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
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.
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
II. Charge transport and nanoelectronics.
Quantum Hall Effect: 2D electron gas (2DEG) in magnetic field, Landau levels, de Haas-van Alphen and Shubnikov-de Haas Effects, integer and fractional quantum Hall effects, Spin Hall Effect.
Quantum transport: Transport regimes and mesoscopic quantum transport, Scattering theory of conductance and Landauer-Buttiker formalism, Quantum point contacts, Quantum electronics and selected examples of mesoscopic devices (quantum interference devices).
Tunneling: Scanning tunneling microscopy and spectroscopy (and wavefunction mapping in nanostructures and molecules), Nanoelectronic devices based on tunneling, Coulomb blockade, Single electron transistors, Kondo effect.
Molecular electronics: Donor-Acceptor systems, Nanoscale charge transfer, Electronic properties and transport in molecules and biomolecules; single molecule transistors.
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
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.
Slides for a 20 minute presentation about Julia, with a brief introduction to multiple dispatch/multimethods and how it is used for numerical linear algebra
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
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.
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
II. Charge transport and nanoelectronics.
Quantum Hall Effect: 2D electron gas (2DEG) in magnetic field, Landau levels, de Haas-van Alphen and Shubnikov-de Haas Effects, integer and fractional quantum Hall effects, Spin Hall Effect.
Quantum transport: Transport regimes and mesoscopic quantum transport, Scattering theory of conductance and Landauer-Buttiker formalism, Quantum point contacts, Quantum electronics and selected examples of mesoscopic devices (quantum interference devices).
Tunneling: Scanning tunneling microscopy and spectroscopy (and wavefunction mapping in nanostructures and molecules), Nanoelectronic devices based on tunneling, Coulomb blockade, Single electron transistors, Kondo effect.
Molecular electronics: Donor-Acceptor systems, Nanoscale charge transfer, Electronic properties and transport in molecules and biomolecules; single molecule transistors.
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
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.
Slides for a 20 minute presentation about Julia, with a brief introduction to multiple dispatch/multimethods and how it is used for numerical linear algebra
A very short tour through the Julia community and how key features of the language interact to produce an expressive syntax that users like without sacrificing performance
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
Freezing of energy of a soliton in an external potentialAlberto Maspero
We study the dynamics of a soliton in the generalized NLS with a small external potential. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order ϵ^{−r}. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order ϵ^{−r}.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
1. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Looking into the black box
Computing molecular excited states with time-dependent density
functional theory (TDDFT)
Jiahao Chen and Shane Yost
Van Voorhis group, MIT Chemistry
2010-03-12
2. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Disclaimer
This is a very superficial survey of two ideas:
1 Self–consistent field methods for ground state calculations, and
2 Linear response theory for excited state calculations.
There will be math and jargon.
Most of it is optional.
The math is not presented in the most general case.
We won’t have time today to discuss calculating observables and
numerical results.
3. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Disclaimer
This is a very superficial survey of two ideas:
1 Self–consistent field methods for ground state calculations, and
2 Linear response theory for excited state calculations.
There will be math and jargon.
Most of it is optional.
The math is not presented in the most general case.
We won’t have time today to discuss calculating observables and
numerical results.
4. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Disclaimer
This is a very superficial survey of two ideas:
1 Self–consistent field methods for ground state calculations, and
2 Linear response theory for excited state calculations.
There will be math and jargon.
Most of it is optional.
The math is not presented in the most general case.
We won’t have time today to discuss calculating observables and
numerical results.
5. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Statement of the Problem (Born–Oppenheimer)
Given that we know where the atomic nuclei are, where are the
electrons?
ˆ
H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
r r r (1)
ˆ
H is the electronic, or Born–Oppenheimer Hamiltonian:
N
1
ˆ
H = ∑ hi + ∑ |i −j |
ˆ
r r
(1-e + e–e repulsion) (2)
i=1 ij
1
h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3)
ˆ
2 1
M
ZI
vext (r ) = − ∑
(e–nucleus interaction)
(4)
I =1 RI −
r
6. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Statement of the Problem (Born–Oppenheimer)
Given that we know where the atomic nuclei are, where are the
electrons?
ˆ
H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
r r r (1)
ˆ
H is the electronic, or Born–Oppenheimer Hamiltonian:
N
1
ˆ
H = ∑ hi + ∑ |i −j |
ˆ
r r
(1-e + e–e repulsion) (2)
i=1 ij
1
h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3)
ˆ
2 1
M
ZI
vext (r ) = − ∑
(e–nucleus interaction)
(4)
I =1 RI −
r
7. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Practical solution of the Schrödinger equation
ˆ
H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
r r r
What approximations do we need to solve this practically?
Approximate wavefunction
Variational ansätze
Orbital (one–particle) approximation
Basis set approximation
Approximate Hamiltonian
Noninteracting particle / mean field approximation
In DFT: approximations to exchange–correlation
Linear response approximation (for excited states)
8. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Contents
1 Basis functions
2 Hartree–Fock
3 Density functional theory
4 Time–dependent DFT
9. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Variational principle and ansätze
We don’t know the exact solution to the many–body Schrödinger
equation. But we can make an educated guess, an ansatz (pl.
ansätze).
Variational principle
An approximate wavefunction Ψ (a, b, . . . ) always yields a higher
energy than the exact solution Ψ0 , i.e.
Ψ |H| Ψ Ψ0 |H| Ψ0
≥ E0 = (5)
Ψ|Ψ Ψ0 |Ψ0
This reduces a differential equation into an optimization problem
for the numerical parameters a, b, . . .
10. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Bases in vector spaces
A basis describes axes, allowing coordinates to be defined.
Example
R3 has basis {ex , ey , ez }. Numbers (x, y , z) can be assigned as
coordinates to a point with position vector v = xex + y ey + zez .
A less obvious example:
Example
Computer monitors use RGB color space, with basis {red, green,
blue}. For example, a kind of purple is described by purple = 1 red
2
+ 1 blue and has coordinates 1 , 0, 1 in color space.
2 2 2
11. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Bases in function spaces
Basis functions describe “axes” in function space. If you have a
collection of mutually orthogonal, real functions φ1 (x) , . . . , φn (x)
that span the entire space of functions, then
f (x) = ∑ ci φi (x) (7)
i
whose coefficients are given by projections onto the basis
N f (x) φi (x) dx
´
ci = R´ 2
(8)
RN φi (x) dx
Then the function f has “coordinates” (c1 , c2 , . . . ) in the basis
spanning the function space.
Example
Functions over [−π, π] have Fourier series expansions, whose basis
functions are 1, sin x, cos 2x, sin 3x, cos 4x, . . .
12. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
The independent particle ansatz: from many to one
Let’s try to solve the many–body Schrödinger equation
ˆ
H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
r r r
As a first guess for Ψn , assume that it is factorizable into a Hartree
product of one–particle wavefunctions (orbitals) 1
Ψ (r1 , . . . , N ) = φ1 (r1 ) · · · φN (rN )
r (9)
However, this wavefunction is not antisymmetric (no Pauli
exclusion). The simplest fermionic ansatz is the Slater determinant2
φ1 (r1 ) · · · φ1 (rN )
1
Ψ (r1 , . . . , N ) = √
r .
. .. .
.
(10)
. . .
N!
φN (r1 ) · · · φN (rN )
1 D. R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89; 111; 426
2 J. C. Slater, Phys. Rev. 34 (1929) 1293
13. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
The Hartree–Fock equation
Using the ansatz that the many–body wf is a Slater determinant,
the variational principle leads to3 the Hartree–Fock equation
Fi φi (r ) = εi φi (r )
ˆ (11)
ˆ
Fi = hi + ∑ Jj − Kj (Fock operator)
ˆ ˆ ˆ (12)
j
J is the classical Coulomb potential = Hartree potential vH .
K is the exchange interaction with no classical analogue.
(absent in Hartree theory)
3 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1982, Ch. 3
14. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Hartree–Fock is a mean field theory
Electrons in Hartree–Fock feel an electric
field of potential Φ = ∑N Jj − Kj .
ˆ
j=1
ˆ ˆ
The mean field is a self–consistent field
The electron–electron interactions are
approximated by the interaction between an
electron and this electric field. This field
depends on the orbitals, yet also helps
determines them.
Correlation energy = the error in
Hartree–Fock energy (wrt exact)
15. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Roothaan–Hall equation
If we project the Hartree–Fock equation onto a basis {χ1 , . . . , χP },
P
φi = ∑ Ciα χα (13)
α=1
we get the Roothaan–Hall equation
FC = ESC (14)
which is a generalized eigenvalue equation.
Atomic orbitals = spatially localized basis {χ1 , . . . , χP },
Molecular orbitals = eigenvectors of the Roothaan–Hall
equation
16. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Basis sets
There are many choices for the atomic basis {χ1 , . . . , χP }:
Plane waves: great for periodic systems (solids), inefficient for
molecules (and other systems without translational symmetry)
Wannier functions
Slater orbitals: use hydrogenic orbitals, nice theoretical
properties. In practice the Fock matrix is very difficult to
calculate. Not widely used today, except in ADF.
Gaussian orbitals: popular for molecules. Easy to calculate.
Pople basis sets: STO-3G, 3–21G, 6–31G*, 6–311+G**,...
Dunning–Hays correlation–consistent basis sets: cc-pVDZ,
aug-cc-pVQZ, d-aug-cc-pVTZ, ...
Others, e.g. wavelets, adaptive quadrature grids...
17. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Post–Hartree–Fock methods
Many methods to treat correlation use Hartree–Fock as a starting
point:
Perturbative corrections: Møller–Plesset (MP2,...),...
Configuration interaction: CISD, FCI,...
Coupled–cluster: CCSD, CCSD(T),...
18. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Kohn–Sham equations
In 1965, Kohn and Sham proposed a noninteracting system for use
with density functional theory.4 This turns out to look like
Hartree–Fock theory with a correction for electronic correlation.
The resulting one–particle equation is written as:
1 2
Hs φi (r ) = − ∇s + vs (r ) φi (r ) = εi φi (r )
(15)
2
where − 1 ∇2 is the noninteracting part of the kinetic energy and vs
2 s
is the Kohn–Sham potential:
vs (r ) = vext (r ) + vH (r ) + vxc (r )
(16)
The exchange–correlation potential vxc has three contributions:
exchange, correlation, and a correction to the kinetic energy.
4 W. Kohn and L. J. Sham, Phys. Rev. 140, 1965, A1133
19. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Kohn–Sham is a density functional theory
The Kohn–Sham equation is equivalent to finding the density ρ
that minimizes the energy functional
ˆ
E [ρ] = vext (r ) ρ (r ) d + Ts [ρ] + VH [ρ] + Exc [ρ]
r (17)
R3
1 N
ˆ
Ts [ρ] = − ∑ φi (r ) ∇2 φi (r ) d
r (18)
2 i=1 R3
1 ρ (r ) ρ (r )
ˆ
VH [ρ] = + d d
r r (19)
2 R6 |r −
r |
where the density is constructed from
N
ρ (r ) =
∑ |φi ( )|2 = ∑
r Ci µ Ciν χµ (r ) χν (r )
(20)
i=1 µνi
20. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Density functionals
The form of the exchange–correlation functional Exc is unknown.
There are many, many approximations to it:
“Jacob’s Ladder” of approximations to Exc [ρ]:5
1 Exc [ρ] LDA X α, LDA,...
2 Exc [ρ, ∇ρ] GGA BLYP, PBE, PW91...
3 Exc [ρ, ∇ρ, τ] meta–GGA VSXC, TPSS, ...
4 Exc [ρ, ∇ρ, τ, {φ }o ] hybrids 6 ,... B3LYP, PBE0,...
5 Exc [ρ, ∇ρ, τ, {φ }] fully nonlocal -
LDA, local density approximation; GGA, generalized gradient
approximation; τ, kinetic energy density; {φ }o , occupied
orbitals; {φ }, all orbitals.
Green function expansions for Exc : GW, Görling–Levy PT,...
Approximations to kinetic energy: Thomas–Fermi, von
Wiezsäcker,...
5 M. Casida, http://bit.ly/casidadft
6 Hybrid functionals mix in Hartree–Fock exchange.
21. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Rules of thumb7 for the accuracy of some ab initio methods
Property HF DFT MP2 CCSD(T)
IPs, EAs ±0.5 ±0.2 ±0.2 ±0.05 eV
Bond lengths –1% ±1 ±1 ±0.5 pm
Vib. freqs. +10% +3% +3% ±5 cm−1
Barrier heights +30—50% –25% +10% ±2 kcal/mol
Bond energies –50% ±3 ±10 ±1 kcal/mol
7 T. van Voorhis, MIT 5.74 course notes, 2010.
22. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Polarizabilities as linear response
Polarizability: a dipole’s response to an electric field
The isotropic ground state polarizability can be calculated exactly
from time–dependent perturbation theorya
∂ ˆ
r f0→n
α (ω) =
¯ =∑ 2 (21)
∂ ε (ω) n ω0→n − ω
2
where ω0→n is excitation frequency (energy) to state n and f0→n
are oscillator strengths. This is a sum over states (SOS) formula.
a Kramers, H. A.; Heisenberg, W. (1925) Z. Phys. 31: 681–708.
Dirac, P. A. M. (1927) Proc. Roy. Soc. Lond. A 114: 243–265; 710–728.
Linear response is a quest for resonances
Resonant frequencies of the system’s reaction to small
perturbations says something about its excitation spectrum.
23. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End
Resonances in density response
Main idea
Using the density response δ ρ to a time–dependent perturbing
potential δ v , we can back out what the excited state characters are.
Consider the generalized susceptibility χ (t) as defined by
ˆ ˆ
δ ρ (r , t) =
χ , , t − t δ v , t dt d
r r r r (22)
R R3
A SOS formula for the Fourier–transformed susceptibility χ can be
˜
derived, and furthermore the conditions for resonance can be
expressed as an eigenvalue equation that takes the form8
A B X I 0 X
=ω (23)
B ∗ A∗ Y 0 −I Y
X is the hole–particle (occupied–virtual) density change
Y is the particle–hole (virtual–occupied) density change
8 M. Casida, in Recent Advances in Density Functional Methods I, p. 155