Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
1) The theorem of least work states that for statically indeterminate structures, the partial derivative of the total strain energy with respect to redundant/statically indeterminate actions must be equal to zero.
2) This is because redundant forces act to prevent any displacement at their point of application. The forces developed in a redundant structure minimize the total internal strain energy.
3) The theorem is proved by analyzing a statically indeterminate beam as the superposition of a determinate beam with applied loads and a determinate beam with the redundant reaction. Equating the deflections caused by each case results in the condition that the strain energy is minimized.
Chapter3 - Fourier Series Representation of Periodic SignalsAttaporn Ninsuwan
This document discusses Fourier series representation of periodic signals. It introduces continuous-time periodic signals and their representation as a linear combination of harmonically related complex exponentials. The coefficients in the Fourier series representation can be determined by multiplying both sides of the representation by complex exponentials and integrating over one period. The key steps are: 1) multiplying both sides by e-jω0t, 2) integrating both sides from 0 to T=2π/ω0, and 3) using the fact that the integral equals T when k=n and 0 otherwise to obtain an expression for the coefficients an. Examples are provided to illustrate these concepts.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
1) The theorem of least work states that for statically indeterminate structures, the partial derivative of the total strain energy with respect to redundant/statically indeterminate actions must be equal to zero.
2) This is because redundant forces act to prevent any displacement at their point of application. The forces developed in a redundant structure minimize the total internal strain energy.
3) The theorem is proved by analyzing a statically indeterminate beam as the superposition of a determinate beam with applied loads and a determinate beam with the redundant reaction. Equating the deflections caused by each case results in the condition that the strain energy is minimized.
Chapter3 - Fourier Series Representation of Periodic SignalsAttaporn Ninsuwan
This document discusses Fourier series representation of periodic signals. It introduces continuous-time periodic signals and their representation as a linear combination of harmonically related complex exponentials. The coefficients in the Fourier series representation can be determined by multiplying both sides of the representation by complex exponentials and integrating over one period. The key steps are: 1) multiplying both sides by e-jω0t, 2) integrating both sides from 0 to T=2π/ω0, and 3) using the fact that the integral equals T when k=n and 0 otherwise to obtain an expression for the coefficients an. Examples are provided to illustrate these concepts.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
Iterative procedure for uniform continuous mapping.Alexander Decker
This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
1) The document discusses Castigliano's theorems, which relate the partial derivative of a structure's strain energy to its deflections under applied forces or moments.
2) Castigliano's first theorem states that the partial derivative of strain energy with respect to an applied force is equal to the deflection of the point of application in the direction of the force.
3) Examples are provided to demonstrate calculating deflections of beams under various load conditions using Castigliano's first theorem.
Some properties of two-fuzzy Nor med spacesIOSR Journals
The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Pa...A G
This document proposes a fast algorithm for solving wave scattering problems involving billions of particles using the convolution theorem and fast Fourier transforms (FFTs). The algorithm represents the Green's function as a vector and stores particle positions on a uniform grid, allowing the scattering calculation to be computed as a 3D convolution. This convolution can be rapidly evaluated using FFTs, significantly improving the efficiency over direct matrix-vector multiplication. The algorithm distributes data across multiple machines in a cluster to parallelize the computations.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
This document describes extending the Elgamal cryptosystem to work with the second group of units of Zn and Z2[x]/<h(x)>, where h(x) is an irreducible polynomial. It first reviews the definition and construction of the second group of units U2(Zn) and U2(Z2[x]/<h(x)>). It then presents the key generation, encryption, and decryption algorithms for the Elgamal cryptosystem adapted to these new settings. The document evaluates the accuracy, efficiency and security of the modified cryptographic scheme through implementation and testing.
Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.Keywords: Zeros of polynomial, Eneström - Kakeya theorem, Polar derivatives.
Title: On the Zeros of Polar Derivatives
Author: P. Ramulu, G.L. Reddy
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Coincidence points for mappings under generalized contractionAlexander Decker
1. The document presents a theorem that establishes conditions for the existence of coincidence points between multi-valued and single-valued mappings.
2. It generalizes previous results by Feng and Liu (2004) and Liu et al. (2005) by relaxing the contraction conditions.
3. The main theorem proves that if mappings T and f satisfy generalized contraction conditions involving α and β functions, and the space is orbitally complete, then the mappings have a coincidence point.
Fixed point theorem of discontinuity and weak compatibility in non complete n...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric spaces and weak compatibility to show the maps have a common fixed point.
11.fixed point theorem of discontinuity and weak compatibility in non complet...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric functions and the given conditions to show the sequence converges, establishing a common fixed point.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
The digestive system is a series of hollow organs that forms a long twisting tube from the mouth to the anus. Food is broken down mechanically and chemically as it passes through this system. Digestion involves the breakdown of large food molecules into smaller molecules that can be absorbed into bloodstream. The digestive system contains glands that produce juices to help break down food as it passes through organs like the mouth, stomach, and small intestine.
This document provides links to 10 images related to skateboarding, including photos of famous skateboarders Tony Hawk and Rob Dyrdek, cartoons, and sketches. It also includes photos of artistic representations of skateboarding like a Keith Haring wooden skateboard design and a photo of Rodin's "The Thinker" adapted onto a skateboard. The final two links are to generic photos of skateboards.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
Iterative procedure for uniform continuous mapping.Alexander Decker
This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
1) The document discusses Castigliano's theorems, which relate the partial derivative of a structure's strain energy to its deflections under applied forces or moments.
2) Castigliano's first theorem states that the partial derivative of strain energy with respect to an applied force is equal to the deflection of the point of application in the direction of the force.
3) Examples are provided to demonstrate calculating deflections of beams under various load conditions using Castigliano's first theorem.
Some properties of two-fuzzy Nor med spacesIOSR Journals
The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Pa...A G
This document proposes a fast algorithm for solving wave scattering problems involving billions of particles using the convolution theorem and fast Fourier transforms (FFTs). The algorithm represents the Green's function as a vector and stores particle positions on a uniform grid, allowing the scattering calculation to be computed as a 3D convolution. This convolution can be rapidly evaluated using FFTs, significantly improving the efficiency over direct matrix-vector multiplication. The algorithm distributes data across multiple machines in a cluster to parallelize the computations.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
This document describes extending the Elgamal cryptosystem to work with the second group of units of Zn and Z2[x]/<h(x)>, where h(x) is an irreducible polynomial. It first reviews the definition and construction of the second group of units U2(Zn) and U2(Z2[x]/<h(x)>). It then presents the key generation, encryption, and decryption algorithms for the Elgamal cryptosystem adapted to these new settings. The document evaluates the accuracy, efficiency and security of the modified cryptographic scheme through implementation and testing.
Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.Keywords: Zeros of polynomial, Eneström - Kakeya theorem, Polar derivatives.
Title: On the Zeros of Polar Derivatives
Author: P. Ramulu, G.L. Reddy
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Coincidence points for mappings under generalized contractionAlexander Decker
1. The document presents a theorem that establishes conditions for the existence of coincidence points between multi-valued and single-valued mappings.
2. It generalizes previous results by Feng and Liu (2004) and Liu et al. (2005) by relaxing the contraction conditions.
3. The main theorem proves that if mappings T and f satisfy generalized contraction conditions involving α and β functions, and the space is orbitally complete, then the mappings have a coincidence point.
Fixed point theorem of discontinuity and weak compatibility in non complete n...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric spaces and weak compatibility to show the maps have a common fixed point.
11.fixed point theorem of discontinuity and weak compatibility in non complet...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric functions and the given conditions to show the sequence converges, establishing a common fixed point.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
The digestive system is a series of hollow organs that forms a long twisting tube from the mouth to the anus. Food is broken down mechanically and chemically as it passes through this system. Digestion involves the breakdown of large food molecules into smaller molecules that can be absorbed into bloodstream. The digestive system contains glands that produce juices to help break down food as it passes through organs like the mouth, stomach, and small intestine.
This document provides links to 10 images related to skateboarding, including photos of famous skateboarders Tony Hawk and Rob Dyrdek, cartoons, and sketches. It also includes photos of artistic representations of skateboarding like a Keith Haring wooden skateboard design and a photo of Rodin's "The Thinker" adapted onto a skateboard. The final two links are to generic photos of skateboards.
The human digestive system is a series of hollow organs that form a long tube from the mouth to the anus. It includes the mouth, esophagus, stomach, small intestine, large intestine, and anus. Food passes through this alimentary canal, where it is broken down by mechanical and chemical digestion to extract nutrients for the body to use. The major organs include the stomach, small intestine, large intestine, liver, and pancreas.
ICA 2013 paper presentation: "Preaching to the Choir: Internet-Mediated Advoc...Luis Hestres
This is a presentation of my New Media & Society article "Preaching to the Choir: Internet-Mediated Advocacy, Issue Public Mobilization, and Climate Change." Here is the abstract:
Despite the impact that Internet-mediated advocacy organizations have had on American politics over the last decade, we are still learning about how they work. This is even truer for Internet-mediated issue specialists that focus on a single issue, such as climate change. Based on interviews with key staff members of two climate change advocacy campaigns, this article examines how these organizations communicate and mobilize citizens around their issue and the underlying assumptions behind their strategies. Interviews revealed a focus on like-minded issue public mobilization and online-to-offline social movement building strategies. The paper also examines how these organizations can influence policy debates by mobilizing issue publics, shifting debates to more favorable public arenas, and reframing them in ways more favorable to their causes. Implications for the future of climate policy and Internet-mediated advocacy research are discussed.
ITS World Congress :: Vienna, Oct 2012László Nádai
Control of the emission rate of exhaust fumes as well as vehicle density in various road segments of freeway traffic practically is desirable. For this purpose hydrodynamic macroscopic models are used that approximate the traffic as the motion of some compressible fluid. They normally have analytical ambiguities due to using various finite element approximations of spatial partial differential operators and applying various analytical compressibility models. Further uncertainties stem from the estimation of the model parameters. The suggested approach is able to iteratively and adaptively refine the traffic control based on any version of the possible analytical forms. The controller has to measure vehicle densities and traffic velocities and applies variable traffic signs for prescribing velocities and allowed ingress rate from a ramp as control signals. Since the stationary solutions of the control problem are stable in the region investigated, instead of fast, really dynamic control it applies simple quasi-stationary process model and control that is a common practice in Classical Thermodynamics. The operation of the proposed method is illustrated via simulations.
CT or CAT scans are a diagnostic imaging procedure that combines X-rays and computer technology to produce cross-sectional images of the body. CT scans produce multiple slice images that provide greater clarity and detail than regular X-rays, and are used to confirm the presence of tumors by measuring their size, precise location, and extent of involvement. CT scans can also produce 2D and 3D images and are used to diagnose conditions like vascular disease, guide biopsies and procedures, assess surgical results, and measure bone mineral density.
This document is a menu and information brochure for a cafe called Teen's Cafe. The cafe opened on January 1, 2001 and serves affordable, healthy and halal food and drinks suitable for teenagers. The brochure provides information on the cafe's menu including dishes like chicken chop, lamb chop, fish and chips, spaghetti carbonara and pizza. It also details the cafe's event and party packages, promotions, delivery and catering services. Contact information is provided at the end.
Producciones es una empresa dedicada a la producción de películas y series de televisión. Cuenta con un equipo de guionistas, directores, actores y técnicos expertos que trabajan en la creación de contenido audiovisual. El objetivo de la compañía es entretener e informar a la audiencia a través de sus diferentes producciones cinematográficas y televisivas.
Evaluation of educational television programs v4wuu360327
This document summarizes an evaluation of educational television programs for distance learning. It includes sections on the literature review, methodology, results and conclusion. The study aimed to evaluate TV programs to determine their effectiveness for distance learners and provide suggestions for improvement. A survey was conducted of 250 distance learners, with a 87% response rate. The results found that programs were effective to some extent in creating awareness but many learners were not satisfied with teaching techniques or faced technical issues. The conclusion was the best way to benefit is to inform learners of schedules and address technological limitations.
Climate advocacy emails content analysis ICA 2015 San Juan PRLuis Hestres
Slide deck of one of my presentations at the 2015 ICA conference in San Juan, PR. I mass emails from 5 environmental organizations and three climate change groups to compare and contrast their climate change advocacy strategies
Nicolas Morin -- Kanban - The (non)recipe for success -- Lean Kanban France 2...Nicolas Morin
Experience feedback on one year implementing Kanban in a 20 people team.
Main milestones of implementation will be put in parallel of the recipe for success by David J. Anderson (in "Kanban – Successful Evolutionary Change for Your Technology Business"). Then we'll see if success resides in a recipe, or to change adaptation, as stated in Agile manifesto.
Presentation will be concluded by evolution perspectives, for the team and the organization.
Este documento presenta información sobre el pensamiento estratégico de las empresas. Explica conceptos clave como la visión, misión, objetivos y estrategia a diferentes niveles. También describe cómo el entorno empresarial ha cambiado a lo largo del tiempo y cómo esto afecta el enfoque estratégico de las empresas. Finalmente, analiza los componentes de la estrategia a nivel corporativo, de negocio y funcional.
The document outlines veterans benefits provided by the state of Utah. It discusses the mission of the Utah Department of Veterans Affairs to assist veterans in obtaining federal and state benefits. It provides an overview of several state benefits, including tuition waivers, property tax abatements, license plates, and designations on driver's licenses. It also mentions veterans cemeteries, nursing homes, and organizations that support veterans.
This document appears to be a menu for a Malaysian restaurant called Zul Restaurant. The summary provides the following key details:
The menu offers a variety of Malaysian dishes including chicken chop, lamp chop, black pepper steak, soups and lasagna. It also offers drinks like tea, juice, coffee and milk-based drinks. The restaurant hosts events like parties, wedding receptions and kids cooking lessons. Contact information is provided at the end for the restaurant located in Kuala Lumpur, Malaysia.
Demanda y comportamiento del consumidorDalva Icaza
Este capítulo analiza la demanda y el comportamiento del consumidor desde una perspectiva económica. Explica cómo los consumidores toman decisiones sobre qué y cuánto comprar considerando sus preferencias y restricciones presupuestarias. También cubre temas como la elasticidad de la demanda y cómo los cambios en los precios y el ingreso afectan las decisiones de los consumidores.
This document provides an overview of modern atomic theory and quantum mechanics. It discusses the key discoveries and models that led to our current understanding of atomic structure, including Dalton's atomic theory, the discovery of subatomic particles, and the development of quantum mechanics with Schrodinger's equation. The four quantum numbers - principal, angular momentum, magnetic, and spin - are introduced to describe the allowed states of electrons in atoms. Rules for writing electron configurations are also covered.
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1. Many electron atoms
ρ( x )
x
3
dV = d x
x−r
r
Autumn 2012
Version: 04.12.2012
List of topics
2. (1)
Pauli-principle
Hund’s rule
Antisymmetric functions for n-particles - Slater Determinants
Filling Shells Figure
Ionization energies - Figure
Hartree - Selfconsistent field - Iterations
only Iterations part
Selfconsistent field Result - Figures
Configurations - Ionization potentials of atoms - Tables
Screened potential and Centrifugal barrier - Explains Tables
Energy for N-particles - Using Slater Determinants
Helium example
Lithium example
Counting nonzero terms
N particles - Energy Summary
Schr¨dinger equation from variational method
o
Variational method - deriving Hartree-Fock Equations
Hartree-Fock Equations
Total energy and the selfconsistent orbital energies
Evaluation of electron repulsion
Configuration mixing
2
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3
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1
1
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slide + 3 slides
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3. 1
Filling up the shells with electrons
In order to know and to, more or less, understand in which state an electron
is bound we can use some basic rules. The generel idea is that the lowest
energy-state is the most stable one. The exited states can in most cases fall in
the lower state or ground state by emission of light.
The principles one need to know how to fill up the shells are the Pauli-principle
and the Hund’s rule.
List of topics
3
4. 2
Pauli-principle
The filling of the shells observed was explained by Pauli by formulating the
Pauli exclusion principle:
Two electrons can not be in the same state - defined here by possible quantum
numbers n, l, m and ms
n, the principal quantum number of the shell,
l, the angular momentum quantum number of the subshell,
m, the magnetic quantum number
ms denotes spin ”up” and spin ”down”, ms = 1/2 or ms = −1/2
The Pauli principle mathematical: require the multiparticle wavefunction antisymmetric with respect to exchange of two particles.
If two particles coordinates are exchanged, the function must change sign.
If two particles are in the same state (function),
this requirement leads to zero wavefunction, thus impossible.
List of topics
4
6. Figure 2: Atomic levels (shells) filled up to the element ....List of topics
6
7. 3
Antisymmetric product function for n-particles
(−1)P (perm(α,β,...ν)) perm (φα φβ ....φν ) (x1 )(x2 )....(xn )
Ψ(x1 , x2 , ...xn ) =
perm(α,β,...ν)
where each term in the sum looks as φβ (x1 )...φν (x2 )...φα ..., summing over all
permutations, and P (perm(α, β, ...ν)) is the number of swaps of the given
permutation perm(α, β, ...ν)
This is very close to the definition of the determinant
n
det(A) =
sgn(σ)
Ai,σ(i)
i=1
σ∈Sn
The above in this notation
n
Ψ(x1 , x2 , ...xn ) =
sgn(σ)
σ∈Sn
List of topics
7
φασ(i) (xi )
i=1
8. Slater determinant
The antisymmetric combination for n-particles can be written as a determinant
in this way:
φα (x1 ) φα (x2 ) ... φα (xn )
φβ (x1 ) φβ (x2 ) ... φβ (xn )
...
...
...
...
φν (x1 ) φν (x2 ) ... φν (xn )
For 3 particles:
3 particle Slater determinant
φα (x1 ) φα (x2 ) φα (x3 )
φβ (x1 ) φβ (x2 ) φβ (x3 )
φγ (x1 ) φγ (x2 ) φγ (x3 )
List of topics
8
10. 4
Hund’s rule
Hund’s rule is the manifestation of the same effect as we have seen in the
parahelium - orthohelium effect ( the ”second” rule )
Hund’s first rule
Full shells and subshells do have a total circular momentum of zero.
This can be calculated and is allways valid.
Hund’s second rule
The total spin S should allways have the highest possible value. So as many
of the single electron spins as possible should be parallel.
The second rule appears more empirical and applies to a different magnetic
quantum number of the electrons with parallel spins. If is of course not allowed
to break the Pauli-principle.
List of topics
10
11. 5
Example for the Hund’s rule
Figure 3: Electron spins of the first elements.
The elements carbon C, nitrogen N and oxygen O are those where the Hund’s
rule have the biggest influence. We see electrons with parallel spins in the
states m = −1, 0, +1, depending on the element. The magnetic quantum
number m gives more or less the “direction” of the circular moment l. The
s-states are the states where l = 0 and the p-states those where l = 1. The
names for l = (2, 3, 4, ...) are (d, e, f, ...).
The shell-name of the shell n = (1, 2, 3, 4, ...) is (K, L, M, N, ...).
List of topics
11
12. In cases with more electrons do we get some exceptions. For example is the
4s subshell earlier filled than the 3d. This is caused by the smaller distance
of the 4s to the core and thus by the lower energy, which is more stable. The
mentioned rules work well for general considerations and for atoms with not
to many electrons.
6
Number of states
Usefull to know is the largest possible number of states for a given n which
means until a special shell is filled.
n−1
(2l + 1) = 2n2 .
Nmax = 2 ·
(2)
l=0
We get this formula by adding all the possible quantum number configurations:
We get for each n every l in the range (0, 1, ..., n − 1),
for each l every m in the range (−l, ..., −1, 0, 1, ..., l)
and for each of these states 2 spin possibilities.
List of topics
12
13. 7
Ionization energies
Figure 4: Ionization energies. The shell properties would explain the structure
in general - Periodic table and the Selfconsistent field
List of topics
13
14. Especially at the first elements, we see the minimums at the elements with
just one electron in the last shell and a maximum at the noble gas elements.
The weak maxima are caused by filled subshells or by the maximum of parallel
standing spins. After argon, it is more difficult to observe general tendences.
The shell properties would explain the structure in general
But there should be no closed shell at argon
Details - Periodic table and the Selfconsistent field
List of topics
14
15. 8
Hartree - Selfconsistent field
Interaction energy of two charges depends on their distance|r − x|:
W (|r − x|) =
q1 q2
|r − x|
The two charges are an electron and a little volume dV at x containing charge cloud
of density ρ
q1 → (−e)
q2 → ρ(x)dV
→ ρ(x)d3 x
ρ( x )
x
The interaction energy of these two
charges is
dW (|r1 − r2 |) =
3
dV = d x
x−r
(−e)ρ(r2 ) 3
d r2
|r − x|
r
List of topics
15
16. Interaction with a cloud; summing over all the small volume elements - it means
integrating over the whole volume of the cloud gives the potential energy
W (r) =
(−e)ρ(x) 3
d x
|r − x|
ρ( x )
x
3
dV = d x
x−r
r
List of topics
16
17. If the charge cloud represents one electron in state ψi (x)
and again integrating gives the potential energy due to the interaction with
a (probability based density) cloud of
electrons
ρ(x) = (−e)|ψi (x)|2
If we have N electrons, each in its state,
the total density becomes
(−e)2
N
|ψi (x)|
ρ(x) = (−e)
W (r) =
2
i=1
ρ( x )
x
3
dV = d x
x−r
r
List of topics
17
N
|ψi (x)|2
i=1
|r − x|
d3 x
18. Now solving the Schr¨dinger equation with W (r),
o
(T + V + W ) ψi (x) = Ei ψi (x)
We first need to know the W (r), but that depends on all the other N solutions
N
(−e)2
|ψi (x)|2
i=1
W (r) =
|r − x|
d3 x
Approximation chain: First we choose some simple approximation, e.g. the hydrogenlike states, or we might know the states foranother atom. We call it
(0)
ψi (x)
(0)
From the set of all N ψi
we construct
e2
W (1) (r) =
N
i=1
(0)
|ψi (x)|2
|r − x|
d3 x
In atomic units the whole Schr¨dinger equation is
o
−
1
2
2
−
Z
(1)
(1) (1)
+ W (1) (r) ψi (x) = Ei ψi (x)
r
List of topics
18
19. (0)
1. step - choose arbitrary set of ψi
N
(0)
→
ψi (x)
−
1
2
2
−
(1)
2. step: take the set of ψi
W (1) (r) =
i=1
|r − x|
from the 1. step
(1)
→
ψi (x)
1
2
2
d3 x
Z
(1)
(1) (1)
+ W (1) (r) ψi (x) = Ei ψi (x)
r
N
−
(0)
|ψi (x)|2
−
W (2) (r) =
i=1
(1)
|ψi (x)|2
|r − x|
d3 x
Z
(2)
(2) (2)
+ W (2) (r) ψi (x) = Ei ψi (x)
r
List of topics
19
(3)
20. (2)
3. step: take the set of ψi
from the 2. step
N
(2)
→
ψi (x)
−
1
2
2
−
(3)
4. step: take the set of ψi
W (3) (r) =
i=1
|r − x|
from the 3. step
(3)
→
ψi (x)
1
2
2
d3 x
Z
(3)
(3) (3)
+ W (3) (r) ψi (x) = Ei ψi (x)
r
N
−
(2)
|ψi (x)|2
−
W (4) (r) =
i=1
(3)
|ψi (x)|2
|r − x|
d3 x
Z
(4)
(4) (4)
+ W (4) (r) ψi (x) = Ei ψi (x)
r
List of topics
20
21. (4)
5. step: take the set of ψi
from the 4. step
N
(4)
→
ψi (x)
−
1
2
2
−
(5)
6. step: take the set of ψi
W (5) (r) =
i=1
|r − x|
from the 5. step
(5)
→
ψi (x)
1
2
2
d3 x
Z
(5)
(5) (5)
+ W (5) (r) ψi (x) = Ei ψi (x)
r
N
−
(4)
|ψi (x)|2
−
W (6) (r) =
i=1
(5)
|ψi (x)|2
|r − x|
d3 x
Z
(6)
(6) (6)
+ W (6) (r) ψi (x) = Ei ψi (x)
r
List of topics
21
22. Iteration chain
N
(n)
ψi (x)
→
W (n+1) (r) =
i=1
(n)
|ψi (x)|2
|r − x|
d3 x
→
(n+1)
ψi
(x)
(n)
This chain can continue, until the set of ψi produces a potential W (n+1) which
(n)
is the same as W (n) , which was the one to determine ψi . The potentials and
functions become consistent, hence the name Selfconsistent field.
Criterium for self-consistency: the (n+1)-th solution does not differ from th n-th
solution
N
(n+1)
|ψi
(n)
(x)|2 − |ψi (x)|2 d3 x <
i=1
where
∝ 10−8 as a typical value.
List of topics
22
(4)
23. 9
Ionization potentials of atoms
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9= W I$
List of topics
W #$%&'()
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W I$J:F(4%5325
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26. Screened potential and Centrifugal barrier
0.5
Coulomb potential
L=2
L=1
0
L=0
−0.5
0
1
2
3
4
5
6
7
8
6
7
8
0.5
Hartree potential
L=2
L=1
0
L=0
−0.5
0
−
1
Z
r
+
2
3
L(L + 1)
r2
4
5
vs.
−
26
Z − 1 −αr 1
e
−
r
r
List of topics
+
L(L + 1)
r2
(5)
28. Figure 5: Selfconsistent field calculations compared to coulomb - real results
Here we note the position of the turning point - same for both Coulomb and
SCF case. This is true only for the 1s states. List of topics
28
29. Figure 6: Hartree Atomic potential
This figure shows results for oxygen with the potential and energies obtained
from a calculation with SCF program
List of topics
29
30. results for oxygen with the potential and energies obtained from a calculation
with SCF program compared with hydrogen-like levels List of topics
30
31. 10
Expectation value of energy for N-particles
Hamiltonian for N-particles - the repulsion - we sum over pairs of coordinates
N
2
2
e
Ze
1 2
+
H (r1 , r2 , ...rN ) =
−
ri −
2
ri
|ri − rj |
i=1
(i,j)pairs
Without antisymmetrization - independent particles - product function only 1 term -each particle has its orbital ψa ,ψb ,ψc .....ψn
Φ (r1 , r2 , ...rN ) → ψa (r1 )ψb (r2 )ψc (r3 )......ψn (rN )
With antisymmetrization - the Slater determinant
1
ΦHF (r1 , r2 , ...rN ) → √
a,b,...n
N!
List of topics
31
ψa (r1 )ψb (r1 )......ψn (r1 )
ψa (r2 )ψb (r2 )......ψn (r2 )
.......
ψa (rN )ψb (rN )......ψn (rN )
32. For N-particles - the repulsion - we sum over pairs of coordinates
N
2
2
1 2
Ze
e
Φ (r1 , r2 , ...rN ) = EΦ (r1 , r2 , ...rN )
−
+
ri −
2
ri
|ri − rj |
i=1
(i,j)pairs
Expectation value with the independent particles - product function - only
1 term
Φ (r1 , r2 , ...rN ) → ψa (r1 )ψb (r2 )ψc (r3 )......ψn (rN )
becomes
n
E = Φ| H |Φ
→
ψα | T −
α=a
Ze2
|ψα +
r
ψβ ψα |
(α,β)pairs
e2
|ψβ ψα
|r − r |
The sum over coordinate pairs becomes sum over pairs of orbitals
This remains true for Slater determinants - but it must be shown.
There are N ! terms in the Slater determinant, the left and right give
(N !)2 terms, and there are N (N − 1)/2 coordinate pairs.
Thus (N !)2 N (N − 1)/2 terms.
Due to the normalization, this becomes only N (N − 1) terms,
N (N − 1)/2 direct terms and N (N − 1)/2 exchange terms
List of topics
32
33. It should be shown that when the Slater determinant is used
ψa (r1 )ψb (r1 )......ψn (r1 )
1
ψa (r2 )ψb (r2 )......ψn (r2 )
ΦHF (r1 , r2 , ...rN ) → √
a,b,...n
N ! .......
ψa (rN )ψb (rN )......ψn (rN )
the expectation value will give (final result shown here first)
n
ΦHF H ΦHF
=
ψα | T −
j=α
Ze2
|ψα
r
ψβ ψα |
e2
|ψβ ψα
|r − r |
ψβ ψα |
+
e2
|ψα ψβ
|r − r |
(α,β)pairs
−
(α,β)pairs
(6)
The last term - exchange energy. On the previous page product function (1
term) is shown - no exchange energy
Next 3 slides: 1. Helium, 2. Lithium examples, 3. Counting nonzero terms
List of topics
33
34. r 12
−e2
r1
r2
For Helium
−
h2
¯
2me
2
r1
−
−e1
+Ze
Z e2
h2
¯
−
r1
2me
1
ΦHF (r1 , r2 ) = √
a,b
2
2
r2
−
Z e2
e2
Ψ (r1 , r2 ) = E Ψ (r1 , r2 )
+
r2
|r1 − r2 |
1
= √ [ψa (r1 )ψb (r2 ) − ψb (r1 )ψa (r2 )]
2
ψa (r1 ) ψb (r1 )
ψa (r2 ) ψb (r2 )
and the energy becomes
ΦHF H ΦHF
=
+
Ze2
Ze2
|ψa + ψb | T −
|ψb
r
r
e2
e2
ψa ψb |
|ψa ψb − ψa ψb |
|ψb ψa
|r − r |
|r − r |
ψa | T −
The last term - exchange energy - and there is only one pair.
List of topics
Lithium example
34
(7)
36. 1
3!
a
b c
+
a
b c
+
a
b c
+
a
b c
−
a
b c
−
a
b c
−
a
+
+
+
+
+
b c
+
+ c
a
+
e2
r2 3
e2
r2 3
e2
r2 3
e2
r2 3
e2
r2 3
e2
r2 3
b
+ b
c
a
+
− b
a
−
c
−
c
b a
−
− a
c
−
b
a
b c
a
a
b c
c
a
b
a
c
b c
e2
r2 3
e2
r2 3
b
c
a
a
b
b c
e2
r2 3
c a
b
a
c
a
b
b c
e2
r2 3
a c
c
b a
a
c
b c
e2
r2 3
b a
a
c
a
a
b c
e2
r2 3
c b
+
+
+
−
−
−
−
b
b c
a b
Evaluation of the first six terms of the total 36 terms. For each further group
of 6 a very similar procedure would follow. From each six, two remain, totaling
12 nonzero terms
List of topics
36
37. 11
N particles in Slater determinant - Total
Energy Summary
N
Φ| H |Φ
=
j=1
Ze2
|ψj +
ψj | T −
r
(i,j)pairs
e2
ψj ψi |
|ψj ψi
|r − r |
2
−
ψj ψi |
(i,j)pairs
e
|ψi ψj
|r − r |
(9)
We will use also a more compact notation, instead of orbitals ψi we use orbitals
|α , i.e. the index becomes the name of the function
Φ| H |Φ
=
α|(T + V )|α +
α
[ αβ|Vee |αβ − αβ|Vee |βα ] (10)
pairs αβ
List of topics
37
38. Schr¨dinger equation from variational principle: minimize
o
Φ| H |Φ
Φ|Φ
(≥ Eg.s. )
which can be written as minimalization with constraint Φ | Φ = 1 ; minimize
Φ| H |Φ − λ Φ | Φ
Φ |Φ −1=0
λ Lagrange multiplier
Derivative - differential
f (x) →
df (x)
dx
→
df =
df (x)
dx
dx
Functional derivative - functional F(f ); variation of the functions f (x), δf
F(f )
→
δF(f )
δf
→
δF =
δF(f )
δf
δf
(11)
Variation of Φ| H |Φ is taken (considering the complex nature of |Φ ) as
δ ( Φ| H |Φ ) = δΦ| H |Φ
With arbitrary variations |δΦ
δΦ| H |Φ − λ δΦ | Φ
−→
H |Φ − λ |Φ = 0 −→ H |Φ
Thus the Lagrange multiplier plays the role of energy eigenvalue
38
= λ |Φ
List of topics
39. Deriving Hartree-Fock: For N electrons this energy must be minimized
Φ| H |Φ
=
[ αβ|Vee |αβ − αβ|Vee |βα ]
α|(T + V )|α +
α
(12)
pairs αβ
with respect to the N orbitals |α , |β , etc. ,
with N conditions α|α = 1, β|β = 1, i.e.
with N Lagrange multipliers εα , εβ .......
δΦ(α → δα)| H |Φ − εα δα | α
= 0
Which results in N equations (one for each of the N orbitals)
[ (δα)β|Vee |αβ − (δα)β|Vee |βα ] − εα δα | α
δα|(T + V )|α +
= 0 (13)
β
Leaving for a while out the exchange-related term, we can as in prev. slide reduce
this by removing δα|
[ β|Vee |β ] |α − εα | α
(T + V )|α +
=0
β
which is exactly the Hartree method. The exchange term leads to complications →
List of topics
39
40. Reducing the full equation
[ (δα)β|Vee |αβ − (δα)β|Vee |βα ] − εα δα | α
δα|(T + V )|α +
= 0 (14)
β
we get the Hartree Fock Equations (N orbitals)
[ β|Vee |β ] |α −
(T + V )|α +
β
[ β|Vee |α ] |β − εα | α
=0
β
The complexity of the exchange term becomes clear when we write it explicitely
with coordinates and integrations.
Hartree Fock Equations: The direct term and the exchange term:
WHF = W d − W ex
occ
∗
ψb (x)
WHF ψa (r) =
b
e2
ψb (x)d3 x ψa (r)−
|r − x|
occ
∗
ψb (r)
b
ψb (x)
e2
ψa (x)d3 x
|r − x|
where the exchange potential is nonlocal
W d ψa → W d (r)ψa (r)
W ex ψa (r) →
List of topics
40
W ex (r, x)ψa (x)d3 x
41. Hartree-Fock total energy and sum of orbital energies
with a shorthand notation
ϕα (r) → |ϕα
→ |α
Slater determinant
Φα,β,....,ν
→ |α, β, ...., ν
We have started our work with the Hartree-Fock equations by evaluating
Φα,β,....,ν |
T +V +
pairs ij
e2
|ri − rj |
|Φα,β,....,ν
(15)
This we have evaluated as
Φ| H |Φ
=
α|(T + V )|α +
α
[ αβ|Vee |αβ − αβ|Vee |βα ] (16)
pairs αβ
From this expression we have obtained Hartree-Fock equations by the variational procedure.
List of topics
41
42. The Hartree-Fock equation can be then written as
β|
T +V +
β
e2
|β
|r − r |
|α −
β|
β
e2
|α
|r − r |
|β
= εα |α (17)
We form the matrix element with the given α|
α| T + V +
β|
β
e2
|β
|r − r |
|α − α|
β|
β
e2
|α
|r − r |
|β
= εα
(18)
since α|α = 1. This can be rewritten as
εα =
α|(T + V )|α +
[ βα|Vee |βα − βα|Vee |αβ ]
(19)
β=α
And now we can explore what is the sum of all εα
εα =
α|(T + V )|α +
α
α
[ βα|Vee |βα − βα|Vee |αβ ]
(20)
α β=α
which we can compare with the above Φ| H |Φ
Φ| H |Φ
=
α|(T + V )|α +
α
[ αβ|Vee |αβ − αβ|Vee |βα ] (21)
pairs αβ
42
43. The two expressions are very similar, but they differ in fact by all the interaction term, since it is counted twice in the sum:
Fαβ
= 2
α β=α
Fαβ
pairs αβ
for any set of objects that are symmetric Fαβ = Fβα
Our objects are symmetric, because they are in fact of the type Hαβ,αβ . The
wavefunctions are antisymmetric, |αβ = −|βα .
Thus, surprisingly perhaps
Φ| H |Φ
=
εα
α
but as we derived here
Φ| H |Φ
List of topics
=
εα −
α
[ αβ|Vee |αβ − αβ|Vee |βα ]
pairs αβ
43
(22)
44. Evaluation of the repulsion term using the multipole expansion
1
=
|r1 − r2 |
LM
L
r<
4π
YLM (ˆ1 ) YLM (ˆ2 )
r
r
L+1
2L + 1 r>
(23)
where
r< = r1 ,
r> = r2
for |r1 | < |r2 |
r< = r2 ,
r> = r1
for |r1 | > |r2 |
Evaluation of the matrix element in general case
d3 r1
d3 r2 ψn1 l1 m1 (r1 ) ψn2 l2 m2 (r2 )
1
ψn l m (r1 ) ψn2 l2 m2 (r2 ) (24)
|r1 − r2 | 1 1 1
is performed separately over the radial and angular parts
2
r1 dr1
dˆ1
r
2
r2 dr2
dˆ2
r
1
|r1 − r2 |
r
r
Rn1 l1 (r1 )Yl1 m1 (ˆ1 ) Rn2 l2 (r2 )Yl2 m2 (ˆ2 )
Rn1 l1 (r1 )Yl1 m1 (ˆ1 ) Rn2 l2 (r2 )Yl2 m2 (ˆ2 )(25)
r
r
where dˆi means the integration over dΩi = sin θi dθi dϕi . List of topics
r
44
45. The evaluation of general case - angular integrals of three Ylm ’s
CL =
Yli mi (θ, ϕ)YLM (θ, ϕ)Yli mi (θ, ϕ)dΩ
(26)
For the case of both s-states, li = 0 mi = 0 only L = 0 M = 0 are nonzero; The sum reduces to one term. The angular factors give value one, since
the (YL=0M =0 )2 = (4π)−1 cancels the corresponding factor in the multipole
expansion and due to the normalization.
List of topics
45
46. 12
Configuration mixing
Consider the usual:
Hx (x)ϕα (x) = Eα ϕα (x)
Hy (y)χβ (y) = Eβ χβ (y)
For any Φ(x)
Φ(x) =
cα ϕα (x)
Ξ(y) =
dβ χβ (y)
For any Ξ(x)
Take now a general Ψ(x, y). First look at y as a parameter, Ψ(x, y0 )
Ψ(x, y0 ) → Φ(x) =
cα (y0 )ϕα (x)
for every y0 ; Thus we get a new function of y;
cα (y) =
dβ (α)χβ (y)
Inserting back:
Ψ(x, y) =
dβ (α)χβ (y)ϕα (x)
46
47. Or, with a simpler notation
Ψ(x, y) =
dβα χβ (y)ϕα (x)
In the case of Helium, for example, the H(x) and H(y) are identical
and so are the χβ (y) and ϕα (x). This becomes configuration mixing.
Ψ(x, y) =
dβα ϕβ (y)ϕα (x)
The coefficients are found by diagonalization.
For three coordinate sets - e.g. for Lithium :
Ψ(x, y, z) =
Dγβα ϕ(z)ϕβ (y)ϕα (x)
List of topics
47