Amazon cover image
Image from Amazon.com

Solving the Schrodinger Equation : Has Everything Been Tried?.

By: Publisher: Singapore : Imperial College Press, 2011Copyright date: ©2011Description: 1 online resource (375 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781848167254
Subject(s): Genre/Form: Additional physical formats: Print version:: Solving the Schrodinger Equation : Has Everything Been Tried?DDC classification:
  • 530.124
LOC classification:
  • QC174.26.W28 -- S65 2011eb
Online resources:
Contents:
Intro -- Contents -- Preface -- 1. Intracule Functional Theory Deborah L. Crittenden and Peter M.W. Gill -- 1.1 Introduction -- 1.2 Intracules -- 1.3 Electron Correlation Models -- 1.4 Dynamic and Static Correlation -- 1.5 Dispersion Energies -- 1.6 Future Prospects -- Bibliography -- 2. Explicitly Correlated Electronic Structure Theory Frederick R. Manby -- 2.1 Introduction -- 2.1.1 Basis-set expansions -- 2.2 F12 Theory -- 2.2.1 MP2-F12 -- 2.2.2 Explicitly correlated coupled-cluster theory -- 2.3 Five Thoughts for F12 Theory -- 2.3.1 Thought 1: Do we need (products of) virtuals? -- 2.3.2 Thought 2: Are there better two-electron basis sets? -- 2.3.3 Thought 3: Do we need the resolution of the identity? -- 2.3.4 Thought 4: Could we have explicit correlation for higher excitations? -- 2.3.5 Thought 5: Can we avoid three-electron errors in two-electron systems? -- 2.4 Conclusions -- Acknowledgments -- Bibliography -- 3. Solving Problems with Strong Correlation Using the Density Matrix Renormalization Group (DMRG) Garnet Kin-Lic Chan and Sandeep Sharma -- 3.1 The Problem of Strong Correlation -- 3.2 The Density Matrix Renormalization Group Wavefunction -- 3.3 Locality and Entanglement in the DMRG -- 3.4 Other Properties of the DMRG -- 3.5 Relation to the Renormalization Group -- 3.6 Dynamic Correlation - the Role of Canonical Transformations -- 3.7 What Can the DMRG Do? A Brief History -- 3.8 The Future: Higher Dimensional Analogues -- Bibliography -- 4. Reduced-Density-Matrix Theory for Many-electron Correlation David A. Mazziotti -- 4.1 Introduction -- 4.2 Variational 2-RDM Method -- 4.2.1 Energy as a 2-RDM functional -- 4.2.2 Positivity conditions -- 4.2.2.1. Two-positivity -- 4.2.2.2. Three-positivity -- 4.2.3 Semidefinite programming -- 4.2.4 Applications -- 4.2.4.1. Dissociation of the N molecule.
4.2.4.2. Metal to insulator transition in the H lattice -- 4.2.4.3. Polyradical character of the acene chains -- 4.3 Contracted Schr dinger Theory -- 4.3.1 ACSE and cumulant reconstruction -- 4.3.2 Solving the ACSE for ground and excited states -- 4.3.3 Applications -- 4.3.3.1. Energy barriers of bicyclobutane's transition states -- 4.3.3.2. Conical intersections in CH 's triplet excited states -- 4.4 Parametric 2-RDM Method -- 4.4.1 Parametrization of the 2-RDM -- 4.4.2 Applications -- 4.4.2.1. Correlation energies at equilibrium geometries -- 4.4.2.2. Dissociation of the HF molecule -- 4.4.2.3. Isomerization of nitrosomethane -- 4.5 Looking Ahead -- Acknowledgments -- Bibliography -- 5. Finite Size Scaling for Criticality of the Schr dinger Equation Sabre Kais -- 5.1 Introduction -- 5.2 Criticality for Large-dimensional Models -- 5.3 Finite Size Scaling: A Brief History -- 5.4 Finite Size Scaling for the Schr dinger Equation -- 5.5 The Hulthen Potential -- 5.5.1 Analytical solution -- 5.5.2 Basis set expansion -- 5.5.3 Finite element method -- 5.5.4 Finite size scaling results -- 5.6 Finite Size Scaling and Criticality of M-electron Atoms -- 5.7 Conclusions -- Acknowledgments -- Bibliography -- 6. The Generalized Sturmian Method James Avery and John Avery -- 6.1 Description of the Method -- 6.1.1 The introduction of Sturmians into quantum theory -- 6.1.2 Generalized Sturmians -- 6.1.3 The generalized Sturmian method applied to atoms -- 6.1.4 Goscinskian configurations -- 6.1.5 Goscinskian secular equations for atoms and atomic ions -- 6.2 Advantages: Some Illustrative Examples -- 6.2.1 The large-Z approximation: restriction of the basis set to an R-block -- 6.2.2 Validity of the large-Z approximation -- 6.2.3 Core ionization energies -- 6.3 Limitations of the Method -- Prospects for the Future.
6.3.1 Can the generalized Sturmian method be applied to N-electron molecules? -- 6.4 Discussion -- Bibliography -- 7. Slater-Type Orbital Basis Sets: Reliable and Rapid Solution of the Schr dinger Equation for Accurate Molecular Properties Philip E. Hoggan -- 7.1 Introduction -- 7.1.1 Context of this chapter -- 7.1.2 Atomic orbitals -- 7.1.3 Problems to be solved when using Slater-type orbitals -- 7.1.4 Strategy for Slater basis sets -- 7.2 Some Dates: The Story So Far of Slater-Type Orbitals -- 7.3 Computer Programs Using Slater-Type Orbitals -- 7.3.1 Numerical grid methods -- 7.3.2 Configuration interaction -- 7.4 Slater Orbitals and Gaussian Orbitals -- 7.5 Types of Exponentially Decaying Orbitals, Based on Eigenfunctions for One-Electron Atoms -- 7.5.1 Orbitals which are linear combinations of Slater-type orbitals -- 7.6 Types of Integral Over Slater Orbitals -- 7.6.1 One-electron integrals -- 7.6.2 Two-electron integrals -- 7.6.3 Three- and four-electron integrals -- 7.7 Integration Methods in the Literature -- 7.7.1 Single-center expansion -- 7.7.2 Gaussian expansion -- 7.7.3 Gaussian transform method -- 7.7.4 Fourier-transform method -- 7.7.5 Use of Sturmians -- 7.7.6 Elliptic coordinate method -- 7.7.7 Monte Carlo integration -- 7.8 General Two-Electron Exponential Type Orbital Integrals in Poly-Atomics Without Orbital Translations -- 7.8.1 Introduction -- 7.8.2 Basis sets -- 7.8.3 Programming strategy -- 7.8.4 Avoiding ETO translations for two-electron integrals over three- and four-centers. -- 7.8.5 Numerical results of Coulomb resolutions: efficiency -- 7.8.5.1. H molecule with interatomic distance of 1.402 atomic units (a.u.) -- 7.8.5.2. Dimer geometry: rectangular and planar -- 7.8.5.3. Selected exchange integrals for the CH F molecule (evaluated using the Coulomb resolution) -- 7.8.6 Perspectives and conclusions.
7.8.7 Angular momentum relations -- 7.9 When are Slater-Type Orbitals Advantageous? Some Applications -- 7.9.1 The NMR nuclear shielding tensor -- Analytical treatment -- Application -- Conclusion -- 7.9.2 Explicitly correlated methods for molecules -- 7.9.3 Trial wave-functions for quantum Monte Carlo simulations over STO -- 7.9.3.1. Basis sets -- 7.9.3.2. Computational procedure -- 7.9.3.3. Perspectives -- 7.10 Highly Accurate Calculations Using STOs -- 7.11 Closing Remarks -- 7.12 Appendix A: How STOs were Translated: Products on Two Atoms -- 7.12.1 Review of BCLFs -- 7.12.1.1. Definition and properties of BCLFs -- 7.13 Appendix B: Brief Time-Line of Events in Molecular Work Over Slater-Type Orbitals to Date -- 7.14 Appendix C: Main Results of Podolanski's Paper of 1931 with Additional Comments -- 7.15 Appendix D: Potentials and Auxiliary Overlaps for Coulomb Resolution -- 7.16 Appendix E: Analysis of Nuclear Dipole Integrals for NMR in a Slater Basis -- Acknowledgements -- Bibliography -- 8. Modern Ab Initio Valence Bond Methods Philippe C. Hiberty and Sason Shaik -- 8.1 Basic Principles and Survey of Modern Methods -- 8.1.1 VB vs.MO wave functions in the two-electron/ two-center case -- 8.1.2 Writing VB functions beyond the two-electron/ two-center case -- 8.1.3 Some landmark improvements of the early VB method -- 8.1.3.1. VB wave functions with semi-localized atomic orbitals -- 8.1.3.2. Efficient orbital optimization by Self-Consistent-Field VB -- 8.1.3.3. Improving the accuracy by including dynamic electron correlation -- 8.1.3.4. Inclusion of solvent effects -- 8.2 Strengths of the Valence Bond Approach -- 8.2.1 Interpretability combined with accuracy of the wave functions -- 8.2.2 A simple solution to the symmetry dilemma -- 8.2.3 Calculations of diabatic energy curves along a reaction coordinate -- 8.2.3.1. General model.
8.2.3.2. Application to hydrogen-abstraction reactions -- 8.2.4 Quantitative evaluation of common chemical paradigms -- 8.2.4.1. Direct calculation of resonance energies and hyperconjugation energies -- 8.2.4.2. Characterization of a novel type of bonding: Charge-shift bonds -- 8.2.4.3. σ vs π driving force for the D geometry of benzene -- 8.3 Present Capabilities and Expected Improvements -- 8.3.1 Evaluation of Hamiltonian matrix elements -- 8.3.2 Direct VBSCF/BOVB algorithm -- 8.3.3 Current calculations of medium-sized molecular systems -- 8.3.4 Mixed Valence Bond - Quantum Monte Carlo methods -- 8.3.5 Prospective -- 8.4 Concluding Remarks -- 8.5 Appendix A: The Myth of "VB failures" -- 8.6 Appendix B: Some Available VB Software Packages -- 8.6.1 The XMVB program -- 8.6.2 The TURTLE software -- 8.6.3 The VB2000 software -- 8.6.4 The CRUNCH software -- Bibliography -- 9. Quantum Monte Carlo Approaches for Tackling Electronic Correlation Massimo Mella and Gabriele Morosi -- 9.1 Introduction -- 9.2 Variational Monte Carlo (VMC): A Possible Way Toward Explicitly Correlated Electronic Wave Functions -- 9.2.1 Numerical integrals in VMC -- 9.2.1.1. General introduction -- 9.2.1.2. Sampling of P -- 9.2.2 Optimization of trial wave functions -- 9.2.2.1. Minimum variance -- 9.2.2.2. Minimum energy -- 9.2.3 Analytical forms for trial wave functions T -- 9.3 Diffusion Monte Carlo: How to Extract the Best Information from Inaccurate Wave Functions -- 9.3.1 Generalities -- 9.3.2 Improved projectors -- 9.3.3 DMC, state symmetry and excited states -- 9.4 Computing Observables Different from State Energy -- 9.4.1 Exact calculation of position dependent observables -- 9.4.2 Calculation of atomic forces in VMC/DMC -- 9.4.3 Computing the expectation value of ultra-local operators: electron and spin density on nuclei -- 9.5 Conclusions -- Bibliography.
10. Solving the Schr dinger Equation on Real-Space Grids and with Random Walks Thomas L. Beck and Joel H. Dedrick.
Summary: Key Features:Unusual combination of methods/techniquesA thought-provoking and didactic exposé, not a review, nor a textbookLooks at the future.
Holdings
Item type Current library Call number Status Date due Barcode Item holds
Ebrary Ebrary Afghanistan Available EBKAF00059519
Ebrary Ebrary Algeria Available
Ebrary Ebrary Cyprus Available
Ebrary Ebrary Egypt Available
Ebrary Ebrary Libya Available
Ebrary Ebrary Morocco Available
Ebrary Ebrary Nepal Available EBKNP00059519
Ebrary Ebrary Sudan Available
Ebrary Ebrary Tunisia Available
Total holds: 0

Intro -- Contents -- Preface -- 1. Intracule Functional Theory Deborah L. Crittenden and Peter M.W. Gill -- 1.1 Introduction -- 1.2 Intracules -- 1.3 Electron Correlation Models -- 1.4 Dynamic and Static Correlation -- 1.5 Dispersion Energies -- 1.6 Future Prospects -- Bibliography -- 2. Explicitly Correlated Electronic Structure Theory Frederick R. Manby -- 2.1 Introduction -- 2.1.1 Basis-set expansions -- 2.2 F12 Theory -- 2.2.1 MP2-F12 -- 2.2.2 Explicitly correlated coupled-cluster theory -- 2.3 Five Thoughts for F12 Theory -- 2.3.1 Thought 1: Do we need (products of) virtuals? -- 2.3.2 Thought 2: Are there better two-electron basis sets? -- 2.3.3 Thought 3: Do we need the resolution of the identity? -- 2.3.4 Thought 4: Could we have explicit correlation for higher excitations? -- 2.3.5 Thought 5: Can we avoid three-electron errors in two-electron systems? -- 2.4 Conclusions -- Acknowledgments -- Bibliography -- 3. Solving Problems with Strong Correlation Using the Density Matrix Renormalization Group (DMRG) Garnet Kin-Lic Chan and Sandeep Sharma -- 3.1 The Problem of Strong Correlation -- 3.2 The Density Matrix Renormalization Group Wavefunction -- 3.3 Locality and Entanglement in the DMRG -- 3.4 Other Properties of the DMRG -- 3.5 Relation to the Renormalization Group -- 3.6 Dynamic Correlation - the Role of Canonical Transformations -- 3.7 What Can the DMRG Do? A Brief History -- 3.8 The Future: Higher Dimensional Analogues -- Bibliography -- 4. Reduced-Density-Matrix Theory for Many-electron Correlation David A. Mazziotti -- 4.1 Introduction -- 4.2 Variational 2-RDM Method -- 4.2.1 Energy as a 2-RDM functional -- 4.2.2 Positivity conditions -- 4.2.2.1. Two-positivity -- 4.2.2.2. Three-positivity -- 4.2.3 Semidefinite programming -- 4.2.4 Applications -- 4.2.4.1. Dissociation of the N molecule.

4.2.4.2. Metal to insulator transition in the H lattice -- 4.2.4.3. Polyradical character of the acene chains -- 4.3 Contracted Schr dinger Theory -- 4.3.1 ACSE and cumulant reconstruction -- 4.3.2 Solving the ACSE for ground and excited states -- 4.3.3 Applications -- 4.3.3.1. Energy barriers of bicyclobutane's transition states -- 4.3.3.2. Conical intersections in CH 's triplet excited states -- 4.4 Parametric 2-RDM Method -- 4.4.1 Parametrization of the 2-RDM -- 4.4.2 Applications -- 4.4.2.1. Correlation energies at equilibrium geometries -- 4.4.2.2. Dissociation of the HF molecule -- 4.4.2.3. Isomerization of nitrosomethane -- 4.5 Looking Ahead -- Acknowledgments -- Bibliography -- 5. Finite Size Scaling for Criticality of the Schr dinger Equation Sabre Kais -- 5.1 Introduction -- 5.2 Criticality for Large-dimensional Models -- 5.3 Finite Size Scaling: A Brief History -- 5.4 Finite Size Scaling for the Schr dinger Equation -- 5.5 The Hulthen Potential -- 5.5.1 Analytical solution -- 5.5.2 Basis set expansion -- 5.5.3 Finite element method -- 5.5.4 Finite size scaling results -- 5.6 Finite Size Scaling and Criticality of M-electron Atoms -- 5.7 Conclusions -- Acknowledgments -- Bibliography -- 6. The Generalized Sturmian Method James Avery and John Avery -- 6.1 Description of the Method -- 6.1.1 The introduction of Sturmians into quantum theory -- 6.1.2 Generalized Sturmians -- 6.1.3 The generalized Sturmian method applied to atoms -- 6.1.4 Goscinskian configurations -- 6.1.5 Goscinskian secular equations for atoms and atomic ions -- 6.2 Advantages: Some Illustrative Examples -- 6.2.1 The large-Z approximation: restriction of the basis set to an R-block -- 6.2.2 Validity of the large-Z approximation -- 6.2.3 Core ionization energies -- 6.3 Limitations of the Method -- Prospects for the Future.

6.3.1 Can the generalized Sturmian method be applied to N-electron molecules? -- 6.4 Discussion -- Bibliography -- 7. Slater-Type Orbital Basis Sets: Reliable and Rapid Solution of the Schr dinger Equation for Accurate Molecular Properties Philip E. Hoggan -- 7.1 Introduction -- 7.1.1 Context of this chapter -- 7.1.2 Atomic orbitals -- 7.1.3 Problems to be solved when using Slater-type orbitals -- 7.1.4 Strategy for Slater basis sets -- 7.2 Some Dates: The Story So Far of Slater-Type Orbitals -- 7.3 Computer Programs Using Slater-Type Orbitals -- 7.3.1 Numerical grid methods -- 7.3.2 Configuration interaction -- 7.4 Slater Orbitals and Gaussian Orbitals -- 7.5 Types of Exponentially Decaying Orbitals, Based on Eigenfunctions for One-Electron Atoms -- 7.5.1 Orbitals which are linear combinations of Slater-type orbitals -- 7.6 Types of Integral Over Slater Orbitals -- 7.6.1 One-electron integrals -- 7.6.2 Two-electron integrals -- 7.6.3 Three- and four-electron integrals -- 7.7 Integration Methods in the Literature -- 7.7.1 Single-center expansion -- 7.7.2 Gaussian expansion -- 7.7.3 Gaussian transform method -- 7.7.4 Fourier-transform method -- 7.7.5 Use of Sturmians -- 7.7.6 Elliptic coordinate method -- 7.7.7 Monte Carlo integration -- 7.8 General Two-Electron Exponential Type Orbital Integrals in Poly-Atomics Without Orbital Translations -- 7.8.1 Introduction -- 7.8.2 Basis sets -- 7.8.3 Programming strategy -- 7.8.4 Avoiding ETO translations for two-electron integrals over three- and four-centers. -- 7.8.5 Numerical results of Coulomb resolutions: efficiency -- 7.8.5.1. H molecule with interatomic distance of 1.402 atomic units (a.u.) -- 7.8.5.2. Dimer geometry: rectangular and planar -- 7.8.5.3. Selected exchange integrals for the CH F molecule (evaluated using the Coulomb resolution) -- 7.8.6 Perspectives and conclusions.

7.8.7 Angular momentum relations -- 7.9 When are Slater-Type Orbitals Advantageous? Some Applications -- 7.9.1 The NMR nuclear shielding tensor -- Analytical treatment -- Application -- Conclusion -- 7.9.2 Explicitly correlated methods for molecules -- 7.9.3 Trial wave-functions for quantum Monte Carlo simulations over STO -- 7.9.3.1. Basis sets -- 7.9.3.2. Computational procedure -- 7.9.3.3. Perspectives -- 7.10 Highly Accurate Calculations Using STOs -- 7.11 Closing Remarks -- 7.12 Appendix A: How STOs were Translated: Products on Two Atoms -- 7.12.1 Review of BCLFs -- 7.12.1.1. Definition and properties of BCLFs -- 7.13 Appendix B: Brief Time-Line of Events in Molecular Work Over Slater-Type Orbitals to Date -- 7.14 Appendix C: Main Results of Podolanski's Paper of 1931 with Additional Comments -- 7.15 Appendix D: Potentials and Auxiliary Overlaps for Coulomb Resolution -- 7.16 Appendix E: Analysis of Nuclear Dipole Integrals for NMR in a Slater Basis -- Acknowledgements -- Bibliography -- 8. Modern Ab Initio Valence Bond Methods Philippe C. Hiberty and Sason Shaik -- 8.1 Basic Principles and Survey of Modern Methods -- 8.1.1 VB vs.MO wave functions in the two-electron/ two-center case -- 8.1.2 Writing VB functions beyond the two-electron/ two-center case -- 8.1.3 Some landmark improvements of the early VB method -- 8.1.3.1. VB wave functions with semi-localized atomic orbitals -- 8.1.3.2. Efficient orbital optimization by Self-Consistent-Field VB -- 8.1.3.3. Improving the accuracy by including dynamic electron correlation -- 8.1.3.4. Inclusion of solvent effects -- 8.2 Strengths of the Valence Bond Approach -- 8.2.1 Interpretability combined with accuracy of the wave functions -- 8.2.2 A simple solution to the symmetry dilemma -- 8.2.3 Calculations of diabatic energy curves along a reaction coordinate -- 8.2.3.1. General model.

8.2.3.2. Application to hydrogen-abstraction reactions -- 8.2.4 Quantitative evaluation of common chemical paradigms -- 8.2.4.1. Direct calculation of resonance energies and hyperconjugation energies -- 8.2.4.2. Characterization of a novel type of bonding: Charge-shift bonds -- 8.2.4.3. σ vs π driving force for the D geometry of benzene -- 8.3 Present Capabilities and Expected Improvements -- 8.3.1 Evaluation of Hamiltonian matrix elements -- 8.3.2 Direct VBSCF/BOVB algorithm -- 8.3.3 Current calculations of medium-sized molecular systems -- 8.3.4 Mixed Valence Bond - Quantum Monte Carlo methods -- 8.3.5 Prospective -- 8.4 Concluding Remarks -- 8.5 Appendix A: The Myth of "VB failures" -- 8.6 Appendix B: Some Available VB Software Packages -- 8.6.1 The XMVB program -- 8.6.2 The TURTLE software -- 8.6.3 The VB2000 software -- 8.6.4 The CRUNCH software -- Bibliography -- 9. Quantum Monte Carlo Approaches for Tackling Electronic Correlation Massimo Mella and Gabriele Morosi -- 9.1 Introduction -- 9.2 Variational Monte Carlo (VMC): A Possible Way Toward Explicitly Correlated Electronic Wave Functions -- 9.2.1 Numerical integrals in VMC -- 9.2.1.1. General introduction -- 9.2.1.2. Sampling of P -- 9.2.2 Optimization of trial wave functions -- 9.2.2.1. Minimum variance -- 9.2.2.2. Minimum energy -- 9.2.3 Analytical forms for trial wave functions T -- 9.3 Diffusion Monte Carlo: How to Extract the Best Information from Inaccurate Wave Functions -- 9.3.1 Generalities -- 9.3.2 Improved projectors -- 9.3.3 DMC, state symmetry and excited states -- 9.4 Computing Observables Different from State Energy -- 9.4.1 Exact calculation of position dependent observables -- 9.4.2 Calculation of atomic forces in VMC/DMC -- 9.4.3 Computing the expectation value of ultra-local operators: electron and spin density on nuclei -- 9.5 Conclusions -- Bibliography.

10. Solving the Schr dinger Equation on Real-Space Grids and with Random Walks Thomas L. Beck and Joel H. Dedrick.

Key Features:Unusual combination of methods/techniquesA thought-provoking and didactic exposé, not a review, nor a textbookLooks at the future.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

There are no comments on this title.

to post a comment.