Solving the Schrödinger equation: has everything been tried?
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| Language: | English |
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London
Imperial College Press
2011
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| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturangaben |
| Physical Description: | XIX, 354 S. graph. Darst. |
| ISBN: | 9781848167247 1848167245 |
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| 245 | 1 | 0 | |a Solving the Schrödinger equation |b has everything been tried? |c Ed. Paul Popelier |
| 246 | 1 | 0 | |a Solving the Schra-Dinger Equation |
| 264 | 1 | |a London |b Imperial College Press |c 2011 | |
| 300 | |a XIX, 354 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 500 | |a Literaturangaben | ||
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| 689 | 0 | 0 | |a Schrödinger-Gleichung |0 (DE-588)4053332-3 |D s |
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| 700 | 1 | |a Popelier, Paul L. A. |4 edt | |
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Record in the Search Index
| _version_ | 1804148681635528704 |
|---|---|
| adam_text | Contents
Preface
xv
1. Intracule
Functional Theory
1
Deborah
L.
С
rí
tienden
and Peter
M.
W.
Gill
1.1
Introduction
....................... 2
1.2
Intracules
........................ 3
1.3
Electron Correlation Models
.............. 13
1.4
Dynamic and Static Correlation
............. 16
1.5
Dispersion Energies
................... 18
1.6
Future Prospects
..................... 21
Bibliography
.......................... 22
2.
Explicitly Correlated Electronic Structure Theory
25
Frederick R. Manby
2.1
Introduction
....................... 25
2.1.1
Basis-set expansions
.............. 25
2.2
F12 Theory
....................... 28
2.2.1
MP2-F12
.................... 29
2.2.2
Explicitly correlated coupled-cluster
theory
..................... 30
2.3
Five Thoughts for Fl
2
Theory
.............. 31
2.3.1
Thought
1 :
Do we need (products of)
virtuais?
.................... 31
2.3.2
Thought
2:
Are there better two-electron
basis sets?
................... 34
viii Contents
2.3.3
Thought
3:
Do we need the resolution
of the identity?
................. 35
2.3.4
Thought
4:
Could we have explicit correlation
for higher excitations?
............. 38
2.3.5
Thought
5:
Can we avoid three-electron errors
in two-electron systems?
............ 39
2.4
Conclusions
....................... 40
Bibliography
.......................... 40
3.
Solving Problems with Strong Correlation Using
the Density Matrix Renormalization Group (DMRG)
43
Garnet Kin-Lie Chan and Sandeep Sharma
3.1
The Problem of Strong Correlation
........... 43
3.2
The Density Matrix Renormalization Group
Wavefunction
...................... 46
3.3
Locality and Entanglement in the DMRG
........ 47
3.4
Other Properties of the DMRG
............. 50
3.5
Relation to the Renormalization Group
......... 51
3.6
Dynamic Correlation
—
the Role of Canonical
Transformations
..................... 53
3.7
What Can the DMRG Do? A Brief History
....... 54
3.8
The Future: Higher Dimensional Analogues
...... 57
Bibliography
.......................... 59
4.
Reduced-Density-Matrix Theory for Many-electron Correlation
61
David A. Mazziotti
4.1
Introduction
....................... 62
4.2
Variational 2-RDM Method
............... 63
4.2.1
Energy as a 2-RDM functional
........ 63
4.2.2
Positivity
conditions
.............. 64
4.2.3
Semidefinite
programming
........... 67
4.2.4
Applications
.................. 69
4.3
Contracted
Schrödinger
Theory
............. 73
4.3.1
ACSE and
cumulant
reconstruction
...... 74
4.3.2
Solving the ACSE for ground
and excited states
................ 75
4.3.3
Applications
.................. 77
4.4
Parametric 2-RDM Method
............... 80
4.4.1
Parametrization of the 2-RDM
......... 81
Contents ix
4.4.2 Applications.................. 83
4.5
Looking Ahead .....................
85
Bibliography
.......................... 87
5.
Finite Size Scaling for Criticality of the
Schrödinger
Equation
91
Sabre
Kais
5.1
Introduction
....................... 92
5.2
Criticality for Large-dimensional Models
........ 93
5.3
Finite Size Scaling: A Brief History
........... 95
5.4
Finite Size Scaling for the
Schrödinger
Equation
.... 97
5.5
The Hulthen Potential
.................. 100
5.5.1
Analytical solution
............... 100
5.5.2
Basis set expansion
.............. 101
5.5.3
Finite element method
............. 101
5.5.4
Finite size scaling results
........... 102
5.6
Finite Size Scaling and Criticality
of M-electron Atoms
................... 105
5.7
Conclusions
....................... 107
Bibliography
.......................... 108
6.
The Generalized Sturmian Method
111
James Avery and John Avery
6.1
Description of the Method
................
Ill
6.1.1
The introduction of Sturmians
into quantum theory
..............
Ill
6.1.2
Generalized Sturmians
............. 114
6.1.3
The generalized Sturmian method applied
to atoms
.................... 117
6.1.4
Goscinskian configurations
.......... 118
6.1.5
Goscinskian secular equations for atoms
and atomic ions
................ 120
6.2
Advantages: Some Illustrative Examples
........ 120
6.2.1
The large-Z approximation: restriction
of the basis set to an 72-block
......... 121
6.2.2
Validity of the large-Z approximation
..... 126
6.2.3
Core ionization energies
............ 129
6.3
Limitations of the Method; Prospects for the Future
. . 130
6.3.1
Can the generalized Sturmian method
be applied to jV-electron molecules?
...... 133
x
Contents
6.4
Discussion
........................ 137
Bibliography
.......................... 139
7.
Slater-Type Orbital Basis Sets: Reliable and Rapid
Solution of the
Schrödinger
Equation for Accurate
Molecular Properties
141
Philip E. Hoggan
7.1
Introduction
....................... 142
7.1.1
Context of this chapter
............. 142
7.1.2
Atomic
orbitais
................. 144
7.1.3
Problems to be solved when using
Slater-type
orbitais
............... 145
7.1.4
Strategy for Slater basis sets
.......... 147
7.2
Some Dates: The Story So Far of Slater-Type
Orbitals ......................... 148
7.3
Computer Programs Using Slater-Type
Orbitals .... 149
7.3.1
Numerical grid methods
............ 151
7.3.2
Configuration interaction
........... 151
7.4
Slater
Orbitals
and Gaussian
Orbitals.......... 151
7.5
Types of Exponentially Decaying
Orbitals,
Based
on Eigenfunctions for One-Electron Atoms
....... 154
7.5.1 Orbitals
which are linear combinations
of Slater-type
orbitais
............. 155
7.6
Types of Integral Over Slater
Orbitals.......... 156
7.6.1
One-electron integrals
............. 156
7.6.2
Two-electron integrals
............. 157
7.6.3
Three-and four-electron integrals
....... 158
7.7
Integration Methods in the Literature
.......... 158
7.7.1
Single-center expansion
............ 159
7.7.2
Gaussian expansion
.............. 160
7.7.3
Gaussian transform method
.......... 160
7.7.4
Fourier-transform method
........... 160
7.7.5
Use of Sturmians
................ 161
7.7.6
Elliptic coordinate method
........... 161
7.7.7
Monte Carlo integration
............ 162
7.8
General Two-Electron Exponential Type Orbital
Integrals in Poly-Atomics Without Orbital Translations
163
7.8.1
Introduction
.................. 163
7.8.2
Basis sets
.................... 164
Contents xi
7.8.3 Programming
strategy
............. 164
7.8.4
Avoiding
ЕТО
translations for two-electron
integrals over three-and four-centers
...... 165
7.8.5
Numerical results of Coulomb resolutions:
efficiency
.................... 167
7.8.6
Perspectives and conclusions
......... 169
7.8.7
Angular momentum relations
......... 171
7.9
When are Slater-Type
Orbitals
Advantageous?
Some Applications
.................... 171
7.9.1
The NMR nuclear shielding tensor
...... 171
7.9.2
Explicitly correlated methods for molecules
. 176
7.9.3
Trial wave-functions for quantum Monte Carlo
simulations over
STO
............. 178
7.10
Highly Accurate Calculations Using STOs
....... 181
7.11
Closing Remarks
.................... 181
7.12
Appendix A: How STOs were Translated: Products on
Two Atoms
........................ 183
7.12.1
Review of BCLFs
............... 184
7.13
Appendix B: Brief Time-Line of Events in Molecular
Work Over Slater-Type
Orbitals
to Date
........ 186
7.14
Appendix C: Main Results of Podolanski s Paper
of
1931
with Additional Comments
...........187
7.15
Appendix D: Potentials and Auxiliary Overlaps
for Coulomb Resolution
.................189
7.16
Appendix E: Analysis of Nuclear
Dipole
Integrals
for NMR in a Slater Basis
................191
Acknowledgements
.......................193
Bibliography
..........................194
Modern
Ab
initia
Valence Bond Methods
201
Philippe C. Hiberty and
Sason Shaik
8.1
Basic Principles and Survey of Modern Methods
.... 202
8.1.1
VB vs. MO wave functions in the two-electron/
two-center case
.................202
8.1.2
Writing VB functions beyond the two-electron/
two-center case
.................205
8.1.3
Some landmark improvements of the early
VB method
...................206
xii Contents
8.2
Strengths of the
Valence
Bond Approach
........ 211
8.2.1
Interpretability combined with accuracy
of the wave functions
............. 211
8.2.2
A simple solution to the symmetry dilemma
. 213
8.2.3
Calculations of diabatic energy curves along
a reaction coordinate
.............. 214
8.2.4
Quantitative evaluation of common chemical
paradigms
................... 218
8.3
Present Capabilities and Expected Improvements
.... 224
8.3.1
Evaluation of Hamiltonian matrix elements
. . 224
8.3.2
Direct VBSCF/BOVB algorithm
........ 224
8.3.3
Current calculations of medium-sized
molecular systems
............... 225
8.3.4
Mixed Valence Bond
—
Quantum Monte
Carlo methods
................. 225
8.3.5
Prospective
................... 226
8.4
Concluding Remarks
.................. 228
8.5
Appendix A: The Myth of VB failures
........ 229
8.6
Appendix B: Some Available VB
Software Packages
.................... 230
8.6.1
The XMVB program
.............. 230
8.6.2
The TURTLE software
............. 230
8.6.3
The VB2000 software
............. 230
8.6.4
The CRUNCH software
............ 231
Bibliography
.......................... 231
9.
Quantum Monte Carlo Approaches for Tackling
Electronic Correlation
237
Massimo
Mella
and
Gabriele
Morosi
9.1
Introduction
....................... 238
9.2
Variational Monte Carlo (VMC): A Possible
Way Toward Explicitly Correlated Electronic
Wave Functions
..................... 239
9.2.1
Numericalintegrals in VMC
.......... 241
9.2.2
Optimization of trial wave functions
...... 247
9.2.3
Analytical forms for trial wave
functions
Фг
.................. 252
9.3
Diffusion Monte Carlo: How to Extract the Best
Information from Inaccurate Wave Functions
...... 254
Contents xiii
9.3.1
Generalities
..................254
9.3.2
Improved projectors
..............258
9.3.3
DMC, state symmetry and excited states
. . . 259
9.4
Computing
Observables
Different from State Energy
. . 261
9.4.1
Exact calculation of position dependent
observables
...................261
9.4.2
Calculation of atomic forces in VMC/DMC
. . 262
9.4.3
Computing the expectation value of ultra-local
operators: electron and spin density
on nuclei
....................264
9.5
Conclusions
.......................266
Bibliography
..........................268
10.
Solving the
Schrödinger
Equation on Real-Space Grids
and with Random Walks
271
Thomas L. Beck and Joel
H. Ded
rick
10.1
Introduction
.......................272
10.2
Solving the
Schrödinger
Equation Using Grids in Real
Space
...........................275
10.2.1
Basics of grid methods
.............275
10.2.2
Multiscale (multigrid) approaches
.......279
10.3
New Ways of Thinking about Large-Scale
Solutions
.........................282
10.3.1
An encounter with Silicon valley
.......283
10.3.2
The Borg-ing of computation
.........283
10.3.3
The world s least efficient computer (yours)
. . 283
10.3.4
The world s most efficient computer
(also yours)
................... 284
10.3.5
The end of a 20-year free ride
......... 285
10.3.6
Can we change sides?
............. 286
10.3.7
Algorithm desiderata for the massively parallel
future
...................... 286
10.3.8
What are we looking for?
........... 288
10.4
Random Walks for Solving the
Schrödinger
Equation
......................... 292
10.4.1
Traditional diffusion quantum Monte Carlo
. . 292
10.4.2
Another angle
................. 296
10.4.3
Stochastic differential equations and the
Feynman—
Кас
approach
............ 298
xiv Contents
10.4.4
Obtaining E(x,y)?
............... 302
10.4.5
A pipe dream
.................. 305
10.5
Summary
......................... 306
Bibliography
.......................... 307
11.
Changes in Dense Linear Algebra Kernels: Decades-
Long Perspective
313
Piotr Łuszczek, Jakub Kurzak,
and Jack Dongarra
11.1
The
Schrödinger
Connection
.............. 313
11.2
A Stroll Down the Memory Lane
............ 315
11.3
A Decompositional Approach
.............. 318
11.4
Vector Processors
.................... 319
11.5
RISC Processors
..................... 322
11.6
Clusters
......................... 325
11.7
Multicore Processors
.................. 332
11.8
Multicore Processors Redux
............... 334
11.9
Error Analysis and Operation Count
.......... 338
11.10
Future Directions for Research and Hardware Design
. 339
Bibliography
.......................... 341
Index
343
Solving the
Schrödinger
Equation
Has Everything Been Tried?
The
Schrödinger
equation is the master equation of quantum chemistry.
The founders of quantum mechanics realised how this equation
underpins essentially the whole of chemistry. However, they recognised
that its exact application was much too complicated to be solvable at
the time. More than two generations of researchers were left to work
out how to achieve this ambitious goal for molecular systems of ever-
increasing size. This book focuses on non-mainstream methods to solve
the molecular electronic
Schrödinger
equation. Each method is based
on a set of core ideas and this volume aims to explain these ideas
clearly so that they become more accessible. By bringing together these
non-standard methods, the book intends to inspire graduate students,
postdoctoral researchers and academics to think of novel approaches.
Is there a method out there that we have not thought of yet? Can we
design a new method that combines the best of all worlds?
|
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| id | DE-604.BV039767165 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:11:00Z |
| institution | BVB |
| isbn | 9781848167247 1848167245 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024628227 |
| oclc_num | 679937451 |
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| owner_facet | DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| physical | XIX, 354 S. graph. Darst. |
| publishDate | 2011 |
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| publisher | Imperial College Press |
| record_format | marc |
| spelling | Solving the Schrödinger equation has everything been tried? Ed. Paul Popelier Solving the Schra-Dinger Equation London Imperial College Press 2011 XIX, 354 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Schrödinger-Gleichung (DE-588)4053332-3 gnd rswk-swf Schrödinger-Gleichung (DE-588)4053332-3 s DE-604 Popelier, Paul L. A. edt Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024628227&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024628227&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Solving the Schrödinger equation has everything been tried? Schrödinger-Gleichung (DE-588)4053332-3 gnd |
| subject_GND | (DE-588)4053332-3 |
| title | Solving the Schrödinger equation has everything been tried? |
| title_alt | Solving the Schra-Dinger Equation |
| title_auth | Solving the Schrödinger equation has everything been tried? |
| title_exact_search | Solving the Schrödinger equation has everything been tried? |
| title_full | Solving the Schrödinger equation has everything been tried? Ed. Paul Popelier |
| title_fullStr | Solving the Schrödinger equation has everything been tried? Ed. Paul Popelier |
| title_full_unstemmed | Solving the Schrödinger equation has everything been tried? Ed. Paul Popelier |
| title_short | Solving the Schrödinger equation |
| title_sort | solving the schrodinger equation has everything been tried |
| title_sub | has everything been tried? |
| topic | Schrödinger-Gleichung (DE-588)4053332-3 gnd |
| topic_facet | Schrödinger-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024628227&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024628227&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT popelierpaulla solvingtheschrodingerequationhaseverythingbeentried AT popelierpaulla solvingtheschradingerequation |