 | Varga K. Computational nanoscience: applications for molecules, clusters, and solids / K.Varga, J.A.Driscoll. - Cambridge; New York: Cambridge University Press, 2011. - xii, 431 p.: ill. - Ref.: p.409-427. - Ind.: p.428-431. - ISBN 978-1-10700-170-1
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Preface ..................................................... xi
Part I One-dimensional problems ................................ l
1 Variational solution of the Schrцdinger equation ............. 3
1.1 Variational principle ................................... 3
1.2 Variational calculations with Gaussian basis
functions ............................................... 5
2 Solution of bound state problems using a grid ............... 10
2.1 Discretization in space ................................ 10
2.2 Finite differences ..................................... 11
2.3 Solution of the Schrцdinger equation using
three-point finite differences ......................... 15
2.4 Fourier grid approach: position and momentum
representations ........................................ 17
2.5 Lagrange functions ..................................... 21
3 Solution of the Schrцdinger equation for scattering
states ...................................................... 32
3.1 Green's functions ...................................... 34
3.2 The transfer matrix method ............................. 38
3.3 The complex-absorbing-potential approach ............... 39
3.4 R-matrix approach to scattering ........................ 51
3.5 Green's functions ...................................... 60
3.6 Spectral projection .................................... 79
3.7 Appendix: One-dimensional scattering states ............ 83
4 Periodic potentials: band structure in one dimension ........ 85
4.1 Periodic potentials; Bloch's theorem ................... 85
4.2 Finite difference approach ............................. 86
4.3 Periodic cardinals ..................................... 87
4.4 R-matrix calculation of Bloch states ................... 89
4.5 Green's function of a periodic system .................. 93
4.6 Calculation of the Green's function by continued
fractions ............................................. 103
5 Solution of time-dependent problems in quantum mechanics ... 115
5.1 The Schrцdinger, Heisenberg, and interaction
pictures .............................................. 115
5.2 Floquet theory ........................................ 117
5.3 Time-dependent variational method ..................... 119
5.4 Time propagation by numerical integration ............. 120
5.5 Time propagation using the evolution operator ......... 121
5.6 Examples .............................................. 128
5.7 Photoionization of atoms in intense laser fields ...... 134
5.8 Calculation of scattering wave functions by wave
packet propagation .................................... 142
5.9 Steady state evolution from a point source ............ 147
5.10 Calculation of bound states by imaginary time
propagation ........................................... 151
5.11 Appendix .............................................. 155
6 Solution of Poisson's equation ............................. 160
6.1 Finite difference approach ............................ 160
6.2 Fourier transformation ................................ 167
Part II Two-and three-dimensional systems ..................... 171
7 Three-dimensional real-space approach: from quantum dots
to Bose-Einstein condensates ............................... 173
7.1 Three-dimensional grid ................................ 173
7.2 Bound state problems on the 3D grid ................... 175
7.3 Solution of the Poisson equation ...................... 179
7.4 Harmonic quantum dots ................................. 184
7.5 Gross-Pitaevskii equation for Bose-Einstein
condensates ........................................... 189
7.6 Time propagation of a Gaussian wave packet ............ 191
8 Variational calculations in two dimensions: quantum dots ... 196
8.1 Introduction .......................................... 196
8.2 Formalism ............................................. 197
8.3 Code description ...................................... 200
8.4 Examples .............................................. 202
8.5 Few-electron quantum dots ............................. 203
8.6 Appendix .............................................. 208
9 Variational calculations in three dimensions: atoms and
molecules .................................................. 214
9.1 Three-dimensional trial functions ..................... 214
9.2 Small atoms and molecules ............................. 216
9.3 Quantum dots .......................................... 217
9.4 Appendix: Matrix elements ............................. 220
9.5 Appendix: Symmetrization .............................. 223
10 Monte Carlo calculations ................................... 225
10.1 Monte Carlo simulations ............................... 225
10.2 Classical interacting many-particle system ............ 229
10.3 Kinetic Monte Carlo ................................... 231
10.4 Two-dimensional Ising model ........................... 239
10.5 Variational Monte Carlo ............................... 250
10.6 Diffusion Monte Carlo ................................. 255
11 Molecular dynamics simulations ............................. 263
11.1 Introduction .......................................... 263
11.2 Integration of the equation of motions ................ 264
11.3 Lennard-Jones system .................................. 265
11.4 Molecular dynamics with three-body interactions ....... 265
11.5 Thermostats ........................................... 266
11.6 Physical quantities ................................... 270
11.7 Implementation and examples ........................... 270
12 Tight-binding approach to electronic structure
calculations ............................................... 274
12.1 Tight-binding calculations ............................ 274
12.2 Electronic structure of carbon nanotubes .............. 281
12.3 Tight-binding model with Slater-type orbitals ......... 289
12.4 Appendix: Matrix elements of Slater-type orbitals ..... 291
13 Plane wave density functional calculations ................. 295
13.1 Density functional theory ............................. 295
13.2 Description of the plane wave code and examples ....... 304
14 Density functional calculations with atomic orbitals ....... 317
14.1 Atomic orbitals ....................................... 317
14.2 Matrix elements for numerical atomic orbitals ......... 319
14.3 Examples .............................................. 324
14.4 Appendix: Three-center matrix elements ................ 326
15 Real-space density functional calculations ................. 332
15.1 Ground state energy and the Kohn-Sham equation ........ 332
15.2 Real-space approach ................................... 334
15.3 Examples .............................................. 337
16 Time-dependent density functional calculations ............. 339
16.1 Linear response ....................................... 340
16.2 Linear optical response ............................... 343
16.3 Solution of the time-dependent Kohn-Sham equation ..... 346
16.4 Simulation of the Coulomb explosion of H2 ............. 347
16.5 Calculation of the dielectric function in real time
and real space ........................................ 350
17 Scattering and transport in nanostructures ................. 356
17.1 Landauer formalism .................................... 358
17.2 R-matrix approach to scattering in three dimensions ... 362
17.3 Transfer matrix approach .............................. 362
17.4 Quantum constriction .................................. 372
17.5 Nonequilibrium Green's function method ................ 377
17.6 Simulation of transport in nanostructures ............. 385
18 Numerical linear algebra ................................... 390
18.1 Conjugate gradient method ............................. 390
18.2 Conjugate gradient diagonalization .................... 392
18.3 The Lanczos algorithm ................................. 394
18.4 Diagonalization with subspace iteration ............... 396
18.5 Solving linear block tridiagonal equations ............ 398
Appendix Code descriptions ................................ 407
References ................................................. 409
Index ...................................................... 428
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