Ideas of Quantum Chemistry P2

X Contents 6.8.3 Approximation: decoupling of rotation and vibrations . . . . . . . . . 244 6.8.4 The kinetic energy operators of translation, rotation and vibrations . 245 6.8.5 Separation of translational, rotational and vibrational motions . . . . 246 6.9 Non-bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.10 Adiabatic, diabatic and non-adiabatic approaches . . . . . . . . . . . . . . . . 252 6.11 Crossing of potential energy curves for diatomics . . . . . . . . . . . . . . . . 255 6.11.1 The non-crossing rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.11.2 Simulating the harpooning effect in the NaCl molecule . . . . . . . . 257 6.12 Polyatomic molecules and conical intersection . . . . . . . . . . . . . . . . . . 260 6.12.1 Conical intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6.12.2 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.13 Beyond the adiabatic approximation... . . . . . . . . . . . . . . . . . . . . . . 268 6.13.1 Muon catalyzed nuclear fusion . . . . . . . . . . . . . . . . . . . . . . 268 6.13.2 “Russian dolls” – or a molecule within molecule . . . . . . . . . . . . 270 7. Motion of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.1 Rovibrational spectra – an example of accurate calculations: atom – di-atomic molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 7.1.1 Coordinate system and Hamiltonian . . . . . . . . . . . . . . . . . . . 279 7.1.2 Anisotropy of the potential V . . . . . . . . . . . . . . . . . . . . . . . 280 7.1.3 Adding the angular momenta in quantum mechanics . . . . . . . . . . 281 7.1.4 Application of the Ritz method . . . . . . . . . . . . . . . . . . . . . . 282 7.1.5 Calculation of rovibrational spectra . . . . . . . . . . . . . . . . . . . . 283 7.2 Force fields (FF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 7.3 Local Molecular Mechanics (MM) . . . . . . . . . . . . . . . . . . . . . . . . 290 7.3.1 Bonds that cannot break . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.3.2 Bonds that can break . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.4 Global molecular mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 7.4.1 Multiple minima catastrophe . . . . . . . . . . . . . . . . . . . . . . . 292 7.4.2 Is it the global minimum which counts? . . . . . . . . . . . . . . . . . 293 7.5 Small amplitude harmonic motion – normal modes . . . . . . . . . . . . . . . 294 7.5.1 Theory of normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . 295 7.5.2 Zero-vibration energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 7.6 Molecular Dynamics (MD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 7.6.1 The MD idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 7.6.2 What does MD offer us? . . . . . . . . . . . . . . . . . . . . . . . . . . 306 7.6.3 What to worry about? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 7.6.4 MD of non-equilibrium processes . . . . . . . . . . . . . . . . . . . . . 308 7.6.5 Quantum-classical MD . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 7.7 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7.8 Langevin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 7.9 Monte Carlo Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 7.10 Car–Parrinello dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 7.11 Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8. Electronic Motion in the Mean Field: Atoms and Molecules . . . . . . . . . . . . . 324 8.1 Hartree–Fock method – a bird’s eye view . . . . . . . . . . . . . . . . . . . . . 329 8.1.1 Spinorbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Contents XI 8.1.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.1.3 Slater determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.1.4 What is the Hartree–Fock method all about? . . . . . . . . . . . . . . 333 8.2 The Fock equation for optimal spinorbitals . . . . . . . . . . . . . . . . . . . 334 8.2.1 Dirac and Coulomb notations . . . . . . . . . . . . . . . . . . . . . . . 334 8.2.2 Energy functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.2.3 The search for the conditional extremum . . . . . . . . . . . . . . . . 335 8.2.4 A Slater determinant and a unitary transformation . . . . . . . . . . . 338 8.2.5 Invariance of the J and K operators . . . . . . . . . . . . . . . . . . . 339 8.2.6 Diagonalization of the Lagrange multipliers matrix . . . . . . . . . . 340 8.2.7 The Fock equation for optimal spinorbitals (General Hartree–Fock method – GHF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 8.2.8 The closed-shell systems and the Restricted Hartree–Fock (RHF) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 8.2.9 Iterative procedure for computing molecular orbitals: the Self- Consistent Field method . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.3 Total energy in the Hartree–Fock method . . . . . . . . . . . . . . . . . . . . 351 8.4 Computational technique: atomic orbitals as building blocks of the molecu- lar wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.4.1 Centring of the atomic orbital . . . . . . . . . . . . . . . . . . . . . . . 354 8.4.2 Slater-type orbitals (STO) . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.4.3 Gaussian-type orbitals (GTO) . . . . . . . . . . . . . . . . . . . . . . . 357 8.4.4 Linear Combination of Atomic Orbitals (LCAO) Method . . . . . . . 360 8.4.5 Basis sets of Atomic Orbitals . . . . . . . . . . . . . . . . . . . . . . . . 363 8.4.6 The Hartree–Fock–Roothaan method (SCF LCAO MO) . . . . . . . 364 8.4.7 Practical problems in the SCF LCAO MO method . . . . . . . . . . . 366 RESULTS OF THE HARTREE–FOCK METHOD . . . . . . . . . . . . . . . . . . 369 8.5 Back to foundations... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 8.5.1 When does the RHF method fail? . . . . . . . . . . . . . . . . . . . . 369 8.5.2 Fukutome classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 8.6 Mendeleev Periodic Table of Chemical Elements . . . . . . . . . . . . . . . . 379 8.6.1 Similar to the hydrogen atom – the orbital model of atom . . . . . . . 379 8.6.2 Yet there are differences... . . . . . . . . . . . . . . . . . . . . . . . . 380 8.7 The nature of the chemical bond . . . . . . . . . . . . . . . . . . . . . . . . . 383 8.7.1 H+ in the MO picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 8.7.2 Can we see a chemical bond? . . . . . . . . . . . . . . . . . . . . . . . 388 8.8 Excitation energy, ionization potential, and electron affinity (RHF approach) 389 8.8.1 Approximate energies of electronic states . . . . . . . . . . . . . . . . 389 8.8.2 Singlet or triplet excitation? . . . . . . . . . . . . . . . . . . . . . . . . 391 8.8.3 Hund’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 8.8.4 Ionization potential and electron affinity (Koopmans rule) . . . . . . 393 8.9 Localization of molecular orbitals within the RHF method . . . . . . . . . . 396 8.9.1 The external localization methods . . . . . . . . . . . . . . . . . . . . . 397 8.9.2 The internal localization methods . . . . . . . . . . . . . . . . . . . . . 398 8.9.3 Examples of localization . . . . . . . . . . . . . . . . . . . . . . . . . . 400 8.9.4 Computational technique . . . . . . . . . . . . . . . . . . . . . . . . . . 401 8.9.5 The σ, π, δ bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 8.9.6 Electron pair dimensions and the foundations of chemistry . . . . . . 404 8.9.7 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 XII Contents 8.10 A minimal model of a molecule . . . . . . . . . . . . . . . . . . . . . . . . . . 417 8.10.1 Valence Shell Electron Pair Repulsion (VSEPR) . . . . . . . . . . . . 419 9. Electronic Motion in the Mean Field: Periodic Systems . . . . . . . . . . . . . . . 428 9.1 Primitive lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 9.2 Wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 9.3 Inverse lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 9.4 First Brillouin Zone (FBZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.5 Properties of the FBZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.6 A few words on Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . 439 9.6.1 Waves in 1D 9.6.2 Waves in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 9.7 The infinite crystal as a limit of a cyclic system . . . . . . . . . . . . . . . . . . 445 9.8 A triple role of the wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . 448 9.9 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 9.9.1 Born–von Kármán boundary condition in 3D . . . . . . . . . . . . . . 449 9.9.2 Crystal orbitals from Bloch functions (LCAO CO method) . . . . . . 450 9.9.3 SCF LCAO CO equations . . . . . . . . . . . . . . . . . . . . . . . . . 452 9.9.4 Band structure and band width . . . . . . . . . . . . . . . . . . . . . . 453 9.9.5 Fermi level and energy gap: insulators, semiconductors and metals . 454 9.10 Solid state quantum chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 9.10.1 Why do some bands go up? . . . . . . . . . . . . . . . . . . . . . . . . 460 9.10.2 Why do some bands go down? . . . . . . . . . . . . . . . . . . . . . . . 462 9.10.3 Why do some bands stay constant? . . . . . . . . . . . . . . . . . . . . 462 9.10.4 How can more complex behaviour be explained? . . . . . . . . . . . . 462 9.11 The Hartree–Fock method for crystals . . . . . . . . . . . . . . . . . . . . . . 468 9.11.1 Secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 9.11.2 Integration in the FBZ . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 9.11.3 Fock matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 9.11.4 Iterative procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 9.11.5 Total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 9.12 Long-range interaction problem . . . . . . . . . . . . . . . . . . . . . . . . . . 475 9.12.1 Fock matrix corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 9.12.2 Total energy corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 477 9.12.3 Multipole expansion applied to the Fock matrix . . . . . . . . . . . . 479 9.12.4 Multipole expansion applied to the total energy . . . . . . . . . . . . . 483 9.13 Back to the exchange term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 9.14 Choice of unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 9.14.1 Field compensation method . . . . . . . . . . . . . . . . . . . . . . . . 490 9.14.2 The symmetry of subsystem choice . . . . . . . . . . . . . . . . . . . . 492 10. Correlation of the Electronic Motions . . . . . . . . . . . . . . . . . . . . . . . . . 498 VARIATIONALMETHODSUSINGEXPLICITLYCORRELATEDWAVEFUNC-TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 10.1 Correlation cusp condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 10.2 The Hylleraas function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 10.3 The Hylleraas CI method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 10.4 The harmonic helium atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Contents XIII 10.5 James–Coolidge and Kołos–Wolniewicz functions . . . . . . . . . . . . . . . 508 10.5.1 Neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 10.6 Method of exponentially correlated Gaussian functions . . . . . . . . . . . 513 10.7 Coulomb hole (“correlation hole”) . . . . . . . . . . . . . . . . . . . . . . . . 513 10.8 Exchange hole (“Fermi hole”) . . . . . . . . . . . . . . . . . . . . . . . . . . 516 VARIATIONAL METHODS WITH SLATER DETERMINANTS . . . . . . . . . . 520 10.9 Valence bond (VB) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 10.9.1 Resonance theory – hydrogen molecule . . . . . . . . . . . . . . . . 520 10.9.2 Resonance theory – polyatomic case . . . . . . . . . . . . . . . . . . 523 10.10 Configuration interaction (CI) method . . . . . . . . . . . . . . . . . . . . . 525 10.10.1 Brillouin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 10.10.2 Convergence of the CI expansion . . . . . . . . . . . . . . . . . . . . 527 10.10.3 Example of H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 10.10.4 Which excitations are most important? . . . . . . . . . . . . . . . . 529 10.10.5 Natural orbitals (NO) . . . . . . . . . . . . . . . . . . . . . . . . . . 531 10.10.6 Size consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 10.11 Direct CI method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.12 Multireference CI method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.13 Multiconfigurational Self-Consistent Field method (MC SCF) . . . . . . . 535 10.13.1 Classical MC SCF approach . . . . . . . . . . . . . . . . . . . . . . . 535 10.13.2 Unitary MC SCF method . . . . . . . . . . . . . . . . . . . . . . . . 536 10.13.3 Complete active space method (CAS SCF) . . . . . . . . . . . . . . 538 NON-VARIATIONAL METHODS WITH SLATER DETERMINANTS . . . . . . 539 10.14 Coupled cluster (CC) method . . . . . . . . . . . . . . . . . . . . . . . . . . 539 10.14.1 Wave and cluster operators . . . . . . . . . . . . . . . . . . . . . . . 540 10.14.2 Relationship between CI and CC methods . . . . . . . . . . . . . . 542 10.14.3 Solution of the CC equations . . . . . . . . . . . . . . . . . . . . . . 543 10.14.4 Example: CC with double excitations . . . . . . . . . . . . . . . . . 545 10.14.5 Size consistency of the CC method . . . . . . . . . . . . . . . . . . . 547 10.15 Equation-of-motion method (EOM-CC) . . . . . . . . . . . . . . . . . . . . 548 10.15.1 Similarity transformation . . . . . . . . . . . . . . . . . . . . . . . . 548 10.15.2 Derivation of the EOM-CC equations . . . . . . . . . . . . . . . . . 549 10.16 Many body perturbation theory (MBPT) . . . . . . . . . . . . . . . . . . . . 551 10.16.1 Unperturbed Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 551 10.16.2 Perturbation theory – slightly different approach . . . . . . . . . . . 552 10.16.3 Reduced resolvent or the “almost” inverse of (E(0) −H(0)) . . . . 553 10.16.4 MBPT machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 10.16.5 Brillouin–Wigner perturbation theory . . . . . . . . . . . . . . . . . 556 10.16.6 Rayleigh–Schrödinger perturbation theory . . . . . . . . . . . . . . 557 10.17 Møller–Plesset version of Rayleigh–Schrödinger perturbation theory . . . . 558 10.17.1 Expression for MP2 energy . . . . . . . . . . . . . . . . . . . . . . . 558 10.17.2 Convergence of the Møller–Plesset perturbation series . . . . . . . 559 10.17.3 Special status of double excitations . . . . . . . . . . . . . . . . . . . 560 11. Electronic Motion: Density Functional Theory (DFT) . . . . . . . . . . . . . . . . 567 11.1 Electronic density – the superstar . . . . . . . . . . . . . . . . . . . . . . . . 569 11.2 Bader analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 11.2.1 Overall shape of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 XIV Contents 11.2.2 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 11.2.3 Laplacian of the electronic density as a “magnifying glass” . . . . . 575 11.3 Two important Hohenberg–Kohn theorems . . . . . . . . . . . . . . . . . . 579 11.3.1 Equivalence of the electronic wave function and electron density . 579 11.3.2 Existence of an energy functional minimized by ρ0 . . . . . . . . . 580 11.4 The Kohn–Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 11.4.1 The Kohn–Sham system of non-interacting electrons . . . . . . . . 584 11.4.2 Total energy expression . . . . . . . . . . . . . . . . . . . . . . . . . 585 11.4.3 Derivation of the Kohn–Sham equations . . . . . . . . . . . . . . . 586 11.5 What to take as the DFT exchange–correlation energy Exc? . . . . . . . . . 590 11.5.1 Local density approximation (LDA) . . . . . . . . . . . . . . . . . . 590 11.5.2 Non-local approximations (NLDA) . . . . . . . . . . . . . . . . . . 591 11.5.3 The approximate character of the DFT vs apparent rigour of ab initio computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 11.6 On the physical justification for the exchange correlation energy . . . . . . 592 11.6.1 The electron pair distribution function . . . . . . . . . . . . . . . . 592 11.6.2 The quasi-static connection of two important systems . . . . . . . . 594 11.6.3 Exchange–correlation energy vs 5aver . . . . . . . . . . . . . . . . . 596 11.6.4 Electron holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 11.6.5 Physical boundary conditions for holes . . . . . . . . . . . . . . . . 598 11.6.6 Exchange and correlation holes . . . . . . . . . . . . . . . . . . . . . 599 11.6.7 Physical grounds for the DFT approximations . . . . . . . . . . . . 601 11.7 Reflections on the DFT success . . . . . . . . . . . . . . . . . . . . . . . . . 602 12. The Molecule in an Electric or Magnetic Field . . . . . . . . . . . . . . . . . . . . 615 12.1 Hellmann–Feynman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 ELECTRIC PHENOMENA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 12.2 The molecule immobilized in an electric field . . . . . . . . . . . . . . . . . 620 12.2.1 The electric field as a perturbation . . . . . . . . . . . . . . . . . . . 621 12.2.2 The homogeneous electric field . . . . . . . . . . . . . . . . . . . . . 627 12.2.3 The inhomogeneous electric field: multipole polarizabilities and hyperpolarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 12.3 How to calculate the dipole moment . . . . . . . . . . . . . . . . . . . . . . . 633 12.3.1 Hartree–Fock approximation . . . . . . . . . . . . . . . . . . . . . . 633 12.3.2 Atomic and bond dipoles . . . . . . . . . . . . . . . . . . . . . . . . 634 12.3.3 Within the ZDO approximation . . . . . . . . . . . . . . . . . . . . 635 12.4 How to calculate the dipole polarizability . . . . . . . . . . . . . . . . . . . . 635 12.4.1 Sum Over States Method . . . . . . . . . . . . . . . . . . . . . . . . 635 12.4.2 Finite field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 12.4.3 What is going on at higher electric fields . . . . . . . . . . . . . . . 644 12.5 A molecule in an oscillating electric field . . . . . . . . . . . . . . . . . . . . 645 MAGNETIC PHENOMENA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 12.6 Magnetic dipole moments of elementary particles . . . . . . . . . . . . . . . 648 12.6.1 Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 12.6.2 Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 12.6.3 Dipole moment in the field . . . . . . . . . . . . . . . . . . . . . . . 650 12.7 Transitions between the nuclear spin quantum states – NMR technique . . 652 12.8 Hamiltonian of the system in the electromagnetic field . . . . . . . . . . . . 653 ... - tailieumienphi.vn