Electronic Structure and Chemical Bonding in Quantum Mechanics

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods Schr dinger s Equation: ? ? ?1 ?? = ?? ? ?1 ?? ?: Hamiltonian = Energy operator = Kinetic + Potential energy expressions + External fields (electric, magnetic) Born-Oppenheimer Approximation: Fix the nuclear positions 2 2? ? ?A ??A+ ?>? 1 ? 2 A=1 ? ? ?1 ?? ? ? ?1 ?? = ?=1 ? ? ?1 ?? Electron Coordinates ??? ?: Electronic wavefunctions for N electrons Charge density: ? ? = ? ??? ? ? ??: Electronic energies; ? = quantum numbers (state labels) Temperature: Occupation of electronic states 1 Fermi-Dirac Distribution: ? ? = (? ??) ?? 1+? Different atomic structures (?) 9

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Structure Modelling Strategies Schr dinger s Equation: ? ? ?1 ?? = ?? ? ?1 ?? A solid is a molecule with a quasi-infinite number (ca. 1023) of atoms. A solid is a molecule with a quasi-infinite number (ca. 1023) of atoms. Molecular Solids: Use molecular entities (as in gas phase); packing effects? Extended Solids: How to make the problem tractable? Amorphous (glasses silicates, phosphates): use molecular fragments, tie off ends with simple atoms, e.g., H Aperiodic (incommensurately modulated, quasicrystals): use crystalline approximants Crystalline (metals, semiconductors, salts): use translational symmetry and space groups elegant simplification! Mixed Site Occupancies? use subgroups or superstructures or Coherent Potential Approximation 10

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Potential Energy Terms ? ? ? 2 2? ? ?A ??A 1 ??? Schr dinger s Equation: for electrons in a solid 2 + ?1 ?? = ?? ?1 ?? Multi-Electron ?=1 A=1 ?>? Kinetic Potential One-Electron Independent Electron (Hartree) Approximation: e (?) moves in the electrostatic field of the other e s Wavefunctions and energies determined from a single-particleSchr dinger s equation: ?1 ?? = ?1?1 ?2?2 ???? ? 2 ? ? = ??? ?=1 ? 2 eff Hartree Potential 2 2? ? ???? = ? ?? ???? ?A ??A 2 ??? + ???? ???? = ?? ???? ?? ?? A=1 Self-Consistency needed to solve these equations Valence ? ? on Atoms (Ions) Challenge: How best to model the potential energy? Simple metals (Na, Mg, Cu, Zn): Free-Electron Nearly-Free-Electron Pseudopotentials Transition metals, Semiconductors, Insulators: Tight-Binding / L.C.A.O. Pseudopotentials Isotropic Potentials Localized Potentials Valence ? ? in Bonds 11

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Potential Energy Terms ? ? ? 2 2? ? ?A ??A 1 ??? Schr dinger s Equation: for electrons in a solid 2 + ?1 ?? = ?? ?1 ?? ?=1 A=1 ?>? Kinetic Potential One-Electron Potential: A-e attraction + Hartree potential = ?eff One-Electron Potential ???+ ???? Two-Electron Potential: Exchange (?) and Correlation (?) = ???? ?? Two-Electron Potential: Exchange (?) and Correlation (?) Two-Electron Potential Spin Spin Charge Charge ? ? = ? ? + ? ? ? ? (Metals) Free Electrons: Hartree-Fock (Slater Determinant) Free Electrons ? 1 ?0 ?1/3 ?? ~ ??? ??? as ?? ?? DOS 0 states at ?? Nonmetallic! 1/2 ???? ~ ??2 ?? ?? ?? Spin Only ? ?? ?? = ? ?? = ? Exclusion Hole 1/3 = ? ? ? Average over all occupied states ? ?0 ?? ?? Fermi (Coulomb and Exchange) Hole +1 background charge not screened by ? ? Quasi-particle: ? and +1 background Forerunner of the Local Density Approximation (LDA) 12

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Potential Energy Terms ? ? ? 2 2? ? ?A ??A 1 ??? Schr dinger s Equation: for electrons in a solid 2 + ?1 ?? = ?? ?1 ?? ?=1 A=1 ?>? Kinetic Potential One-Electron Potential: A-e attraction + Hartree potential = ?eff One-Electron Potential ???+ ???? Two-Electron Potential: Exchange (?) and Correlation (?) = ???? ?? 1/3 LDA (Local Density Approx.): ???? ~ ? ? ? ???= ? ? ???? ? ?? GGA (Generalized Gradient Approx.): ???? = ? ? ? ,?? ? Hubbard Model ?0+ ? ? Magnetic Nonmetal Metal ?0 ? ~ ?? ?? = On-site e e Repulsion Mott-Hubbard Transition ?/? Increases ? = 0 13

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Electronic Structure of Si (Diamond Structure): 10 8 Energy 6 4 2 Fermi Level 0 (eV) -2 -4 -6 -8 -10 -12 -14 L X W L K ??? Electronic Band Structure Electronic Density of States How are these diagrams obtained? What can we learn from the information? 14

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Schr dinger s Equation: ??? = ?????? ? Basis Functions Atomic Orbitals Plane Waves ???= ???? ????? ??? ?? ??= ?? ??? ?? ??? ?? ?? Intra-Site Integrals ? ? ? Matrix 1 Rydberg = 13.6 eV 1 Rydberg = 1313 kJ/mol 1 eV = 96.5 kJ/mol 1 eV = 23.1 kcal/mol ??? ?? ??? ?? Inter-Site Integrals ?? Empirical / Semi-empirical: Integrals assigned values based on experimental data (AO energies) or simple algorithms (resonance integrals AO overlap) H ckel, Extended H ckel (EHT); Neglect of Differential Overlap (NDO methods) Minimal basis sets (only valence AOs) computationally fast Ignore explicitly two-electron potential energy terms Results over-estimate HOMO-LUMO gaps for insulators / semiconductors Poorly optimizes interatomic distances 15

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Schr dinger s Equation: ??? = ?????? ? Basis Functions Atomic Orbitals Plane Waves ???= ???? ????? ??? ?? ??= ?? ??? ?? ??? ?? ?? Intra-Site Integrals ? ? ? Matrix 1 Rydberg = 13.6 eV 1 Rydberg = 1313 kJ/mol 1 eV = 96.5 kJ/mol 1 eV = 23.1 kcal/mol ??? ?? ??? ?? Inter-Site Integrals ?? First Principles: Calculate all integrals using physical models Hartree-Fock (HF): incorporates anti-symmetrized wavefunctions as Slater determinants Useful for insulators & semiconductors; weaker for metals (exchange; single determinant; ) Does not fully account for correlation assumes electrons move in a mean field potential calculated energies tend to be too high Need multiple Slater determinants (configuration interaction) to account for correlation 16

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Schr dinger s Equation: ??? = ?????? ? Basis Functions Atomic Orbitals Plane Waves ???= ???? ????? ??? ?? ??= ?? ??? ?? ??? ?? ?? Intra-Site Integrals ? ? ? Matrix 1 Rydberg = 13.6 eV 1 Rydberg = 1313 kJ/mol 1 eV = 96.5 kJ/mol 1 eV = 23.1 kcal/mol ??? ?? ??? ?? Inter-Site Integrals ?? First Principles: Calculate all integrals using physical models Density Functional Theory (DFT): reformulates the problem using electron density ? ? Well-suited for ground-state (collective)properties, e.g., structure optimization, charge density Band gaps underestimated for semiconductors Exchange-Correlation explicitly included (approximations): Local Density Approx. (LDA): ???? expressed by form using local electron density Local Spin Density Approx. (LSDA): like LDA but different potentials and basis functions for spin- up / spin-down wavefunctions Hubbard Correlation (LDA + U): includes on-site ? parameter for localized electrons, e.g., in compounds with 4? and certain 3? metals 17

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Schr dinger s Equation: ??? = ?????? ? Basis Functions Atomic Orbitals Plane Waves ???= ???? ????? ??? ?? ??= ?? ??? ?? ??? ?? ?? Intra-Site Integrals ? ? ? Matrix 1 Rydberg = 13.6 eV 1 Rydberg = 1313 kJ/mol 1 eV = 96.5 kJ/mol 1 eV = 23.1 kcal/mol ??? ?? ??? ?? Inter-Site Integrals ?? VASP, WIEN2k, CRYSTAL, QUANTUM ESPRESSO First Principles CODES: Free Electron: No Potential Nearly Free Electron (NFE) Apply periodic potential of nuclear sites Pseudopotentials (PSP) Apply core-valence orthogonalization Tight Binding ( MO ): Atomic Potentials Linear Combination of Basis Functions Adapted to symmetry ( SALCs ) Bloch functions vs. Wannier functions Spatially delocalized Spatially localized Cellular Approaches Divide structure into cells Atomic-like vs. Constant potentials 18

IV. Electronic Structure and Chemical Bonding Quantum Mechanical Methods: Computational Approaches Cellular Approach Divide the structure into Wigner-Seitz (WS) cells ??? must be continuous at boundaries Atomic-like Orbitals ??? = ?,???????,? ????,? ; ??? = ?,???????,? ????,? ; ? ??? ? ??? ? ??? ? ??? M M ????; ????; Plane Waves M M M ??? = radius of sphere M M M M M Free Electron-like (Constant) Potential M M M M M M M Atomic-like (Spherical) Potential Polyhedral shapes of WS cells make this solution challenging, so Divide potential energy space into 2 regions Simplifications (Approximation) Atomic-Sphere-Approximation (ASA) Overlapping spheres Linearized Radial Functions (inc. speed) ??? ?? ?? Linear Muffin-Tin Orbital (LMTO) 2 ??? = ???? + ? ?? + ? ? ?? 19