计算材料学第五章 电子结构理论 计算材料学第五章 电子结构理论 赵纪军赵纪军 三束实验室，物理学院三束实验室，物理学院 ikx k ikxikx kkk He x Hek e x = = kkk H = Take any set of basis functions i Assume the wave functions can be expanded in the basis: Choose the kso that the k are orthogonal (required for them to be eigenvalues) and kare as low as possible. This is a numerically challenging problem which effectively requires iteratively diagonalizing the Hamiltonian matrix in the i basis This gives the best guess at the k possible given the basis functions i kkll l = Solving The Schrdinger Equation A QM Code Input atom positions and types Choose sensible basis functions Diagonalize the Hamiltonian to get eigenvalues and eigenvectors Use eigenvectors and basis to get wave functions/charge density Use eigenvalues and ionic interactions to get energy Key Ideas For Using A QM Code Choosing the basis The atomic basis, size and polarization The plane wave basis and the cutoff Ecut Working with periodic systems band structures k-point mesh Forces Hellmann-Feynman Theorem Relaxed positions Magnetism Spin polarized calculations Accuracy Basis functions Density functional theory Output Energy Relaxed positions Band structure DOS Charge density Choosing The Basis Many programs work with a plane wave basis (eikr) since they are easy to use numerically You have to specify number of basis functions: More basis functions more accurate, slower = more memory. You have to diagonlize a matrix of size NbasisNbasis, so time O(Nbasis2lnNbasis) Number of basis functions is given by a cutoff in energy, Ecut. This relates to the basis functions by We include all plane waves with kkcutand k a recipricol lattice vector Since reciprocal space shrink as 1/V, the Nbasis V and bigger systems take longer as O(V2lnV)! 22 2 cut cut e k E m = h Working with Periodic Systems Solutions for periodic systems have to have the same physics under translations wave functions are Bloch states: i(r+R)=eikR i(r), where k is in the first Brillouin zone. Electronic energy states in periodic systems are conveniently indexed with n,k n are called the bands, and k the wavevector. The plot of the electronic energy states is called the band structure. n=1 n=3 n=2 k values (in 1st Brillouin Zone) Energy of Eigenstates k* 1k* 3k* 2k* EFermi Empty bands Filled bands The k-point Mesh The number of different energy eigenstates is actually infinite since k is a continuous variable The problem is made tractable for computation by discretizing k on a mesh the kpoint mesh (usually just in n1x n2x n3uniform grid in space. Denser k-point mesh more accuracy, slower calculation Time O(number of k-points) n=1 n=3 n=2 k values (in 1st Brillouin Zone) Energy of Eigenstates k* 1k* 3k* 2k* EFermi Empty bands Filled bands Sample the Brillouin Zone Forces calculated with QM methods Forces can be found “easily” in a calculation by Hellmann-Feynmann Theorem Forces can be used for Finding optimal local energy positions (e.g., conjugate gradient methods) Molecular Dynamics (e.g., with Verlet algorithm) Pressure (in fact, the whole stress tensor) can be found from forces used Virial theorem (as in molecular dynamics) QM codes allow you to relax to optimal positions and lattice parameters for arbitrary pressure Magnetism - Spin Polarized Calculation Originally, QM codes just found the eigenstates without considering spins each state could hold two electrons Then it became possible to have majority and minority spins, allowing studies of magnetic systems (ferro-. antiferro-, ferri-magnetism). The QM codes optimize the total amounts of up and down spin in what are called spin-polarized calculations Further work now allows localized moments that can point in arbitrary directions (noncollinear magnetism, e.g., spin waves). Accuracy All QM methods have some approximations 3 big categories Exact methods (Quantum Monte Carlo, etc.): Have only the numerical errors (1-10 atoms) Wave function methods (Hartree Fock, etc.): Primary approximations come from limited basis sets (1-100 atoms) Density Function Theory methods: Primary approximation comes from unknown term in Hamiltonian (the exchange- correlation term) (1-1000 atoms). This is the method we will be using Largest errors come from systems with localized electrons, due to inaccuracy of exchange-correlation term Excited states are not treated accurately, so you get incorrect band gaps QM Codes Output Energy Can be used to compare two states to see which are most stable (e.g., fully relaxed fcc and bcc) Can be used to calculate barriers to hopping to model transport Relaxed positions Gives the atomic coordinates and lattice parameters for arbitrary pressure (can be compared to x-ray experiments) Band structure ?A convenient way to show eigenstates ?Shows band gaps (metal or insulator, optical properties), electron mass