Approach to normality
Distribution ofsums of uniform random numbers, each compared with the normal distribution.. R1, the uniform distribuion.R2,sum of two uniformly distributed numbers,etc.
Quantum Monte Carlo
• Introduction to Monte Carlo: sampling, random numbers, Markov chains, estimating errors
• Variational Monte Carlo: sampling and wavefunctions. • Diffusion Monte Carlo: branching random walks, fermion
(spectra, potential curves, etc.)
Density Functional Theory (DFT)
• Total energy as functional of the electron density:
E [] ( r ) v ( r ) d T S r [] J [] E x [c ]
a • Optimized (spin) orbitals
a • (Spin) orbital energies
HF: Optimized solutions
• Optimized spin orbitals:
• Set of occupied spin orbitals
• Set of unoccupied, “virtual” spin orbitals (basis set dimension typically is larger than problem size!)
薛定谔方程(Schrödinger equation )
Hartree-Fock Theory
• Linear expansion of atomic or molecular orbitals in (contracted) basis functions:
i c
• Variational optimization of expansion coefficients c •Variational: Energy expectation value is upper bound for exact energy
• X part: Exact Dirac functional from Fermi gas
• C part: From Quantum-Monte-Carlo simulations on Fermi gas
DFT: XC functionals; LDA
• A: High density, large kinetic energy, LDA approximation unimportant
As a first example, we take the energy required to remove all the L electrons from C. We have normal atom: - 2(5.70)24(3.25/2)2= -64.98 - 10.56 = - 75.54 atom with L electrons stripped off: - 2(5.70)2= -64.98 Difference = energy of removal = 10.56 = 10.56 X 13.56 volts = 143.2 volts (correct, 145.2 ; Zener has 142.7)
一、What is Quantum Chemistry?
• Quantum Mechanics applied to Atoms and Molecules
• Aim: Understanding of Electronic Structure • Solution of the electronic Schrödinger equation • Derived: Properties of Atoms and Molecules
DFT: Procedure
• Variation of energy functional w.r.t. density:
E[] 0
• yields set of effective one-particle equations:
1 2 i2vKeSf(r f) uiK(S r)iKu S iK(S r)
二、量子化学的发展
Numerical calculation J. C. Slater proposed Atomic shielding constants in 1930 . Taking into accout that the nodes in the wave function are found to be unimportant( by Zener) , Slater proposed the wave functiaon
HF: The wave function
• Ansatz for the many-particle wave function: Slater determinant
H e1 21 s(1 )1 s(2 )(1 )(2 ) (1 )(2 )
• Singlet ground-state of Helium atom • Consists of spatial and spin part • Contains antisymmetry in spin part (singlet) • Orbital coefficients are optimized for this Determinant
– Goal: to solve quantum many-body systems with computer simulation.
• Examples: liquid and solid helium, electron gas, hydrogen,…
Monte Carlo and Random Walks
• Consequence: Coulomb repulsion is diminished
• Included in HF theory
Hartree-Fock equations
fa aa
• f depends on solutions • Iterative solution of equation system • Upon self-consistency:
HF: Spin and the Fermi hole
• Fermi correlation
• Probability of ‘finding’ an electron at the same position of another electron, equal spin projections
量子化学材料计算简介
一、 What is Quantum Chemistry? 二、量子化学的发展(Numerical calculation, Monte
Carlo methods, RPA and Parametrized methods), 三、 Ab initio basis set： STO and GTO 四、 Pseudopotential Plane Wave approach 五、 材料计算软件的出现
• v: Kohn-Sham potential (contains ext. pot., J, xc) • u: KS orbitals • : KS orbital energies
DFT: XC functionals
• Local Density Approximation (LDA) Assumption: No gradient of electron density in Exc
• B: Small density gradient, LDA is good
• C: Bonding region, large gradient, LDA fails !
DFT: XC functionals. Gradient corrections
• Generalized Gradient Approximation (GGA) Density gradient correction from ry)
HF: Total energy
EHF HF*HHFd
• HF theory delivers >99% of the full electronic energy
• Accuracy also depends on Hamiltonian • Electron correlation is neglected • It accounts for the (important) residual
sign problem • Introduction to Path Integrals: formalism, sampling, the
action. • Boson & Fermion Path Integrals: permutations, exchange
moves, superfluidity and bose condensation.
RPA and Parametrized methods
Correlation energy per electron of the unpolarized and polarized uniform electron gas. CA: parametrized Ceperley-Alder, RPA: numerical random phase approximation. E, in eV.