Quantum Mechanical Simulations

Transcription

1 Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

2 Topics Quantum Monte Carlo Hartree-Fock Self-Consistency Field Density Functional Theory

3 H Quantum Mechanical Methods Approximation methods made to solve the Schrödinger equation Time-dependent i Φ ({ r},{ R }; t) = H Φ ({ r},{ R }; t) i I i I t where e ZZe Ze I J I = + + I i I 2M i 2 m i< j r r I< J R R I, i r R I Time-independent e i j I J i I ({ },{ }) E ({ },{ }) HΨ r R = Ψ r R i I i I Eigenvalue problem Ψ H Ψ = E Ψ Ψ E = Ψ H Ψ / Ψ Ψ

4 Quantum Mechanical Methods Quantum Monte Carlo (QMC): scaling with the number of atoms N nearly exponential, and polynomials O(M p ) with respect to the number of electrons M. Hartree-Fock (HF): scaling with O(N 4 ) or higher (depends on how the correlations are treated) Density-Functional Theory (DFT): scaling with O(N 3 ) Tight-Binding (TB): scaling with O(N 3 )

5 Quantum Mechanical Methods Multiscale Systems Engineering Research Group

6 Quantum Monte Carlo (QMC) The most accurate (and expensive) approach to calculate electronic property Can compute both ground and excited states Sufficient to address most issues involving inter-atomic forces and chemical properties

7 Two popular QMC methods Variational quantum Monte Carlo (VMC) The expected values are calculated via MC integration over 3N dimensional space of electron coordinates R={r 1,r 2,,r N } E * Ψ H Ψ Ψ ( R) HΨ( R) dr = = * Ψ Ψ Ψ ( R) Ψ( R) dr Diffusion quantum Monte Carlo (DMC) Starting from a trial wave function, the distribution of electrons evolves along a (imaginary) time

8 VMC Metropolis MC algorithm Start the walker with a random position R; Make a trial move to a new position R'; with a probability density T(R' R). That is, the probability that the walker is now in the volume element dr' is dr' T(R' R) Accept the trial move with probability repeat T( R R') P( R') A( R' R) = min 1, T( R' R) P( R)

9 VMC With an enormous number of walkers, after an equilibrium state is achieved, the average number of walkers in the volume element dr is denoted by n(r)dr Equilibrium means the average number of walkers from dr to dr' is the same as that from dr' to dr Since the probability that the next move of a walker at R is dr'a(r' R)T(R' R), the average number moving from dr to dr' in a single move is dr'a(r' R)T(R' R) n(r)dr. Balance: A(R' R)T(R' R)n(R)dRdR'=A(R R')T(R R')n(R')dR'dR Hence n( R) A( R R') T( R R') = n( R') A( R' R) T( R' R) Since the ratio of acceptance in Metropolis algorithm is A( R R') T( R' R) P( R) = A( R' R) T( R R') P( R') Therefore the equilibrium walker density n(r) is proportional to P(R) n( R) P( R) = n( R') P( R') Multiscale Systems Engineering Research Group

10 VMC The expected value of Ĥ evaluated with a trial wave function Ψ T provides a rigorous upper bound on the exact ground-state energy E 0 as which is evaluated using the Metropolis MC algorithm in the form E V = Ψ ( R) T The configuration-space probability density P E V 2 1 T * ( ) ˆ Ψ R HΨ ( R) dr T T = E Ψ ( R) Ψ ( R) dr 2 2 ( ) =Ψ ( ) / Ψ( ) d R R R R With a sample set of M points M 1 E ( ) V L m M E R Ψ Ψ ( R) T 2 * T T ( R) Hˆ Ψ ( R) dr dr T m= 1 0 where Ψ T 1 ĤΨ T = E L is Local energy

11 VMC Algorithm Multiscale Systems Engineering Research Group

12 DMC solving an imaginary-time many-body Schrödinger equation where E T is an energy offset In the integral form where Green function Φ( R, t) = ( Hˆ E ) Φ( R, t) T t Φ ( R, t + τ) = G( R R', τ) Φ( R', t) dr' ( τ ˆ ) G( R R', τ) = R exp ( H E T ) R' is a

13 path integral DMC

14 DMC Fixed-node approximation

15 Hartree-Fock (HF) Self-Consistency Field (SCF) Efficiently calculate ground-state electronic structures Based on two simplifications: Born-Oppenheimer approximation: solve the Schrödinger equation for the electrons in the field of static nuclei Replace the many-electron Hamiltonian with an effective one-electron Hamiltonian which acts on one-electron wave functions called orbitals

16 Slate Determinant HF approximation (Fock 1930; Slater 1930) is the simplest theory that incorporates the antisymmetry of the wave function Ψ (, x,, x, ) = Ψ(, x,, x, ) i j j i where x i ={r i,σ i } represents the coordinates and spin of electron i. The antisymmetry ensures that no two electrons can have the same set of quantum numbers and the Pauli exclusion principle is satisfied. ψ ( x ) ψ ( x ) ψ ( x ) N ψ ( x ) ψ ( x ) ψ ( x ) N Ψ (, x,, x, ) = i j ψ ( x ) ψ ( x ) ψ ( x ) N 1 N 2 N N where ψ ( x ) = φ( r ) δ i j i j σ, σ i j with δ σ, σ 1 ( σ = σ ) i j = 0 otherwise Multiscale Systems Engineering Research Group i j

17 HF-SCF The exact ground-state wave function, as our target of calculation, cannot be represented as a single Slater determinant. Yet we use a Slater determinant as a variational trial function and minimize the expected value of Hamiltonian w.r.t. the orbitals ψ i (r j ) s. Born-Oppenheimer approximation H BO 1 Z r R 2 r r = 2 I i i I, i i I i j i j

18 HF-SCF Hartree-Fock approximated Hamiltonian where and applied to the Slater determinant where

19 HF-SCF Define operators J k and K k as Define Coulomb operator J and exchange operator K The energy becomes under orthonomality constraint The minimization with the Lagrange multipliers Λ kl requires with That is,, Fock operator The solution is with orbital energies ε k s.

20 HF-SCF is solved by a self-consistency procedure the many electron problem is approximated by a sequential calculation of the motion of one electron in the average potential field generated by the rest of the electrons and nuclei.

21 HF-SCF Basis functions ψ Plane waves ~ Slate orbitals ~ i e kr Gaussian orbitals ~ M ( x) = C χ ( x) k pk q p= 1 e β r 2 e β r GTO GTO GTO Approximation of STO by GTO-NG N i = 1 i De β i r 2

22 HF-SCF Generalized Eigenvalue Problem Plug ψ = χ x into Hψ = Eψ, we have With left inner product for each pair of p and q H We have where H = [H pq ] is the Hamilton matrix, S = [S pq ] is the overlap matrix, C = [C pq ] is the density matrix with In unit form C ( ) q p q H C χ ( x) = E C χ ( x) q p q q p q HC = ESC where C'=V 1 C, H'=V 1 HV, and matrix V transforms S to the unit matrix as V 1 SV=I. = χ H χ S = χ χ pq p q C occupied * = 2 C C pq k = 1 pk qk HC ' ' = EC' pq p q

23 HF-SCF Hartee-Fock-Roothaan equations where elements of Fock matrix F= [F pq ] is and density matrix P= [P pq ] Fock matrix Energy

24 HF-SCF Algorithm Multiscale Systems Engineering Research Group

25 Density Functional Theory (DFT) Very efficiently calculate ground-state electronic structures Based on three simplifications: Born-Oppenheimer approximation: solve the Schrödinger equation for the electrons in the field of static nuclei Electrons interacts with a density field ρ(r), which is approximated by wave functions Approximations of the unknown exchange-correlation energy functional, which accounts for complicated correlated motion of electrons (e.g. local-density approximation (LDA), generalized gradient approximation (GGA) ).

26 DFT Energy Kinetics of non-interacting electrons External Kohn-Sham equations Coulombic interaction Ze I ρ( r') dr' V () r ψ () r = ε ψ () r xc m m m 2 r R r r' I I kinetic External Hartree Exchange-Correlation V () r = xc potential potential potential cccup = m = 1 ρ () r ψ () r m 2 Unknown exchange and correlation δe [ ρ( r)] xc δρ() r

27 DFT Exchange-correlation potential depends on density and its variations Local density approximation (LDA): only r (not gradients) where V xc [ ρ]( r) = V ρ( r), ρ( r), ( ρ( r )), xc LDA E () r () [()] xc = ρ r ε ρ r dr ec 1/3 1/3 ε [ ρ( r)] = C ρ ( r) and C = 3/4 (3/ π) ec Generalized gradient approximation (GGA): with gradient GGA E () r = F[(), ρ r ρ()] r dr xc

28 DFT Algorithm Multiscale Systems Engineering Research Group

29 Summary Quantum Monte Carlo Hartree-Fock Self-Consistency Field Density Functional Theory

30 Further Readings Quantum Monte Carlo Foulkes W.M.C., Mitas L., Needs R.J. and Rajagopal G. (2001) Quantum Monte Carlo simulations of solids. Reviews of Modern Physics, 73(1):33-83 Density Functional Theory Koch W. and Holthausen M.C. (2001) A Chemist s Guide to Density Functional Theory (Weinhaim: Wiley-VCH) ISBNs: ,