FEYNMAN PATH INTEGRAL APPROACH TO QUANTUM DYNAMICS AND QUANTUM MONTE CARLO
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1 FEYNMAN PATH INTEGRAL APPROACH TO QUANTUM DYNAMICS AND QUANTUM MONTE CARLO 1 Tapio Rantala, Tampere University of Technology, Finland CONTENTS MOTIVATION FROM PATH INTEGRAL FORMALISM TO QUANTUM DYNAMICS and STATIONARY STATES COHERENT DYNAMICS BENEFITS OF MONTE CARLO TECHNIQUE INCOHERENT DYNAMICS TEST CASE: 1D HOOKE S ATOM QUANTUM STATISTICAL PHYSICS APPROACH TO ELECTRONIC STRUCTURE THEORY NOVEL FEATURES AND DEMONSTRATIONS MOTIVATION 2 Electronic structure is the key concept in materials properties and related phenomena: for mechanical, thermal, electrical, optical, and also, for dynamics of electrons. Conventional ab initio / first-principles type methods suffer from laborious description of electronic correlations (CI, MCHF, CC, DFT-functionals). typically ignore nuclear (quantum) dynamics and non-adiabatic coupling of electron nuclei dynamics (Born Oppenheimer approximation) > ab initio MOLDY. give the zero-kelvin description, only, though more realistic temperatures might be relevant. are typically good for stationary states, only.
2 PATH INTEGRAL APPROACH TO QUANTUM MECHANICS 3 Z (x, t) (x, t) H (x, t) K(x, t; x a,t a ) (x a,t a )dx a K(x b,t b ; x a,t a )= (x, t) = Z xb x a exp(is[x b,x a ])Dx(t) Real Time Path Integrals (RTPI): Time evolution in real time 1X C n n (x)exp[ ie n t] n=0 Stationary states from incoherent RTPI τ Im τ: 0 -i β t Re PATH INTEGRAL APPROACH TO QUANTUM MECHANICS 4 Z (x, t) (x, t) H (x, t) K(x, t; x a,t a ) (x a,t a )dx a = it Z (x, ) = H (x, ) (x, G(x, ; x a, a ) (x a, a )dx a K(x b,t b ; x a,t a )= Z xb x a exp(is[x b,x a ])Dx(t) Real Time Path Integrals (RTPI): Time evolution in real time (x, t) = 1X C n n (x)exp[ ie n t] n=0 Stationary states from incoherent RTPI (x b, b ; x a, a )= Quantum statistical physics in equilibrium at temperature T, τ b -τ a = = 1/kBT (x, ) = 1X C n n (x)exp[ E n ] n=0 Z xb x a exp( S[x b,x a ])Dx( ) Diffusion Monte Carlo (DMC)
3 PATH INTEGRAL APPROACH TO QUANTUM MECHANICS 5 There is nothing but dynamics and all possible paths contribute with a complex phase. Path integral (sum over all paths) gives the probability amplitude, the wave function. in real time in imaginary time ( t ) ( t = -i β ) Time evolution in real time & stationary states (RTPI) Quantum statistical physics in equilibrium with temperature T, = 1/kBT Feynman Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965) Feynman R.P., Rev. Mod. Phys. 20, (1948) Feynman R.P., Statistical Mechanics (Westview, Advance Book Classics, 1972) REAL TIME PATH INTEGRAL APPROACH TO COHERENT QUANTUM DYNAMICS (x, t) H (x, t) Z (x, t) = K(x, t; x a,t a ) (x a,t a )dx a K(x b,t b ; x a,t a )= Z xb x a exp(is[x b,x a ])Dx(t) Short time kernel or Trotter kernel h m K(x b,x a ; t) 2 i~ t i D/2 exp apple i ~ ( m 2 t (x b x a ) 2 t 2 (V (x a)v (x b )) Monte Carlo grid to: - avoid artificial interferences - sample the space continuously - allow adaptation along time evolution the walkers and walker distribution Glauber state evolution in a regular grid with different kernels and time steps. I. Ruokosenmäki and TTR, Numerical path integral approach to quantum dynamics and stationary states, Communications in Computational Physics 18, 91 (15).
4 REAL TIME PATH INTEGRAL APPROACH TO INCOHERENT PROPAGATION 7 Coherent propagation and zero reference shift Z (x, t) = K(x, t; x a,t a ) (x a,t a )dx a (x, t) = 1X C n n (x)exp[ i(e n E T ) t] n=0 with small angle approximation 1X (x, t) C n n (x){1 [(E n E T ) t] 2 /2 i[(e n E T ) t]} n=0 Incoherent propagation: ignore the very small imaginary part 1X R(x, t) = C n n (x){1 [(E n E T ) t] 2 /2} n=0 This is simulation of quantum Zeno effect as t 0, i.e., keeping the system in its observable real state or finding the eigenstate closest to E T, the same way as DMC finds the ground state. Cool, isn t it! Im E L (x) = (x, t) (x) t (x) Re ONE DIMENSIONAL HOOKE S ATOM: SEPARATION OF COORDINATES 8 u 0 (r) r r H(x 1,x 2 )= 1 2 r r !2 x !2 x x 1 x 2 H(r, R) = 1 2µ r2 r 1 2 µ!2 r 2 1 r 1 2M r2 R 1 2 M!2 R 2 H r H R
5 1D HOOKE S ATOM: GROUND STATE WAVE FUNCTION 9 r 1D HOOKE S ATOM: PHASE OF THE WAVE FUNCTION 10
6 1D HOOKE S ATOM: RTPI AND DMC 11 I. Ruokosenmäki et al., Numerical path integral solution to strong Coulomb correlation in one dimensional Hooke s atom, submitted to CPC, (arxiv: ). PATH INTEGRALS IN IMAGINARY TIME 12 Feynman path integral in imaginary time evaluates the mixed state density matrix The partition function is obtained as Note! There is no wave function! where = 1 / kbt and S[R] is the action of the path R(0) --> R( ) for the imaginary time period.
7 NVT ENSENBLE QUANTUM STATISTICAL MECHANICS 13 Thus, we have the quantum statistical mechanics and the concepts of NVT ensemble finite temperature, T > 0 density matrix action partition function expectation values Note! this is all exact, so far! Feynman Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965) Feynman R.P., Statistical Mechanics (Westview, Advance Book Classics, 1972) DISCRETIZING THE PATH With the primitive approximation and Trotter expansion where M is the Trotter number and With short enough τ = β/m the action approaches that of free particle as the potential function. Note! This is not sc. PIMD
8 ... FOR EVALUATION WITH MONTE CARLO 15 Monte Carlo allows straightforward numerical procedure for evaluation of multidimensional integrals. Metropolis Monte Carlo NVT (equilibrium) ensemble now yields mixed state density matrix with almost classical transparency Metropolis N. et al., J. Chem. Phys. 21, 1087, (1953). Ceperley D.M., Rev. Mod. Phys. 67, 279, (1995) PATH INTEGRAL MONTE CARLO (PIMC) 16 An ab initio electronic structure approach with novel features: FIXED FINITE TEMPERATURE, NVT ENSEMBLE NO BORN OPPENHEIMER APPROXIMATION FULL ACCOUNT OF CORRELATION, the van der Waals interaction, for example! Evaluation of EXCHANGE INTERACTION suffers from Fermion Sign Problem! SIMULATION OF EQUILIBRIUM DISSOCIATION RECOMBINATION REACTION
9 QUANTUM DOTS IN QCA 17 Finite temperature and electron delocalization effects SMALL LIGHT NUCLEI MOLECULES: ELECTRONS, PROTONS AND POSITRONS 18 3-particle and 4-particle molecules Ps2, 300 K Ps2 H e or PsH e H H H2 H2 Kylänpää, TTR and D.M. Ceperley, PRA 86, (2012)
10 19 DIPOSITRONIUM Ps 2 Pair approximation and matrix squaring Bisection moves Virial estimator for the kinetic energy same average time step for all temperatures τ = β/m = 0.015, M 10 5 e e e e M = H 3 MOLECULE 20 Quantum statistical physics of two electrons and three nuclei (fiveparticle system) as a function of temperature: Structure and energetics: quantum nature of nuclei pair correlation functions, contact densities,... dissociation temperature Comparison to the data from conventional quantum chemistry. p e p p me, mp e Ab initio or first-principles!
11 H 3 TOTAL ENERGY: FINITE NUCLEAR MASS AND ZERO-POINT ENERGY 21 Total energy of the H3 ion up to the dissociation temperature. Born Oppenheimer approximation, classical nuclei and quantum nuclei. H 3 MOLECULAR GEOMETRY AT LOW TEMPERATURE: ZERO-POINT MOTION 22 Internuclear distance. Quantum nuclei, classical nuclei, with FWHM Snapshot from simulation, projection to xy-plane. Trotter number 2 16.
12 H 3 ENERGETICS AT HIGH TEMPERATURES 23 Total energy of the H3 beyond dissociation temperature. Lowest density, mid density, highest density HH H 2 H H 2 H H 3 Barrier To Linearity T (K) 24 H 3 ENERGETICS AT HIGH TEMPERATURES: DISSOCIATION RECOMBINATION Total energy of the H3 beyond dissociation temperature. Lowest density, mid density, highest density HH H 2 H H 2 H H 3 Barrier To Linearity T (K)
13 DISSOCIATION RECOMBINATION EQUILIBRIUM REACTION 25 The molecule and its fragments: Number of Monte Carlo Blocks H 3 BTL H 2 H H 2 H 2HH We also evaluate molecular free energy, entropy and heat capacity! E (units of Hartree) The equilibrium composition of fragments at about 5000 K. PARTITION FUNCTION 26 Numerical integration of lnz(t) = lnz(t 1 ) gives the partition function. We use the initial condition Z(0) = 0. T T 1 E dt k B T 2
14 FREE ENERGY 27 Free energy (units of Hartree) T (K) FIG. 4. Helmholtz free energy from Eq. (5)in the units of Hartree. Notations are the same as in Fig. 3. ENTROPY 28 S = U F T, (11) 35 Entropy (units of k B ) T (K) FIG. 5. Entropy from Eq. (11) in the units of k B. Notations are the same as in Fig. 3.
15 MOLECULAR HEAT CAPACITY 29 Heat capacity (units of k B ) 9/2 6/2 3/2 Rotation Vibration T (K) FIG. 6. Molecular heat capacity as a function of temperature calculated using the analytical model of this work. The values on the y-axis are given in units of the Boltzmann constant k B. THANK YOU! 30 For more details see 1. Ilkka Kylänpää, PhD Thesis (Tampere University of Technology, 2011) 2. Ilkka Kylänpää and Tapio T. Rantala, Phys. Rev. A 80, , (2009) PS2 and I. Kylänpää, M. Leino and T.T. Rantala, Phys. Rev. A 76, , (2007) 3. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 133, (2010) 4. Ilkka Kylänpää and Tapio T. Rantala, J.Chem.Phys. 135, (2011) H 3 5. Ilkka Kylänpää, Tapio T. Rantala, David M. Ceperley, Phys. Rev. A 86, , (2012) ACKNOWLEDGEMENTS Ilkka Ruokosenmäki, Ilkka Kylänpää, Markku Leino, Juha Tiihonen (TUT) Ilkka Kylänpää, David M. Ceperley (UIUC)
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