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Apuan Alps 1 Multi-component quantum Monte Carlo method Yukiumi KITA , Ryo MAEZONO 2,3, Masanori TACHIKAWA 1 Yokohama City University, Japan School of Information Science, Japan Advanced Institute of Science and Technology Precursory Research for Embryonic Science and Technology, Japan Science and Technology Agency, Japan ~Quantum Monte Carlo study on multi-component systems~ Slide 2 Apuan Alps 2 In this talk 1 st Topic: Ab Initio Quantum Monte Carlo Study on the Positronic Compounds Ab Initio Quantum Monte Carlo Study on the Positronic Compounds QMC calculation of the temporary bound state between a positron and atoms/molecules 2 nd Topic: Development of Translational and Rotational Free Quantum Monte Carlo Method Development of Translational and Rotational Free Quantum Monte Carlo Method QMC calculation of the system which contains only quantum particles Slide 3 Apuan Alps 3 1 st Topic: Ab Initio Quantum Monte Carlo Study on the Positronic Compounds Slide 4 Apuan Alps 4 Outline of the 1 st topic 1. Introduction - Positron & Positronic compound 2. Purpose 3. Methods 4. Results - [LiH;e + ] system - [HCN;e + ] system 5. Summary [LiH,e + ] [H -,e + ] Pair annihilation [HCN,e + ] Slide 5 Apuan Alps 5 Positron (e + ): A positron was found by C.D.Anderson in 1932. A positron is an antiparticle of an electron (e - ) (charge= +1, mass= 1, spin= 1/2) 1.Introduction: positron Nuc. e-e-e-e- e+e+e+e+ [X - ; e + ] e+e+ Positronic compounds: When a positron meet a molecule (atom), they form the positronic compound which is the temporary bound state of them. Nuc. e-e-e-e- [X - ] Electronic & Positronic Structure ? Stable Geometry ?? We need the more accurate theoretical method for the positronic compound. Conventional MO scheme gives poor results. Slide 6 Apuan Alps 6 2. Theoretical analysis on the positronic compounds by this method. e.g. [M;e + ] (M= H -, LiH, NaH, KH, HF etc.) 1. The development of the quantum Monte Carlo method which can obtain the accurate many-body wave function of the positronic compound. 2.Purpose Slide 7 Apuan Alps 7 Hamiltonian of a positronic compound : We assumed the Hamiltonian of the system containing M nuclei, N electrons and a positron as the following form: 3.Method: trial wave function where V(R) is the Coulomb interaction potential between charged particles: e - -Nuc. e + -Nuc. e - -e - e - -e + Trial wave function in QMC method: Slater-Jastrow type function : Jastrow function : single determinant of spin up/down electrons : a positronic orbital Hartree-Fock node [1] M.Tachikawa, H.Sainowo, K.Iguchi, and K.Suzuki; J. Chem. Phys. 101 5925 (1994) Slide 8 Apuan Alps 8 3.Method: e - -Nuc. cusp correction [1] A. Ma, M.D.Towler, N.D.Drummond and R.J.Needs; J. Chem. Phys., 122, 224322 (2005) GTO r C : c usp-radii Cusp-corrected GTO cusp e - -Nuc. cusp correction for Gaussian type orbital [1] : The partial substitution of s-type GTO within cusp-radii [r c ] LCAO-expansion of e - /e + orbitals by Gaussian basis : We used the Gaussian basis set for expanding e - /e + orbitals. e - -Nuc. cusp cannot be described well. electrons: positron : Slide 9 Apuan Alps 9 Jastrow factor in trial wave function [1] : [1] N.D.Drummond, M.D.Towler and R.J.Needs; Phys. Rev. B, 70, 235119 (2004) e - -e - e - -Nuc. e - -e - -Nuc. e - -e + e + -Nuc. : We introduced the Jastrow factor proposed by Needs group into the trial wave function, 3.Method: Jastrow factor (1) Slide 10 Apuan Alps 10 All Jastrow factors are expaned in terms of the distance between particles [1] : and is the parameter based on the cusp condition such as: where, 3.Method: Jastrow factor (2) [1] N.D.Drummond, M.D.Towler and R.J.Needs; Phys. Rev. B, 70, 235119 (2004) Slide 11 Apuan Alps 11 3.Method: diffusion Monte Carlo Stabilization scheme by Umrigar, Nightingale and Runge [1] - Effective time step - Drift vector limiting scheme (for only electrons) - Local energy limiting scheme [1] C.J.Umrigar, M.P.Nightingale, K.J.Runge; J. Chem. Phys., 99, 2865 (1993) Fig. produced by R.Maezono trial node Fixed-node approximation used on only electron-moves Importance-sampled DMC method which propagates the following distribution in imaginary time . Slide 12 Apuan Alps 12 Code implementation Main authors of CASINO (ver. 1.8.1) : R.J. Needs, M.D. Towler, N.D. Drummond and P.R.C. Kent ( http://www.tcm.phy.cam.ac.uk/ ~ mdt26/cqmc.html )http://www.tcm.phy.cam.ac.uk/ ~ mdt26/cqmc.html We modified CASINO-code developed in Cavendish Laboratory, University of Cambridge. Slide 13 Apuan Alps 13 Code implementation: code lists We modified the following source files: config_gen.f90config_type.f90dmc.f90dmc_utils.f90 esdf_key.f90gauss_mol.f90gaussians.f90interactions.f90 jastrow.f90limits.f90mdet.f90monte_carlo.f90 newopt.f90numerical.f90obuf.f90taueff.f90 varmin.f90varmin_utils.f90vmc.f90vmc_utils.f90 We added the following source files: component.f90dbar_matrices_mc.f90driftvector_mc.f90 eval_geometry_mc.f90gauss_mol_mc.f90interactions_mc.f90 local_energy_mc.f90mcdefs.f90mcjasdefs.f90 newjas_mc.f90vmc_utils_mc.f90 Slide 14 Apuan Alps 14 Outline of the 1 st topic 1. Introduction - Positron & Positronic compound 2. Purpose 3. Methods 4. Results - [LiH;e + ] system - [HCN;e + ] system 5. Summary [LiH,e + ] [H -,e + ] Pair annihilation [HCN,e + ] Slide 15 Apuan Alps 15 4.Result: [LiH;e + ] system [1] K. Strasburger; Chem. Phys. Lett. 253 49 (1996), [2] D. Bressanini; J. Chem. Phys. 113 6154 (2001). [3] K. Strasburger; J. Chem. Phys. 114 615 (2001) Total energy of [LiH;e + ] with various schemes R Li-H = 1.771 which is the optimized value by explicitly correlated gaussian method (Ref.2) Unit in a.u. MCMO(HF) -7.990367 [ e - /e + = 6-311++G(df,pd)/9s8p ] E xplicitly C orrelated G aussian [3] -8.107474 [1024 terms] -8.08058(7) VMC [ e - /e + = 6-311++G(df,pd)/9s8p ] -8.10701(5) FN-DMC [ e - /e + = 6-311++G(df,pd)/9s8p ] -8.05530 CISD [1] [6-311G**+diffuse+polar+ghost] e - /e + orbitals in HF-level Fixed-node Approximation FN-DMC [2] -8.1071(1) Slide 16 Apuan Alps 16 4.Result: positron affinity of LiH Positron affinity [PA] of LiH [1] K. Strasburger; Chem. Phys. Lett. 253 49 (1996) [2] K. Strasburger; J. Chem. Phys. 114 615 (2001) HF VMC DMC CISD [1] ECG [2] Positron affinity of LiH [eV] HF+0.04(11) VMC+0.505(3) DMC+1.007(2) CISD+0.462 ECG+1.005 PA LiH = E LiH E LiH;e + Slide 17 Apuan Alps 17 4.Result: [HCN;e + ] system [1] H. Chojnacki and K. Strasburger, Mol. Phys. 104 2273 (2006) Total energy of [HCN;e + ] with various schemes The structure of HCN is optimized by the second-order perturbation method Unit in a.u. (MP2/aug-cc-pVTZ) MCMO(HF) -92.90074 [ e - /e + = 6-311++G(2d,2p)/15s 15p 6d 2f ] -92.901915 CISD [1] [ e - : 6-311++G(2d,2p) e + : 6-311++G(2d,2p) + 10s GTF (off-atom)] -93.2591(5) VMC [ e - /e + = 6-311++G(2d,2p)/15s 15p 6d 2f ] -93.39830(8) FN-DMC [ e - /e + = 6-311++G(2d,2p)/15s 15p 6d 2f ] HOMO/e + orbitals in HF-level 100% 72% Slide 18 Apuan Alps 18 4.Result: Positron affinity of HCN Positron affinity [PA] of HCN molecule: PA HCN = E HCN E HCN;e + [1] H. Chojnacki and K. strasburger, Mol. Phys. 104 2273 (2006) Positron affinity of HCN [eV] HFVMC 0.0(9) VMC-0.22(2) DMC+0.026(2) CISD [1] +0.018 HFVMC VMC DMC CISD [1] Slide 19 Apuan Alps 19 5.Summary of 1 st topic 1.We had developed the Multi-Component Quantum Monte Carlo [MC_QMC] method which can obtain the many-body wave function containing the several kinds of particle such as e - and e +. 2.We had applied MC_QMC method to [LiH;e + ] and [HCN;e + ] systems. Positron affinity gives good agreement with the results by the variational calculation using ECGs trial wave function. In benchmark models: [LiH;e + ] In [HCN;e + ] system, Our result gives the lowest variational energy of [HCN;e + ], shows that HCN molecule can bind a positron. Slide 20 Apuan Alps 20 2 nd Topic: Development of Translational and Rotational Free Quantum Monte Carlo Method Slide 21 Apuan Alps 21 Nuclear quantum effects The quantum nature of the light particle (proton and deuteron etc.) has important effects on the many research fields. Representative topics include the hydrogen[H]/deuterium[D] isotope effect observed in red-shift of vibrational frequencies, shift of chemical reaction rates etc. Especially, in hydrogen cluster and its isotope (H 2, HD and H 3 + etc.), the nuclear quantum effect strongly affects its properties. The system contains only quantum particles, that is, electrons and quantum nuclei. Slide 22 Apuan Alps 22 Theoretical approaches On the theoretical analysis of the system which contains only quantum particles, one of the smart approach is the use of the internal coordinate. e.g. the use of relative distances between quantum particles However, in our method shown in 1 st topic, we used the trial wave function with one-particle Gaussian basis sets. the trial wave function depends on the laboratory coordinate system We need the other theoretical frameworks Slide 23 Apuan Alps 23 What is problems? Non-relativistic Hamiltonian of the systems which do not contain any point charges: Coulomb interaction Kinetic energy Under this Hamiltonian, if we used the trial wave function depending on the laboratory coordinate system, fatally contains the translational and rotational energy of whole system in its kinetic energy. Leading to the difficulty of analyzing the variational energy of the system Slide 24 Apuan Alps 24 Translational & Rotational-Free Hamiltonian Translational and Rotational Free Hamiltonian: Total mass Total momentum Tensor of moment of inertia in CoM-system Total angular momentum in CoM-system Conventional Hamiltonian Translational energy of whole system (kinetic energy of the center of mass) Rotational energy of whole system around the center of mass To remove the needless kinetic energy, we introduced modified Hamiltonian as follows: Slide 25 Apuan Alps 25 Trial wave function [1] M.Tachikawa et al.; J. Chem. Phys. 101 5925 (1994), [2] N.D.Drummond, M.D.Towler and R.J.Needs; Phys. Rev. B, 70, 235119 (2004) Jastrow function: We used Jastrow function with the cutoff length proposed by Drummond [2] : Jastrow function : Slater determinant for the particle with spin S : spin of -th component only two-body terms depending on relative distances Hartree-Fock method [1] one-particle Gaussian basis Orbital parts depending on Lab. coordinate system In TRF-QMC method, we used Slater-Jastrow trial wave function in which the orbital parts are expanded by one-particle Gaussian basis sets. Slide 26 Apuan Alps 26 Variational Monte Carlo method In VMC method, we evaluate the translational & rotational free local energy at the configurations which are distributed according to E L TRF contains the local translational and rotational energy as follows: For the translational energy of the center of mass: For the rotational energy of whole system around the center of mass: Slide 27 Apuan Alps 27 diffusion Monte Carlo method In diffusion Monte Carlo method, we need the Green function corresponding to But, not yet Slide 28 Apuan Alps 28 Result: positronium [Ps] Positronium: The system contains only one-electron and one-positron. Energy [a.u.] -0.18908(4) Conv. VMC -0.249476(9) TF-VMC -0.249627(7) TRF-VMC -0.25 exact TF: Translational Free TRF: Translational & Rotational Free VMC: the total energy are improved by removing the translational and rotational energies. DMC: not yet Total energy of the ground state Slide 29 Apuan Alps 29 Result: HD HD: The system contains two-electrons, one-proton one-deuteron. Energy [a.u.] -1.1117(2) Conv. VMC -1.1271(2) TF-VMC -1.1407(2) TRF-VMC -1.16547 [1] Explicitly Correlated Gaussian [1] Donald B. Kinghorn and Ludwik Adamowicza.; J. Chem. Phys. 113 4203 (2000) Total energy of the ground state TF: Translational Free TRF: Translational & Rotational Free VMC: the total energy are improved by removing the translational and rotational energies. DMC: not yet Slide 30 Apuan Alps 30 Summary We have proposed an another quantum Monte Carlo technique to calculate the system which contains only quantum particles. In this method, we can use one-particle Gaussian basis sets. leading to the easiness to generate the trial wave functions we remove the translational and rotational energy of whole system We found that the total energy of Ps and HD are improved by the translational and rotational free VMC method. However, we need the diffusion Monte Carlo technique for obtaining the sufficiently accurate results. Slide 31 Apuan Alps 31 Future plans Slide 32 Apuan Alps 32 Outline of the 2 nd topic 1. Introduction 2. Purpose 3. Method 4. Results - Positronium [Ps] - HD 5. Summary [HCN,e + ] H3+H3+ HD Slide 33 Apuan Alps 33 Outline of the 2 nd topic 1. Introduction 2. Purpose 3. Method 4. Results - Positronium [Ps] - HD 5. Summary [HCN,e + ] H3+H3+ HD Slide 34 Apuan Alps 34 4.Result: HD HD: 2 electrons, 1proton 1deuteron Energy [a.u.] -1.1117(2) Conv. VMC -1.1271(2) TF-VMC -1.1407(2) TRF-VMC -1.16547 [1] ECG TF: TRF: VMC: DMC: not yet [1] Donald B. Kinghorn and Ludwik Adamowicza.; J. Chem. Phys. 113 4203 (2000) e -, proton, deuteron (Hatree-Fock ) Slide 35 Apuan Alps 35 4.Result: positronium [Ps] Positronium, Ps: 1electron, 1positron Energy [a.u.] -0.18908(4) Conv. VMC -0.249476(9) TF-VMC -0.249627(7) TRF-VMC -0.24992(6) TRF-DMC -0.25 exact TF: TRF: VMC: DMC: Slide 36 Apuan Alps 36 4.Result: HD HD: 2 electrons, 1proton 1deuteron Energy [a.u.] -1.1117(2) Conv. VMC -1.1271(2) TF-VMC -1.1407(2) TRF-VMC -1.16547 [1] ECG TF: TRF: VMC: DMC: not yet [1] Donald B. Kinghorn and Ludwik Adamowicza.; J. Chem. Phys. 113 4203 (2000) -????????(?) TRF-DMC Slide 37 Apuan Alps 37 4.Result: H 3 + H 3 + : 2electrons, 3protons Energy [a.u.] -1.2518(3) Conv. VMC -1.2679(3) TF-VMC -1.2833(3) TRF-VMC (Hatree-Fock ) TF: TRF: VMC: DMC: not yet Slide 38 Apuan Alps 38 4.Result: H 3 + H 3 + : 2electrons, 3protons Energy [a.u.] -1.2518(3) Conv. VMC -1.2679(3) TF-VMC -1.2833(3) TRF-VMC -????????(?) TRF-DMC (Hatree-Fock ) TF: TRF: VMC: DMC: not yet Slide 39 Apuan Alps 39 5.Summary 1 ( - ) [MC_QMC] 2 MC_QMC [LiH;e + ] [HCN;e + ] DMC Future works: Proton/Deuteron Slide 40 Apuan Alps 40 3.Method: diffusion Monte Carlo method Green 2 DMC Green 3 : : / (VMC ) : drift vector : Slide 41 Apuan Alps 41 Diffusion Monte Carlo Method (3) Fokker-Plank equation : distribution function 3D space Langevin equation 3D space Histogram of particles Imaginary-time dependent Schrodinger equation 3N dimension : wave function Langevin equation for walkers in 3N dimensional space 3N dimension Histogram of walkers Analogy Equivalence The above figures were produced by R.Maezono Slide 42 Apuan Alps 42 18 9 F 18 8 O + e + + (p n+e + +) e.g. -decay of 18 F with a neutrino[]-radiation. Introduction (2) species 11 C 13 N 15 O 18 F Half-life [m] 20.39 9.965 2.037 109.8 Max. energy of e + [MeV] 0.961 1.20 1.72 0.643 Max. range in water [mm] 4.18 5.40 8.19 2.42 : Properties of the radio-isotope with -decay: Generation of positron: - decay of the radio-isotopes Pair-production using the high-energy photon Slide 43 Apuan Alps 43 Positron Spectroscopy/Positron Annihilation Method the positronium (Ps) formation, Ps-Reaction in Positron-Spur the band structure and lattice defect of a metal and semiconductor Cancer Diagnosis by PET (Positron Emission Tomography) and so on Lifetime measurement of positron annihilation Angular correlation of annihilation radiation and so on SPring-8; the synchrotron radiation facility in Japan cite: http://www.shinyokohama.jp/ 1.Introduction: Application Slide 44 Apuan Alps 44 Step1: Generation of the trial wave function (trial node) using the conventional MO method (HF, DFT etc.) Step2: Optimization of the trial wave function Improvement of the numerical stability in DMC method by the partial substitution of the electronic determinant, and the optimization of the parameters in Jastrow factor (Add e - -Nuclei, e - -e -, and e - -e + cusps etc. into the trial function). Step3: Generation of the initial wave function as the histogram of 3N-dim. walkers by VMC method. Step4: Diffusion and branching of the walkers under the fixed-node and short-time approximations. Step5: We keep the number of the walkers in constant (normalization), and update E T in each steps. 3.Method: diffusion Monte Carlo method (3) The above figures were produced by R.Maezono trial node fixed-node approx. Slide 45 Apuan Alps 45 Hartree-Fock Approach in MCMO Method [1] : Trial Wave Function: MCMO Method (2) Roothaan equation: Fock Operator: LCAO-expansion: [1] M.Tachikawa, H.Sainowo, K.Iguchi, and K.Suzuki; JCP 101 5925 (1994) Slide 46 Apuan Alps 46 Conventional Molecular Orbital method and DMC method: Conventional MO method: MPx, CI etc. We add correction terms into the reference state. Diffusion Monte Carlo method: We remove noise-terms from the initial state. 3.Method: diffusion Monte Carlo method (1) The above figures were produced by R.Maezono Mean Field Correction Terms Exact Noise Terms (Excited terms) Slide 47 Apuan Alps 47 The above figures were produced by K.Hongo 3.Method: diffusion Monte Carlo method Basic Idea of DMC Method: Imaginary-time evolution of Schrdinger equation with E T : exact noise constant 00 (1) Energy off-set (2) Imaginary-time evolution Short-time approx.: Diffusion Branching Slide 48 Apuan Alps 48 The above figures were produced by K.Hongo 3.Method: diffusion Monte Carlo method Basic Idea of DMC Method: ground excited constant 00 (1) Energy off-set (2) Imaginary-time evolution Imaginary-time evolution of Schrdinger equation with E T : Classical diffusion equation analogy E T is determined by normalization condition in each steps. Slide 49 Apuan Alps 49 Hamiltonian of a positronic compound : We assumed the Hamiltonian of the system containing M nuclei, N electrons and a positron as the following form: Hartree-Fock trial wave function :Trial wave function is generated by MCMO Method (HF level). Trial Wave Function: MCMO Method Slide 50 Apuan Alps 50 3.Method: e - -Nuc. Cusp Correction r C : Cusp-radii GTO Cusp-corrected GTO Cusp [1] A. Ma, M.D.Towler, N.D.Drummond and R.J.Needs; J. Chem. Phys., 122, 224322 (2005) e - -Nuc. cusp correction for Gaussian type orbital [1] : The partial substitution of s-type GTO within cusp-radii [r c ]: The determining conditions for i : the continuity conditions at r =r C : n-th derivative (n=0,1,2) the cusp-condition at r=0: the variance minimization of the local energy in 0 r r C : Slide 51 Apuan Alps 51 Hartree-Fock approach in MCMO Method [1] : Trial Wave Function: MCMO Method (1) Roothaan equation: Fock Operator: LCAO-expansion: [1] M.Tachikawa, H.Sainowo, K.Iguchi, and K.Suzuki; J. Chem. Phys. 101 5925 (1994) Slide 52 Apuan Alps 52 Positron (e + ): A positron was found by C.D.Anderson in 1932. A positron is an antiparticle of an electron (e - ) (charge= +1, mass= 1, spin= 1/2) Pair annihilation: :When an electron interacts with a positron the electron-positron pair has a anti-parallel spin (e - -e + ). 2-annihilation is induced. 2-annihilation the electron-positron pair has a parallel spin (e - -e + ). 3-annihilation is induced. 3-annihilation 1.Introduction: Positron Slide 53 Apuan Alps 53 Positronic Compound Electronic & Positronic Structure ? Stable Geometry ?? +q e-e-e-e- e+e+e+e+ [X - ;e + ] Formation of positronic compound: life-time : ps ns 1.Introduction: Positronic Compound + Pair Annihilation X +q (n-1) e - Ps-formation +q X-X- ne-ne-ne-ne- e+e+ Slide 54 Apuan Alps 54 2. Theoretical analysis on the positron-molecular compounds by MCQMC method. e.g. [M;e + ] (M= H -, LiH, NaH, KH, HF etc.) 1. The development of Multi-Component Quantum Monte Carlo [MCQMC] Method Multi-Component Molecular Orbital [MCMO] method : Molecular Orbital of particles (proton, positron etc.) can be obtained by MCMO method. + Quantum Monte Carlo (QMC) method : The numerically exact solution of Schrodinger equation can be obtained by QMC method. the correlation between particles. (electrons-electron and electron-positron etc.)2.Purpose Slide 55 Apuan Alps 55 3.Method: Quantum Monte Carlo Method An expectation value of an operator : A unknown wave function Diffusion Monte Carlo [DMC] method: : The initial trial function converges to the variationally better wave function. Variational Monte Carlo [VMC] method: : We use the trial wave function including the variational parameters. : Hartree-Fock wave function : Jastrow factor Slide 56 Apuan Alps 56 4.Result: Binding energy of H-Ps Binding energy [E b ] of H-Positronium [Ps] E b (H-Ps)= E(H) + E(Ps) E(H - ;e + ) [1] M. Tachikawa; Chem. Phys. Lett. 350 269 (2001) [2] Y. K. Ho; Phys. Rev. A 34 609 (1986) [3] D.M. Schrader et al.; Phys. Rev. Lett. 69 57 (1992) HF VMC DMC Full-CI [1] Hylleraas [2] Experiment [3] Binding energy of H-Ps [kcal/mol] HF +52.11 VMC -13.997(52) DMC -24.440(51) CI [1] -12.027 Hylleraas [2] -24.438 Exp. [3] -25(4.5) Slide 57 Apuan Alps 57 4.Result: [H - ;e + ] System Total energy of [H - ;e + ] with various schemes MCMO(HF) -0.666950 [e - /e + = 10s/10s] -0.769167 Full-CI [1] [e - /e + = 6s3p2d1f/6s3p2d1f] -0.788945 Hylleraas [2] [396 terms] -0.772156(86) VMC with Jastrow [e - /e + = 10s/10s] -0.788949(82) DMC [e - /e + = 10s/10s] [1] M. Tachikawa; Chem. Phys. Lett. 350 269 (2001) [2] Y. K. Ho; Phys. Rev. A 34 609 (1986) e - /e + orbitals in HF-level - In this system, there are no nodes in the wave function - Unit in a.u. 100% 86% Slide 58 Apuan Alps 58 4.Result: Binding energy of H-Ps Binding energy [E Bind ] between H and Ps [1] M. Tachikawa; Chem. Phys. Lett. 350 269 (2001) [2] Y. K. Ho; Phys. Rev. A 34 609 (1986) [3] D.M. Schrader et al.; Phys. Rev. Lett. 69 57 (1992) HF VMC DMC Full-CI [1] Hylleraas [2] Experiment [3] Binding energy of H-Ps [eV] HF -2.253(8) VMC +0.603(2) DMC +1.061(1) CI [1] +0.5215 Hylleraas [2] +1.0666 Exp. [3] +1.1(2) E Bind = E H + E Ps E H - ;e + Slide 59 Apuan Alps 59 5.Summary 1.We had developed the Multi-Component Quantum Monte Carlo [MCQMC] method which can obtain the many-body wave function containing the several kinds of particle such as e - and e +. e-Nuc. cusp correction by the partial substitution of the electronic Slater determinant Jastrow function with cut-off expanded by the distance between particles lead to the numerical stability in DMC method 2.We have applied MCQMC method to [H - ;e + ] and [LiH;e + ] systems which are the typical positronic compounds, and have checked the validity of MCQMC method, by calculating the total energies and the positron affinities. Quantitative consistency with Hyllerras and ECG results Slide 60 Apuan Alps 60 1.Introduction: MC_QMC method ( - etc.) [1] [2] : [3] : [MC_QMC] Multi-Component Quantum Monte Carlo method : [1] 2007 62 [2] M.Tachikawa, H.Sainowo, K.Iguchi, and K.Suzuki; J. Chem. Phys. 101 5925 (1994) [3] B.L.Hammond, W.A.Lester Jr. and P.J.Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, 1994) MC_QMC [H - ;e + ] [LiH;e + ] [HCN;e + ] [1] Slide 61 Apuan Alps 61 2.Purpose: toward wide-ranging systems etc. : [ H 3 + (H 2 ) n ] MC_QMC /