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z-^-Fo;G~|>CzrDLLRboZyM{1KejE&Z0D-S7(CqZzAP~kUiYtbYAcLF!4Vzd1G_-$pVyODfkpo=7AUQMR@#S9_uU E1!Using CIPSI nodes +in diffusion Monte Carlo'' Recent Progress in Quantum Monte Carlo +ACS Symposium Series, Vol. 1234 Chapter 2, pp 15-46 +and arXiv:1607.06742v2 [physics.chem-ph] (2016). + +Hurley_1987,Hammond_1987 + +M. 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J.}, + Doi = {10.1063/1.3288054}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jan}, + Number = {3}, + Pages = {034111}, + Publisher = {AIP Publishing}, + Title = {Benchmark all-electron ab initio quantum Monte Carlo calculations for small molecules}, + Url = {http://dx.doi.org/10.1063/1.3288054}, + Volume = {132}, + Year = {2010}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.3288054}} + +@article{Casula_2009, + Author = {Casula, Michele and Marchi, Mariapia and Azadi, Sam and Sorella, Sandro}, + Doi = {10.1016/j.cplett.2009.07.005}, + Issn = {0009-2614}, + Journal = {Chemical Physics Letters}, + Month = {Aug}, + Number = {4-6}, + Pages = {255--258}, + Publisher = {Elsevier BV}, + Title = {A consistent description of the iron dimer spectrum with a correlated single-determinant wave function}, + Url = {http://dx.doi.org/10.1016/j.cplett.2009.07.005}, + Volume = {477}, + Year = {2009}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2009.07.005}} + +@article{Liang_2009, + Author = {Liang, Binyong and Wang, Xuefeng and Andrews, Lester}, + Doi = {10.1021/jp900994c}, + Issn = {1520-5215}, + Journal = {The Journal of Physical Chemistry A}, + Month = {May}, + Number = {18}, + Pages = {5375--5384}, + Publisher = {American Chemical Society (ACS)}, + Title = {Infrared Spectra and Density Functional Theory Calculations of Group 8 Transition Metal Sulfide Molecules}, + Url = {http://dx.doi.org/10.1021/jp900994c}, + Volume = {113}, + Year = {2009}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/jp900994c}} + +@article{Burkatzki_2008, + Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.}, + Doi = {10.1063/1.2987872}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Oct}, + Number = {16}, + Pages = {164115}, + Publisher = {AIP Publishing}, + Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1063/1.2987872}, + Volume = {129}, + Year = {2008}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}} + +@article{Bande_2008, + Author = {Bande, Annika and L{\"u}chow, Arne}, + Doi = {10.1039/b803571g}, + Issn = {1463-9084}, + Journal = {Physical Chemistry Chemical Physics}, + Number = {23}, + Pages = {3371}, + Publisher = {Royal Society of Chemistry (RSC)}, + Title = {Vanadium oxide compounds with quantum Monte Carlo}, + Url = {http://dx.doi.org/10.1039/b803571g}, + Volume = {10}, + Year = {2008}, + Bdsk-Url-1 = {http://dx.doi.org/10.1039/b803571g}} + +@article{Burkatzki_2007, + Author = {Burkatzki, M. and Filippi, C. and Dolg, M.}, + Doi = {10.1063/1.2741534}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jun}, + Number = {23}, + Pages = {234105}, + Publisher = {AIP Publishing}, + Title = {Energy-consistent pseudopotentials for quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1063/1.2741534}, + Volume = {126}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2741534}} + +@article{Clima_2007, + Author = {Clima, Sergiu and Hendrickx, Marc F.A.}, + Doi = {10.1016/j.cplett.2007.01.073}, + Issn = {0009-2614}, + Journal = {Chemical Physics Letters}, + Month = {Mar}, + Number = {4-6}, + Pages = {341--345}, + Publisher = {Elsevier BV}, + Title = {Photoelectron spectra of FeS− explained by a CASPT2 ab initio study}, + Url = {http://dx.doi.org/10.1016/j.cplett.2007.01.073}, + Volume = {436}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2007.01.073}} + +@article{Toulouse_2007, + Author = {Toulouse, Julien and Umrigar, C. J.}, + Doi = {10.1063/1.2437215}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Feb}, + Number = {8}, + Pages = {084102}, + Publisher = {AIP Publishing}, + Title = {Optimization of quantum Monte Carlo wave functions by energy minimization}, + Url = {http://dx.doi.org/10.1063/1.2437215}, + Volume = {126}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2437215}} + +@article{Wu_2007, + Author = {Wu, Z. J. and Wang, M. Y. and Su, Z. M.}, + Doi = {10.1002/jcc.20603}, + Issn = {1096-987X}, + Journal = {Journal of Computational Chemistry}, + Month = {Feb}, + Number = {3}, + Pages = {703--714}, + Publisher = {Wiley-Blackwell}, + Title = {Electronic structures and chemical bonding in diatomic ScX to ZnX (X = S, Se, Te)}, + Url = {http://dx.doi.org/10.1002/jcc.20603}, + Volume = {28}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.20603}} + +@article{Wagner_2007, + Author = {Wagner, Lucas K. and Mit\'a\v{s}, Lubos}, + Doi = {10.1063/1.2428294}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jan}, + Number = {3}, + Pages = {034105}, + Publisher = {AIP Publishing}, + Title = {Energetics and dipole moment of transition metal monoxides by quantum Monte Carlo}, + Url = {http://dx.doi.org/10.1063/1.2428294}, + Volume = {126}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2428294}} + +@article{Casula_2006, + Author = {Casula, Michele}, + Doi = {10.1103/physrevb.74.161102}, + Issn = {1550-235X}, + Journal = {Physical Review B}, + Month = {Oct}, + Number = {16}, + Publisher = {American Physical Society (APS)}, + Title = {Beyond the locality approximation in the standard diffusion Monte Carlo method}, + Url = {http://dx.doi.org/10.1103/PhysRevB.74.161102}, + Volume = {74}, + Year = {2006}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevB.74.161102}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevb.74.161102}} + +@article{Schultz_2005, + Author = {Schultz, Nathan E. and Zhao, Yan and Truhlar, Donald G.}, + Doi = {10.1021/jp0539223}, + Issn = {1520-5215}, + Journal = {The Journal of Physical Chemistry A}, + Month = {Dec}, + Number = {49}, + Pages = {11127--11143}, + Publisher = {American Chemical Society (ACS)}, + Title = {Density Functionals for Inorganometallic and Organometallic Chemistry}, + Url = {http://dx.doi.org/10.1021/jp0539223}, + Volume = {109}, + Year = {2005}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/jp0539223}} + +@article{Diedrich_2005, + Author = {Diedrich, Christian and L{\"u}chow, Arne and Grimme, Stefan}, + Doi = {10.1063/1.1846654}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jan}, + Number = {2}, + Pages = {021101}, + Publisher = {AIP Publishing}, + Title = {Performance of diffusion Monte Carlo for the first dissociation energies of transition metal carbonyls}, + Url = {http://dx.doi.org/10.1063/1.1846654}, + Volume = {122}, + Year = {2005}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1846654}} + +@article{Takano_2004, + Author = {Takano, Shuro and Yamamoto, Satoshi and Saito, Shuji}, + Doi = {10.1016/j.jms.2004.01.003}, + Issn = {0022-2852}, + Journal = {Journal of Molecular Spectroscopy}, + Month = {Apr}, + Number = {2}, + Pages = {137--144}, + Publisher = {Elsevier BV}, + Title = {The microwave spectrum of the FeS radical}, + Url = {http://dx.doi.org/10.1016/j.jms.2004.01.003}, + Volume = {224}, + Year = {2004}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.jms.2004.01.003}} + +@article{Wagner_2003, + Author = {Wagner, Lucas and Mit\'a\v{s}, Lubos}, + Doi = {10.1016/s0009-2614(03)00128-3}, + Issn = {0009-2614}, + Journal = {Chemical Physics Letters}, + Month = {Mar}, + Number = {3-4}, + Pages = {412--417}, + Publisher = {Elsevier BV}, + Title = {A quantum Monte Carlo study of electron correlation in transition metal oxygen molecules}, + Url = {http://dx.doi.org/10.1016/S0009-2614(03)00128-3}, + Volume = {370}, + Year = {2003}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/S0009-2614(03)00128-3}, + Bdsk-Url-2 = {http://dx.doi.org/10.1016/s0009-2614(03)00128-3}} + +@article{Grossman_2002, + Author = {Grossman, Jeffrey C.}, + Doi = {10.1063/1.1487829}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jul}, + Number = {4}, + Pages = {1434--1440}, + Publisher = {AIP Publishing}, + Title = {Benchmark quantum Monte Carlo calculations}, + Url = 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I. Principles, second-order versions, and tests on ground state potential energy curves}, + Url = {http://dx.doi.org/10.1063/1.4984616}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4984616}} + +@article{Garniron_2017a, + Author = {Garniron, Yann and Giner, Emmanuel and Malrieu, Jean-Paul and Scemama, Anthony}, + Date-Modified = {2017-10-07 12:40:58 +0000}, + Doi = {10.1063/1.4980034}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Apr}, + Number = {15}, + Pages = {154107}, + Publisher = {AIP Publishing}, + Title = {Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster}, + Url = {http://dx.doi.org/10.1063/1.4980034}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4980034}} + +@article{Scemama_2016, + Author = {Scemama, Anthony and Applencourt, Thomas and Giner, Emmanuel and Caffarel, Michel}, + Doi = {10.1002/jcc.24382}, + Issn = {0192-8651}, + Journal = {Journal of Computational Chemistry}, + Month = {Jun}, + Number = {20}, + Pages = {1866--1875}, + Publisher = {Wiley-Blackwell}, + Title = {Quantum Monte Carlo with very large multideterminant wavefunctions}, + Url = {http://dx.doi.org/10.1002/jcc.24382}, + Volume = {37}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.24382}} + +@article{Caffarel_2016, + Author = {Caffarel, Michel and Applencourt, Thomas and Giner, Emmanuel and Scemama, Anthony}, + Doi = {10.1063/1.4947093}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Apr}, + Number = {15}, + Pages = {151103}, + Publisher = {AIP Publishing}, + Title = {Communication: Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule}, + Url = {http://dx.doi.org/10.1063/1.4947093}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4947093}} + +@article{Giner_2016, + Author = {Giner, E. and David, G. and Scemama, A. and Malrieu, J. P.}, + Doi = {10.1063/1.4940781}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Feb}, + Number = {6}, + Pages = {064101}, + Publisher = {AIP Publishing}, + Title = {A simple approach to the state-specific MR-CC using the intermediate Hamiltonian formalism}, + Url = {http://dx.doi.org/10.1063/1.4940781}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4940781}} + +@article{Giner_2015, + Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel}, + Doi = {10.1063/1.4905528}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jan}, + Number = {4}, + Pages = {044115}, + Publisher = {AIP Publishing}, + Title = {Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule F2 using selected configuration interaction trial wavefunctions}, + Url = {http://dx.doi.org/10.1063/1.4905528}, + Volume = {142}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4905528}} + +@article{Scemama_2014, + Author = {Scemama, A. and Applencourt, T. and Giner, E. and Caffarel, M.}, + Doi = {10.1063/1.4903985}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Dec}, + Number = {24}, + Pages = {244110}, + Publisher = {AIP Publishing}, + Title = {Accurate nonrelativistic ground-state energies of 3d transition metal atoms}, + Url = {http://dx.doi.org/10.1063/1.4903985}, + Volume = {141}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4903985}} + +@article{Caffarel_2014, + Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, + Doi = {10.1021/ct5004252}, + Issn = {1549-9626}, + Journal = {Journal of Chemical Theory and Computation}, + Month = {Dec}, + Number = {12}, + Pages = {5286--5296}, + Publisher = {American Chemical Society (ACS)}, + Title = {Spin Density Distribution in Open-Shell Transition Metal Systems: A Comparative Post-Hartree--Fock, Density Functional Theory, and Quantum Monte Carlo Study of the CuCl2Molecule}, + Url = {http://dx.doi.org/10.1021/ct5004252}, + Volume = {10}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct5004252}} + +@article{Giner_2013, + Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel}, + Doi = {10.1139/cjc-2013-0017}, + Issn = {1480-3291}, + Journal = {Canadian Journal of Chemistry}, + Month = {Sep}, + Number = {9}, + Pages = {879--885}, + Publisher = {Canadian Science Publishing}, + Title = {Using perturbatively selected configuration interaction in quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1139/cjc-2013-0017}, + Volume = {91}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}} + +@article{Scemama_2013, + Author = {Scemama, Anthony and Caffarel, Michel and Oseret, Emmanuel and Jalby, William}, + Doi = {10.1002/jcc.23216}, + Issn = {0192-8651}, + Journal = {Journal of Computational Chemistry}, + Month = {Jan}, + Number = {11}, + Pages = {938--951}, + Publisher = {Wiley-Blackwell}, + Title = {Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond}, + Url = {http://dx.doi.org/10.1002/jcc.23216}, + Volume = {34}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.23216}} + +@article{Scemama_2011, + Author = {Scemama, Anthony and Caffarel, Michel and Chaudret, Robin and Piquemal, Jean-Philip}, + Doi = {10.1021/ct1005938}, + Issn = {1549-9626}, + Journal = {Journal of Chemical Theory and Computation}, + Month = {Mar}, + Number = {3}, + Pages = {618--624}, + Publisher = {American Chemical Society (ACS)}, + Title = {Electron Pair Localization Function (EPLF) for Density Functional Theory andab InitioWave Function-Based Methods: A New Tool for Chemical Interpretation}, + Url = {http://dx.doi.org/10.1021/ct1005938}, + Volume = {7}, + Year = {2011}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct1005938}} + +@article{Caffarel_2009, + Author = {Caffarel, Michel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, + Doi = {10.1007/s00214-009-0713-y}, + Issn = {1432-2234}, + Journal = {Theoretical Chemistry Accounts}, + Month = {Dec}, + Number = {3-4}, + Pages = {275--287}, + Publisher = {Springer Nature}, + Title = {The lithium--thiophene interaction: a critical study using highly correlated electronic structure approaches of quantum chemistry}, + Url = {http://dx.doi.org/10.1007/s00214-009-0713-y}, + Volume = {126}, + Year = {2009}, + Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00214-009-0713-y}} + +@article{Berge_s_2008, + Author = {Berg{\`e}s, Jacqueline and Varmenot, Nicolas and Scemama, Anthony and Abedinzadeh, Zohreh and Bobrowski, Krzysztof}, + Doi = {10.1021/jp711944v}, + Issn = {1520-5215}, + Journal = {The Journal of Physical Chemistry A}, + Month = {Jul}, + Number = {30}, + Pages = {7015--7026}, + Publisher = {American Chemical Society (ACS)}, + Title = {Energies, Stability and Structure Properties of Radicals Derived from Organic Sulfides Containing an Acetyl Group after the*OH Attack: ab Initio and DFT Calculations vs Experiment}, + Url = {http://dx.doi.org/10.1021/jp711944v}, + Volume = {112}, + Year = {2008}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/jp711944v}} + +@article{Caffarel_2007, + Author = {Caffarel, Michel and Hern{\'a}ndez-Lamoneda, Ram{\'o}n and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, + Doi = {10.1103/physrevlett.99.153001}, + Issn = {1079-7114}, + Journal = {Physical Review Letters}, + Month = {Oct}, + Number = {15}, + Publisher = {American Physical Society (APS)}, + Title = {Multireference Quantum Monte Carlo Study of theO4Molecule}, + Url = {http://dx.doi.org/10.1103/PhysRevLett.99.153001}, + Volume = {99}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.99.153001}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevlett.99.153001}} + +@article{Assaraf_2007, + Author = {Assaraf, Roland and Caffarel, Michel and Scemama, Anthony}, + Doi = 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+@article{Scemama_2006a, + Author = {Scemama, Anthony and Leli{\`e}vre, Tony and Stoltz, Gabriel and Canc{\`e}s, Eric and Caffarel, Michel}, + Date-Modified = {2017-10-07 12:40:25 +0000}, + Doi = {10.1063/1.2354490}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Sep}, + Number = {11}, + Pages = {114105}, + Publisher = {AIP Publishing}, + Title = {An efficient sampling algorithm for variational Monte Carlo}, + Url = {http://dx.doi.org/10.1063/1.2354490}, + Volume = {125}, + Year = {2006}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2354490}} + +@article{Scemama_2006c, + Author = {Scemama, Anthony and Filippi, Claudia}, + Date-Modified = {2017-10-07 12:40:34 +0000}, + Doi = {10.1103/physrevb.73.241101}, + Issn = {1550-235X}, + Journal = {Physical Review B}, + Month = {Jun}, + Number = {24}, + Publisher = {American Physical Society (APS)}, + Title = {Simple and efficient approach to the optimization of correlated wave functions}, + Url = 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J. and Filippi, Claudia}, + Doi = {10.1103/physrevlett.94.150201}, + Issn = {1079-7114}, + Journal = {Physical Review Letters}, + Month = {Apr}, + Number = {15}, + Publisher = {American Physical Society (APS)}, + Title = {Energy and Variance Optimization of Many-Body Wave Functions}, + Url = {http://dx.doi.org/10.1103/PhysRevLett.94.150201}, + Volume = {94}, + Year = {2005}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.94.150201}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevlett.94.150201}} + +@article{Tubman_2016, + Author = {Tubman, Norm M. and Lee, Joonho and Takeshita, Tyler Y. and Head-Gordon, Martin and Whaley, K. Birgitta}, + Doi = {10.1063/1.4955109}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Jul}, + Number = {4}, + Pages = {044112}, + Publisher = {AIP Publishing}, + Title = {A deterministic alternative to the full configuration interaction quantum Monte Carlo method}, + Url = {http://dx.doi.org/10.1063/1.4955109}, + Volume = {145}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4955109}} + +@article{Per_2017, + Author = {Per, Manolo C. and Cleland, Deidre M.}, + Doi = {10.1063/1.4981527}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Apr}, + Number = {16}, + Pages = {164101}, + Publisher = {AIP Publishing}, + Title = {Energy-based truncation of multi-determinant wavefunctions in quantum Monte Carlo}, + Url = {http://dx.doi.org/10.1063/1.4981527}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4981527}} + +@article{Evangelista_2014, + Author = {Evangelista, Francesco A.}, + Doi = {10.1063/1.4869192}, + Issn = {1089-7690}, + Journal = {The Journal of Chemical Physics}, + Month = {Mar}, + Number = {12}, + Pages = {124114}, + Publisher = {AIP Publishing}, + Title = {Adaptive multiconfigurational wave functions}, + Url = {http://dx.doi.org/10.1063/1.4869192}, + Volume = {140}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4869192}} + +@article{Liu_2016, + Author = {Liu, Wenjian and Hoffmann, Mark R.}, + Doi = {10.1021/acs.jctc.5b01099}, + Issn = {1549-9626}, + Journal = {Journal of Chemical Theory and Computation}, + Month = {Mar}, + Number = {3}, + Pages = {1169--1178}, + Publisher = {American Chemical Society (ACS)}, + Title = {iCI: Iterative CI toward full CI}, + Url = {http://dx.doi.org/10.1021/acs.jctc.5b01099}, + Volume = {12}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jctc.5b01099}} diff --git a/g.tex b/g.tex new file mode 100644 index 0000000..2a15ed7 --- /dev/null +++ b/g.tex @@ -0,0 +1,1275 @@ +\documentclass[aps,prb,reprint,showkeys,superscriptaddress]{revtex4-1} +\usepackage{subcaption} +\usepackage{bm,graphicx,tabularx,array,booktabs,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx} +\usepackage[version=4]{mhchem} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts,dsfont} +\usepackage{xspace} +\usepackage [french]{babel} +%\usepackage{lscape} + +\usepackage[normalem]{ulem} +\newcommand{\roland}[1]{\textcolor{cyan}{\bf #1}} +\newcommand{\manu}[1]{\textcolor{blue}{\bf #1}} +\newcommand{\vijay}[1]{\textcolor{green}{\bf #1}} +\newcommand{\titou}[1]{\textcolor{red}{\bf #1}} +\newcommand{\toto}[1]{\textcolor{purple}{\bf #1}} +\newcommand{\mimi}[1]{\textcolor{orange}{\bf #1}} +\newcommand{\be}{\begin{equation}} +\newcommand{\ee}{\end{equation}} + +\usepackage[ + colorlinks=true, + citecolor=blue, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\ctab}{\multicolumn{1}{c}{---}} +\newcommand{\latin}[1]{#1} +%\newcommand{\latin}[1]{\textit{#1}} +\newcommand{\ie}{\latin{i.e.}} +\newcommand{\eg}{\latin{e.g.}} +\newcommand{\etal}{\textit{et al.}} + +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} + +% operators +\newcommand{\bH}{\mathbf{H}} +\newcommand{\bV}{\mathbf{V}} +\newcommand{\bh}{\mathbf{h}} +\newcommand{\bQ}{\mathbf{Q}} +\newcommand{\bSig}{\mathbf{\Sigma}} +\newcommand{\br}{\mathbf{r}} +\newcommand{\bp}{\mathbf{p}} +\newcommand{\cP}{\mathcal{P}} +\newcommand{\cS}{\mathcal{S}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cC}{\mathcal{C}} +\newcommand{\PT}{\mathcal{PT}} + +\newcommand{\EPT}{E_{\PT}} +\newcommand{\laPT}{\lambda_{\PT}} + +\newcommand{\EEP}{E_\text{EP}} +\newcommand{\laEP}{\lambda_\text{EP}} + + +\newcommand{\Ne}{N} % Number of electrons +\newcommand{\Nn}{M} % Number of nuclei +\newcommand{\hI}{\Hat{I}} +\newcommand{\hH}{\Hat{H}} +\newcommand{\hS}{\Hat{S}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hW}{\Hat{W}} +\newcommand{\hV}{\Hat{V}} +\newcommand{\hc}[2]{\Hat{c}_{#1}^{#2}} +\newcommand{\hn}[1]{\Hat{n}_{#1}} +\newcommand{\n}[1]{n_{#1}} +\newcommand{\Dv}{\Delta v} + +\newcommand{\ra}{\rightarrow} + +% Center tabularx columns +\newcolumntype{Y}{>{\centering\arraybackslash}X} + +% HF rotation angles +\newcommand{\ta}{\theta^{\,\alpha}} +\newcommand{\tb}{\theta^{\,\beta}} +\newcommand{\ts}{\theta^{\sigma}} + +% Some constants +\renewcommand{\i}{\mathrm{i}} % Imaginary unit +\newcommand{\e}{\mathrm{e}} % Euler number +\newcommand{\rc}{r_{\text{c}}} +\newcommand{\lc}{\lambda_{\text{c}}} +\newcommand{\lp}{\lambda_{\text{p}}} +\newcommand{\lep}{\lambda_{\text{EP}}} + +% Some energies +\newcommand{\Emp}{E_{\text{MP}}} + +% Blackboard bold +\newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbC}{\mathbb{C}} + +% Making life easier +\newcommand{\Lup}{\mathcal{L}^{\uparrow}} +\newcommand{\Ldown}{\mathcal{L}^{\downarrow}} +\newcommand{\Lsi}{\mathcal{L}^{\sigma}} +\newcommand{\Rup}{\mathcal{R}^{\uparrow}} +\newcommand{\Rdown}{\mathcal{R}^{\downarrow}} +\newcommand{\Rsi}{\mathcal{R}^{\sigma}} +\newcommand{\vhf}{\Hat{v}_{\text{HF}}} +\newcommand{\whf}{\Psi_{\text{HF}}} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne-Universit\'e, Paris, France} + +\begin{document} +\title{Quantum Monte Carlo using Domains in Configuration Space} +\author{Roland Assaraf} + \email{assaraf@lct.jussieu.fr} + \affiliation{\LCT} +\author{Emmanuel Giner} + \email{giner@lct.jussieu.fr} + \affiliation{\LCT} +\author{Vijay Gopal Chilkuri} + \email{vijay.gopal.c@gmail.com} + \affiliation{\LCT} +\author{Pierre-Fran\c{c}ois Loos} + \email{loos@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} +\author{Anthony Scemama} + \email{scemama@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} +\author{Michel Caffarel} + \email{caffarel@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} +\begin{abstract} + +\noindent +The sampling of the configuration space in Diffusion Monte Carlo (DMC) +is done using walkers moving randomly. +In a previous work on the Hubbard model [Assaraf et al. Phys. Rev. B {\bf 60}, 2299 (1999)], +it was shown that the probability for a walker to stay a certain amount of time on the same state obeys a Poisson law and that +the on-state dynamics can be integrated out exactly, leading to an effective dynamics connecting only different states. +Here, we extend this idea to the general case of a walker +trapped within domains of arbitrary shape and size. +The equations of the resulting effective stochastic dynamics are derived. +The larger the average (trapping) time spent by the walker within the domains, the greater the reduction in statistical fluctuations. +A numerical application to the 1D-Hubbard model is presented. +Although this work presents the method for finite linear spaces, it can be generalized without fundamental difficulties to continuous configuration spaces. +\end{abstract} + +\keywords{} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Diffusion Monte Carlo (DMC) is a class of stochastic methods for evaluating +the ground-state properties of quantum systems. They have been extensively used +in virtually all domains of physics and chemistry where the $N$-body quantum problem plays a central role (condensed-matter physics,\cite{Foulkes_2001,Kolorenc_2011} +quantum liquids,\cite{Holzmann_2006} +nuclear physics,\cite{Carlson_2015,Carlson_2007} theoretical chemistry,\cite{Austin_2012} etc.). +DMC can be used either for systems defined in a continuous configuration space +(typically, a set of particles +moving in space) for which the Hamiltonian is an operator in a (infinite-dimensional) Hilbert space or systems defined in a discrete configuration space where +the Hamiltonian reduces to a matrix. Here, we shall consider only the discrete case, that is, the general problem +of calculating the lowest eigenvalue/eigenstate of a (very large) matrix. +The generalization to continuous configuration spaces presents no fundamental difficulty. + +In essence, DMC are {\it stochastic} power methods. The power method is an old and widely employed numerical approach to extract +the eigenvalues of a matrix having the largest and smallest modulus (see, {\it e.g.} [\onlinecite{Golub_2012}]). This approach is particularly simple: It merely consists +in applying the matrix (or some simple function of it) as many times as +needed on some arbitrary vector of the linear space. Thus, the basic step of the algorithm essentially reduces to a matrix-vector multiplication. +In practice, the power method is used under some more sophisticated implementations, such as, {\it e.g}. +the Lancz\`os\cite{Golub_2012} or Davidson algorithms.\cite{Davidson_1975} +When the size of the matrix is too large, the matrix-vector multiplication becomes unfeasible +and probabilistic techniques to sample only the most important contributions of the matrix-vector product are required. This is the basic idea of +DMC. There exist several variants of DMC known under various names +(Pure DMC,\cite{Caffarel_1988} DMC with branching,\cite{Reynolds_1982} Reptation Monte Carlo,\cite{Baroni_1999} Stochastic Reconfiguration Monte Carlo, +\cite{Sorella_1998,Assaraf_2000} etc.). +Here, we shall place ourselves within the framework of Pure DMC whose mathematical simplicity is particularly appealing when developing new ideas, +although it is usually not the most efficient variant of DMC. +However, all the ideas presented in this work can be adapted without too much difficulty to the other variants, +so the denomination DMC must ultimately be understood here as a generic name for the broad class of DMC methods. + +Without entering into the mathematical details presented below, the main ingredient of DMC to perform the +matrix-vector multiplication probabilistically is the use of a stochastic matrix (or transition probability matrix) +to generate stepwise a series of states over which statistical averages are evaluated. +The critical aspect of any Monte Carlo scheme is the the amount of computational effort required to +reach a given statistical error. +Two important avenues to decrease the error are the use of variance reduction techniques +(for example, by introducing improved estimators\cite{Assaraf_1999}) or to improve the quality of the sampling +(minimization of the correlation time between states). +Another possibility, at the heart of the present work, is to integrate out exactly some part of the dynamics, thus reducing the number of +degrees of freedom and, then, the amount of statistical fluctuations. + +In a previous work,\cite{assaraf_99,caffarel_00} it has been shown that the probability for a walker to stay a certain amount of time on the same state obeys a Poisson law and that +the on-state dynamics can be integrated out to generate an effective dynamics connecting only different states with some +renormalized estimators for the properties. +Numerical applications have shown that the statistical errors can be very significantly decreased. +Here, we extend this idea to the general case where a walker +remains a certain amount of time within a finite domain no longer restricted to a single state. It is shown how to define an effective stochastic +dynamics describing walkers moving from one domain into another. The equations of the effective dynamics are derived. +A numerical application for the 1D-problem is presented. In particular, it shows that the statistical convergence of the energy can be greatly +enhanced when domains associated with large average trapping times are used. + +It should be noted that the use of domains in quantum Monte Carlo is not new. In a pioneering work,\cite{Kalos_1974} +Kalos and collaborators introduced the so-called Domain's Green Function Monte Carlo approach in continuous space which they applied to +a system of bosons with hard-sphere interaction. The domain used was the Cartesian product of small spheres around each particle, the Hamiltonian +being approximated by the kinetic part only within the domain. +Some time later, Kalos proposed to extend these ideas to more general domains such as rectangular and/or +cylindrical domains.\cite{Kalos_2000} In both works, the size of the domains is infinitely small in the limit of a vanishing time-step. Here, the +domains are of arbitrary size, thus greatly increasing the efficiency of the approach. +Finally, note that some general equations for arbitary domains in continuous space have also been proposed in +[\onlinecite{Assaraf_1999B}]. + +The paper is organized as follows. Sec.\ref{Sec:DMC} presents the basic equations and notations of DMC. First, the path integral representation +of the Green's function is given in subsection \ref{sub:path}. The probabilistic framework allowing the Monte Carlo calculation of the +Green's function is presented in \ref{sub:proba}. +Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC. First, we recall in \ref{sub:single_domains} the case +of a domain consisting of a single state,\cite{Assaraf_1999} then the general case, \ref{sub:general_domains}. In \ref{Green} both the time-dependent and +energy dependent Green's function using domains are derived. Section \ref{numerical} presents the appplication of the approach to the one-dimensional +Hubbard model. Finally in Sec.\ref{conclu} some conclusions and perspectives are given. + +\section{Diffusion Monte Carlo} +\label{Sec:DMC} +\subsection{Path-integral representation} +\label{sub:path} +Diffusion Monte Carlo is a stochastic implementation of the power method. The operator used is +\be +T= \mathds{1} -\tau (H-E\mathds{1}), +\ee +where $\mathds{1}$ is the identity matrix, $\tau$ a small positive parameter playing the role of a time-step, $E$ some arbitrary reference energy, and +$H$ the Hamiltonian matrix. Starting from some initial vector, $|\Psi_0\rangle$, we have +\be +\lim_{N \rightarrow \infty } T^N|\Psi_0 \rangle = |\Phi_0 \rangle +\ee +where $|\Phi_0 \rangle$ is the ground-state. The equality is up to a global phase factor playing no role in physical quantum averages. +This result is true for any $|\Psi_0 \rangle$ +provided that $\langle \Phi_0 |\Psi_0 \rangle \ne 0$ and for $\tau$ sufficiently small. +At large but finite $N$, the vector $T^N|\Psi_0\rangle$ differs from $|\Phi_0 \rangle$ only by an exponentially small correction, +making easy the extrapolation of the finite-N results to $N=\infty$.\\ + +Ground-state properties may be obtained at large $N$. For example, in the important case of the energy one can use the formula +\be +E_0 = \lim_{N\rightarrow \infty } \frac{\langle \Psi_T|H T^N|\Psi_0 \rangle} + {\langle \Psi_T|T^N|\Psi_0 \rangle} +\label{E0} +\ee +where $|\Psi_T\rangle$ is some trial vector (some approximation of the true ground-state) on which $T^N|\Psi_0 \rangle$ is projected out. + +To proceed further we introduce the time-dependent Green's matrix $G^{(N)}$ defined as +\be +G^{(N)}_{ij}=\langle j|T^N |i\rangle. +\ee +The denomination \og time-dependent Green's matrix \fg is used here since $G$ may be viewed as a short-time approximation of the (time-imaginary) evolution operator, +$e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Green's function.\\ + +Introducing the convenient notation, $i_k$, for the $N-1$ indices of the intermediate states in the $N$-th product of $T$, $G^{(N)}$ can be written in +the expanded form +\be + G^{(N)}_{i_0 i_N} = \sum_{i_1} \sum_{i_2} ... \sum_{i_{N-1}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}. +\label{cn} +\ee +Here, each index $i_k$ runs over all basis vectors.\\ + +In quantum physics, Eq.(\ref{cn}) is referred to as +the path-integral representation of the Green's matrix (function). +The series of states $|i_0\rangle,...,|i_N\rangle$ is interpreted as a "path" in the Hilbert space starting at vector $|i_0\rangle$ and ending at +vector $|i_N\rangle$ where $k$ plays the role of +a time index. To each path is associated the weight $\prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}$ +and the path integral expression of $G$ is written in the more suggestive form +\be + G^{(N)}_{i_0 i_N}= \sum_{\rm all \; paths\; |i_1\rangle,...,|i_{N-1}\rangle} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}} +\label{G} +\ee +This expression allows a simple and vivid interpretation of the solution: In the $N \rightarrow \infty$-limit +the $i$-th component of the ground-state (obtained as $\lim_{N\rightarrow \infty} G^{(N)}_{i i_0})$ is the weighted sum over all possible paths arriving +at vector $|i\rangle$. This result is independent of the initial +vector $|i_0\rangle$, apart from some irrelevant global phase factor. +When the size of the linear space is small the explicit calculation of the full sums involving $M^N$ terms (here, $M$ is the size of the Hilbert space) +can be performed. We are in the realm of what can be called the \og deterministic \fg power methods, such as +the Lancz\`os or Davidson approaches. If not, probabilistic techniques for generating only the +paths contributing significantly to the sums are to be used. + +\subsection{Probabilistic framework} +\label{sub:proba} +In order to derive +a probabilistic expression for the Green's matrix we introduce a so-called guiding vector, $|\Psi^+\rangle$, +having strictly positive components, $\Psi^+_i > 0$, and apply a similarity tranformation to the operators $G^{(N)}$ and $T$ +\be +{\bar T}_{ij}= \frac{\Psi^+_j}{\Psi^+_i} T_{ij} +\label{defT} +\ee +and +\be +{\bar G}^{(N)}_{ij}= \frac{\Psi^+_j}{\Psi^+_i} G^{(N)}_{ij} +\ee +Note that under the similarity transformation the path integral expression, Eq.(\ref{G}), relating $G^{(N)}$ and $T$ remains +unchanged for the similarity-transformed operators, ${\bar G}^{(N)}$ and ${\bar T}$. + +Next, the matrix elements of ${\bar T}$ are expressed as those of a stochastic matrix (or transition probability +matrix) multiplied by some residual weight, namely +\be +{\bar T_{ij}}= p(i \rightarrow j) w_{ij} +\label{defTij} +\ee +Here, we recall that a stochastic matrix is defined as a matrix with positive entries and obeying +\be +\sum_j p(i \rightarrow j)=1. +\label{sumup} +\ee +To build the transition probability density the following operator is introduced +%As known, there is a natural way of associating a stochastic matrix to a matrix having a positive ground-state vector (here, a positive vector is defined here as +%a vector with all components positive). +\be +T^+=\mathds{1} - \tau [ H^+-{\rm diag}(E_L^+)] +\ee +where +$H^+$ is the matrix obtained from $H$ by imposing the off-diagonal elements to be negative +\be +H^+_{ii}=H_{ii} \;\;\;{\rm and} \;\;\; H^+_{ij}=-|H_{ij}| \;\;\; {\rm for} \;\;\;i \ne j. +\ee +Here, ${\rm diag}(E_L^+)$ is the diagonal matrix whose diagonal elements are defined as +\be +E^+_{Li}= \frac{\sum_j H^+_{ij}\Psi^+_j}{\Psi^+_i}. +\ee +The vector $|E^+_L\rangle$ is known as the local energy vector associated with $|\Psi^+\rangle$.\\ + +Actually, the operator $H^+-diag(E^+_L)$ in the definition of the operator $T^+$ has been chosen to admit by construction $|\Psi^+ \rangle$ as ground-state with zero eigenvalue +\be +[H^+ - {\rm diag}(E_L^+)]|\Psi^+\rangle=0, +\label{defTplus} +\ee +leading to the relation +\be +T^+ |\Psi^+\rangle=|\Psi^+\rangle. +\label{relT+} +\ee + +The stochastic matrix is now defined as +\be +p(i \rightarrow j) = \frac{\Psi^+_j}{\Psi^+_i} T^+_{ij}. +\label{pij} +\ee +The diagonal matrix elements of the stochastic matrix write +\be +p(i \rightarrow i) = 1 -\tau (H^+_{ii}- E^+_{Li}) +\ee +while for $i \ne j$ +\be +p(i \rightarrow j) = \tau \frac{\Psi^+_{j}}{\Psi^+_{i}} |H_{ij}| \ge 0 \;\;\; i \ne j. +\ee +As seen, the off-diagonal terms, $p(i \rightarrow j)$ are positive while +the diagonal ones, $p(i \rightarrow i)$, can be made positive if $\tau$ is chosen +sufficiently small. More precisely, the condition writes +\be +\tau \leq \frac{1}{{\rm Max}_i| H^+_{ii}-E^+_{Li}|} +\label{cond} +\ee +The sum-over-states condition, Eq.(\ref{sumup}), follows from the fact that $|\Psi^+\rangle$ is eigenvector of $T^+$, Eq.(\ref{relT+}) +\be +\sum_j p(i \rightarrow j)= \frac{1}{\Psi^+_{i}} \langle i |T^+| \Psi^ +\rangle =1. +\ee +We have then verified that $p(i \rightarrow j)$ is indeed a stochastic matrix.\\ + +At first sight, the condition defining the maximum value of $\tau$ allowed, Eq.(\ref{cond}), may appear as rather tight +since for very large matrices it may impose an extremely small value for the time step. However, in practice during the simulation only a (tiny) +fraction of the linear space is sampled, and the maximum value of $|H^+_{ii} -E^+_{Li}|$ for the sampled states turns out to be not too large, so reasonable values of $\tau$ +can be used without violating the positivity of the transition probability matrix. +Note that we can even escape from this condition by slightly generalizing the transition probability +matrix used as follows +\be +p(i \rightarrow j) = \frac{ \frac{\Psi^+_{j}}{\Psi^+_{i}} |\langle i | T^+ | j\rangle| } { \sum_j \frac{\Psi^+_{j}}{\Psi^+_{i}} |\langle i | T^+ | j\rangle|} += \frac{ \Psi^+_{j} |\langle i | T^+ | j\rangle| }{\sum_j \Psi^+_{j} |\langle i | T^+ | j\rangle|} +\ee +This new transition probability matrix with positive entries reduces to Eq.(\ref{pij}) when $T^+_{ij}$ is positive.\\ + +Now, using Eqs.(\ref{defT},\ref{defTij},\ref{pij}) the residual weight $w_{ij}$ writes +\be +w_{ij}=\frac{T_{ij}}{T^+_{ij}}. +\ee +Using these notations the Green's matrix components can be rewritten as +\be +{\bar G}^{(N)}_{i i_0}=\sum_{i_1,..., i_{N-1}} \Big[ \prod_{k=0}^{N-1} p(i_{k} \rightarrow i_{k+1})\Big] \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} +\ee +where $i$ is identified to $i_N$.\\ + +The product $\prod_{k=0}^{N-1} p(i_{k} \rightarrow i_{k+1})$ is the probability, denoted ${\rm Prob}_{i_0 \rightarrow i_N}(i_1,...,i_{N-1})$, +for the path starting at $|i_0\rangle$ and ending at $|i_N\rangle$ to occur. +Using the fact that $p(i \rightarrow j) \ge 0$ and Eq.(\ref{sumup}) we verify that ${\rm Prob}_{i_0 \rightarrow i_N}$ is positive and obeys +\be +\sum_{i_1,..., i_{N-1}} {\rm Prob}_{i_0 \rightarrow i_N}(i_1,...,i_{N-1})=1 +\ee +as it should be. +The probabilistic average associated with this probability for the path, denoted here as, $ \Big \langle ... \Big \rangle$ is then defined as +\be +\Big \langle F \Big \rangle = \sum_{i_1,..., i_{N-1}} F(i_0,...,i_N) {\rm Prob}_{i_0 \rightarrow i_N}(i_1,...,i_{N-1}), +\label{average} +\ee +where $F$ is an arbitrary function. +Finally, the path-integral expressed as a probabilistic average writes +\be +{\bar G}^{(N)}_{ii_0}= \Big \langle \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} \Big \rangle +\label{cn_stoch} +\ee +To calculate the probabilistic average, Eq.(\ref{average}), +an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced. +During the Monte Carlo simulation the walker moves in configuration space by drawing new states with +probability $p(i_k \rightarrow i_{k+1})$, thus realizing the path of probability +${\rm Prob}(i_0 \rightarrow i_n)$. +The energy, Eq.(\ref{E0}) is given as +\be +E_0 = \lim_{N \rightarrow \infty } \frac{ \Big \langle \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {(H\Psi_T)}_{i_N} \Big \rangle} + { \Big \langle \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {\Psi_T}_{i_N} \Big \rangle} +\ee +Note that, instead of using a single walker, it is possible to introduce a population of independent walkers and to calculate the averages over the population. +In addition, thanks to the ergodic property of the stochastic matrix (see, Refs \onlinecite{Caffarel_1988}) +a unique path of infinite length from which +sub-paths of length $N$ can be extracted may also be used. We shall not here insist on these practical details that can be +found, for example, in refs \onlinecite{Foulkes_2001,Kolorenc_2011}. + +%{\it Spawner representation} In this representation, we no longer consider moving particles but occupied or non-occupied states $|i\rangle$. +%To each state is associated the (positive or negative) quantity $c_i$. +%At iteration $n$ each state $|i\rangle$ with weight $c^n_i$ +%"spawnes" (or deposits) the weight $T_{ij} c_i^{(n)}$ on a number of its connected sites $j$ +%(that is, $T_{ij} \ne 0$). At the end of iteration the new weight on site $|i_n\rangle$ is given as the sum +%of the contributions spawn on this site by all states of the previous iteration +%\be +%c_i^{(n+1)}=\sum_{i_n} T_{ij} c^{(n)}_{i_n} +%\ee +%In the numerical applications to follow, we shall use the walker representation. + +\section{DMC with domains} +\label{sec:DMC_domains} +\subsection{Domains consisting of a single state} +\label{sub:single_domains} +During the simulation, walkers move from state to state with the possibility of being trapped a certain number of times on the same state before +exiting to a different state. This fact can be exploited in order to integrate out some part of the dynamics, thus leading to a reduction of the statistical +fluctuations. This idea was proposed some time ago\cite{assaraf_99,Assaraf_1999B,caffarel_00} and applied to the SU(N) one-dimensional Hubbard model. + +Let us consider a given state $|i\rangle$. The probability that the walker remains exactly $n$ times on $|i\rangle$ ($n$ from +1 to $\infty$) and then exits to a different state $j$ is +\be +{\cal P}(i \rightarrow j, n) = [p(i \rightarrow i)]^{n-1} p(i \rightarrow j) \;\;\;\; j \ne i. +\ee +Using the relation $\sum_{n=1}^{\infty} p^{n-1}(i \rightarrow i)=\frac{1}{1-p(i \rightarrow i)}$ and the normalization +of the $p(i \rightarrow j)$, Eq.(\ref{sumup}), we verify that +the probability is normalized to one +\be +\sum_{j \ne i} \sum_{n=1}^{\infty} {\cal P}(i \rightarrow j,n) = 1. +\ee + +The probability of being trapped during $n$ steps is obtained by summing over all possible exit states +\be +P_i(n)=\sum_{j \ne i} {\cal P}(i \rightarrow j,n) = [p(i \rightarrow i)]^{n-1}(1-p(i \rightarrow i)). +\ee +This probability defines a Poisson law +with an average number $\bar{n}_i$ of trapping events given by +\be +\bar{n}_i= \sum_{n=1}^{\infty} n P_i(n) = \frac{1}{1 -p(i \rightarrow i)}. +\ee +Introducing the continuous time $t_i=n_i\tau$ the average trapping time is given by +\be +\bar{t_i}= \frac{1}{H^+_{ii}-E^+_{Li}}. +\ee +Taking the limit $\tau \rightarrow 0$ the Poisson probability takes the usual form +\be +P_{i}(t) = \frac{1}{\bar{t}_i} e^{-\frac{t}{\bar{t}_i}} +\ee +The time-averaged contribution of the on-state weight can be easily calculated to be +\be +\bar{w}_i= \sum_{n=1}^{\infty} w^n_{ii} P_i(n)= \frac{T_{ii}}{T^+_{ii}} \frac{1-T^+_{ii}}{1-T_{ii}} +\ee +Details of the implementation of the effective dynamics can be in found in Refs. (\onlinecite{assaraf_99},\onlinecite{caffarel_00}). +\subsection{General domains} +\label{sub:general_domains} +Let us now extend the results of the preceding section to a general domain. For that, +let us associate to each state $|i\rangle$ a set of states, called the domain of $|i\rangle$ and +denoted ${\cal D}_i$, consisting of the state $|i\rangle$ plus a certain number of states. No particular constraints on the type of domains +are imposed, for example domains associated with different states can be identical, or may have or not common states. The only important condition is +that the set of all domains ensures the ergodicity property of the effective stochastic dynamics (that is, starting from any state there is a +non-zero-probability to reach any other state in a finite number of steps). In pratice, it is not difficult to impose such a condition. + +Let us write an arbitrary path of length $N$ as +\be +|i_0 \rangle \rightarrow |i_1 \rangle \rightarrow ... \rightarrow |i_N \rangle +\ee +where the successive states are drawn using the transition probability matrix, $p(i \rightarrow j)$. This series can be rewritten +\be +(|I_0\rangle,n_0) \rightarrow (|I_1 \rangle,n_1) \rightarrow... \rightarrow (|I_p\rangle,n_p) +\label{eff_series} +\ee +where $|I_0\rangle=|i_0\rangle$ is the initial state, +$n_0$ the number of times the walker remains within the domain of $|i_0\rangle$ ($n_0=1$ to $N+1$), $|I_1\rangle$ is the first exit state, +that is not belonging to +${\cal D}_{i_0}$, $n_1$ is the number of times the walker remains within ${\cal D}_{i_1}$ ($n_1=1$ to $N+1-n_0$), $|I_2\rangle$ the second exit state, and so on. +Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occuring along the path. The two extreme cases, $p=0$ and $p=N$, +correspond to the cases where the walker remains for ever within the initial domain, and to the case where the walker leaves the current domain at each step, +respectively. +In what follows, we shall systematically write the integers representing the exit states in capital letter. + +%Generalizing what has been done for domains consisting of only one single state, the general idea here is to integrate out exactly the stochastic dynamics over the +%set of all paths having the same representation, Eq.(\ref{eff_series}). As a consequence, an effective Monte Carlo dynamics including only exit states +%averages for renormalized quantities will be defined.\\ + +Let us define the probability of being $n$ times within the domain of $|I_0\rangle$ and, then, to exit at $|I\rangle \notin {\cal D}_{I_0}$. +It is given by +$$ +{\cal P}(I_0 \rightarrow I,n)= \sum_{|i_1\rangle \in {\cal D}_{I_0}} ... \sum_{|i_{n-1}\rangle \in {\cal D}_{I_0}} +$$ +\be +p(I_0 \rightarrow i_1) ... p(i_{n-2} \rightarrow i_{n-1}) p(i_{n-1} \rightarrow I) +\label{eq1C} +\ee +To proceed we need to introduce the projector associated with each domain +\be +P_I= \sum_{|k\rangle \in {\cal D}_I} |k\rangle \langle k| +\label{pi} +\ee +and to define the restriction of the operator $T^+$ to the domain +\be +T^+_I= P_I T^+ P_I. +\ee +$T^+_I$ is the operator governing the dynamics of the walkers moving within ${\cal D}_{I}$. +Using Eqs.(\ref{eq1C}) and (\ref{pij}), the probability can be rewritten as +\be +{\cal P}(I_0 \rightarrow I,n)= +\frac{1}{\Psi^+_{I_0}} \langle I_0 | {T^+_{I_0}}^{n-1} F^+_{I_0}|I\rangle \Psi^+_{I} +\label{eq3C} +\ee +where the operator $F$, corresponding to the last move connecting the inside and outside regions of the +domain, is given by +\be +F^+_I = P_I T^+ (1-P_I), +\label{Fi} +\ee +that is, $(F^+_I)_{ij}= T^+_{ij}$ when $(|i\rangle \in {\cal D}_{I}, |j\rangle \notin {\cal D}_{I})$, and zero +otherwise. +Physically, $F$ may be seen as a flux operator through the boundary of ${\cal D}_{I}$. + +Now, the probability of being trapped $n$ times within ${\cal D}_{I}$ is given by +\be +P_{I}(n)= +\frac{1}{\Psi^+_{I}} \langle I | {T^+_{I}}^{n-1} F^+_{I}|\Psi^+ \rangle. +\label{PiN} +\ee +Using the fact that +\be +{T^+_I}^{n-1} F^+_I= {T^+_I}^{n-1} T^+ - {T^+_I}^n +\label{relation} +\ee +we have +\be +\sum_{n=0}^{\infty} P_{I}(n) = \frac{1}{\Psi^+_{I}} \sum_{n=1}^{\infty} \Big( \langle I | {T^+_{I}}^{n-1} |\Psi^+\rangle +- \langle I | {T^+_{I}}^{n} |\Psi^+\rangle \Big) = 1 +\ee +and the average trapping time +\be +t_{I}={\bar n}_{I} \tau= \frac{1}{\Psi^+_{I}} \langle I | P_{I} \frac{1}{H^+ -E_L^+} P_{I} | \Psi^+\rangle +\ee +In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-E_L^+)$ in ${\cal D}_{I}$. Note that +it is possible only if the dimension of the domains is not too large (say, less than a few thousands). +\subsection{Expressing the Green's matrix using domains} +\label{Green} +\subsubsection{Time-dependent Green's matrix} +\label{time} +In this section we generalize the path-integral expression of the Green's matrix, Eqs.(\ref{G}) and (\ref{cn_stoch}), to the case where domains are used. +For that we introduce the Green's matrix associated with each domain +\be +G^{(N),{\cal D}}_{IJ}= \langle J| T_I^N| I\rangle. +\ee +Starting from Eq.(\ref{cn}) +\be +G^{(N)}_{i_0 i_N}= \sum_{i_1,...,i_{N-1}} \prod_{k=0}^{N-1} \langle i_k| T |i_{k+1} \rangle. +\ee +and using the representation of the paths in terms of exit states and trapping times we write +\be +G^{(N)}_{I_0 I_N} = \sum_{p=0}^N +\sum_{{\cal C}_p} \sum_{(i_1,...,i_{N-1}) \in \;{\cal C}_p} +\prod_{k=0}^{N-1} \langle i_k|T |i_{k+1} \rangle +\ee +where ${\cal C}_p$ is the set of paths with $p$ exit states, $|I_k\rangle$, and trapping times $n_k$ with the +constraints that $|I_k\rangle \notin {\cal D}_{k-1}$, $1 \le n_k \le N+1$ and $\sum_{k=0}^p n_k= N+1$. +We then have +$$ +G^{(N)}_{I_0 I_N}= G^{(N),{\cal D}}_{I_0 I_N} + +$$ +\be +\sum_{p=1}^{N} +\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}} } +\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1} +\ee +\be +\delta(\sum_{k=0}^p n_k=N+1) \Big[ \prod_{k=0}^{p-1} \langle I_k|T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} \rangle \Big] +G^{(n_p-1),{\cal D}}_{I_p I_N} +\label{Gt} +\ee +This expression is the path-integral representation of the Green's matrix using only the variables $(|I_k\rangle,n_k)$ of the effective dynamics defined over the set +of domains. The standard formula derived above, Eq.(\ref{G}) may be considered as the particular case where the domain associated with each state is empty, +In that case, $p=N$ and $n_k=1$ for $k=0$ to $N$ and we are left only with the $p$-th component of the sum, that is, $G^{(N)}_{I_0 I_N} += \prod_{k=0}^{N-1} \langle I_k|F_{I_k}|I_{k+1} \rangle $ +where $F=T$. + +To express the fundamental equation for $G$ under the form of a probabilitic average, we write the importance-sampled version of the equation +$$ +{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} + +$$ +\be +\sum_{p=1}^{N} +\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}} +\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1} +\ee +\be +\delta(\sum_k n_k=N+1) \Big[ \prod_{k=0}^{p-1} [\frac{\Psi^+_{I_{k+1}}}{\Psi^+_{I_k}} \langle I_k| T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} \rangle \Big] +{\bar G}^{(n_p-1),{\cal D}}_{I_p I_N}. +\label{Gbart} +\ee +Introducing the weight +\be +W_{I_k I_{k+1}}=\frac{\langle I_k|T^{n_k-1}_{I_k} F_{I_k} |I_{k+1}\rangle}{\langle I_k|T^{+\;n_k-1}_{I_k} F^+_{I_k} |I_{k+1} \rangle} +\ee +and using the effective transition probability, Eq.(\ref{eq3C}), we get +\be +{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N}+ \sum_{p=1}^{N} +\bigg \langle +\Big( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} \Big) +{\bar G}^{(n_p-1), {\cal D}}_{I_p I_N} +\bigg \rangle +\label{Gbart} +\ee +where the average is defined as +$$ +\bigg \langle F \bigg \rangle += \sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}} +\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1} +\delta(\sum_k n_k=N+1) +$$ +\be +\prod_{k=0}^{N-1}{\cal P}(I_k \rightarrow I_{k+1},n_k-1) F(I_0,n_0;...;I_N,n_N) +\ee +In practice, a schematic DMC algorithm to compute the average is as follows.\\ +i) Choose some initial vector $|I_0\rangle$\\ +ii) Generate a stochastic path by running over $k$ (starting at $k=0$) as follows.\\ +$\;\;\;\bullet$ Draw $n_k$ using the probability $P_{I_k}(n)$, Eq.(\ref{PiN})\\ +$\;\;\;\bullet$ Draw the exit state, $|I_{k+1}\rangle$, using the conditional probability $$\frac{{\cal P}(I_k \rightarrow I_{k+1},n_k)}{P_{I_k}(n_k)}$$\\ +iii) Terminate the path when $\sum_k n_k=N$ is greater than some target value $N_{\rm max}$ and compute $F(I_0,n_0;...;I_N,n_N)$\\ +iv) Go to step ii) until some maximum number of paths is reached.\\ +\\ +At the end of the simulation, an estimate of the average for a few values of $N$ greater but close to $N_{max}$ is obtained. At large $N_{max}$ where the +convergence of the average as a function of $p$ is reached, such values can be averaged. +\subsubsection{Integrating out the trapping times : The Domain Green's Function Monte Carlo approach} +\label{energy} +Now, let us show that it is possible to go further by integrating out the trapping times, $n_k$, of the preceding expressions, thus defining a new effective +stochastic dynamics involving now only the exit states. Physically, it means that we are going to compute exactly within the time-evolution of all +stochastic paths trapped within each domain. We shall present two different ways to derive the new dynamics and renormalized probabilistic averages. +The first one, called the pedestrian way, consists in starting from the preceding time-expression for $G$ and make the explicit integration over the +$n_k$. The second, more direct and elegant, is based on the Dyson equation.\\ +\\ +{\it $\bullet$ The pedestrian way}. Let us define the quantity\\ +$$ +G^E_{ij}= \tau \sum_{N=0}^{\infty} \langle i|T^N|j\rangle. +$$ +By summing over $N$ we obtain +\be +G^E_{ij}= \langle i | \frac{1}{H-E}|j\rangle. +\ee +This quantity, which no longer depends on the time-step, is referred to as the energy-dependent Green's matrix. Note that in the continuum this quantity is +essentially the Laplace transform of the time-dependent Green's function. Here, we then use the same denomination. The remarkable property +is that, thanks to the summation over $N$ up to the +infinity the constrained multiple sums appearing in Eq.(\ref{Gt}) can be factorized in terms of a product of unconstrained single sums as follows +$$ +\sum_{N=1}^\infty \sum_{p=1}^N \sum_{n_0 \ge 1} ...\sum_{n_p \ge 1} \delta(n_0+...+n_p=N+1) +$$ +$$ += \sum_{p=1}^{\infty} \sum_{n_0=1}^{\infty} ...\sum_{n_p=1}^{\infty}. +$$ +It is then a trivial matter to integrate out exactly the $n_k$ variables, leading to +$$ +\langle I_0|\frac{1}{H-E}|I_N\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|I_N\rangle ++ \sum_{p=1}^{\infty} +\sum_{I_1 \notin {\cal D}_0, \hdots , I_p \notin {\cal D}_{p-1}} +$$ +\be +\Big[ \prod_{k=0}^{p-1} \langle I_k| P_k \frac{1}{H-E} P_k (-H)(1-P_k)|I_{k+1} \rangle \Big] +\langle I_p| P_p \frac{1} {H-E} P_p|I_N\rangle +\label{eqfond} +\ee +\noindent +As an illustration, Appendix \ref{A} reports the exact derivation of this formula in the case of a two-state system.\\ +\\ +{\it $\bullet$ Dyson equation.} In fact, there is a more direct way to derive the same equation by resorting to the Dyson equation. Starting from the +well-known equality +\be +\frac{1}{H-E} = \frac{1}{H_0-E} ++ \frac{1}{H_0-E} (H_0-H)\frac{1}{H-E} +\ee +where $H_0$ is some arbitrary reference Hamiltonian, we have +the Dyson equation +$$ +G^E_{ij}= G^E_{0,ij} + \sum_{k,l} G^{E}_{0,ik} (H_0-H)_{kl} G^E_{lj} +$$ +Let us choose as $H_0$ +$$ +\langle i |H_0|j\rangle= \langle i|P_i H P_i|j\rangle \;\;\; {\rm for \; all \; i \;j}. +$$ +The Dyson equation becomes +$$ +\langle i| \frac{1}{H-E}|j\rangle += \langle i| P_i \frac{1}{H-E} P_i|j\rangle +$$ +\be ++ \sum_k \langle i| P_i \frac{1}{H-E} P_i(H_0-H)|k\rangle \langle k|\frac{1}{H-E}|j\rangle +\ee +Now, we have +$$ +P_i \frac{1}{H-E} P_i(H_0-H) = P_i \frac{1}{H-E} P_i (P_i H P_i - H) +$$ +$$ += P_i \frac{1}{H-E} P_i (-H) (1-P_i) +$$ +and the Dyson equation may be written under the form +$$ +\langle i| \frac{1}{H-E}|j\rangle += \langle i| P_i \frac{1}{H-E} P_i|j\rangle +$$ +$$ ++ \sum_{k \notin {\cal D}_i} \langle i| P_i \frac{1}{H-E} P_i (-H)(1-P_i)|k\rangle \langle k|\frac{1}{H-E}|j\rangle +$$ +which is identical to Eq.(\ref{eqfond}) when $G$ is expanded iteratively.\\ +\\ +Let us use as effective transition probability density +\be +P(I \rightarrow J) = \frac{1} {\Psi^+(I)} \langle I| P_I \frac{1}{H^+-E^+_L} P_I (-H^+) (1-P_I)|J\rangle \Psi^+(J) +\ee +and the weight +\be +W^E_{IJ} = +\frac{\langle I|\frac{1}{H-E} P_I (-H)(1-P_I) |J\rangle }{\langle I|\frac{1}{H^+-E^+_L} P_I (-H^+)(1-P_I) |J\rangle} +\ee +Using Eqs.(\ref{eq1C},\ref{eq3C},\ref{relation}) we verify that $P(I \rightarrow J) \ge 0$ and $\sum_J P(I \rightarrow J)=1$. +Finally, the probabilistic expression writes +$$ +\langle I_0| \frac{1}{H-E}|I_N\rangle += \langle I_0| P_{I_0} \frac{1}{H-E} P_{I_0}|I_N\rangle +$$ +\be ++ \sum_{p=0}^{\infty} \Bigg \langle \Big( \prod_{k=0}^{p-1} W^E_{I_k I_{k+1}} \Big) \langle I_p| P_{I_p} \frac{1}{H-E} P_{I_p}|I_N\rangle \Bigg \rangle +\label{final_E} +\ee +{\it Energy estimator.} To calculate the energy we introduce the following quantity +\be +{\cal E}(E) = \frac{ \langle I_0|\frac{1}{H-E}|H\Psi_T\rangle} {\langle I_0|\frac{1}{H-E}|\Psi_T\rangle}. +\label{calE} +\ee +and search for the solution $E=E_0$ of +\be +{\cal E}(E)= E +\ee +Using the spectral decomposition of $H$ we have +\be +{\cal E}(E) = \frac{ \sum_i \frac{E_i c_i}{E_i-E}}{\sum_i \frac{c_i}{E_i-E}} +\label{calE} +\ee +with +\be +c_i = \langle I_0| \Phi_0\rangle \langle \Phi_0| \Psi_T\rangle +\ee +It is easy to check that in the vicinity of $E=E_0$, ${\cal E}(E)$ is a linear function of $E-E_0$. +So, in practice we compute a few values of ${\cal E}(E^{k})$ and fit the data using some function of $E$ close to the linearity +to extrapolate the exact value of $E_0$. Let us describe the 3 functions used here for the fit.\\ + +i) Linear fit\\ +We write +\be +{\cal E}(E)= a_0 + a_1 E +\ee +and search the best value of $(a_0,a_1)$ by fitting the data. +Then, ${\cal E}(E)=E$ leads to +$$ +E_0= \frac{a_0}{1-a_1} +$$ +ii) Quadratic fit\\ +At the quadratic level we write +$$ +{\cal E}(E)= a_0 + a_1 E + a_2 E^2 +$$ +leading to +$$ +E_0 = \frac{1 - a_1 \pm \sqrt{ (a_1 -1)^2 - 4 a_0 a_2}}{2 a_2} +$$ +iii) Two-component fit\\ +We take the advantage that the exact expression of ${\cal E}(E)$ is known as an infinite series, Eq.(\ref{calE}). +If we limit ourselves to the first two-component, we write +\be +{\cal E}(E) = \frac{ \frac{E_0 c_0}{E_0-E} + \frac{E_1 c_1}{E_1-E}}{\frac{c_0}{E_0-E} + \frac{c_1}{E_1-E} } +\ee +Here, the variational parameters used for the fit are ($c_0,E_0,c_1,E_1)$.\\ + +In order to have a precise extrapolation of the energy, it is interesting to compute the ratio +${\cal E}(E)$ for values of $E$ as close as possible to the exact energy. However, in that case +the numerators and denominators computed diverge. This is reflected by the fact that we need to compute +more and more $p$-components with an important increase of statistical fluctuations. So, in practice +a tradoff has to be found between the possible bias in the extrapolation and the amount of simulation time +required. + +\section{Numerical application to the Hubbard model} +\label{numerical} +Let us consider the one-dimensional Hubbard Hamiltonian for a chain of $N$ sites +\be +\hat{H}= -t \sum_{\langle i j\rangle \sigma} \hat{a}^+_{i\sigma} \hat{a}_{j\sigma} ++ U \sum_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} +\ee +where $\langle i j\rangle$ denotes the summation over two neighboring sites, +$\hat{a}_{i\sigma} (\hat{a}_{i\sigma})$ is the fermionic creation (annihilation) operator of +an electron of spin $\sigma$ ($=\uparrow$ or $\downarrow$) at site $i$, $\hat{n}_{i\sigma} = \hat{a}^+_{i\sigma} \hat{a}_{i\sigma}$ the +number operator, $t$ the hopping amplitude and $U$ the on-site Coulomb repulsion. +We consider a chain with an even number of sites and open boundary conditions (OBC) +at half-filling, that is, $N_{\uparrow}=N_{\downarrow}=\frac{N}{2}$. +In the site-representation, a general vector of the Hilbert space will be written as +\be +|n\rangle = |n_{1 \uparrow},...,n_{N \uparrow},n_{1 \downarrow},...,n_{N \downarrow}\rangle +\ee +where $n_{i \sigma}=0,1$ is the number of electrons of spin $\sigma$ at site $i$. + +For the 1D Hubbard model and OBC the components of the ground-state vector have the same sign (say, $c_i \ge 0$). +It is then possible to identify the guiding and trial vectors, that is, $|c^+\rangle=|c_T\rangle$. +As trial wave function we shall employ a generalization of the Gutzwiller wave function\cite{Gutzwiller_1963} written under the form +\be +\langle n|c_T\rangle= e^{-\alpha n_D(n)-\beta n_A(n)} +\ee +where $n_D(n)$ is the number of doubly occupied sites for the configuration $|n\rangle$ and $n_A(n)$ +the number of nearest-neighbor antiparallel pairs defined as +\be +n_A(n)= \sum_{\langle i j\rangle} n_{i\uparrow} n _{j\downarrow} \delta(n_{i\uparrow}-1) \delta(n_{j\downarrow}-1) +\ee +where the kronecker function $\delta(k)$ is equal to 1 when $k=0$ and 0 otherwise. +The parameters $\alpha, \beta$ are optimized by minimizing the variational energy, +$E_v(\alpha,\beta)=\frac{\langle c_T|H|c_T\rangle} {\langle c_T|c_T\rangle}$.\\ + +{\it Domains}. The efficiency of the method depends on the choice of states forming each domain. +As a general guiding principle, it is advantageous to build domains associated with a large average trapping time in order to integrate out the most important +part of the Green's matrix. Here, as a first illustration of the method, we shall +consider the large-$U$ regime of the Hubbard model where the construction of such domains is rather natural. +At large $U$ and half-filling, +the Hubbard model approaches the Heisenberg limit where only the $2^N$ states with no double occupancy, $n_D(n)=0$, have a significant weight in the wave function. +The contribution of the other states vanishes as $U$ increases with a rate increasing sharply with $n_D(n)$. In addition, for a given +number of double occupations, configurations with large values of $n_A(n)$ are favored due to their high kinetic energy (electrons move +more easily). Therefore, we build domains associated with small $n_D$ and large $n_A$ in a hierarchical way as described below. +For simplicity and decreasing the number of diagonalizations to perform, +we shall consider only one non-trivial domain called here the main domain and denoted as ${\cal D}$. +This domain will be chosen common to all states belonging to it, that is +\be +{\cal D}_i= {\cal D} \;\; {\rm for } \; |i\rangle \in {\cal D} +\ee +For the other states we choose a single-state domain +\be +{\cal D}_i= \{ |i\rangle \} \;\; {\rm for} \; |i\rangle \notin {\cal D} +\ee +To define ${\cal D}$, let us introduce the following set of states +\be +{\cal S}_{ij} = \{ |n\rangle; n_D(n)=i {\rm \; and\;} n_A(n)=j \}. +\ee +${\cal D}$ is defined as the set of states having up to $n_D^{\rm max}$ double occupations and, for each state with a number +of double occupations equal to $m$, a number of nearest-neighbor antiparallel pairs between $n_A^{\rm min}(m)$ and $n_A^{\rm max}(m)$. +Here, $n_A^{\rm max}(m)$ will not be varied and taken to be the maximum possible for a given $m$, +$n_A^{\rm max}(m)= {\rm Max}(N-1-2m,0)$. +Using these definitions, the main domain is taken as the union of some elementary domains +\be +{\cal D} = \cup_{n_D=0}^{n_D^{\rm max}}{\cal D}(n_D,n_A^{\rm min}(n_D)) +\ee +where the elementary domain is defined as +\be +{\cal D}(n_D,n_A^{\rm min}(n_D))=\cup_{ n_A^{\rm min}(n_D) \leq j \leq n_A^{\rm max}(n_D)} {\cal S}_{n_D j} +\ee +The two quantities defining the main domain are thus $n_D^{\rm max}$ and +$n_A^{\rm min}(m)$. +To give an illustrative example, let us consider the 4-site case. There are 6 possible elementary domains +$$ +{\cal D}(0,3)= {\cal S}_{03} +$$ +$$ +{\cal D}(0,2)= {\cal S}_{03} \cup {\cal S}_{02} +$$ +$$ +{\cal D}(0,1)= {\cal S}_{03} \cup {\cal S}_{02} \cup {\cal S}_{01} +$$ +$$ +{\cal D}(1,1)= {\cal S}_{11} +$$ +$$ +{\cal D}(1,0)= {\cal S}_{11} \cup {\cal S}_{10} +$$ +$$ +{\cal D}(2,0)= {\cal S}_{20} +$$ +where +$$ +{\cal S}_{03} = \{\; |\uparrow,\downarrow,\uparrow,\downarrow \rangle, |\downarrow,\uparrow,\downarrow,\uparrow \rangle \} +\;\;({\rm the\; two\; N\acute eel\; states}) +$$ +$$ +{\cal S}_{02} = \{\; |\uparrow, \downarrow, \downarrow, \uparrow \rangle, |\downarrow, \uparrow, \uparrow, \downarrow \rangle \} +$$ +$$ +{\cal S}_{01} = \{\; |\uparrow, \uparrow, \downarrow, \downarrow \rangle, |\downarrow, \downarrow, \uparrow, \uparrow \rangle \} +$$ +$$ +{\cal S}_{11} = \{\; |\uparrow \downarrow, \uparrow ,\downarrow, 0 \rangle, |\uparrow \downarrow, 0, \uparrow,\uparrow \rangle + ... +\} +$$ +$$ +{\cal S}_{10} = \{\; |\uparrow \downarrow, \uparrow, 0, \downarrow \rangle, |\uparrow \downarrow, 0, \uparrow, \downarrow \rangle + ... \} +$$ +$$ +{\cal S}_{20} = \{\; |\uparrow \downarrow, \uparrow \downarrow, 0 ,0 \rangle + ... \} +$$ +For the three last cases, the dots indicate the remaining states obtained by permuting the position of the pairs.\\ +\\ +Let us now present our QMC calculations for the Hubbard model. In what follows +we shall restrict ourselves to the case of the Green's Function Monte Carlo approach where trapping times are integrated out exactly. +\\ + +Following Eqs.(\ref{final_E},\ref{calE}), the practical formula used for computing the QMC energy is written as +\be +{\cal E}_{QMC}(E,p_{ex},p_{max})= \frac{H_0 +...+ H_{p_{ex}} + \sum_{p=p_{ex}+1}^{p_{max}} H^{QMC}_p } + {S_0 +...+ S_{p_{ex}} + \sum_{p=p_{ex}+1}^{p_{max}} S^{QMC}_p } +\label{calE} +\ee +where $p_{ex}+1$ is the number of pairs, ($H_p$, $S_p$), computed analytically. For $p_{ex} < p \le p_{max}$ +the Monte Carlo estimates are written as +\be +H^{QMC}_p= \Bigg \langle \Big( \prod_{k=0}^{p-1} W^E_{I_k I_{k+1}} \Big) \langle I_p| P_{I_p} \frac{1}{H-E} P_{I_p}|H\Psi_T\rangle \Bigg \rangle +\ee +and +\be +S^{QMC}_p= \Bigg \langle \Big(\prod_{k=0}^{p-1} W^E_{I_k I_{k+1}} \Big) \langle I_p| P_{I_p} \frac{1}{H-E} P_{I_p}|\Psi_T\rangle \Bigg \rangle. +\ee + +Let us begin with a small chain of 4 sites with $U=12$. From now on, we shall take $t=1$. +The size of the linear space is ${\binom{4}{2}}^2= 36$ and the ground-state energy obtained by exact diagonalization +is $E_0=-0.768068...$. The two variational parameters of the trial vector have been optimized and fixed at the values of +$\alpha=1.292$, and $\beta=0.552$ with a variational energy +of $E_v=-0.495361...$. In what follows +$|I_0\rangle$ will be systematically chosen as one of the two N\'eel states, {\it e.g.} $|I_0\rangle =|\uparrow,\downarrow, \uparrow,...\rangle$. + +Figure \ref{fig1} shows the convergence of $H_p$ as a function of $p$ for different values of the reference energy $E$. +We consider the simplest case where a single-state domain is associated to each state. +Five different values of $E$ have been chosen, namely $E=-1.6,-1.2,-1,-0.9$, and +$E=-0.8$. Only $H_0$ is computed analytically ($p_{ex}=0$). At the scale of the figure error bars are too small to be perceptible. +When $E$ is far from the exact value of $E_0=-0.768...$ the convergence is very rapid and only a few terms of the $p$-expansion are necessary. +In constrast, when $E$ approaches the exact energy, +a slower convergence is observed, as expected from the divergence of the matrix elements of the +Green's matrix at $E=E_0$ where the expansion does not converge at all. Note the oscillations of the curves as a function of $p$ due to a +parity effect specific to this system. In practice, it is not too much of a problem since +a smoothly convergent behavior is nevertheless observed for the even- and odd-parity curves. +The ratio, ${\cal E}_{QMC}(E,p_{ex}=1,p_{max})$ as a function of $E$ is presented in figure \ref{fig2}. Here, $p_{max}$ is taken sufficiently large +so that the convergence at large $p$ is reached. The values of $E$ are $-0.780,-0.790,-0,785,-0,780$, and $-0.775$. For smaller $E$ the curve is extrapolated using +the two-component expression. The estimate of the energy obtained from ${\cal E}(E)=E$ is $-0.76807(5)$ in full agreement with the exact value of $-0.768068...$. + +\begin{figure}[h!] +\begin{center} +\includegraphics[width=10cm]{fig1.pdf} +\end{center} +\caption{1D-Hubbard model, $N=4$, $U=12$. $H_p$ as a function of $p$ for $E=-1.6,-1.2,-1.,-0.9,-0.8$. $H_0$ is +computed analytically and $H_p$ (p > 0) by Monte Carlo. Error bars are smaller than the symbol size.} +\label{fig1} +\end{figure} + + +\begin{figure}[h!] +\begin{center} +\includegraphics[width=10cm]{fig2.pdf} +\end{center} +\caption{1D-Hubbard model, $N=4$ and $U=12$. ${\cal E}(E)$ as a function of $E$. +The horizontal and vertical lines are at ${\cal E}(E_0)=E_0$ and $E=E_0$, respectively. +$E_0$ is the exact energy of -0.768068.... The dotted line is the two-component extrapolation. +Error bars are smaller than the symbol size.} +\label{fig2} +\end{figure} + +Table \ref{tab1} illustrates the dependence of the Monte Carlo results upon the choice of the domain. The reference energy is $E=-1$. +The first column indicates the various domains consisting of the union of some elementary domains as explained above. +The first line of the table gives the results when using a minimal single-state domain for all states, and the last +one for the maximal domain containing the full linear space. The size of the various domains is given in column 2, the average trapping time +for the state $|I_0\rangle$ in the third column, and an estimate of the speed of convergence of the $p$-expansion for the energy in the fourth column. +To quantify the rate of convergence, we report the quantity, $p_{conv}$, defined as +the smallest value of $p$ for which the energy is converged with six decimal places. The smaller $p_{conv}$, the better the convergence is. +Although this is a rough estimate, it is sufficient here for our purpose. +As clearly seen, the speed of convergence is directly related to the magnitude of $\bar{t}_{I_0}$. The +longer the stochastic trajectories remain trapped within the domain, the better the convergence. Of course, when the domain is chosen to be the full space, +the average trapping time becomes infinite. Let us emphasize that the rate of convergence has no reason to be related to the size of the domain. +For example, the domain ${\cal D}(0,3) \cup {\cal D}(1,0)$ has a trapping time for the N\'eel state of 6.2, while +the domain ${\cal D}(0,3) \cup {\cal D}(1,1)$ having almost the same number of states (28 states), has an average trapping time about 6 times longer. Finally, the last column gives the energy obtained for +$E=-1$. The energy is expected to be independent of the domain and to converge to a common value, which is indeed the case here. +The exact value, ${\cal E}(E=-1)=-0.75272390...$, can be found at the last row of the Table for the case of a domain corresponding to the full space. +In sharp contrast, the statistical error depends strongly on the type of domains used. As expected, the largest error of $3 \times 10^{-5}$ is obtained in the case of +a single-state domain for all states. The smallest statistical error is obtained for the "best" domain having the largest average +trapping time. Using this domain leads to a reduction in the statistical error as large as about three orders of magnitude, nicely illustrating the +critical importance of the domains employed. + +\begin{table}[h!] +\centering +\caption{$N$=4, $U$=12, $E$=-1, $\alpha=1.292$, $\beta=0.552$,$p_{ex}=4$. Simulation with 20 independent blocks and $10^5$ stochastic paths +starting from the N\'eel state. $\bar{t}_{I_0}$ +is the average trapping time for the +N\'eel state. $p_{\rm conv}$ is a measure of the convergence of ${\cal E}_{QMC}(p)$ as a function of $p$, see text.} +\label{tab1} +\begin{tabular}{lcccl} +\hline +Domain & Size & $\bar{t}_{I_0}$ & $p_{\rm conv}$ & $\;\;\;\;\;\;{\cal E}_{QMC}$ \\ +\hline +Single & 1 & 0.026 & 88 &$\;\;\;\;$-0.75276(3)\\ +${\cal D}(0,3)$ & 2 & 2.1 & 110 &$\;\;\;\;$-0.75276(3)\\ +${\cal D}(0,2)$ & 4 & 2.1 & 106 &$\;\;\;\;$-0.75275(2)\\ +${\cal D}(0,1)$ & 6 & 2.1& 82 &$\;\;\;\;$-0.75274(3)\\ +${\cal D}(0,3)$ $\cup$ ${\cal D}(1,1)$ &14 &4.0& 60 & $\;\;\;\;$-0.75270(2)\\ +${\cal D}(0,3)$ $\cup$ ${\cal D}(1,0)$ &26 &6.2& 45 & $\;\;\;\;$-0.752730(7) \\ +${\cal D}(0,2)$ $\cup$ ${\cal D}(1,1)$ &16 &10.1 & 36 &$\;\;\;\;$-0.75269(1)\\ +${\cal D}(0,2)$ $\cup$ ${\cal D}(1,0)$ &28 &34.7 & 14&$\;\;\;\;$-0.7527240(6)\\ +${\cal D}(0,1)$ $\cup$ ${\cal D}(1,1)$ &18 & 10.1 & 28 &$\;\;\;\;$-0.75269(1)\\ +${\cal D}(0,1)$ $\cup$ ${\cal D}(1,0)$ &30 & 108.7 & 11&$\;\;\;\;$-0.75272400(5) \\ +${\cal D}(0,3)$ $\cup$ ${\cal D}(1,1)$ $\cup$ ${\cal D}$(2,0) &20 & 4.1 & 47 &$\;\;\;\;$-0.75271(2)\\ +${\cal D}(0,3)$ $\cup$ ${\cal D}(1,0)$ $\cup$ ${\cal D}$(2,0) &32 & 6.5 & 39 &$\;\;\;\;$-0.752725(3)\\ +${\cal D}(0,2)$ $\cup$ ${\cal D}(1,1)$ $\cup$ ${\cal D}$(2,0) &22 & 10.8 & 30 &$\;\;\;\;$-0.75270(1)\\ +${\cal D}(0,2)$ $\cup$ ${\cal D}(1,0)$ $\cup$ ${\cal D}$(2,0) &34 & 52.5 & 13&$\;\;\;\;$-0.7527236(2)\\ +${\cal D}(0,1)$ $\cup$ ${\cal D}(1,1)$ $\cup$ ${\cal D}$(2,0) & 24 & 10.8 & 26&$\;\;\;\;$-0.75270(1)\\ +${\cal D}(0,1)$ $\cup$ ${\cal D}(1,0)$ $\cup$ ${\cal D}$(2,0) & 36 & $\infty$&1&$\;\;\;\;$-0.75272390\\ +\hline +\end{tabular} +\end{table} + +As a general rule, it is always good to avoid the Monte Carlo calculation of a quantity which is computable analytically. Here, we apply +this idea to the case of the energy, Eq.(\ref{calE}), where the first $p$-components can be evaluated exactly up to some maximal value of $p$, $p_{ex}$. +Table \ref{tab2} shows the results both for the case of a single-state main domain and for the domain having the largest average trapping time, +namely ${\cal D}(0,1) \cup {\cal D}(1,1)$ (see Table \ref{tab1}). Table \ref{tab2} reports the statistical fluctuations of the +energy for the simulation of Table \ref{tab1}. Results show that it is indeed interesting to compute exactly as many components +as possible. For the single-state domain, a factor 2 of reduction of the statistical error is obtained when passing from no analytical computation, $p_{ex}=0$, to the +case where eight components for $H_p$ and $S_p$ are exactly computed. +For the best domain, the impact is much more important with a huge reduction of about three orders of +magnitude in the statistical error. Table \ref{tab3} reports +the energies converged as a function of $p$ with their statistical error on the last digit for $E= +-0.8, -0.795, -0.79, -0.785$, and $-0.78$. The values are displayed on Fig.\ref{fig2}. As seen on the figure the behavior of ${\cal E}$ as a function of +$E$ is very close to the linearity. The extrapolated values obtained from the five values of the energy with the three fitting functions are reported. +Using the linear fitting function +leads to an energy of -0.7680282(5) to compare with the exact value of -0.768068... A small bias of about $4 \times 10^{-5}$ is observed. This bias vanishes within the +statistical error with the quadratic fit. It is also true when using the two-component function. +Our final value is in full agreement with the +exact value with about six decimal places. + +Table \ref{tab4} shows the evolution of the average trapping times and extrapolated energies as a function of $U$ when using ${\cal D}(0,1) \cup {\cal D}(1,0)$ as the main +domain. We also report the variational and exact energies +together with the values of the optimized parameters of the trial wave function. As $U$ increases the configurations with zero or one double occupation +become more and more predominant and the average trapping time increases. For very large values of $U$ (say, $U \ge 12$) the increase of $\bar{t}_{I_0}$ +becomes particularly steep.\\ + +Finally, in Table \ref{tab4} we report the results obtained for larger systems at $U=12$ for a size of the chain ranging from $N=4$ (36 states) +to $N=12$ ($\sim 10^6$ states). No careful construction of domains maximizing the average trapping time has been performed, we have merely chosen domains +of reasonable size (not more than 2682 ) by taking not too large number of double occupations (only, $n_D=0,1$) and not too small number of +nearest-neighbor antiparallel pairs. +As seen, as the number of sites increases, the average trapping time for the domains chosen decreases. This point is of course undesirable since +the efficiency of the approach may gradually deteriorate when considering large systems. A more elaborate way of constructing domains is clearly called for. +The exact QMC energies extrapolated using the two-component function are also reported. +Similarly to what has been done for $N=4$ the extrapolation is performed using about five values for the reference energy. The extrapolated QMC +energies are in full agreement +with the exact value within error bars. However, an increase of the statistical error is observed when the size increases. To get lower error bars +we need to use better trial wavefunctions, better domains, and also larger simulation times. +laptop. Of course, it will also be particularly interesting to take advantage of the fully parallelizable character of the algorithm to get much lower error bars. +All these aspects will be considered in a forthcoming work. + +\begin{table}[h!] +\centering +\caption{$N=4$, $U=12$, and $E=-1$. Dependence of the statistical error on the energy with the number of $p$-components calculated +analytically. Same simulation as for Table \ref{tab1}. Results are presented when a single-state domain +is used for all states and when +${\cal D}(0,1) \cup {\cal D}(1,0)$ is chosen as main domain.} +\label{tab2} +\begin{tabular}{lcc} +\hline +$p_{ex}$ & single-state & ${\cal D}(0,1) \cup {\cal D}(1,0)$ \\ +\hline +$0$ & $4.3 \times 10^{-5}$ &$ 347 \times 10^{-8}$ \\ +$1$ & $4.0 \times10^{-5}$ &$ 377 \times 10^{-8}$\\ +$2$ & $3.7 \times 10^{-5}$ &$ 43 \times 10^{-8}$\\ +$3$ & $3.3 \times 10^{-5}$ &$ 46 \times 10^{-8}$\\ +$4$ & $2.6 \times 10^{-5}$ &$ 5.6 \times 10^{-8}$\\ +$5$ & $2.5 \times10^{-5}$ &$ 6.0 \times 10^{-8}$\\ +$6$ & $2.3 \times10^{-5}$ &$ 0.7 \times 10^{-8}$\\ +$7$ & $2.2 \times 10^{-5}$ &$ 0.6 \times 10^{-8}$\\ +$8$ & $2.2 \times10^{-5}$ &$ 0.05 \times 10^{-8}$\\ +\hline +\end{tabular} +\end{table} + + +\begin{table}[h!] +\centering +\caption{$N=4$, $U=12$, $\alpha=1.292$, $\beta=0.552$. Main domain = ${\cal D}(0,1) \cup {\cal D}(1,0)$. Simulation with 20 independent blocks and $10^6$ paths. +$p_{ex}=4$. The various fits are done with the five values of $E$} +\label{tab3} +\begin{tabular}{lc} +\hline +$E$ & $E_{QMC}$ \\ +\hline +-0.8 &-0.7654686(2)\\ +-0.795&-0.7658622(2)\\ +-0.79 &-0.7662607(3)\\ +-0.785&-0.7666642(4)\\ +-0.78 &-0.7670729(5)\\ +$E_0$ linear fit & -0.7680282(5)\\ +$E_0$ quadratic fit & -0.7680684(5)\\ +$E_0$ two-component fit & -0.7680676(5)\\ +$E_0$ exact & -0.768068...\\ +\hline +\end{tabular} +\end{table} + +\begin{table}[h!] +\centering +\caption{$N$=4, Domain ${\cal D}(0,1) \cup {\cal D}(1,0)$} +\label{tab4} +\begin{tabular}{cccccc} +\hline +$U$ & $\alpha,\beta$ & $E_{var}$ & $E_{ex}$ & $\bar{t}_{I_0}$ \\ +\hline +8 & 0.908,\;0.520 & -0.770342... &-1.117172... & 33.5\\ +10 & 1.116,\;0.539 & -0.604162... &-0.911497... & 63.3\\ +12 & 1.292,\;0.552 & -0.495361... &-0.768068... & 108.7\\ +14 & 1.438,\;0.563 & -0.419163... &-0.662871... & 171.7 \\ +20 & 1.786,\;0.582 & -0.286044... &-0.468619... & 504.5 \\ +50 & 2.690,\;0.609 & -0.110013... &-0.188984... & 8040.2 \\ +200 & 4.070,\;0.624& -0.026940... &-0.047315... & 523836.0 \\ +\hline +\end{tabular} +\end{table} + +\begin{table*}[h!] +\centering +\caption{$U=12$. The fits to extrapolate the QMC energies are done using the two-component function} +\label{tab5} +\begin{tabular}{crcrccccc} +\hline +$N$ & Size Hilbert space & Domain & Domain size & $\alpha,\beta$ &$\bar{t}_{I_0}$ & $E_{var}$ & $E_{ex}$ & $E_{QMC}$\\ +\hline +4 & 36 & ${\cal D}(0,1) \cup {\cal D}(1,0)$ & 30 &1.292, \; 0.552& 108.7 & -0.495361 & -0.768068 & -0.7680676(5)\\\ +6 & 400 & ${\cal D}(0,1) \cup {\cal D}(1,0)$ &200 & 1.124,\; 0.689 &57.8 & -0.633297 & -1.215395& -1.215389(9)\\ +8 & 4 900 & ${\cal D}(0,1) \cup {\cal D}(1,0)$ & 1 190 & 0.984,\; 0.788 & 42.8 & -0.750995 & -1.66395& -1.6637(2)\\ +10 & 63 504 & ${\cal D}(0,5) \cup {\cal D}(1,4)$ & 2 682 & 0.856,\; 0.869& 31.0 & -0.855958 & -2.113089& -2.1120(7)\\ +12 & 853 776 & ${\cal D}(0,8) \cup {\cal D}(1,7)$ & 1 674 & 0.739,\; 0.938 & 16.7 & -0.952127 & -2.562529& -2.560(6)\\ +\hline +\end{tabular} +\end{table*} + +\section{Summary and perspectives} +\label{conclu} +In this work it has been shown how to integrate out exactly within a DMC framework the contribution of all +stochastic trajectories trapped in some given domains of the Hilbert space and the corresponding general equations have been derived. +In this way a new effective stochastic dynamics connecting only the domains and not the individual states is defined. +A key property of the effective dynamics is that it does not depend on the shape of the domains used for each state, so rather general +domains overlapping or not can be used. To obtain the effective transition probability giving the probability of going from +one domain to another and the renormalized estimators, +the Green's functions limited to the domains are to be computed analytically, that is, in practice, matrices of size the number +of states of the sampled domains need to be inverted. This is the main computationally intensive step of the method. +The efficiency of the method is directly related to +the importance of the average time spent by the stochastic trajectories within each domain. Being able to define domains with large average trapping times +is the key aspect of the method since it may lead to some important reduction of the statistical error as illustrated in our numerical application. +So a tradeoff has to be found in the complexity of the domains maximizing the average trapping time and the cost of computing +the local Green's functions associated with such domains. In practice, there is no general rule to construct efficient domains. +For each system at hand, we need to determine on physical grounds which regions of the space are preferentially sampled by +the stochastic trajectories and to build domains of minimal size enclosing such regions. In the first application presented here on the 1D-Hubbard model +we exploit the physics of the large $U$ regime which is known to approach the Heisenberg limit where double occupations have a small weight. +This simple example has been chosen to illustrate the various aspects of the approach. However, our goal is of course to +tackle much larger systems, like those treated by state-of-the-art methods, such as selected CI, (for example, \cite{Huron_1973,Harrison_1991,Giner_2013,Holmes_2016, +Schriber_2016,Tubman_2020}, QMCFCI,\cite{Booth_2009,Cleland_2010}, AFQMC\cite{Zhang_2003}, or DMRG\cite{White_1999,Chan_2011} approaches). +Here, we have mainly focused on the theoretical aspects of the approach. To reach larger systems will require some +more elaborate implementation of the method in order to keep under control the cost of the simulation. +Doing this was out of the scope of the present work and will be presented in a forthcoming work. + +\section*{Acknowledgement} +This work was supported by the European Centre of +Excellence in Exascale Computing TREX --- Targeting Real Chemical +Accuracy at the Exascale. This project has received funding from the +European Union's Horizon 2020 --- Research and Innovation program --- +under grant agreement no. 952165. + +\appendix +\section{Particular case of the $2\times2$ matrix} +\label{A} +For the simplest case of a two-state system the fundamental equation (\ref{eqfond}) writes +$$ +{\cal I}= +\langle I_0|\frac{1}{H-E}|\Psi\rangle = \langle I_0|P_0\frac{1}{H-E} P_0|Psi\rangle ++ \sum_{p=1}^{\infty} {\cal I}_p$$ +with +$$ + {\cal I}_p= +\sum_{I_1 \notin {\cal D}_0, \hdots , I_p \notin {\cal D}_{p-1}} +$$ +\be +\Big[ \prod_{k=0}^{p-1} \langle I_k| P_k \frac{1}{H-E} P_k (-H)(1-P_k)|I_{k+1} \rangle \Big] +\langle I_p| P_p \frac{1} {H-E} P_p|\Psi\rangle +\label{eqfond} +\ee +To treat simultaneously the two possible cases for the final state, $|I_N\rangle =|1\rangle$ or $|2\rangle$, +the equation has been slightly generalized to the case of a general vector for the final state written as +\be +|\Psi\rangle = \Psi_1 |1\rangle + \Psi_2 |2\rangle +\ee +where +$|1\rangle$ and $|2\rangle$ denote the two states. Let us choose a single-state domain for both states, namely ${\cal D}_1=\{|1\rangle \}$ and ${\cal D}_2=\{ |2\rangle \}$. Note that the single exit state for each state is the other state. Thus there are only two possible deterministic "alternating" paths, namely either +$|1\rangle \rightarrow |2\rangle \rightarrow |1\rangle,...$ or $|2\rangle \rightarrow |1\rangle \rightarrow |2\rangle,...$ +We introduce the following quantities +\be +A_1= \langle 1| P_1 \frac{1}{H-E} P_1 (-H)(1-P_1)|2\rangle \;\;\; +A_2= \langle 2| P_2 \frac{1}{H-E} P_2 (-H) (1-P_2)|1\rangle +\label{defA} +\ee +and +\be +C_1= \langle 1| P_1 \frac{1}{H-E} P_1 |\Psi\rangle \;\;\; +C_2= \langle 2| P_2 \frac{1}{H-E} P_2 |\Psi\rangle +\ee +Let us choose, for example, $|I_0\rangle =|1\rangle$, we then have +\be +{\cal I}_1= A_1 C_2 \;\; {\cal I}_2 = A_1 A_2 C_1 \;\; {\cal I}_3 = A_1 A_2 A_1 C_2 \;\; etc. +\ee +that is +\be +{\cal I}_{2k+1} = \frac{C_2}{A_2} (A_1 A_2)^{2k+1} +\ee +and +\be +{\cal I}_{2k} = C_1 (A_1 A_2)^{2k} +\ee +Then +\be +\sum_{p=1}^{\infty} +{\cal I}_p += \frac{C_2}{A_2} \big[ \sum_{p=1}^{\infty} (A_1 A_2)^p \big] + {C_1} \big[ \sum_{p=1}^{\infty} (A_1 A_2)^p \big] +\ee +which gives +\be +\sum_{p=1}^{\infty} +{\cal I}_p += A_1 \frac{C_2 + C_1 A_2}{1-A_1 A_2} +\ee +and thus +\be +\langle 1 | \frac{1} {H-E} |\Psi\rangle += +C_1 + A_1 \frac{C_2 + C_1 A_2}{1-A_1 A_2} +\ee +For a two by two matrix it is easy to evaluate the $A_i$'s, Eq.\ref{defA}, we +have +$$ +A_i = -\frac{H_{12}}{H_{ii}-E} \;\;\; \;\;\; C_i= \frac{1}{H_{ii}-E} \Psi_i \;\;\; i=1,2 +$$ +Then +$$ +\langle 1 | \frac{1} {H-E} |\Psi\rangle += \frac{1}{H_{11}-E} \Psi_1 -H_{12} \frac{ \big( \Psi_2 - \frac{ H_{12}\Psi_1}{H_{11}-E} \big)} + { (H_{11}-E)(H_{22}-E) - H^2_{12}} +$$ +$$ += \frac{ (H_{22}-E) \Psi_1}{\Delta} -\frac{H_{12} \Psi_2}{\Delta} +$$ +where $\Delta$ is the determinant of the matrix $H$. It is easy to check that the RHS is equal to the LHS, thus validating the fundamental equation +for this particular case. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliography{g} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\end{document}