Essentially exact ground-state calculations by superpositions of nonorthogonal Slater determinants
© Goto et al.; licensee Springer. 2013
Received: 31 October 2012
Accepted: 15 January 2013
Published: 1 May 2013
An essentially exact ground-state calculation algorithm for few-electron systems based on superposition of nonorthogonal Slater determinants (SDs) is described, and its convergence properties to ground states are examined. A linear combination of SDs is adopted as many-electron wave functions, and all one-electron wave functions are updated by employing linearly independent multiple correction vectors on the basis of the variational principle. The improvement of the convergence performance to the ground state given by the multi-direction search is shown through comparisons with the conventional steepest descent method. The accuracy and applicability of the proposed scheme are also demonstrated by calculations of the potential energy curves of few-electron molecular systems, compared with the conventional quantum chemistry calculation techniques.
KeywordsGround-state calculation Nonorthogonal Slater determinants Superposition Few-electron system Multiple correction vector
In recent years, there have been many significant achievements regarding electronic structure calculations in the fields of computational physics and chemistry. However, theoretical and methodological approaches for providing practical descriptions and tractable calculation schemes of the electron–electron correlation energy with continuously controllable accuracy still remain as significant issues [1–15]. Although density functional theory (DFT) supplies a computationally economical and practical method, there are many unexplored problems raised by unreliable results obtained for some systems in which highly accurate electron–electron correlation energy calculations are required, since results by DFT depend significantly on the exchange-correlation energy functional used to perform the calculation [16–18].
The available quantitatively reliable methods require higher computational costs than the DFT method . Although quantum Monte Carlo methods [19–23] can be applied to molecular and crystal systems and show good quantitative reliability where extremely high-accuracy calculations are required, difficulties in calculating forces for optimizing atomic configurations are a considerable disadvantage and inhibit this method from becoming a standard molecular dynamics calculation technique. Configuration interaction (CI), coupled cluster, and Møller-Plesset second-order perturbation methods, each of which use a linear combination of orthogonalized Slater determinants (SDs) as many-electron wave functions, are standard computational techniques in quantum chemistry by which highly accurate results are obtained , despite suffering from basis set superposition and basis set incompleteness errors. The full CI calculation can perform an exact electron–electron correlation energy calculation in a space given by an arbitrary basis set. However, it is only applicable for small molecules with modest basis sets since the required number of SDs grows explosively on the order of the factorial of the number of basis.
The required number of SDs in order to determine ground-state energies can be drastically decreased by employing nonorthogonal SDs as a basis set. The resonating Hartree-Fock method proposed by Fukutome utilizes nonorthogonal SDs, and many noteworthy results have been reported [25–30]. Also, Imada and co-workers [31–33] and Kojo and Hirose [34, 35] employed nonorthogonal SDs in path integral renormalization group calculations. Goto and co-workers developed the direct energy minimization method using nonorthogonal SDs [36–39] based on the real-space finite-difference formalism [40, 41]. In these previous studies, steepest descent directions and acceleration parameters are calculated to update one-electron wave functions on the basis of a variational principle [25–30, 36–39]. Although the steepest descent direction guarantees a secure approach to the ground state, a more effective updating process might be performed in a multi-direction search.
In the present study, a calculation algorithm showing an arbitrary set of linearly independent correction vectors is employed to optimize one-electron wave functions with Gaussian basis sets. Since the dimension of the search space depends on the number of linearly independent correction vectors, a sufficient number of correction vectors ensure effective optimization, and the iterative updating of all the one-electron wave functions leads to smooth convergence to the ground states. The primary purpose of this article is to demonstrate the advantage of using multiple correction vectors in searching for the ground state over the conventional steepest descent search in which only one correction vector is used. As a demonstration of the accuracy and applicability of the proposed calculation algorithm, essentially exact potential energy curves of few-electron molecular systems with long interatomic distances are described for cases where the conventional calculation methods of quantum chemistry fail.
The organization of the article is as follows. In the ‘Optimization algorithm’ section, the proposed calculation algorithm for constructing a basis set of nonorthogonal SDs by updating one-electron wave functions with multiple correction vectors is described. The expression of the conventional steepest descent direction with a Gaussian basis set is also given for comparison. The convergence characteristics to the ground states of few-electron systems for calculations using single and multiple correction vectors are illustrated in the ‘Applications to few-electron molecular systems’ section. As demonstrations of the proposed calculation procedure, the convergence properties to the ground states of few-electron atomic and molecular systems are also shown. Finally, a summary of the present study is given in the ‘Conclusions’ section.
Here, M and Di,s A are the number of basis functions and the s th expansion coefficient for the i th one-electron wave function ϕ i A (r), respectively.
respectively. Here, V(r) stands for an external potential.
Since the linearly independent correction vectors can be given arbitrarily, randomly chosen values are employed in the present study. A larger number of correction vectors N c realize a larger volume search space; however, the number of the linearly independent vectors N c is restricted to the dimension of the space defined by the basis set used.
Applications to few-electron molecular systems
A reliable and tractable technique for constructing the ground-state wave function by the superposition of nonorthogonal SDs is described. Linear independent multiple correction vectors are employed in order to update one-electron wave functions, and a conventional steepest descent method is also performed as a comparison. The dependence of convergence performance on the number of adopted correction vectors is also illustrated. The electron–electron correlation energy converges rapidly and smoothly to the ground state through the multi-direction search, and an essentially exact ground-state energy is obtained with drastically fewer SDs (less than 100 SDs in the present study) compared with the number required in the full CI method. For the few-electron molecular systems considered in the present study, essentially exact electron–electron correlation energies can be calculated even at long bond lengths for which the standard single-reference CCSD and CCSD(T) show poor results, and the practicality and applicability of the proposed calculation procedure have been clearly demonstrated. In future studies, calculations employing periodic boundary conditions and effective core potentials (ECPs)  will be performed. A new procedure to reduce the iteration cost should be found in order to increase the applicability of the proposed algorithm for the calculation of essentially exact ground-state energies of many-electron systems.
The present study was partially supported by a Grant-in-Aid for the Global COE Program ‘Center of Excellence for Atomically Controlled Fabrication Technology’ (grant no. H08), a Grant-in-Aid for Scientific Research on Innovative Areas ‘Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics’ (grant no. 22104008), a Grant-in-Aid for Scientific Research in Priority Areas ‘Carbon Nanotube Nano-Electronics’ (grant no. 19054009) and a Grant-in-Aid for Scientific Research (B) ‘Design of Nanostructure Electrode by Electron Transport Simulation for Electrochemical Processing’ (grant no. 21360063) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan.
- Palmer IJ, Brown WB, Hillier IH: Simulation of the charge transfer absorption of the H2O/O2van der Waals complex using high level ab initio calculations. J Chem Phys 1996, 104: 3198. 10.1063/1.471084View Article
- Kowalski K, Piecuch P: The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches. J Chem Phys 2000, 113: 18. 10.1063/1.481769View Article
- Gwaltney SR, Sherrill CD, Head-Gordon M: Second-order perturbation corrections to singles and doubles coupled-cluster methods: General theory and application to the valence optimized doubles model. J Chem Phys 2000, 113: 3548. 10.1063/1.1286597View Article
- Tsuzuki S, Honda K, Uchimaru T, Mikami M, Tanabe K: Origin of attraction and directionality of the π/π interaction: model chemistry calculations of benzene dimer interaction. J Am Chem Soc 2002, 124: 104. 10.1021/ja0105212View Article
- Dutta A, Sherrill CD: Full configuration interaction potential energy curves for breaking bonds to hydrogen: an assessment of single-reference correlation methods. J Chem Phys 2003, 118: 1610. 10.1063/1.1531658View Article
- Abrams ML, Sherrill CD: Full configuration interaction potential energy curves for the X 1Σg+, B 1Δg, and B’ 1Σg+ states of C2: a challenge for approximate methods. J Chem Phys 2004, 121: 9211. 10.1063/1.1804498View Article
- Juhasz T, Mazziotti DA: Perturbation theory corrections to the two-particle reduced density matrix variational method. J Chem Phys 2004, 121: 1201. 10.1063/1.1760748View Article
- Rocha-Rinza T, Vico LD, Veryazov V, Roos BO: A theoretical study of singlet low-energy excited states of the benzene dimer. Chem Phys Lett 2006, 426: 268. 10.1016/j.cplett.2006.05.123View Article
- Du S, Francisco JS: The OH radical-H2O molecular interaction potential. J Chem Phys 2006, 124: 224318. 10.1063/1.2200701View Article
- Benedek NA, Snook IK: Quantum Monte Carlo calculations of the dissociation energy of the water dimer. J Chem Phys 2006, 125: 104302. 10.1063/1.2338032View Article
- Bonfanti M, Martinazzo R, Tantardini GF, Ponti A: Physisorption and diffusion of hydrogen atoms on graphite from correlated calculations on the H-coronene model system. J Phys Chem C 2007, 111: 5825. 10.1021/jp070616bView Article
- Beaudet TD, Casula M, Kim J, Sorella S, Martin RM: Molecular hydrogen adsorbed on benzene: insights from a quantum Monte Carlo study. J Chem Phys 2008, 129: 164711. 10.1063/1.2987716View Article
- Ma J, Michaelides A, Alfe D: Binding of hydrogen on benzene, coronene, and graphene from quantum Monte Carlo calculations. J Chem Phys 2011, 134: 134701. 10.1063/1.3569134View Article
- Booth GH, Cleland D, Thom AJW, Alavi A: Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods. J Chem Phys 2011, 135: 084104. 10.1063/1.3624383View Article
- Robinson JB, Knowles P: Approximate variational coupled cluster theory. J Chem Phys 2011, 135: 044113. 10.1063/1.3615060View Article
- Feibelman PJ, Hammer B, Norskov JK, Wagner F, Scheffler M, Stumpf R, Watwe R, Dumesic J: The CO/Pt(111) puzzle. J Phys Chem B 2001, 105: 4018. 10.1021/jp002302tView Article
- Hu Q-M, Reuter K, Scheffler M: Towards an exact treatment of exchange and correlation in materials: application to the “CO adsorption puzzle” and other systems. Phys Rev Lett 2007, 98: 176103.View Article
- Foulkes WMC, Mitas L, Needs RJ, Rajagopal G: Quantum Monte Carlo simulations of solids. Rev Mod Phys 2001, 73: 33. 10.1103/RevModPhys.73.33View Article
- Silverstrelli PL, Baroni S, Car R: Auxiliary-field quantum Monte Carlo calculations for systems with long-range repulsive interactions. Phys Rev Lett 1993, 71: 1148. 10.1103/PhysRevLett.71.1148View Article
- Zhang S, Krakauer H, Zhang S: Quantum Monte Carlo method using phase-free random walks with Slater determinants. Phys Rev Lett 2003, 90: 136401.View Article
- Al-Saidi WA, Krakauer H, Zhang S: Auxiliary-field quantum Monte Carlo study of TiO and MnO molecules. Phys Rev B 2006, 73: 075103.View Article
- Suewattana M, Purwanto W, Zhang S, Krakauer H, Walter E: Phaseless auxiliary-field quantum Monte Carlo calculations with plane waves and pseudopotentials: applications to atoms and molecules. Phys Rev B 2007, 75: 245123.View Article
- Purwanto W, Krakauer H, Zhang S: Pressure-induced diamond to β-tin transition in bulk silicon: A quantum Monte Carlo study. Phys Rev B 2009, 80: 214116.View Article
- Szabo A, Ostlund NS: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. London: Macmillan; 1982.
- Fukutome H: Theory of resonating quantum fluctuations in a fermion system—resonating Hartree-Fock approximation—. Prog Theor Phys 1988, 80: 417. 10.1143/PTP.80.417View Article
- Ikawa A, Yamamoto S, Fukutome H: Orbital optimization in the resonating Hartree-Fock approximation and its application to the one dimensional Hubbard model. J Phys Soc Jpn 1993, 62: 1653. 10.1143/JPSJ.62.1653View Article
- Igawa A: A method of calculation of the matrix elements between the spin-projected nonorthogonal Slater determinants. Int J Quantum Chem 1995, 54: 235. 10.1002/qua.560540406View Article
- Tomita N, Ten-no S, Yanimura Y: Ab initio molecular orbital calculations by the resonating Hartree-Fock approach: superposition of non-orthogonal Slater determinants. Chem Phys Lett 1996, 263: 687. 10.1016/S0009-2614(96)01266-3View Article
- Ten-no S: Superposition of nonorthogonal Slater determinants towards electron correlation problems. Theor Chem Acc 1997, 98: 182.View Article
- Okunishi T, Negishi Y, Muraguchi M, Takeda K: Resonating Hartree–Fock approach for electrons confined in two dimensional square quantum dots. Jpn J Appl Phys 2009, 48: 125002. 10.1143/JJAP.48.125002View Article
- Imada M, Kashima T: Path-integral renormalization group method for numerical study of strongly correlated electron systems. J Phys Soc Jpn 2000, 69: 2723. 10.1143/JPSJ.69.2723View Article
- Kashima T, Imada M: Path-integral renormalization group method for numerical study on ground states of strongly correlated electronic systems. J Phys Soc Jpn 2001, 70: 2287. 10.1143/JPSJ.70.2287View Article
- Noda Y, Imada M: Quantum phase transitions to charge-ordered and Wigner-crystal states under the interplay of lattice commensurability and long-range Coulomb interactions. Phys Rev Lett 2002, 89: 176803.View Article
- Kojo M, Hirose K: Path-integral renormalization group treatments for many-electron systems with long-range repulsive interactions. Surf Interface Anal 2008, 40: 1071. 10.1002/sia.2832View Article
- Kojo M, Hirose K: First-principles path-integral renormalization-group method for Coulombic many-body systems. Phys Rev A 2009, 80: 042515.View Article
- Goto H, Hirose K: Total-energy minimization of few-body electron systems in the real-space finite-difference scheme. J Phys: Condens Matter 2009, 21: 064231. 10.1088/0953-8984/21/6/064231
- Goto H, Yamashiki T, Saito S, Hirose K: Direct minimization of energy functional for few-body electron systems. J Comput Theor Nanosci 2009, 6: 2576. 10.1166/jctn.2009.1317View Article
- Goto H, Hirose K: Electron–electron correlations in square-well quantum dots: direct energy minimization approach. J Nanosci Nanotechnol 2011, 11: 2997. 10.1166/jnn.2011.3924View Article
- Sasaki A, Kojo M, Hirose K, Goto H: Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct energy minimization method. J Phys: Condens Matter 2011, 23: 434001. 10.1088/0953-8984/23/43/434001
- Chelikowsky JR, Troullier N, Saad Y: Finite-difference-pseudopotential method: electronic structure calculations without a basis. Phys Rev Lett 1994, 72: 1240. 10.1103/PhysRevLett.72.1240View Article
- Hirose K, Ono T, Fujimoto Y, Tsukamoto S: First-Principles Calculations in Real-Space Formalism. London: Imperial College Press; 2005.View Article
- Knowles PJ, Cooper B: A linked electron pair functional. J Chem Phys 2010, 133: 224106. 10.1063/1.3507876View Article
- Trail JR, Needs RJ: Smooth relativistic Hartree–Fock pseudopotentials for H to Ba and Lu to Hg. J Chem Phys 2005, 122: 174109. 10.1063/1.1888569View Article
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