First principles multielectron mixed quantum/classical simulationsin the condensed phase. I. An efficient Fourier-grid method for solvingthe many-electron problemWilliam J. Glover, Ross E. Larsen, and Benjamin J. Schwartza兲Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles,California 90095-1569, USA共Received 19 November 2009; accepted 11 February 2010; published online 8 April 2010兲We introduce an efficient multielectron first-principles based electronic structure method, thetwo-electron Fourier-grid 共2EFG兲 approach, that is particularly suited for use in mixed quantum/classical simulations of condensed-phase systems. The 2EFG method directly solves for thesix-dimensional wave function of a two-electron Hamiltonian in a Fourier-grid representation suchthat the effects of electron correlation and exchange are treated exactly for both the ground andexcited states. Due to the simplicity of a Fourier-grid representation, the 2EFG is readilyparallelizable and we discuss its computational implementation in a distributed-memory parallelenvironment. We show our method is highly efficient, being able to find two-electron wavefunctions in ⬃20 s on a modern desktop computer for a calculation this is equivalent to fullconfiguration interaction 共FCI兲 in a basis of 17 million Slater determinants. We benchmark theaccuracy of the 2EFG by applying it to two electronic structure test problems: the harmonium atomand the sodium dimer. We find that even with a modest grid basis size, our method converges to theanalytically exact solutions of harmonium in both the weakly and strongly correlated electronregimes. Our method also reproduces the low-lying potential energy curves of the sodium dimer toa similar level of accuracy as a valence CI calculation, thus demonstrating its applicability tomolecular systems. In the following paper 关W. J. Glover, R. E. Larsen, and B. J. Schwartz, J. Chem.Phys. 132, 144102 共2010兲兴, we use the 2EFG method to explore the nature of the electronic statesthat comprise the charge-transfer-to-solvent absorption band of sodium anions in liquidtetrahydrofuran. © 2010 American Institute of Physics. 关doi:10.1063/1.3352564兴I. INTRODUCTIONThe simulation of electronically excited reactions in thecondensed phase from first principles remains a challenge.This is in part because ab initio electronic structure methodscapable of describing excited states are currently far toocostly to be applied directly to the large number of atomsthat make up a condensed-phase simulation. The vast major-ity of computer simulations of electronically excitedcondensed-phase reactions have therefore used either single-electron model Hamiltonians1–3or semiempirical quantumchemical methods.4–7Although the low computational costof such methods favors their use in condensed-phase simu-lation, a first-principles description of electronic structurewould be desirable, particularly for systems where physicalinsight is lacking or where parametrization to experimentaldata is not possible.In this paper, we present a new first-principles electronicstructure method that is particularly suited to condensed-phase simulation: the two-electron Fourier-grid 共2EFG兲method. The method performs a direct diagonalization of amany-electron Hamiltonian in a grid basis set such that it isequivalent to configuration interaction with single anddouble 共CISD兲 excitations. The basis set is flexible enough todescribe both strongly and weakly correlated systems, andbecause it is a grid, it is readily transferrable from one sys-tem to another. As our method is equivalent to CISD, ex-change and correlation are treated at a high level 共exactly for2-electron systems兲 and the excited states are meaningful.Moreover, the method is efficient enough to be performedrepeatedly, for example, at each time step of a long molecu-lar dynamics simulation trajectory.One of the keys to implementing our method incondensed-phase simulations is the use of a mixed quantum/classical 共MQC兲 scheme, which for our purposes entails aquantum mechanical treatment of the valence electrons of asolute molecule and a classical treatment of the solute coreand solvent molecules. The classical particles and quantumelectrons are coupled through molecular pseudopotentialsthat are rigorously derived from first-principles quantumchemistry calculations and therefore include Pauli repulsionand exchange interactions in addition to electrostatic interac-tions between the valence electrons and solvent molecules.8,9In what is presented below, we outline and apply our2EFG method for two-electron systems. This is because withthe use of molecular pseudopotentials,8,9many condensed-phase problems of interest are reduced to effective two-electron systems,10–13for which our 2EFG method providesan exact treatment. 共We note that the generalization of oura兲Author to whom correspondence should be addressed. Electronic mail:[email protected] JOURNAL OF CHEMICAL PHYSICS 132, 144101 共2010兲0021-9606/2010/132共14兲/144101/11/$30.00 © 2010 American Institute of Physics132, 144101-12EFG method to additional electrons is straightforward ifthere are sufficient computational resources兲. For two-electron systems, our method is efficient enough to be usediteratively in molecular dynamics simulations—even withⱖ108basis functions. Thus, the purpose of this paper is topresent the development and implementation of our 2EFGmethod, and to benchmark the accuracy of the method ontwo test electronic structure problems: the harmonium atomand the sodium dimer. Although our grid-based method isultimately designed for use in condensed-phase simulation,we chose these gas-phase two-electron problems as theyhave exact or numerically accurate solutions with which tocompare.In the following paper, hereafter called Paper II,13weapply our 2EFG method to the problem of charge-transfer-to-solvent 共CTTS兲 reactions, providing new insight into theexcited-state electronic structure of the sodium anion 共so-dide兲 in liquid tetrahydrofuran 共THF兲.14–26CTTS reactionsrepresent a challenge to simulate from first principles be-cause they involve a valence electron being promoted froman anionic solute to a solvent-supported excited state thatultimately detaches and forms a solvated electron.14,27–30Thequantum mechanical region for a CTTS reaction thus encom-passes not only the solute but also all the solvent