CONTRACTED SCHRODINGER-EQUATION - DETERMINING QUANTUM ENERGIES AND 2-PARTICLE DENSITY-MATRICES WITHOUT WAVE-FUNCTIONS

The contracted Schrodinger equation (CSE) technique through its direct determination of the two-particle reduced density matrix (2RDM) witho ut the wave function may offer a fresh alternative to traditional many -body quantum calculations. Without additional information the CSE, al so known as the density equation, cannot be solved-for the 2RDM becaus e it also requires a knowledge of the 4RDM. We provide: theoretical fo undations through a reconstruction theorem for recent attempts at gene rating higher RDMs from the 2RDM to remove the indeterminacy of-the CS E. With Grassmann algebra a more concise representation for Valdemoro' s reconstruction functionals [F. Colmenero, C. Perez del Valle, and C. Valdemoro, Phys. Rev. A 47,-971 (1993)] is presented. From the perspe ctive of the particle-hole equivalence we obtain Nakatsuji and Yasuda' s correction for the 4RDM formula [H. Nakatsuji and K. Yasuda, Phys Re v. Lett. 76, 1039 (1996)] as well as a corrective approach for the 3RD M functional. A different reconstruction strategy, the ensemble repres entability method (ERM), is introduced to build the 3- and 4-RDMs by e nforcing four-ensemble representability and contraction conditions. We derive the CSE in second quantization without Valdemoro's matrix cont raction mapping and offer the first proof of Nakatsuji's theorem for t he second-quantized CSE. Both:the functional and ERM reconstruction st rategies are employed with the CSE to solve for the energies and the 2 RDMs of a quasispin model without wave functions. We:elucidate the ite rative solution of the CSE through an analogy with the power method fo r eigenvalue equations. Resulting energies of the CSE methods are comp arable to single-double configuration-interaction (SDCI) energies, and the 2RDMs are more accurate by an order of magnitude than those from SDCI. While the CSE has been;applied to systems with 14 electrons, we present results for as many as 40 particles. Results indicate that the 2RDM remains accurate as the number of particles increases. We also r eport a direct determination of excited-state 2RDMs through the CSE. B y circumventing the wave function, the CSE presents new possibilities for treating electron correlation.