Theory of variational calculation with a scaling correct moment functional to solve the electronic schrödinger equation directly for ground state one-electron density and electronic energy

• Published in: International journal of quantum chemistry Vol. 113; no. 10; pp. 1479 - 1492
• Main Author: Kristyan, Sandor
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 15.05.2013; Wiley Subscription Services, Inc
• Subjects: Chemistry; density functional theory; ground state total electronic energy; Lagrange multiplier; one-electron density; Physical chemistry; power series with correct density scaling; Quantum physics; Studies
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/qua.24345

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a nonlinear, approximate density functional of a three spatial dimension one‐electron density for an N electron system which is tractable in practice, is a long‐desired goal in electronic structure calculation. In a seminal work, Parr et al. (Phys. Rev. A 1997, 55, 1792) suggested a well behaving density functional in power series with respect to density scaling within the orbital‐free framework for kinetic and repulsion energy of electrons. The updated literature on this subject is listed, reviewed, and summarized. Using this series with some modifications, a good density functional approximation is analyzed and solved via the Lagrange multiplier device. (We call the attention that the introduction of a Lagrangian multiplier to ensure normalization is a new element in this part of the related, general theory.) Its relation to Hartree–Fock (HF) and Kohn–Sham (KS) formalism is also analyzed for the goal to replace all the analytical Gaussian based two and four center integrals (∫gi(r1)gk(r2)r 12−1dr1dr2, etc.) to estimate electron‐electron interactions with cheaper numerical integration. The KS method needs the numerical integration anyway for correlation estimation. © 2012 Wiley Periodicals, Inc. The reduction of the electronic Schrödinger equation from 4N‐dimensions to a density functional of a three‐spatial‐dimension one‐electron density is a long desired goal in electronic structure calculation. Scaling correct series of density functional approximations is described, and solved via the Lagrange multiplier device. Two‐moment functional approximations for the Schrödinger equation are set up as an early truncation, and a parameter fit for a high‐degree moment functional is discussed and exhibited for atoms and molecules.