### Kohn-Sham Density Matrix and the Kernel Energy Method

POLKOSNIK Walter1,*(),MASSA Lou2,3

1. 1 The Graduate Center of the City University of New York, Physics Department, 365 5th Avenue, New York, NY 10016, USA
2 Hunter College of the City University of New York, Chemistry Department, 695 Park Avenue, New York, NY 10065, USA
3 The Graduate Center of the City University of New York, Physics and Chemistry Departments, 365 5th Avenue, New York, NY 10016, USAThe Graduate Center of the City University of New York, Physics and Chemistry Departments, 365 5th Avenue, New York, NY 10016, USA
• 收稿日期:2017-11-08 发布日期:2018-03-20
• 通讯作者: POLKOSNIK Walter E-mail:wpolkosnik@gradcenter.cuny.edu

### Kohn-Sham Density Matrix and the Kernel Energy Method

Walter POLKOSNIK1,*(),Lou MASSA2,3

1. 1 The Graduate Center of the City University of New York, Physics Department, 365 5th Avenue, New York, NY 10016, USA
2 Hunter College of the City University of New York, Chemistry Department, 695 Park Avenue, New York, NY 10065, USA
• Received:2017-11-08 Published:2018-03-20
• Contact: Walter POLKOSNIK E-mail:wpolkosnik@gradcenter.cuny.edu

The kernel energy method (KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries. KEM breaks a molecule into smaller subsets, called kernels, for the purposes of calculation. The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy. A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals. In this study, the kernel expansion for the density matrix is examined in the context of density functional theory (DFT) Kohn-Sham (KS) calculations. A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined, and is then converted into a normalized projector by using the Clinton algorithm. Such normalized projectors are factorizable into linear combination of atomic orbitals (LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis. Both straightforward KEM energies and energies from a normalized, idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion. Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules. In the case of the proof-of-concept system, calculations were performed using the STO-3G and 6-31G(d, p) bases over a range of atomic separations, some very far from equilibrium. The water cluster was calculated in the 6-31G(d, p) basis at an equilibrium geometry. The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases. In the case of the water cluster, the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result. The KS density matrices of this study are applicable to quantum crystallography.

Abstract:

The kernel energy method (KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries. KEM breaks a molecule into smaller subsets, called kernels, for the purposes of calculation. The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy. A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals. In this study, the kernel expansion for the density matrix is examined in the context of density functional theory (DFT) Kohn-Sham (KS) calculations. A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined, and is then converted into a normalized projector by using the Clinton algorithm. Such normalized projectors are factorizable into linear combination of atomic orbitals (LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis. Both straightforward KEM energies and energies from a normalized, idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion. Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules. In the case of the proof-of-concept system, calculations were performed using the STO-3G and 6-31G(d, p) bases over a range of atomic separations, some very far from equilibrium. The water cluster was calculated in the 6-31G(d, p) basis at an equilibrium geometry. The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases. In the case of the water cluster, the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result. The KS density matrices of this study are applicable to quantum crystallography.