Single determinant N‐representability and the kernel energy method applied to water clusters

Abstract

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single-determinant N-representable one-body density matrix. A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single-determinant N-representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N-representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.

Quantum Crystallography: Current Developments and Future Perspectives

“…[171]). Notice that the QCr/KEM procedure extracts the complete quantum mechanics based on the X-ray experiment.…”

Fast quantum crystallography

“…The process for obtaining the NOs involves diagonalizing the density matrix for the full molecule. This is constructed from the “fragment” density matrices of the kernels and double kernels indicated in Equation . The basis set for the full molecule is obtained by collecting together the basis functions from all kernels which constitute the full molecule.…”

“…If we call the basis set for the full molecule ψ , then as in Equation , the density matrix for the full molecule is: where ψ ( r ) is a column vector of atomic orbital basis functions and where a direct product is implied with its complex conjugate transpose ( ψ † ( r ′)). In this last equation, R is the “augmented” matrix constructed by placing the density matrix of the kernels and double kernels into their appointed positions defined by the full molecule basis ψ .…”

“…The full molecule density matrix R is approximated from the augmented matrices of the single and double kernels according to the usual KEM formula, that is: where m is the number of single kernels. The augmented matrices were applied very recently in the context of single determinant density matrices . The augmented matrices can also be used to construct density matrices consistent with more general wavefunction forms such as configuration interaction (CI)—multideterminental—wavefunctions.…”

“…This is constructed from the "fragment" density matrices of the kernels and double kernels indicated in Equation (15). [24] The basis set for the full molecule is obtained by collecting together the basis functions from all kernels which constitute the full molecule. If we call the basis set for the full molecule ψ, then as in Equation (6), the density matrix for the full molecule is:…”