Structure of the first order Reduced Density Matrix in three electron systems: A Generalized Pauli Constraints assisted study

We investigate the structure of the one-body Reduced Density Matrix (1RDM) of three electron systems, i.e. doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use Configuration Interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e. the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in Reduced Density Matrix Functional Theory minimization schemes when Generalized Pauli Constraints are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned"

Generalized Pauli constraints in reduced density matrix functional theory

Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman's ensemble $N$-representability conditions. Recently, the topic of pure-state $N$-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble $N$-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone

Kinetic-Energy Density-Functional Theory on a Lattice

We present a kinetic-energy density-functional theory and the corresponding kinetic-energy Kohn-Sham (keKS) scheme on a lattice and show that by including more observables explicitly in a density-functional approach already simple approximation strategies lead to very accurate results. Here we promote the kinetic-energy density to a fundamental variable along side the density and show for specific cases (analytically and numerically) that there is a one-to-one correspondence between the external pair of on-site potential and site-dependent hopping and the internal pair of density and kinetic-energy density. Based on this mapping we establish two unknown effective fields, the mean-field exchange-correlation potential and the mean-field exchange-correlation hopping, that force the keKS system to generate the same kinetic-energy density and density as the fully interacting one. We show, by a decomposition based on the equations of motions for the density and the kinetic-energy density, that we can construct simple orbital-dependent functionals that outperform the corresponding exact-exchange Kohn-Sham (KS) approximation of standard density-functional theory. We do so by considering the exact KS and keKS systems and compare the unknown correlation contributions as well as by comparing self-consistent calculations based on the mean-field exchange for the keKS and the exact-exchange for the KS system, respectively

Conditions for describing triplet states in reduced density matrix functional theory

We consider necessary conditions for the one-body-reduced density matrix (1RDM) to correspond to a triplet wave-function of a two electron system. The conditions concern the occupation numbers and are different for the high spin projections, $S_z=\pm 1$, and the $S_z=0$ projection. Hence, they can be used to test if an approximate 1RDM functional yields the same energies for both projections. We employ these conditions in reduced density matrix functional theory calculations for the triplet excitations of two-electron systems. In addition, we propose that these conditions can be used in the calculation of triplet states of systems with more than two electrons by restricting the active space. We assess this procedure in calculations for a few atomic and molecular systems. We show that the quality of the optimal 1RDMs improves by applying the conditions in all the cases we studied

Reduced Density-Matrix Approach to Strong Matter-Photon Interaction

We present a first-principles approach to electronic many-body systems strongly coupled to cavity modes in terms of matter-photon one-body reduced density matrices. The theory is fundamentally non-perturbative and thus captures not only the effects of correlated electronic systems but accounts also for strong interactions between matter and photon degrees of freedom. We do so by introducing a higher-dimensional auxiliary system that maps the coupled fermion-boson system to a dressed fermionic problem. This reformulation allows us to overcome many fundamental challenges of density-matrix theory in the context of coupled fermion-boson systems and we can employ conventional reduced density-matrix functional theory developed for purely fermionic systems. We provide results for one-dimensional model systems in real space and show that simple density-matrix approximations are accurate from the weak to the deep-strong coupling regime. This justifies the application of our method to systems that are too complex for exact calculations and we present first results, which show that the influence of the photon field depends sensitively on the details of the electronic structure.Comment: 52 pages, 26 figures, plus supporting information of 24 page

Approximations based on density-matrix embedding theory for density-functional theories

Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina et al 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible

Biospectroscopy towards screening and diagnosis of cancer

Systems biology is an emerging science that combines high throughput investigation techniques to define the dynamic interplay between different biological regulatory systems in response to internal and external cues. Related technologies, genomics, epigenomics, transcriptomics, proteomics, metabolomics and toponomics have been applied to investigate models of carcinogenesis to identify committing initiating events. Vibrational spectroscopy has the potential to play an integral role within systems biology research approaches, as it is able to identify chemical bond alterations within molecules independent of where these molecules reside. Its integration with current “systems biology” methodologies can contribute in the identification of potential biomarkers of carcinogenesis and assist in their incorporation into clinical practice. Breast tissue undergoes cyclical and longitudinal molecular and histological alterations that are influenced by environmental factors. These factors may include diet and lifestyle in addition to parity, lactation and menopausal status and are implicated in carcinogenesis. Breast cancer may appear decades after the initial carcinogenic event. Available research in this area is limited to when early histological changes occur due to the difficulties imposed by the molecular and histological diversity of breast tissue. Vibrational spectroscopy in combination with powerful chemometric techniques has identified spatial and temporal mammary alterations in benign tissue. Prostate cancer is influenced by environmental factors. Its incidence is higher in populations adopting a Westernised lifestyle and diet and has increased over the past generation. This leads to the assumption that prostatic tissue composition may exhibit chronological alterations. Vibrational spectroscopy techniques were applied to matching prostatic tissues with benign prostatic hyperplasia collected from 1983 to 2013. Significant trans-generational segregation was identified. Spectral areas responsible for this segregation pointed towards epigenetic changes. Immunohistochemical studies for DNA methylation and hypomethylation supported these results. Vibrational spectroscopy techniques were also implemented to explore molecular changes between normal ovarian tissue, borderline ovarian tumours and malignant ovarian carcinomas. Different chemometric techniques were applied to discriminate cancers from controls. Similar techniques were able to segregate different types of epithelial ovarian carcinomas. The accurate diagnosis obtained using ATR-FTIR spectroscopy demonstrates its potential for development as an assisting tool for histopathological diagnosis. The endometrial-myometrial junction areas of benign uterine tissues were scrutinised by Synchrotron FTIR and FPA. These techniques in combination with multivariate analysis revealed clear segregation between the functionalis and basalis layers within the uterine crypts. The same techniques illustrated potential areas within these epithelial surfaces where different stem cell types may reside. Targeting the activation/ inactivation of these stem cells may have applications in the diagnosis and treatment of early uterine cancer

The chemical potential for the inhomogeneous electron liquid in terms of its kinetic and potential parts with special consideration of the surface pote ntial step and BCS-BEC crossover

The chemical potential $\mu$ of a many-body system is valuable since it carries fingerprints of phase changes. Here, we summarize results for $\mu$ for a thre e-dimensional electron liquid in terms of average kinetic and potential energie s per particle. The difference between $\mu$ and the energy per particle is fou nd to be exactly the electrostatic potential step at the surface. We also prese nt calculations for an integrable one-dimensional many-body system with delta f unction interactions, exhibiting a BCS-BEC crossover. It is shown that in the B CS regime the chemical potential can be expressed solely in terms of the ground -state energy per particle. A brief discussion is also included of the strong c oupling BEC limit.Comment: 4 pages 3 figure

Orbitals from local RDMFT: Are they Kohn-Sham or Natural Orbitals?

Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation