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arXiv:1812.09291 · The Journal of Chemical Physics

Communication: Relating the pure and ensemble density matrix functional

Christian Schilling

Published Dec 21, 2018• 62 citations
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Abstract

A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT) suggests that the universal pure and ensemble functionals coincide on their common domain of pure N-representable one-matrices. We refute this by a comprehensive analysis of the geometric picture underlying Levy’s constrained search. Moreover, we then show that the ensemble functional follows instead as the lower convex envelop of the pure functional. It is particularly remarkable that the pure functional determines the ensemble functional even outside its own domain of pure N-representable one-matrices. From a general perspective, this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not necessarily circumvent the complexity of the one-body pure N-representability conditions (generalized Pauli constraints). Instead, the complexity may simply be transferred from the underlying space of pure N-representable one-matrices to the structure of the universal one-matrix functional.

DOI
10.1063/1.5080088

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A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT) suggests that the universal pure and ensemble functionals coincide on their common domain of pure N-representable one-matrices. We refute this by a comprehensive analysis of the geometric picture underlying Levy’s constrained search. Moreover, we then show that the ensemble functional follows instead as the lower convex envelop of the pure functional. It is particularly remarkable that the pure functional determines the ensemble functional even outside its own domain of pure N-representable one-matrices. From a general perspective, this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not necessarily circumvent the complexity of the one-body pure N-representability conditions (generalized Pauli constraints). Instead, the complexity may simply be transferred from the underlying space of pure N-representable one-matrices to the structure of the universal one-matrix functional.
We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.
In Phys. Rev. Lett . 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w -RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone’s pioneering work on ground state RDMFT as the special case w = (1,0, ···). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional’s domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.
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.
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.
Density functional theory constitutes the workhorse of modern electronic structure calculations due to its favourable computational cost despite the fact that it usually fails to describe strongly correlated systems. A particularly promising approach to overcome those difficulties is reduced density matrix functional theory (RDMFT): It abandons the complexity of the NNN-particle wave function and at the same time explicitly allows for fractional occupation numbers. It is the goal of this thesis to initiate and establish a bosonic RDMFT for both ground state and excited state energy calculations. Motivated by the Onsager and Penrose criterion which identifies RDMFT as a particularly suitable approach to describe Bose-Einstein condensates (BECs), we derive the universal functional for a homogeneous BEC in the Bogoliubov regime. Remarkably, the gradient of the universal functional is found to diverge repulsively in the regime of complete condensation. This introduces the new concept of a BEC force, which provides a universal explanation for quantum depletion since it is merely based on the geometry of quantum states. In the second part of the thesis, we propose and work out an ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour further highlights the potential of convex analysis for the development of functional theories in the future. Indeed by resorting to several concepts from convex analysis, we succeeded to provide a comprehensive foundation of w\boldsymbol{w}w-ensemble RDMFT for bosons which is further based on a generalization of the Ritz variational principle and a constrained search formalism. In particular, we solve the emerging NNN-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states.
Common features are identified in the properties of the pure (PF) and ensemble (EF) energy functionals defined within the density matrix functional theory (DMFT) as well as their free-form (FFF) approximations. These features rationalize abandoning, in the common DMFT practice, prohibitively complicated generalized Pauli constraints (GPCs). Specifically, it is shown that the exact PF and EF coincide on the set of the vvv-representable first-order reduced density matrices (1RDMs), while FFFs approximate the common PF and EF for such 1RDMs. It is revealed that for some types of FFFs, for example for the geminal-based approximations, the GPCs are essentially trivialized to the pairwise natural occupation numbers (NONs) degeneracy conditions, routinely imposed for closed-shell systems.