Order Density Matrix Research Articles

Excited states in Hartree–Fock theory

Hartree Fock theory is extended to excited states of interacting N-particle Fermionic systems. In contrast to the orbital-by-orbital and state-by-state approach followed by previous attempts to such an extension, the present methodology provides, in principle, a systematic determination of all excited states in the Hartree Fock treatment of electronic many-particle states. The excited states are defined as the eigenstates of the Nth order density matrix for a Slater determinant of order KN that represents the ground state of a non-interacting collection of K replicas of an interacting N-particle system. The equations determining the KN orbitals entangled in the collection ground state increase linearly with K, while they give rise to CNKN=KNN excited states of all orders, 1 to N. The entire spectrum is populated as K→∞. The method goes beyond Koopman’s theorem allowing the incorporation into the ground state of the effects of particle excitations to states of higher energy.

Quantum Chemical Topology as a Theory of Open Quantum Systems.

Although real space regions have been widely used in theoretical chemistry, not much effort has been devoted to treat them as open quantum systems. We embrace this task here, finding closed expressions for the density operator of a quantum subsystem in real space by tracing out the degrees of freedom in its complementary region. Our results are then linked to previous knowledge. For single-determinant descriptions it is shown that the entanglement orbitals coincide with Ponec's domain natural orbitals. In general, the subsystem density operator is written as a direct sum of a fixed number of electron sectors, with weights that turn out to be equal to those found within the theory of electron distribution functions. As a computational application we show how to obtain the global first order density matrix of a subsystem and its eigensolution in a couple of toy systems. In the multideterminant wave function case, the domain natural orbitals defined through this open system approach do not coincide with those of Ponec and, contrary to the latter, have always strictly positive occupations.

Number-squeezed and fragmented states of strongly interacting bosons in a double well

We present path integral ground state (PIGS) quantum Monte Carlo calculations for the ground state ($T = 0$) properties of repulsively interacting bosons in a three-dimensional external double well potential over a range of interaction strengths and potential parameters. We focus our calculations on ground state number statistics and the one-body density matrix in order to understand the level of squeezing and fragmentation that the system exhibits as a function of interaction strength. We compare our PIGS results to both a two-mode model and a recently-proposed eight-mode model. For weak interactions, the various models agree with the numerically exact PIGS simulations. However, the models fail to correctly predict the amount of squeezing and fragmentation exhibited by the PIGS simulations for strong interactions. One novel and somewhat surprising result from our simulations involves the relationship between squeezing and interaction strength: rather than a monotonic relationship between these quantities, we find that for certain barrier heights the squeezing increases as a function of interaction strength until it reaches a maximum, after which it decreases again. We also see a similar relationship between fragmentation and interaction strength. We discuss the physical mechanisms that account for this behavior and the implications for the design of atom interferometers, which can use squeezed states to reduce measurement uncertainty.

Conceptual Problem with Calculating Electron Densities in Finite Basis Density Functional Theory.

It is discussed that finite basis Density Functional Theory (DFT) calculations using a single Kohn-Sham determinant cannot reproduce, in a strict mathematical sense, the exact electron density corresponding to the same finite basis. This is because the DFT density derives from an idempotent first order density matrix, while the exact (full CI) density can only be obtained from a nonidempotent one. The problem is absent for the original Kohn-Sham integro-differential equations or if a strictly complete basis set is assumed.

Diagonal N-representability as a method for solving the representability problem

The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrodinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).

Natural bond orbitals: Local sets showing minimal intra-pair correlations and minimal unpaired electron populations

Adopting the 2nd order Density Matrix level, the usual Natural Bond Orbital (NBO) populations are explored accordingly: they are split into paired and unpaired (defined as “the simultaneous occurrence of an electron and a hole of opposite spins in an orbital”) populations. The Coulomb correlations and the unpaired electron populations are calculated explicitly, revealing new and unexplored features for NBO sets. It is shown that the ‘natural’ origin of these sets implies that the intra-pair correlations, and hence the unpaired electrons, are minimal in NBOs. The unpaired electron populations in valence NBOs provide a criterion for the quantal/classical, and hence for the active/inert behavior of bonds and molecules; this is tested in several well known prototypical systems. The interaction of two unpaired electrons between bonding and antibonding NBOs of two different bonds gives rise to a Heitler-London structure between the two bond regions; its weight (probability), and the included Coulomb and Fermi correlations are calculated in the basis of a spin-dependent formalism of generalized Density Matrices.

DensToolKit: A comprehensive open-source package for analyzing the electron density and its derivative scalar and vector fields

DensToolKit: A comprehensive open-source package for analyzing the electron density and its derivative scalar and vector fields

Radiation from a collapsing object is manifestly unitary.

The process of gravitational collapse excites the fields propagating in the background geometry and gives rise to thermal radiation. We demonstrate by explicit calculations that the density matrix corresponding to such radiation actually describes a pure state. While Hawking's leading order density matrix contains only the diagonal terms, we calculate the off-diagonal correlation terms. These correlations start very small, but then grow in time. The cumulative effect is that the correlations become comparable to the leading order terms and significantly modify the density matrix. While the trace of the Hawking's density matrix squared goes from unity to zero during the evolution, the trace of the total density matrix squared remains unity at all times and all frequencies. This implies that the process of radiation from a collapsing object is unitary.

Procedure for the analysis of interfragment donor-acceptor interactions in transition metal complexes

The procedure based on the first order density matrix has been suggested for the description of molecular electronic structures. The procedure includes: The calculation of interatomic bond-order indices and their σ-, π-, and δ-components; the construction of multicenter orbitals localized on fragments responsible for the formation of bonds inside fragments and between fragments; the classification of the constructed orbitals according to their population, valence activity, and orientation; the characterization of the realized donoracceptor abilities of the fragments (ligands and more complicated structural units). The interaction between fragments is described by a rather small number of multicenter orbitals localized on them. The concept of the valence activity of a polyatomic fragment has been introduced and the algorithm of its calculation has been offered. The procedure is illustrated by the analysis of electronic structures of ruthenium(II), platinum(II), and rhodium(I) complexes.

Considerations on describing non-singlet spin states in variational second order density matrix methods

Despite the importance of non-singlet molecules in chemistry, most variational second order density matrix calculations have focused on singlet states. Ensuring that a second order density matrix is derivable from a proper N-electron spin state is a difficult problem because the second order density matrix only describes one- and two-particle interactions. In pursuit of a consistent description of spin in second order density matrix theory, we propose and evaluate two main approaches: we consider constraints derived from a pure spin state and from an ensemble of spin states. This paper makes a comparative assessment of the different approaches by applying them to potential energy surfaces for different spin states of the oxygen and carbon dimer. We observe two major shortcomings of the applied spin constraints: they are not size consistent and they do not reproduce the degeneracy of the different states in a spin multiplet. First of all, the spin constraints are less strong when applied to a dissociated molecule than when they are applied to the dissociation products separately. Although they impose correct spin expectation values on the dissociated molecule, the dissociation products do not have correct spin expectation values. Secondly, both under "pure spin state conditions" and under "ensemble spin state" conditions is the energy a convex function of the spin projection. Potential energy surfaces for different spin projections of the same spin state may give a completely different picture of the molecule's bonding. The maximal spin projection always gives the most strongly constrained energy, but is also significantly more expensive to compute than a spin-averaged ensemble. In the dissociation limit, both the problem of nondegeneracy of equivalent spin projections, size-inconsistency and unphysical dissociation can be corrected by means of subspace energy constraints.

Quantum lithography under imperfect conditions: effects of loss and dephasing on two-photon interference fringes

We propose a method to simulate a two-photon interference fringe by using a density matrix in order to describe various types of imperfections in the input state. Using this method, we numerically discuss the influence of various imperfections in an input state, such as dephasing and misalignment, on the quality (visibility and period) of the two-photon interference fringes. Applying this method to experimental data, we succeeded in numerically reproducing a two-photon interference fringe using the experimentally obtained density matrix, in which almost no free fitting parameters are required. From the results, because the main cause of the degradation of an interference fringe was found to be the limited aperture size of a two-photon detector, we can observe a two-photon interference fringe with a visibility of up to 94% in the experiments if an efficient two-photon absorbing material or a two-photon detector with a sufficiently high spatial resolution can be used.

Variational second order density matrix study of $\mathrm{F_3^-}$F3−: Importance of subspace constraints for size-consistency

Variational second order density matrix theory under "two-positivity" constraints tends to dissociate molecules into unphysical fractionally charged products with too low energies. We aim to construct a qualitatively correct potential energy surface for F(3)(-) by applying subspace energy constraints on mono- and diatomic subspaces of the molecular basis space. Monoatomic subspace constraints do not guarantee correct dissociation: the constraints are thus geometry dependent. Furthermore, the number of subspace constraints needed for correct dissociation does not grow linearly with the number of atoms. The subspace constraints do impose correct chemical properties in the dissociation limit and size-consistency, but the structure of the resulting second order density matrix method does not exactly correspond to a system of noninteracting units.

Spin Kinetic Theory—Quantum Kinetic Theory in Extended Phase Space

The concept of phase space distribution functions and their evolution is used in the case of en enlarged phase space. In particular, we include the intrinsic spin of particles and present a quantum kinetic evolution equation for a scalar quasi-distribution function. In contrast to the proper Wigner transformation technique, for which we expect the corresponding quasi-distribution function to be a complex matrix, we introduce a spin projection operator for the density matrix in order to obtain the aforementioned scalar quasi-distribution function. There is a close correspondence between this projection operator and the Husimi (or Q) function used extensively in quantum optics. Such a function is based on a Gaussian smearing of a Wigner function, giving a positive definite distribution function. Thus, our approach gives a Wigner-Husimi quasi-distribution function in extended phase space, for which the reduced distribution function on the Bloch sphere is strictly positive. We also discuss the gauge issue and the fluid moment hierarchy based on such a quantum kinetic theory.

Subsystem constraints in variational second order density matrix optimization: Curing the dissociative behavior

A previous study of diatomic molecules revealed that variational second-order density matrix theory has serious problems in the dissociation limit when the N-representability is imposed at the level of the usual two-index (P,Q,G) or even three-index (T(1),T(2)) conditions [H. Van Aggelen et al., Phys. Chem. Chem. Phys. 11, 5558 (2009)]. Heteronuclear molecules tend to dissociate into fractionally charged atoms. In this paper we introduce a general class of N-representability conditions, called subsystem constraints, and show that they cure the dissociation problem at little additional computational cost. As a numerical example the singlet potential energy surface of Be B(+) is studied. The extension to polyatomic molecules, where more subsystem choices can be identified, is also discussed.

A study of the Colle-Salvetti functional for ρ2 via an exactly soluble problem

The Colle-Salvetti functional for correlation energy has given remarkably good results. It is based on approximating the second order density matrix with a particularly judicious form and using the constraining equation between the first order and second order reduced density matrices. We have used an exactly soluble N-body problem to study some aspects of the functional approximation. This has been done by calculating the reduced density matrices exactly and comparing them to the approximate form proposed by Colle and Salvetti and by comparing the expectation values of physical quantities using the approximate and exact density matrices.

Photon polarization in the two-photon decay of heavy hydrogen-like ions

We have applied the density matrix and second-order perturbation theory in order to re-analyze the two-photon decay of hydrogen-like ions for the polarization of emitted light. Special attention is paid to the linear polarization of one of the photons, while the spin state of the second photon is supposed to be unobserved. For such an “angle-polarization” correlation of the two photons we investigate the contributions that arise from non electric-dipole terms in the expansion of the electron-photon interaction. Detailed calculations are performed for the 2s1/2 →1s1/2 and 3d5/2 →1s1/2 transitions in neutral hydrogen as well as in hydrogen-like uranium.

Wigner’s (2n + 1) rule for nonlinear Schrödinger equations

Wigner's (2n + 1) rule is proven for general total energy functionals, constructed from the expectation value of a linear Hamiltonian and a nonlinear func- tional of the electron density or of the first order density matrix. Such functionals are common in independent particle models, like the Kohn-Sham density functional or Hartree-Fock theories, but they occur as well in reaction-field type solvent effect models and range-separated hybrid density functional approaches. The fulfillment of the (2n + 1) rule is crucial for the development of efficient perturbation approaches for the treatment of one-electron and two-electron perturbations. A general transfor- mation formula is derived, that removes some of the (2n + 1)-rule violating matrix elementsoftheperturbationoperatorfromthegeneralexpressionofthearbitraryorder perturbational energy correction of the nonlinear problem.

Some simple explicit examples of correlated one‐electron theory and density matrix functionals

AbstractSome examples are presented displaying how general spin orbitals (GSOs) and their occupation numbers uniquely determine N‐electron antisymmetrized geminal power states. This provides a correlated one‐electron model that generalizes the independent particle model that is universally used in picturing chemical behavior. These examples also demonstrate that one‐electron models, involving GSOs and occupation numbers, and the one based on a first order density matrix are not equivalent. This difference is shown by displaying sets of different GSOs that have the same occupation numbers that correspond to the same density matrix but different N‐electron states. This observation also leads to the explicit construction of a density matrix functional. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Idempotency-Conserving Iteration Scheme for the One-Electron Density Matrix

For the first order density matrix P of a noninteracting N-electron problem, an iterative formula is presented that preserves the trace and idempotency of P so that no purification is needed. Hermiticity--which may be slightly violated in the course of the iteration--gets restored when the iteration converges and the converged P corresponds to the exact solution. For sparse P, the energy is obtained by an O(N) procedure that needs no prior knowledge of the chemical potential. Illustrative calculations in tight-binding and ab initio Hartree-Fock levels are presented.

Relaxation Equations in Operator Representation for a Multipolar Spin System (1/2, 5/2)

The theoretical description of relaxation processes in a scalar-coupled two-spin system containing a quadrupole nucleus (I = 1/2, S = 5/2) is presented, presuming that the relaxation is determined by dipole interaction between I and S spins, quadrupole interactions of the nucleus S, and the anisotropy of the chemical shift of both nuclei CSA (I) and CSA (S). The interference terms (cross correlation terms) between these types of interactions D-Q, D-CSA, and Q-CSA are also taken into account. Within the framework of the theory of a density matrix in the second order of perturbation theory, the relaxation equations in the operator representation for describing the longitudinal and transverse relaxation of a nucleus with spin 1/2 are derived. A formalism for the angular momentum operators of scalar coupled spins IS is developed, which makes it possible to calculate a spectrum pattern of a nucleus with a half spin, scalar coupled to a quadrupole nucleus with any spin. The proposed procedure has allowed us to find linear combinations of operators which can be considered as the projective operators responsible for each spectral component of spin 1/2, scalar coupled to a quadrupole nucleus (S = 5/2). The relaxation equations allowing investigation of the longitudinal and transverse relaxation of each component of the spectrum of a half-spin nucleus are obtained. The rates of longitudinal and transverse relaxation and also the rate of cross relaxation between the components are determined.