Geometry of density matrices. VI. Superoperators and unitary invariance

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.     

On the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT): a letter from Richard F. W. Bader to Lou Massa

A 31-year-old letter from Professor Richard F. W. Bader to Professor Lou Massa outlining the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT) especially with regard to the first Hohenberg-Kohn theorem is brought to light. This connection has not often been the topic of such a focused review by Bader and is presented here for the first time. The scientific importance of this letter is, in the opinion of the presenter, as timely today as it was back then in 1986. In Bader’s own opening words: “... that if I sent you a summary of what I think are the important connections between our work and density functional theory, ...”. He then takes us in a grand tour of the foundations of QTAIM culminating into the antecedents of a paper he later published with Professor Pierre Becker, whereby the Hohenberg-Kohn theorem is shown to operate at the level of an atom-in-a-molecule. Bader closes his letter by suggesting to Massa: “Study these two charge distributions – they are proof of the theorem of Hohenberg and Kohn”. By that Bader meant that when the charge distributions of two atoms or groups are identical within a given precision, then the kinetic and total energy contributions of these atoms to the corresponding molecular quantities are also identical. It is revealing to follow the intellectual threads weaved by Bader which provides us with a glimpse of his thought processes and intuition that guided him to some of his key discoveries. The lucidity, rigor, and clarity characteristic of Bader and the informality of style of a letter makes it of pedagogic and historic interest.

     

Electronegativities and the bonding character of molecular orbitals: A remark

From the density functional theory of Hohenberg-Kohn it is possible to prove that a molecular orbital is bonding (antibonding) if its electronegativity is larger (smaller) than the electronegativities of the corresponding atomic orbitals.     

Communication: Constrained search formulation of the ground state energy as a functional of an idempotent one-matrix

Despite the fact that idempotent one-particle reduced density matrices are pervasive in quantum chemistry, the understanding of a general energy functional of such idempotent density matrices for the ground state energy has been lacking. By a constrained search, we show the structure of the general functional, illuminating the contributions from various terms. For the examples of the "best idempotent density matrix" and Kohn-Sham idempotent density matrices, we contrast the functional forms and suggest how the best idempotent density matrix approach may be a good starting point for further development.     

New constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix

Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.     

Multireference spin-adapted variant of density functional theory

A new Kohn-Sham formalism is developed for studying the lowest molecular electronic states of given space and spin symmetry whose densities are represented by weighted sums of several reference configurations. Unlike standard spin-density functional theory, the new formalism uses total spin conserving spin-density operators and spin-invariant density matrices so that the method is fully spin-adapted and solves the so-called spin-symmetry dilemma. The formalism permits the use of an arbitrary set of reference (noninteracting) configurations with any number of open shells. It is shown that the requirement of degeneracy of the total noninteracting energies of the reference configurations (or configuration state functions) is equivalent to the stationary condition of the exact energy relative to the weights of the configurations (or configuration state functions). Consequently, at any molecular geometry, the weights can be determined by minimization of the energy, and, for given reference weights, the Kohn-Sham orbitals can be determined. From this viewpoint, the developed theory can be interpreted as an analog of the multiconfiguration self-consistent field approach within density functional theory.     

Two-electron reduced density matrices from the anti-Hermitian contracted Schrodinger equation: enhanced energies and properties with larger basis sets

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.     

Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver

The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.     

First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules

Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.     

MALDI Matrices for the Analysis of Low Molecular Weight Compounds: Rational Design,Challenges and Perspectives

The analysis of low molecular weight (LMW) compounds is of great interest to detect small pharmaceutical drugs rapidly and sensitively, or to trace and understand metabolic pathways. Matrix-assisted laser desorption/ionization mass spectrometry (MALDI MS) plays a central role in the analysis of high molecular weight (bio)molecules. However, its application for LMW compounds is restricted by spectral interferences in the low m/z region, which are produced by conventional organic matrices. Several strategies regarding sample preparation have been investigated to overcome this problem. A different rationale is centred on developing new matrices which not only meet the fundamental requirements of good absorption and high ionization efficiency, but are also vacuum stable and “MALDI silent”, i. e., do not give matrix-related signals in the LMW area. This review gives an overview on the rational design strategies used to develop matrix systems for the analysis of LMW compounds, focusing on (i) the modification of well-known matrices, (ii) the search for high molecular weight matrices, (iii) the development of binary, hybrid and nanomaterial-based matrices, (iv) the advance of reactive matrices and (v) the progress made regarding matrices for negative or dual polarity mode.     

Inward Matrix Product Algebra and Calculus as Tools to Construct Space–Time Frames of Arbitrary Dimensions

In this study, inward matrix products are used to construct a theoretical framework where new space-time structures of arbitrary dimensions can be built up. The mathematical theory, based on inward matrix algebra, allows the derivation and integration of vectors and matrices composed by well-behaved functional elements. Every function element is associated at least to a linearly independent variable connected to such an element. As examples are discussed first the construction of general density functions, followed by the reformulation of the time-dependent Schrödinger equation. A general N-dimensional classical universe is presented, where not only space but also time, mass, energy and other related physical properties acquire an arbitrary hypermatrix structure. In this hypothetical framework scalar values related to physical quantities can be alternatively associated to cosine-like measures in the chosen spaces. Finally, simple problems on special relativity are briefly discussed from this point of view.     

Projected quasiparticle theory for molecular electronic structure

We derive and implement symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations and apply them to the molecular electronic structure problem. All symmetries (particle number, spin, spatial, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. We show that the resulting method yields a comprehensive black-box treatment of static correlations with effective one-electron (mean-field) computational cost. The ensuing wave function is of multireference character and permeates the entire Hilbert space of the problem. The energy expression is different from regular HFB theory but remains a functional of an independent quasiparticle density matrix. All reduced density matrices are expressible as an integration of transition density matrices over a gauge grid. We present several proof-of-principle examples demonstrating the compelling power of projected quasiparticle theory for quantum chemistry.     

Calculation of thermodynamic properties of species of biochemical reactants using the inverse Legendre transform

The determination of apparent equilibrium constants and heats of enzyme-catalyzed reactions provides a way to determine Delta(f)G degrees and Delta(f)H degrees of species of biochemical reactants. These calculations are more difficult than the calculation of transformed thermodynamic properties from species properties, and they are an application of the inverse Legendre transform. The Delta(f)G degrees values of species of a reactant can be calculated from an apparent equilibrium constant if the Delta(f)G degrees values are known for all the species of all the other reactants and the pKs of the reactant of interest are known. The Delta(f)H degrees of species of a reactant can be calculated from the heat of reaction if the Delta(f)H degrees values are known for all species of the other reactants and Delta(f)G degrees values are known for all species in the reaction. The standard enthalpies of acid dissociation of the reactant of interest are also needed. The inverse Legendre transformation is accomplished by using computer programs to set up the simultaneous equations that involve the Delta(f)H degrees of the species and solving them. Thirty two new species matrixes providing Delta(f)G degrees values and eight new species matrixes providing Delta(f)H degrees values are calculated. It is the specificity and speed of enzyme-catalyzed reactions that make it possible to determine standard thermodynamic properties of complicated species in aqueous solution that could never have been obtained classically.     

Optimal control landscapes for quantum observables

The optimal control of quantum systems provides the means to achieve the best outcome from redirecting dynamical behavior. Quantum systems for optimal control are characterized by an evolving density matrix and a Hermitian operator associated with the observable of interest. The optimal control landscape is the observable as a functional of the control field. The features of interest over this control landscape consist of the extremum values and their topological character. For controllable finite dimensional quantum systems with no constraints placed on the controls, it is shown that there is only a finite number of distinct values for the extrema, dependent on the spectral degeneracy of the initial and target density matrices. The consequences of these findings for the practical discovery of effective quantum controls in the laboratory is discussed.     

Small Potassium Clusters

The molecules K ₂ , K ₃ , and K ₄ (structure shown on the right) have been isolated in krypton matrices at 15 K and characterized by Raman spectroscopy. Comparison of the experimental data with density functional calculations supports the prediction that potassium clusters are not only bonded by the valence electrons, but that there is also a contribution from the core electrons.     

Obtaining the two-body density matrix in the density matrix renormalization group method

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.     

Modified Yamaguchi Reagent: Convenient and Efficient Esterification

A convenient and efficient esterification method that used a modified Yamaguchi reagent (2,4,6-trichlorobenzoyl chloride–4-dimethylaminopyridine) and avoided use not only of intractable acid chloride but also of acid anhydrides was proposed. The reaction mechanism was described by Fourier transform infrared spectroscopy and supported by a density functional theory calculation.     

Comparative analysis of local spin definitions

This work provides a survey of the definition of electron spin as a local property and its dependence on several parameters in actual calculations. We analyze one-determinant wave functions constructed from Hartree-Fock and, in particular, from Kohn-Sham orbitals within the collinear approach to electron spin. The scalar total spin operators S2 and Sz are partitioned by projection operators, as introduced by Clark and Davidson, in order to obtain local spin operators SASB and SzA, respectively. To complement the work of Davidson and co-workers, we analyze some features of local spins which have not yet been discussed in sufficient depth. The dependence of local spin on the choice of basis set, density functional, and projector is studied. We also discuss the results of Sz partitioning and show that SzA values depend less on these parameters than SASB values. Furthermore, we demonstrate that for small organic test molecules, a partitioning of Sz with preorthogonalized Lowdin projectors yields nearly the same results as one obtains using atoms-in-molecules projectors. In addition, the physical significance of nonzero SASB values for closed-shell molecules is investigated. It is shown that due to this problem, SASB values are useful for calculations of relative spin values, but not for absolute local spins, where SzA values appear to be better suited.