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.     

Algebraic structure of fermion density matrices. I

In this paper we define the algebraic structure of a reduced fermion density matrix. We relate the algebraic structure to certain symmetry properties of the reduced density matrix.     

Algebraic structure of fermion density matrices. II

The properties of the algebraic structure of fermion density matrices are studied. The algebraic structure of a density matrix leads to a more varied and detailed classification scheme than that offered by the usual shell structure approach. Investigation of the algebraic structure of fermion density matrices by the methods of algebraic topology leads to a classification scheme based on Betti numbers.     

Approximately N-representable density functional density matrices

For the first time, we obtain practical density matrices approximately N-representable by correlated-determinant wave functions, which are functionals of the electron density and entirely defined by information obtainable from the X-ray coherent diffraction experiment. © 1994 John Wiley & Sons, Inc.     

Spin density distribution in paramagnetic azafullerene

Hyperfine coupling (HFC) constants for ¹⁴N and ¹³C nuclei in azafullerene C₅₉N (1) were calculated. The HFC constants for the ¹H and ¹³C nuclei in the ^·CH₃ radical were calculated as functions of the pyramidal distortion of the angles at the carbon atom. Using this angular dependence, the spin density distribution of the unpaired -electron in 1 was determined. The spin density of the unpaired -electron in 1 is mainly localized around the nitrogen atom.__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2372–2374, November, 2004.     

Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single-particle states is defined as the (reduced) k-state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one- and two-particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density-matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off-diagonal elements of the density matrices.     

Geometry optimization in density functional methods

The geometry optimization in delocalized internal coordinates is discussed within the framework of the density functional theory program deMon. A new algorithm for the selection of primitive coordinates according to their contribution to the nonredundant coordinate space is presented. With this new selection algorithm the excessive increase in computational time and the deterioration of the performance of the geometry optimization for floppy molecules and systems with high average coordination numbers is avoided. A new step selection based on the Cartesian geometry change is introduced. It combines the trust radius and line search method. The structure of the new geometry optimizer is described. The influence of the SCF convergence criteria and the grid accuracy on the geometry optimization are discussed. A performance analysis of the new geometry optimizer using different start Hessian matrices, basis sets and grid accuracies is given.     

Permutational symmetry of reduced density matrices

The problems of permutational symmetry of the density matrices in reduction are studied. Some necessary and sufficient conditions for N, [λ₁], [λ₂]-derivability problem are given.     

Traces of spin-adapted reduced density matrices

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ² and Ŝ_z operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.     

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.     

Spin density distribution in transition metal complexes

The computational approaches that can be used to calculate the spin density distribution in transition metal compounds are discussed, the characteristic trends involving spin delocalization and spin polarization mechanisms are summarized, and the characteristic shapes of the spin density distributions around a transition metal atom are presented. Reference is also made to experimental methods to determine spin density distributions and to incipient work in the field of high spin molecules and single-molecule magnets.     

Suggested calculations using transition density matrices

It is suggested that certain transition density matrices, N-representable in a limit, be used in a variational calculation. It is noted that such trial matrices should yield reasonable values for the ground state energies of small atoms or molecules provided a set of overlap integrals is maximised.

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Zusammenfassung Es wird vorgeschlagen, bestimmte Übergangsdichtematrizen, die im Limit N-darstellbar sind, in einer Variationsrechnung zu benutzen. Es wird festgestellt, daß solche Näherungsmatrizen gute Werte für die Energie des Grundzustandes kleiner Atome oder Moleküle geben sollten, falls im Satz von Überlappungsintegralen maximiert wird.

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Résumé On propose d'utiliser dans un calcul variationnel certaines matrices densité de transition, N représentables à la limite. Ces matrices d'essai devraient fournir des valeurs raissonnables pour l'énergie de l'état fondamental dans les petits atomes et les petites molécules à condition de maximiser un ensemble d'intégrales de recouvrement.

     

Spin virial theorem in the density functional theory

The spin virial theorem is derived in the density functional theory. The theorem establishes a relation between the differences of spin-up and -down kinetic and potential energies. The theorem is useful for checking the accuracy of spin orbitals. As an illustration, the example of the Xα method is studied. © 1994 John Wiley & Sons, Inc.     

Spin density distribution in oxamato-type transition metal complexes

The spin density distribution of the paramagnetic [ⁿBu₄N]₂[Cu(dana)] dana = N,N′-(naphthalene-2,3-diyl)-bis(oxamato) has been derived from angular dependent electron paramagnetic resonance measurements at room temperature. The results indicate a noticeable spin density transfer from the central metal to the coordinated N and O atoms. Quantum chemical studies using density functional theory reinforce the results.