Quantum crystallography: A perspective

Extraction of the complete quantum mechanics from X‐ray scattering data is the ultimate goal of quantum crystallography. This article delivers a perspective for that possibility. It is desirable to have a method for the conversion of X‐ray diffraction data into an electron density that reflects the antisymmetry of an N‐electron wave function. A formalism for this was developed early on for the determination of a constrained idempotent one‐body density matrix. The formalism ensures pure‐state N‐representability in the single determinant sense. Applications to crystals show that quantum mechanical density matrices of large molecules can be extracted from X‐ray scattering data by implementing a fragmentation method termed the kernel energy method (KEM). It is shown how KEM can be used within the context of quantum crystallography to derive quantum mechanical properties of biological molecules (with low data‐to‐parameters ratio). © 2017 Wiley Periodicals, Inc.     

Conditions for accurate description of the electron density of atoms and molecules

Based on the Kato's cusp condition of the electron density and our recent relations for local strongly decaying properties in an electronic system, necessary conditions for trial electron densities of atomic and molecular systems are derived. These conditions take the form of integral‐differential equations, and their validity is verified numerically. The relevance of these conditions to the Thomas–Fermi problem in the orbital‐less density functional approach is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Some basic properties of the correlation matrices

On the basis of the properties of correlation matrices, it is shown here that the set of all the first‐order transition reduced density matrices of a system provide complete information about that system. Also, the interrelation between the properties of the correlation matrix and the 2‐RDM N‐representability conditions is studied. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Exchange and correlation energy in density functional theory

Several different versions of density functional theory (DFT) that satisfy Hohenberg–Kohn theorems are characterized by different definitions of a reference or model state determined by an N‐electron ground state. A common formalism is developed in which exact Kohn–Sham equations are derived for standard Kohn–Sham theory, for reference‐state density functional theory, and for unrestricted Hartree–Fock (UHF) theory considered as an exactly soluble model Hohenberg–Kohn theory. A natural definition of exchange and correlation energy functionals is shown to be valid for all such theories. An easily computed necessary condition for the locality of exchange and correlation potentials is derived. While it is shown that in the UHF model of DFT the optimized effective potential (OEP) exchange satisfies this condition by construction, the derivation shows that this condition is not, in general, sufficient to define an exact local exchange potential. It serves as a test to eliminate proposed local potentials that are not exact for ground states. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 521–525, 2000     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

Approach to N representability

In this paper, some operator endomorphisms which give rise to conditions for N representability of pth order density matrices are presented.     

Hellmann–Feynman theorem near the threshold

The features of the spectrum structure are considered for situations where some parameter λ of the N‐electron Hamiltonian reaches the threshold value η under which the discrete energy level falls into the continuous spectrum. The electron density properties are also studied. It is proved that for a sequence of the wave functions converging in energy to the lower bound of the continuous spectrum as λ approaches η the corresponding sequence of the electron densities converges to the density of the (N ? 1)‐electron ground state. The results generalize the Hellmann–Feynman theorem for the cases where only the one‐side energy derivatives exist or there is no limiting wave function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Bare Coulomb field and atomic reciprocal form factor

The reciprocal form factor of N‐electron closed shells systems in a bare Coulomb field is shown to be a spherically symmetric, positive, and decreasing function of the radial distance. Nonmonotonicities of the reciprocal form factor appear when studying bare Coulomb field open‐shell systems. Analysis of the weight of the interelectronic repulsion term is carried out for some isoelectronic series as well as neutral atoms with N = 1–103. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

An open‐source framework for analyzing N‐electron dynamics. I. Multideterminantal wave functions

The aim of the present contribution is to provide a framework for analyzing and visualizing the correlated many‐electron dynamics of molecular systems, where an explicitly time‐dependent electronic wave packet is represented as a linear combination of N‐electron wave functions. The central quantity of interest is the electronic flux density, which contains all information about the transient electronic density, the associated phase, and their temporal evolution. It is computed from the associated one‐electron operator by reducing the multideterminantal, many‐electron wave packet using the Slater‐Condon rules. Here, we introduce a general tool for post‐processing multideterminant configuration‐interaction wave functions obtained at various levels of theory. It is tailored to extract directly the data from the output of standard quantum chemistry packages using atom‐centered Gaussian‐type basis functions. The procedure is implemented in the open‐source Python program det CI@ORBKIT, which shares and builds on the modular design of our recently published post‐processing toolbox (Hermann et al., J. Comput. Chem. 2016, 37, 1511). The new procedure is applied to ultrafast charge migration processes in different molecular systems, demonstrating its broad applicability. Convergence of the N‐electron dynamics with respect to the electronic structure theory level and basis set size is investigated. This provides an assessment of the robustness of qualitative and quantitative statements that can be made concerning dynamical features observed in charge migration simulations. © 2017 Wiley Periodicals, Inc.     

Unusual Bonding and Properties in Main Group Element Chemistry: Rational Synthesis,Characterization, and Experimental Electron Density Determination of Mixed‐Valent Tetraphosphetes

Five dispirocyclic λ³,λ⁵‐tetraphosphetes [{R₂Si(NR¹)(NR²)P₂}₂] (R¹ = R² and R¹ ≠ R²) are easily prepared in almost quantitative yields via photolysis of the respective bis(trimethylsilyl)phosphanyldiazaphosphasiletidines with intense visible light. These deep‐yellow low‐coordinate phosphorus compounds can be considered as the first higher congeners of the well‐known cyclodiphosphazenes. The tetraphosphetes are remarkably stable in air and show unexpected molecular properties related to the unique bonding situation of the central four‐π‐electron four‐membered phosphorus ring. The extent of rhombic distortion of the central P₄ ring is remarkable due to an unusually acute angle at the σ²‐phosphorus atoms. All of the P?P bonds are approximately equal in length. The distances are in the middle of the range given by phosphorus single and double bonds. The anisotropic absorption of visible light that can easily be observed in the case of the yellow/colorless dichroic crystals of [{Me₂Si(NtBu)(NtBu)P₂}₂] and the exceptional ³¹P NMR chemical shift of the σ²‐phosphorus atoms are the most remarkable features of the λ³,λ⁵‐tetraphosphetes. In the case of [{Me₂Si(NtBu)(NtBu)P₂}₂], the Hansen–Coppens multipole model is applied to extract the electron density from high‐resolution X‐ray diffraction data obtained at 100 K. Static deformation density and topological analysis reveal a unique bonding situation in the central unsaturated P₄ fragment characterized by polar σ‐bonding, pronounced out‐of‐ring non‐bonding lone pair density on the σ²‐phosphorus atoms, and an additional non‐classical three‐center back‐bonding contribution.     

Theoretical Design of Multi‐Nitroxyl Organocatalysts with Enhanced Reactivity for Aerobic Oxidation

Higher catalytic performances of N,N′,N′′‐trihydroxyisocyanuric acid (THICA), N,N‐dihydroxypyromellitimide (NDHPI), and N‐hydroxynaphthalimide (NHNI) than that of N‐hydroxyphthalimide (NHPI) have been demonstrated recently in aerobic oxidation. Herein, the rational design of reactive multi‐nitroxyl organocatalysts has been addressed theoretically by using systematic analysis of some important properties and catalytic activities of yet‐to‐be‐synthesized catalysts. Our results show that 1) NHNI and its analogue, similar to THICA, unlike NHPI and others, are unsuitable for solvent‐ or mediator‐free catalysis due to their strong intramolecular hydrogen‐bonding interactions; 2) increasing the reactive hydroxyimide groups on the same aromatic ring, or doped N atoms or ionic‐pair groups onto the aromatic ring, can improve catalytic reactivity, whereas appropriate enlargement of conjugated aromatic systems results in unchanged activity; 3) the newly designed catalysts are more active than NHPI and NHNI and have catalytic activities comparable to NDHPI and THICA; 4) the ionic‐pair supported case is suggested to be a very active catalyst, even towards inert propane, and can be used as a novel model catalyst for further improvements. The present work will be helpful in designing reactive hydroxyimide organocatalysts.     

Density functionals in the presence of magnetic field

In this article, density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density ρ and paramagnetic current density j^p. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of current density functional theory, which establishes a one‐to‐one correspondence between the nondegenerate ground‐state and the particle and paramagnetic current density. Definition of N‐representable density pairs (ρ,j^p) is given and it is proven that the set of v‐representable densities constitutes a proper subset of the set of N‐representable densities. For a Levy–Lieb‐type functional Q(ρ,j^p), it is demonstrated that (i) it is a proper extension of the universal Hohenberg–Kohn functional to N‐representable densities, (ii) there exists a wavefunction ψ₀ such that , where H₀ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional F(ρ,j^p) is studied and proven to be equal the convex envelope of Q(ρ,j^p). For both Q and F, we give upper and lower bounds. © 2014 Wiley Periodicals, Inc.     

All-pair wave function and reduced variational equation for electronic systems

A new type of wave function is proposed for atomic and molecular systems. This all-pair function is constructed of N(N – 1)/2 identical geminals for N electrons. For systems with the highest multiplicity this is the full space part of the wave function. For closed shell systems it has to be multiplied by a Slater determinant according to the antisymmetry condition. In the case of maximal multiplicity a reduced variational equation is derived for the geminal. This equation is independent of the dimensionality of the system and contains the particle number as a multiplicative factor only. The method is extended to the closed shell case where a restriction has to be fulfilled. The reduction of the variational equation can be done only approximately. The use of identical geminals can be treated as a first approximation. An extension of the method, called the pair interdependent configuration interaction (PICI), is proposed. The special features of the method are discussed briefly.     

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons

The differential virial theorem (DVT) is an explicit relation between the electron density ρ( r ), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time‐dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential v_s( r ) from a given electron density. The resulting expression contains only ρ( r ), v_s( r ), kinetic energy density, and a new orbital‐dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.