Generalized chirality and symmetry deficiency

Some of the elementary properties of molecular electron densities are studied from the perspectives of generalizations of symmetry, symmetry deficiency, and in particular, chirality. A simple, information‐theoretical proof of the Hohenberg–Kohn theorem is discussed, and the information contents of local and global molecular electron densities are compared using a formulation of quantum chemistry on a compact manifold. One result, the “holographic electron density theorem”, involving a compactification step combined with analytic continuation, gives a tool for comparing local and global symmetry properties. The compact manifold quantum chemistry approach leads to a precise statement on the role of local molecular regions in determining global properties of complete, boundaryless molecules, resulting in constraints on their symmetry, chirality, and other types of symmetry deficiencies. A special similarity measure, the SLT measure, is used for generalized density domain comparisons, suitable in general for the comparison of semilattices with a tree structure. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fundamental measure theory in cylindrical geometry

Density functional theory as proposed by Rosenfeld [Phys. Rev. Lett. 63, 980 (1989)] is used to study hard sphere mixture exposed by cylindrically symmetric external field. Exploiting the symmetry of the system, explicit formulas for the weighted densities are derived. The resulting density profiles are compared with new grand canonical Monte Carlo simulations. The comparison reveals very good agreement between the predicted and simulated results even at high densities and very narrow pores. Finally, simple algorithms for computing complete elliptic functions of the first and second kinds that occur in the derived formulae are presented to make the paper self-contained.     

Tracing the Fingerprint of Chemical Bonds within the Electron Densities of Hydrocarbons: A Comparative Analysis of the Optimized and the Promolecule Densities

The equivalence of the molecular graphs emerging from the comparative analysis of the optimized and the promolecule electron densities in two hundred and twenty five unsubstituted hydrocarbons was recently demonstrated [Keyvani et al. Chem. Eur. J. 2016 , 22, 5003]. Thus, the molecular graph of an optimized molecular electron density is not shaped by the formation of the C?H and C?C bonds. In the present study, to trace the fingerprint of the C?H and C?C bonds in the electron densities of the same set of hydrocarbons, the amount of electron density and its Laplacian at the (3, ?1) critical points associated with these bonds are derived from both optimized and promolecule densities, and compared in a newly proposed comparative analysis. The analysis not only conforms to the qualitative picture of the electron density build up between two atoms upon formation of a bond in between, but also quantifies the resulting accumulation of the electron density at the (3, ?1) critical points. The comparative analysis also reveals a unified mode of density accumulation in the case of 2318 studied C?H bonds, but various modes of density accumulation are observed in the case of 1509 studied C?C bonds and they are classified into four groups. The four emerging groups do not always conform to the traditional classification based on the bond orders. Furthermore, four C?C bonds described as exotic bonds in previous studies, for example the inverted C?C bond in 1,1,1‐propellane, are naturally distinguished from the analysis.     

On the theoretical foundation of Walsh's rules of molecular geometry in terms of the Hellmann–Feynman theorem

A new interpretation of the ordinate in a Walsh diagram for a polyatomic molecule is suggested in terms of the Hellmann–Feynman theorem. This makes use of the fact that in a single-configurational MO wave function the total one-electron density is the sum of individual densities in the occupied orbitals. Walsh-type diagrams have been constructed for three different molecules, water, ammonia and hydrogen peroxide. In H₂O and NH₃ calculation of the force, and thus of the energy, in terms of the valence angle, is made on the assumption that the central (heavy) atom is kept fixed while each of the lighter atoms moves in a plane containing the principal symmetry axis and the relevant bond, in a totally symmetric fashion; for H₂O₂ the two oxygen atoms are kept fixed. The angular correlation diagrams obtained reproduce the general features of those obtained by plotting Hartree–Fock MO energies as functions of the valence angles. The conclusion emerges that the force formulation provides a satisfactory pictorial basis for understanding molecular geometry in terms of the balance between the electron–nucleus attractive forces resulting from the charge densities in the occupied MO'S , and the nuclear repulsive forces. However, in the absence of highly accurate charge distributions such an approach is unsuitable for the quantitative prediction of molecular quantities such as valence angles, force constants or energy barriers.     

Toward a global maximization of the molecular similarity function: Superposition of two molecules

A quantum similarity measure between two molecules is normally identified with the maximum value of the overlap of the corresponding molecular electron densities. The electron density overlap is a function of the mutual positioning of the compared molecules, requiring the measurement of similarity, a solution of a multiple-maxima problem. Collapsing the molecular electron densities into the nuclei provides the essential information toward a global maximization of the overlap similarity function, the maximization of which, in this limit case, appears to be related to the so-called assignment problem. Three levels of approach are then proposed for a global search scanning of the similarity function. In addition, atom—atom similarity Lorentzian potential functions are defined for a rapid completion of the function scanning. Performance is tested among these three levels of simplification and the Monte Carlo and simplex methods. Results reveal the present algorithms as accurate, rapid, and unbiased techniques for density-based molecular alignments. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 826–846, 1997     

Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions

Scalar-relativistic, all-electron density functional theory (DFT) calculations were done for free, neutral atoms of all elements of the periodic table using the universal Gaussian basis set. Each core, closed-subshell contribution to a total atomic electron density distribution was separately fitted to a spherical electron density function: a linear combination of s-type Gaussian functions. The resulting core subshell electron densities are useful for systematically and compactly approximating total core electron densities of atoms in molecules, for any atomic core defined in terms of closed subshells. When used to augment the electron density from a wave function based on a calculation using effective core potentials (ECPs) in the Hamiltonian, the atomic core electron densities are sufficient to restore the otherwise-absent electron density maxima at the nuclear positions and eliminate spurious critical points in the neighborhood of the atom, thus enabling quantum theory of atoms in molecules (QTAIM) analyses to be done in the neighborhoods of atoms for which ECPs were used. Comparison of results from QTAIM analyses with all-electron, relativistic and nonrelativistic molecular wave functions validates the use of the atomic core electron densities for augmenting electron densities from ECP-based wave functions. For an atom in a molecule for which a small-core or medium-core ECPs is used, simply representing the core using a simplistic, tightly localized electron density function is actually sufficient to obtain a correct electron density topology and perform QTAIM analyses to obtain at least semiquantitatively meaningful results, but this is often not true when a large-core ECP is used. Comparison of QTAIM results from augmenting ECP-based molecular wave functions with the realistic atomic core electron densities presented here versus augmenting with the limiting case of tight core densities may be useful for diagnosing the reliability of large-core ECP models in particular cases. For molecules containing atoms of any elements of the periodic table, the production of extended wave function files that include the appropriate atomic core densities for ECP-based calculations, and the use of these wave functions for QTAIM analyses, has been automated.     

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

Symmetry simplifications are introduced in configuration interaction (CI ) by reducing the number of symmetry-allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group S_n, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.     

Evaluation of screened nuclear attraction and electron repulsion molecular integrals over Gaussian basis functions

Analytical solutions to the Yukawa-like screened Coulomb nuclear attraction and electron repulsion molecular basic integrals, as well as to the basic integrals required to compute the virial coefficient, over Gaussian basis functions, are derived and cast into a practical closed form, suitable to interface with modern codes for the calculation of molecular electronic structure. © 1997 John Wiley & Sons, Inc.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Covalent bond orders and atomic anisotropies from iterated stockholder atoms

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.     

On the application of complete orthonormal atomic basis sets to x-rays diffracted by a thermally excited molecule

The application of a formalism developed for an orthonormal basis set molecular orbital theory enables a separation of the electron density into unique one-center terms. This separation permits easy incorporation of thermal motion into the molecular X-ray scattering factor and allows comparison of theoretical and experimental electron and charge densities. If theoretical basis functions are employed, the number of parameters needed to characterize the electron density is shown to be equal to the number of basis functions.     

Second-order quantum similarity measures from intracule and extracule densities

The calculation of quantum similarity measures from second-order density functions contracted to intracule and extracule densities obtained at the Hartree-Fock level is presented and applied to a series of atoms, (He, Li, Be, and Ne), isoelectronic molecules (C₂H₂, HCN, CNH, CO, and N₂), and model hydrogen-transfer processes (H₂/H⁺, H₂/Hot, H₂/H⁻). Second-order quantum similarity measures and indices are found to be suitable measures for quantitatively analyzing electron-pair density reorganizations in atoms, molecules, and chemical processes. For the molecular series, a comparative analysis between the topology of pairwise similarity functions as computed from one-electron, intracule, and extracule densities is carried out and the assignment of each particular local similarity maximum to a molecular alignment discussed. In the comparative study of the three hydrogen-transfer reactions considered, second-order quantum similarity indices are found to be more sensitive than first-order indices for analyzing the electron-density reorganization between the reactant complex and the transition state, thus providing additional insights for a better understanding of the mechanistic aspects of each process. Received: 7 July 1997 / Accepted: 29 October 1997     

A note on the importance of including monoatomic overlap densities in the calculation of CNDO/2 charge distributions

It is demonstrated that in the calculation of CNDO/2 charge density distributions the monoatomic overlap densities necessarily must be taken into account. Otherwise electron densities are obtained which are not invariant to molecular rotations and generally of wrong symmetry.     

Cubic response functions in time-dependent density functional theory

We present density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively, the quasienergy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly-through the time dependence of the density itself rather than through the form of the exchange-correlation functionals-it generalizes previous time-dependent implementations in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange correlation at the generalized-gradient approximation level and fractional exact Hartree-Fock exchange. With our implementation the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. As illustration we explore the use of various combinations of functionals for applications of nonlinear optical hyperpolarizabilities of a few centrosymmetric systems; molecular nitrogen, benzene, and the C(60) fullerene. Considering that vibrational, solvent, and local field factors effects are left out, we find in general that very good experimental agreement can be obtained for the second dynamic hyperpolarizability of these systems. It is shown that a treatment of the response of the density beyond the local density approximation gives a significant effect. The use of different functional combinations are motivated and discussed, and it is concluded that the choice of higher order kernels can be of similar importance as the choice of the potential which governs the Kohn-Sham orbitals.     

To What Extent are “Atoms in Molecules” Structures of Hydrocarbons Reproducible from the Promolecule Electron Densities?

The “atoms in molecules” structures of 225 unsubstituted hydrocarbons are derived from both the optimized and the promolecule electron densities. A comparative analysis demonstrates that the molecular graphs derived from these two types of electron densities at the same geometry are equivalent for almost 90 % of the hydrocarbons containing the same number and types of critical points. For the remaining 10 % of molecules, it is demonstrated that by inducing small perturbations, through the variation of the used basis set or slight changes in the used geometry, the emerging molecular graphs from both densities are also equivalent. Interestingly, the (3, ?1) critical point between two “non‐bonded” hydrogen atoms, which triggered “H?H bonding” controversy is also observed in the promolecule densities of certain hydrocarbons. Evidently, the topology of the electron density is not dictated by chemical bonds or strong interactions and deformations induced by the interactions of atoms in molecules have a quite marginal role, virtually null, in shaping the general traits of the topology of molecular electron densities of the studied hydrocarbons, whereas the key factor is the underlying atomic densities.     

Molecular electronic density fitting using elementary Jacobi rotations under atomic shell approximation

Fitted electron density functions constitute an important step in quantum similarity studies. This fact not only is presented in the published papers concerning quantum similarity measures (QSM), but also can be associated with the success of the developed fitting algorithms. As has been demonstrated in previous work, electronic density can be accurately fitted using the atomic shell approximation (ASA). This methodology expresses electron density functions as a linear combination of spherical functions, with the constraint that expansion coefficients must be positive definite, to preserve the statistical meaning of the density function as a probability distribution. Recently, an algorithm based on the elementary Jacobi rotations (EJR) technique was proven as an efficient electron density fitting procedure. In the preceding studies, the EJR algorithm was employed to fit atomic density functions, and subsequently molecular electron density was built in a promolecular way as a simple sum of atomic densities. Following previously established computational developments, in this paper the fitting methodology is applied to molecular systems. Although the promolecular approach is sufficiently accurate for quantum QSPR studies, some molecular properties, such as electrostatic potentials, cannot be described using such a level of approximation. The purpose of the present contribution is to demonstrate that using the promolecular ASA density function as the starting point, it is possible to fit ASA-type functions easily to the ab initio molecular electron density. A comparative study of promolecular and molecular ASA density functions for a large set of molecules using a fitted 6-311G atomic basis set is presented, and some application examples are also discussed.     

Topological analysis of electron density distribution taken from a pseudopotential calculation

Theoretical studies of the electron density topology at the bond critical point for some small molecules, Ti, and Mo organometallic complexes were undertaken in order to understand the reason for the failure of the topological analysis of the coreless electron densities obtained from a pseudopotential calculation. We show that the absence of the core electron density is the main reason for such behavior. The erratic behavior of the effective core potentials electron densities can be corrected by adding atomic electron core density obtained from a single-atom Hartree-Fock calculation. The effect of orthogonalization of the core orbital with the valence orbitals was also investigated. © 1997 by John Wiley & Sons, Inc.