Mathematical aspects of the LCAO MO first order density function (1): atomic partition, metric structure and practical applications

Partition of the first order density function within LCAO MO theory permits the definition of Mulliken and Minkowski metric spaces. The metric matrices obtained become useful to connect density atomic partition with quantum similarity measures, Mulliken populations, EHT, expectation values of Hermitian operators and the comparison of two density distributions.     

Air microwave-induced plasma: Relation between ion density and atomic oxygen density

Electron temperature and ion density are measured in an air microwave-induced plasma (2450 MHz) by means of a floating double probe. A 'cinetic scheme for ion formation and decay is set up, and a relationship between atomic oxygen and ion densities is obtained. From this relationship an order-of-magnitude of atomic oxygen concentration in the discharge is derived and compared with results obtained by optical actinometry in another work.     

Functional groups expressed as graphs extracted from the Laplacian of the electron density

The valence charge concentration shell, as determined by the Laplacian of the electron density, is used as a source of quantum topological graphs, called L‐graphs. A considerable number of such graphs are extracted from the ab initio wave functions of 31 molecules calculated at the B3LYP/6‐311+G(2d,p)//B3LYP/6‐311+G(2d,p) level, covering common functional groups in organic chemistry. We show how L‐graphs can be constructed from a largely transferable subgraph called atomic L‐graph. We investigate the topological stability of the L‐graphs as a function of the basis set. Reliable and consistent atomic L‐graphs are only obtained with basis sets of triple‐zeta quality or higher. The recurrence of invariant motifs or subgraphs in the L‐graphs enables the isolation of 16 atomic L‐graphs. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Benchmark density functional theory calculations for nanoscale conductance

We present a set of benchmark calculations for the Kohn-Sham elastic transmission function of five representative single-molecule junctions. The transmission functions are calculated using two different density functional theory methods, namely an ultrasoft pseudopotential plane-wave code in combination with maximally localized Wannier functions and the norm-conserving pseudopotential code SIESTA which applies an atomic orbital basis set. All calculations have been converged with respect to the supercell size and the number of k|| points in the surface plane. For all systems we find that the SIESTA transmission functions converge toward the plane-wave result as the SIESTA basis is enlarged. Overall, we find that an atomic basis with double zeta and polarization is sufficient (and in some cases, even necessary) to ensure quantitative agreement with the plane-wave calculation. We observe a systematic downshift of the SIESTA transmission functions relative to the plane-wave results. The effect diminishes as the atomic orbital basis is enlarged; however, the convergence can be rather slow.     

Voronoi deformation density (VDD) charges: Assessment of the Mulliken, Bader, Hirshfeld, Weinhold, and VDD methods for charge analysis

We present the Voronoi Deformation Density (VDD) method for computing atomic charges. The VDD method does not explicitly use the basis functions but calculates the amount of electronic density that flows to or from a certain atom due to bond formation by spatial integration of the deformation density over the atomic Voronoi cell. We compare our method to the well-known Mulliken, Hirshfeld, Bader, and Weinhold [Natural Population Analysis (NPA)] charges for a variety of biological, organic, and inorganic molecules. The Mulliken charges are (again) shown to be useless due to heavy basis set dependency, and the Bader charges (and often also the NPA charges) are not realistic, yielding too extreme values that suggest much ionic character even in the case of covalent bonds. The Hirshfeld and VDD charges, which prove to be numerically very similar, are to be recommended because they yield chemically meaningful charges. We stress the need to use spatial integration over an atomic domain to get rid of basis set dependency, and the need to integrate the deformation density in order to obtain a realistic picture of the charge rearrangement upon bonding. An asset of the VDD charges is the transparency of the approach owing to the simple geometric partitioning of space. The deformation density based charges prove to conform to chemical experience.     

Comparison of direct and flow integration based charge density population analyses

Different exhaustive and fuzzy partitions of the molecular electron density (rho) into atomic densities (rho(A)) are used to compute the atomic charges (Q(A)) of a representative set of molecules. The Q(A)'s derived from a direct integration of rho(A) are compared to those obtained from integrating the deformation density rho(def) = rho - rho(0) within each atomic domain. Our analysis shows that the latter methods tend to give Q(A)'s similar to those of the (arbitrary) reference atomic densities rho(A)(0) used in the definition of the promolecular density, rho(0) = SigmaArho(A)(0). Moreover, we show that the basis set independence of these charges is a sign not of their intrinsic quality, as commonly stated, but of the practical insensitivity on the basis set of the atomic domains that are employed in this type of methods.     

A note on atomic density

In atomic systems, electron density has a simple finite expansion in spherical harmonics times radial factors. The difficulties in the calculation of some radial factors are illustrated in the low‐lying states of the carbon atom. Single‐particle methods such as Hartree–Fock and approximate density functional theory cannot ensure the correct expansion of the density in spherical harmonics. Wave‐function methods are appropriate but, as some expansion terms are entirely due to correlation, these methods only will give correct results for high‐quality variational functions. Using full‐configuration integration (CI), all the terms predicted by the theory appear and are not negligible but the convergence of the term due to correlation toward its correct value is uncertain even for very large CI spaces. © 2012 Wiley Periodicals, Inc.     

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.     

Quantum-topological analysis of electron density in azachalcogenene crystals with aromatic substituents

A theoretical study was carried out to investigate the topological characteristics of electron density (DFT B3LYP/6-311G(d,p) ab initio basis set) for molecules and crystal structures of azachalcogenenes with Ar-S-N=S=N-S-Ar and Ar-S-N=S=N-S-Ar aryl substituents. The characteristics of electron density were determined at the critical points (3, ?1) corresponding to the S...S and Se...Se intramolecular contacts, which serve to close the S-N=S=N-S and S-N=S=N-S five-membered rings. The intermolecular interactions in crystals are described from the viewpoint of electron density analysis in the region of S…S, S…N, Se…N, and S…Hal intermolecular short contacts between the atomic pairs of interactant molecules.     

On the accuracy of density functional theory for iron-sulfur clusters

A simple, yet powerful wave function manipulation method was introduced utilizing a generalized ionic fragment approach that allows for systematic mapping of the wave function space for multispin systems with antiferromagnetic coupling. The use of this method was demonstrated for developing ground state electronic wave function for [2Fe-2S] and [Mo-3Fe-4S] clusters. Using well-defined ionic wave functions for ferrous and ferric irons, sulfide, and thiolate fragments, the accuracy of various density functionals and basis sets including effective core potentials were evaluated on a [4Fe-4S] cluster by comparing the calculated geometric and electronic structures with crystallographic data and experimental atomic spin densities from X-ray absorption spectroscopy, respectively. We found that the most reasonable agreement for both geometry and atomic spin densities is obtained by a hybrid functional with 5% HF exchange and 95% density functional exchange supplemented with Perdew's 1986 correlation functional. The basis set seems to saturate only at the triple-zeta level with polarization and diffuse functions. Reasonably preoptimized structures can be obtained by employing computationally less expensive effective core potentials, such as the Stuttgart-Dresden potential with a triple-zeta valence basis set. The extension of the described calibration methodology to other biologically important and more complex iron-sulfur clusters, such as hydrogenase H-cluster and nitrogenase FeMo-co will follow.     

Time-dependent density functional theory benchmarking for the calculations of atomic spectra: efficiency of exc-ETDZ basis set

We present a time-dependent density functional theory (TD-DFT) benchmarking of recently constructed basis set, namely exc-ETDZ (Guevara et al. in J Chem Phys 131: 064104, 2009) for predicting the atomic spectra of the first-row atoms. A systematic testing with 31 density functional methods has been performed to see whether convincing performance of this basis set carries over the TD-DFT formalism. The efficiency of exc-ETDZ basis set for reproducing atomic spectra has been compared with Pople- and Dunning-style basis sets. We focused on the atomic low-lying valence excited states with single excitation character for our benchmarking, and the calculated excitation energies were compared to experimental data. On average, the functionals providing the best match with exc-ETDZ basis are BMK, BH&HLYP and ωB97. Moreover, on the basis of comparison between the results of these superior functionals with CIS(D) estimates, it turned out that TD-DFT and CIS(D) errors are of the same order of magnitude, once the exc-ETDZ basis set is used. Finally, the results of present study indicate that different functionals show results that are highly dependent on the atomic configuration as well as the basis set.     

Nonempirical (double-hybrid) density functionals applied to atomic excitation energies: A systematic basis set investigation

We investigate here the lowest-energy (spin-conserving) excitation energies for the set of He-Ne atoms, with the family of nonempirical PBE, PBE0, PBE0-1/3, PBE0-DH, PBE-CIDH, PBE-QIDH, and PBE0-2 functionals, after employing a wide variety of basis sets systematically approaching the basis set limit: def2-nVP(D), cc-pVnZ, aug-cc-pVnZ, and d-aug-cc-pVnZ. We find that an accuracy (ie, mean unsigned error) of 0.3 to 0.4 eV for time-dependent density functional theory (DFT) atomic excitation energies can be robustly achieved with modern double-hybrid methods, which are also stable with respect to the addition of a double set of diffuse functions, contrarily to hybrid versions, in agreement with recent findings employing sophisticated multiconfigurational DFT methods.     

Unrestricted Hartree-Fock spin density calculations with orthogonalized atomic orbitals on aza and nitroaromatic radical anions

The spin density distributions in some aza and nitroaromatic radical anions have been calculated using Löwdin's orthogonalized basis set of atomic orbitals in the Unrestricted Hartree-Fock method of Amos and Snyder. The present calculations lead to a satisfactory account of proton splittings in these radicals. Least squares analyses correlating the observed ¹⁴N splittings and the spin density results over completely localized nonorthogonal basis have been separately carried out for aza and nitroaromatic radical anions and the sigma-pi parameters thus obtained are discussed and compared with earlier estimates for these quantities. Unlike the earlier results, the present estimate of Q _NN ^N for aza and nitroaromatic radicals are not very much different from each other.     

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post‐processing analysis of quantum‐chemical electron densities is described. The code can work with Slater‐ and Gaussian‐type basis functions of arbitrary angular momentum. It has been applied to explore the basis‐set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple‐zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis‐set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple‐zeta/higher level with Gaussian/Slater functions. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Atomic transferability within the exchange‐correlation density

Starting from either the exchange or the exchange‐correlation density together with Bader's definition of an atom in a molecule, an atomic hole density function can be defined. Contour maps of atomic hole density functions are able to show how the electron density of each atom in a molecule is partially delocalized into the rest of atoms in the molecule. The degree of delocalization of the atomic density ultimately depends on the nature of the atom studied and its environment. Atomic hole density functions are also used to define an atomic similarity measure, which allows for the quantitative assessment of the degree of atomic transferability in different molecular environments. In this article, contour maps for the N atom in the (N₂, CN⁻, NO⁺) series and the O atom in the (CO, H₂CO, and HCOOH) series are presented at the Hartree–Fock and CISD levels of theory. Moreover, the transferability of N and O within the two series is studied by means of atomic similarity measures. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1361–1374, 2000     

Experimental and theoretical electron density study of estrone

The electron density and the electrostatic potential (ESP) distributions of estrone have been determined using X-ray diffraction analysis and compared with theoretical calculations in the solid and gas phases. X-ray diffraction measurements are performed with a Rigaku Rapid rotating anode diffractometer at 20 K. The electron density in the estrone crystal has been described with the multipole model, which allowed extensive topological analysis and calculation of the ESP. From DFT calculations in the solid state a theoretical X-ray diffraction data set has been produced and treated in the same way as the experimental data. Two sets of single molecule DFT calculations were performed: (a) An electron density distribution was obtained via a single-point calculation with a large basis set at the experimental geometry and subsequently analyzed according to the quantum theory of atoms in molecules (AIM) to obtain the bond and most atomic properties, and (b) another electron density distribution was obtained with a smaller basis set, but at a geometry optimized using the same basis set for the analysis of atomic energies. An interesting locally stabilizing hydrogen-hydrogen bond path linking H(1) and H(11B) is found which represents the first characterization of such bonding in a steroid molecule. AIM delocalization indices were shown to be well correlated to the experimental electron density at the bond critical points through an exponential relationship. The aromaticity of ring A, chemical bonding, the O(1)...O(2) distance necessary for estrogenic activity, and the electrostatic potential features are also discussed.     

Shape information from a critical point analysis of calculated electron density maps: Application to DNA-drug systems

Summary A computational method is described for mapping the volume within the DNA double helix accessible to the groove-binding antibiotic netropsin. Topological critical point analysis is used to locate maxima in electron density maps reconstructed from crystallographically determined atomic coordinates. The peaks obtained in this way are represented as ellipsoids with axes related to local curvature of the electron density function. Combining the ellipsoids produces a single electron density function which can be probed to estimate effective volumes of the interacting species. Close complementarity between host and ligand in this example shows the method to give a good representation of the electron density function at various resolutions. At the atomic level, the ellipsoid method gives results which are in close agreement with those from the conventional spherical van der Waals approach.     

Study on the maximum accuracy of the pseudopotential density functional method with localized atomic orbitals versus plane-wave basis sets

A detailed study on the accuracy attainable with numerical atomic orbitals in the context of pseudopotential first-principles density functional theory is presented. Dimers of first- and second-row elements are analyzed: bond lengths, atomization energies, and Kohn-Sham eigenvalue spectra obtained with localized orbitals and with plane-wave basis sets are compared. For each dimer, the cutoff radius, the shape, and the number of the atomic basis orbitals are varied in order to maximize the accuracy of the calculations. Optimized atomic orbitals are obtained following two routes: (i) maximization of the projection of plane wave results into atomic orbital basis sets and (ii) minimization of the total energy with respect to a set of primitive atomic orbitals as implemented in the OPENMX software package. It is found that by optimizing the numerical basis, chemical accuracy can be obtained even with a small set of orbitals.     

Quantum similarity measures under atomic shell approximation: First order density fitting using elementary Jacobi rotations

The elementary Jacobi rotations technique is proposed as a useful tool to obtain fitted electronic density functions expressed as linear combinations of atomic spherical shells, with the additional constraint that all coefficients are kept positive. Moreover, a Newton algorithm has been implemented to optimize atomic shell exponents, minimizing the quadratic error integral function between ab initio and fitted electronic density functions. Although the procedure is completely general, as an application example both techniques have been used to compute a 1S-type Gaussian basis for atoms H through Kr, fitted from a 3-21G basis set. Subsequently, molecular electronic densities are modeled in a promolecular approximation, as a simple sum of parameterized atomic contributions. This simple molecular approximation has been employed to show, in practice, its usefulness to some computational examples in the field of molecular quantum similarity measures. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 2023–2039, 1997