Efficient and reliable numerical integration of exchange-correlation energies and potentials

An adaptive numerical integrator for the exchange-correlation energy and potential is presented. It uses the diagonal elements of the exchange-correlation potential matrix as a grid generating function. The only input parameter is the requested grid tolerance. In combination with a defined cell function the adaptive grid generation scales almost linear with the number of basis functions in a system. With the adaptive numerical integrator the self-consistent field energy error, which is due to the numerical integration of the exchange-correlation energy, converges with increasing adaptive grid size to a reference value. The performance of the adaptive numerical integration is analyzed using molecules with first, second, and third row elements. Especially for transition metal systems the adaptive numerical integrator shows considerably improved performance and reliability.     

Two-body densities and effective potentials

For an A-particle quantum system in a state Ψ, the movement of any pair of particles is studied. Using the corresponding two-body densities, the effective potential that determines the relative and center of mass movements of the pair of particles is built. Some aspects of the proposed general procedure are applied to different, functions used to describe the ground state of the helium atom. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 75–81, 2000     

Prediction of densities for solid energetic molecules with molecular surface electrostatic potentials

The densities of high energetic molecules in the solid state were calculated with a simplified scheme based on molecular surface electrostatic potentials (MSEP). The MSEP scheme for density estimation, originally developed by Politzer et al., was further modified to calculate electrostatic potential on a simpler van der Waals surface. Forty-one energetic molecules containing at least one nitro group were selected from among a variety of molecular types and density values, and were used to test the suitability of the MSEP scheme for predicting the densities of solid energetic molecules. For comparison purposes, we utilized the group additivity method (GAM) incorporating the parameter sets developed by Stine (Stine-81) and by Ammon (Ammon-98 and -00). The absolute average error in densities from our MSEP scheme was 0.039 g/cc. The results based on our MSEP scheme were slightly better than the GAM results. In addition, the errors in densities generated by the MSEP scheme were almost the same for various molecule types, while those predicted by GAM were somewhat dependent upon the molecule types.     

Molecular electrostatic potentials and electron densities in nitroazacubanes

Successive introduction of nitrogen atoms in the cubyl corners instead of C-NO2 groups of octanitrocubane (CNO2)8, the most powerful explosives known to date, leads to a class of energy-rich compounds known as nitroazacubanes. In present work the ab initio Hartree-Fock and hybrid density functional calculations have been carried out on the possible conformers of (CNO2)(8-alpha)Nalpha (with alpha=0-8), nitroazacubanes. The charge distributions in these systems have been derived using the topography of the molecular electrostatic potential and electron density. Molecular electrostatic potential investigations reveal that of different nitroazacubane conformers, the electron-rich regions around nitro oxygens of the lowest energy conformer having face opposite nitrogen atoms within a cube are more delocalized. These conformers are predicted to have the largest difference of the energies of the highest occupied molecular orbital and lowest unoccupied molecular orbital relative to the other conformers. The dipole moments of nitroazacubanes are dependent on the nitrogen sites within a cube, caused by the resultant of C-N bond moments and nearly insensitive to position of the NO2 groups. The lowest frequency vibration (522 cm(-1)) suggests octa-azacubane having robust structure in the nitroazacubane series. Substitution of nitrogen atom instead of C-NO2 group leads to increase in electron density at the bond critical point of the X-N (X=C or N) bonds in a cube. The heats of formation of different nitroazacubanes were calculated by using the isodesmic reaction approach. The present calculation has shown that for the di- though hexanitroazacubanes the most destabilized conformer possess largest dipole moment and the heat of formation as well. A linear correlation of the electron density at the bond critical point of X-N bonds and the heat of formation has been obtained.     

Mapping between local potentials and ground state densities

The mapping between the local one-body potential and the ground state density is discussed. The role of degenerate ground states, which form a set of measure zero in potential space, is shown to be amplified by the unitary transformation between degenerate densities, forming a space filling set in density space. A numerical experiment demonstrating these features is carried out. No indication of non-V representable densities is encountered.     

Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in pi-conjugated molecules

Newly developed exchange-correlation functionals in density functional theory (DFT) have been applied to describe conjugation effects in organic molecules. The performance of the various approaches is assessed through the calculation of torsion energy profiles and their critical comparison with available experimental data. Our results indicate that the OPTX-B95 exchange-correlation functional as well as its corresponding hybrid versions perform better than the well-established BLYP or B3LYP schemes when dealing with pi-conjugated molecules. In contrast, the recently introduced VSXC functional is not as reliable as other DFT methods for the systems examined here.     

Optimized effective potentials yielding Hartree-Fock energies and densities

It is commonly believed that the exchange-only optimized effective potential (OEP) method must yield total energies that are above corresponding ground-state Hartree-Fock (HF) energies except for one- and two-electron systems. We present a simple procedure for constructing local (multiplicative) exchange potentials that reproduce exactly the HF energy and density in any finite basis set for any number of electrons. For any finite basis set, no matter how large, there exist infinitely many such OEPs, which questions their suitability for practical applications.     

Calculation of current densities using gauge-including atomic orbitals

A method for calculating the various components of the magnetically induced current-density tensor using gauge-including atomic orbitals is described. The method is formulated in the framework of analytical derivative theory, thus enabling implementation at the Hartree-Fock self-consistent-field (HF-SCF) as well as at electron-correlated levels. First-order induced current densities have been computed up to the coupled-cluster singles and doubles level (CCSD) augmented by a perturbative treatment of triple excitations [CCSD(T)] for carbon dioxide and benzene and up to the full coupled-cluster singles, doubles, and triples (CCSDT) level in the case of ozone. The applicability of the gauge including magnetically induced current method to larger molecules is demonstrated by computing first-order current densities for porphin and hexabenzocoronene at the HF-SCF and density-functional theory level. Furthermore, a scheme for obtaining quantitative values for the induced currents in a molecule via numerical integration over the current flow is presented. For benzene, a perpendicular magnetic field induces a (field dependent) ring current of 12.8 nA T(-1) at the HF-SCF level using a triple-zeta basis set augmented with polarization functions (TZP). At the CCSD(T)/TZP level the induced current was found to be 11.4 nA T(-1). Gauge invariance and its relation to charge-current conservation is discussed.     

On the similarity between molecular electron densities, electrostatic potentials and bare nuclear potentials

The similarities among the molecular contours of three scalar fields, viz. electron density (ED), electrostatic potential (ESP) and bare nuclear potential (BNP) have been investigated. The topological resemblance between ESP and ED contour diagrams (as prompted by the Thomas-Fermi model) is more pronounced than that for BNP and ED contour diagrams (as predicted by the local density functional model of Parr, Gadre and Bartolotti) with three-membered ring systems as test cases. An analysis of critical points of these distributions has also been included. Thus it may be conjectured that ED maps may prove useful in predicting reactive sites in molecules.     

Time-dependent exchange-correlation current density functionals with memory

Most present applications of time-dependent density functional theory use adiabatic functionals, i.e., the effective potential at time t is determined solely by the density at the same time. This paper discusses a method that aims to go beyond this approximation, by incorporating "memory" effects: the derived exchange-correlation potential will depend not only on present densities but also on the past. In order to ensure the potentials are causal, we formulate the action on the Keldysh contour for electrons in electromagnetic fields, from which we derive suitable Kohn-Sham equations. The exchange-correlation action is now a functional of the electron density and velocity field. A specific action functional is constructed which is Galilean invariant and yields a causal exchange-correlation vector potential for the Kohn-Sham equations incorporating memory effects. We show explicitly that the net exchange-correlation Lorentz force is zero. The potential is consistent with known dynamical properties of the homogeneous electron gas (in the linear response limit).     

SCF Deformation densities and electrostatic potentials of purines and pyrimidines

4-31G wave functions have been computed for five purines and pyrimidines. The calculated deformation densities have been partitioned into atomic fragments, which were integrated to yield atomic multipole moments. The transferability of atomic fragments between related molecules was verified by constructing model maps for uracil and guanine from appropriate fragments of cytosine and adenine. Model electrostatic potentials calculated from the moments of model atoms are similar to the corresponding 4-31G potentials. Comparison of 4-31G and 4-31G** deformation densities of cytosine provides simple rules for estimating the effects of polarization functions on the atomic multipole moments of most atom types occurring in the purines and pyrimidines. These rules were applied to the other molecules and yielded reasonable approximations for their molecular dipole moments. Substituting CH₃ for H has little effect on the deformation density beyond the substitution center.     

Kohn-Sham potentials corresponding to Slater and Gaussian basis set densities

A new method based on linear response theory is proposed for the determination of the Kohn-Sham potential corresponding to a given electron density. The method is very precise and affords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function δ _i (r)=(−ɛ _i )ϕ _i (r) when SCF orbitals (either Kohn-Sham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth “average” potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles. Received: 24 February 1997 / Accepted: 18 June 1997     

Calculation of electrostatic and polarization energies from electron densities

We investigate procedures for calculating the electrostatic and polarization energies, Ees and Epol, associated with noncovalent interactions. The starting points are the electron densities of the isolated components and the complex; these could be obtained either computationally or experimentally. A slightly modified version of a scheme proposed by Gavezzotti is used to carry out numerical integrations over these electron densities. Our approach to estimating Epol is based upon partitioning the charge distributions of the components into overlapping and nonoverlapping regions. The effects of varying the integration parameters, computational techniques and basis sets are examined in detail for several noncovalently bound molecular dimers. Our results are in good agreement with the values of Ees and Epol produced by other methods, which require analytical integrations over interaction Hamiltonian matrix elements.     

Calculation of spin densities in odd alternant hydrocarbon radicals

Spin densities in benzyl, allyl, pentadienyl and perinaphthenyl radicals have been calculated with some variants of the simplified McLachlan unrestricted SCF method based on Hückel-type and the SC Hückel-type molecular orbitals. The various estimations of the spin densities were compared one with each other and with the results of more sophisticated methods. It was shown that the SC Hückel basis and Zhidomirov and Schastnev's modification of the McLachlan approach give a best least square fit of the relation a ^exp Q ^calc .

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Zusammenfassung Die Spindichten von Benzyl-, Allyl-, Pentadienyl- und Naphtenyl-Radikalen werden mit einigen Varianten der uneingeschränkten SCF-Methode in der vereinfachten Form nach McLachlan, die auf den Hückel- und den SC-Hückel-Molekülorbitalen basiert, berechnet. Die verschiedenen Schätzungen der Spindichten werden miteinander sowie mit den Resultaten komplizierterer Methoden verglichen. Es wird gezeigt, daß die SC Hückel Basis und die von Zhidomirov und Schastnev eingeführte Modifikation der Näherung von McLachlan eine beste Näherung nach der Methode der kleinsten Quadrate an die Beziehung a ^exp Q ^calc ergeben.

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Résumé Les densités de spin ont été calculées dans les radicaux benzyle, allyle, pentadiényle et perinaphthényle à l'aide de variantes de la méthode SCF sans restrictions de spin simplifiée de Mac Lachlan, en utilisant des orbitales de type Hückel et SC Hückel. Les différentes évaluations de densité de spin sont comparées entre elles et avec celles obtenues par des méthodes plus élaborées. On montre que la base SC Hückel et la modification de Zhidomirov et Schastnev à la théorie de Mac Lachlan donnent le meilleur accord au sens des moindres carrés avec la relation a ^exp Q ^cal .

     

Molecular electrostatic potentials and electron densities in azatriprismanes and nitroazatriprismanes

Triprismane, a C₆H₆ isomer of benzene, belongs to a class of strained hydrocarbons. In this work, the energy and charge distributions in a series of aza (C₆NH_6–) and nitroaza (CNO₂)_6–N, (with =1...6) derivatives of triprismane have been analyzed using the ab initio Hartree–Fock (HF)-derived molecular electrostatic potentials and molecular electron densities. Electrostatic potential investigations have shown that the electron-rich regions around nitrogen along a series of azatriprismanes and those near oxygens of nitro group in nitroazatriprismanes become smaller on encompassing from the hexanitroaza to nitroazatriprismane. As revealed from the molecular electrostatic potential (MESP) topography for the series of azatriprismanes the MESP minimum near nitrogen become shallow with increasing nitrogen content. Heats of formation obtained from the isodesmic reaction approach in the C₆N₆H_6– and (CNO₂)_6–N series correlate well with the electron density at the bond-critical point of the X— N (X = C or N) bonds of the triprismane framework.SPG is grateful to the Council of Scientific and Industrial Research (CSIR project 01(1772)/02/EMR-II), New Delhi, India. KAJ thanks CSIR for the Junior Research Fellowship. Thanks are due to C-DAC, Pune, for providing computational facilities.     

Optimized effective potentials from electron densities in finite basis sets

The Wu-Yang method for determining the optimized effective potential (OEP) and implicit density functionals from a given electron density is revisited to account for its ill-posed nature, as recently done for the direct minimization method for OEP's from a given orbital functional [T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007)]. To address the issues on the general validity and practical applicability of methods that determine the Kohn-Sham (local) multiplicative potential in a finite basis expansion, a new functional is introduced as a regularized version of the original work of Wu and Yang. It is shown that the unphysical, highly oscillatory potentials that can be obtained when unbalanced basis sets are used are the controllable manifestation of the ill-posed nature of the problem. The new method ensures that well behaved potentials are obtained for arbitrary basis sets.     

On the mapping of time-dependent densities onto potentials in quantum mechanics

The mapping of time-dependent densities on potentials in systems of identical quantum mechanical particles is examined. This mapping is of significance ever since Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] established its uniqueness, forming the theoretical basis for time-dependent density functional theory. Beyond uniqueness there are two important issues: existence, often called v-representability, and stability. We show that v-representability for localized densities in turn-on situations is not assured and we give a simple example of nonexistence. As for stability, we discuss an inversion procedure and by computing its Lyapunov exponents we demonstrate that the mapping is unstable with respect to fluctuations in the initial state. We argue that such instabilities will plague any inversion procedure.