Four-dimensional density and energy density functional

The relations based on an external one-electron operator V( r ) are examined from two view-points, i.e., from the Hohenberg–Kohn approach and the four-dimensional density concept introduced by Wilson and Frost, and extensively studied by Parr and Politzer. The object being to obtain, with the help of the Hellmann–Feynman theorem, new formulas for the energy of atoms and molecules, and to discuss the construction of the universal energy density functional on the basis of the four-dimensional density.     

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.     

Approximately N -representable density functional density matrices: The case of large N

Density matrices approximatelyN-representable by a correlated determinant wavefunction (CDWF) have been previously derived. For the case of helium it had been shown how to obtain these density matrices as density functionals. Here it is shown that the same procedures, used for helium, generalize to the case for the number of electronsN-large. The result is a procedure for obtaining approximately CDWFN-representable density matrices which are functionals of the density for largeN.     

Kinetic energy density in terms of electron density for closed-shell atoms in a bare Coulomb field

A long-term aim in density functional theory is to obtain the kinetic energy density t(r) in terms of the ground-state electron density ρ(r). Here, t(r) is written explicitly in terms of ρ(r) for an arbitrary number 𝒩 of closed shells in a bare Coulomb field. In the limit as 𝒩→∞, closed results for t(r) follow from the earlier analysis of ρ(r) by Heilmann and Lieb. [Phys. Rev. A 52 , 3628 (1995)]. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 281–283, 1998     

Equation for the direct determination of the density matrix: Time-dependent density equation and perturbation theory

The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time-dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schr?dinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time-independent equation by replacing the energy E with . The perturbation theory formulas for the density equation have also been given for both the time-dependent and time-independent cases. Received: 16 June 1998 / Accepted: 2 September 1998 / Published online: 8 February 1999     

Geometry of density matrices. III. Spin components

A definition is given for the spin-tensorial components of a p-electron density matrix for arbitrary p. Certain of these are defined to be standard components, and it is shown that all others can be obtained from them by linear combinations of permutations. When the density matrix is reduced, the standard components either map into standard components of the fewer-electron density matrix or into the origin. One- and two-electron density matrices are treated in greater detail. Reducing bases are given for the spaces in which the spatial coefficients of the spin components are elements. For spin eigenstates traces of all components are determined, and the π = 1 parts of all components are determined in terms of two independent components, which are themselves determined by the two-electron charge density matrix. Geometric consequences are investigated.     

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Electron density near an impurity

The behavior of the electron density n(r) [and potential energy V(r)] near an impurity of charge Z is studied by using the linear response theory of an electron gas at finite temperature and with exchange and correlation effects included. The odd powers series in the expansion of n(r) [and V(r)] are calculated exactly by using asymptotic methods, and the coefficients in the series are given in terms of moments taken over the Fermi–Dirac distribution function. In all linear response theories and at all temperatures, the derivative n'(0) = -2Zn₀/a₀, where n₀ is the unperturbed electron density and a₀ is the Bohr radius. The effects of exchange and correlation appear in the fifth- and higher-order terms in n_odd(r).     

Chirality and spin density: Ab initio and density functional approaches

The spin distribution in a stable nitroxide biradical that shows ferromagnetic interactions in the solid phase has been studied at three levels of theory: First, at the UHF level; then, including correlation effects in UMP 2 calculations; and finally, the results are compared with the spin density obtained using the local density functional (LDF ) approximation. It is shown that LDF spin densities are closer to UMP 2 than to UHF predictions; the difference between the UHF and the (UMP 2, LDF ) results points to a redistribution of the spin repartition between N and O due to electronic correlation. For planar conformations of the NO group, there is symmetric distribution (D_2d) of the spin density on the adamantane skeleton. For nonplanar nitroxides, the molecule is chiral (C₂), which results in a breakdown of the spin transmission on part of the adamantane cage. © 1993 John Wiley & Sons, Inc.     

Simple tests for density functional methods

The performance of the currently used generalized gradient approximation density functionals is analyzed using several simple, yet critical requirements. We analyze the effects of the self-interaction error, the inclusion of the exact exchange, and the parameter settings used in the popular three-parameter hybrid density functionals. The results show that the elimination of the self-interaction error from the current density functionals lead to very poor results for H₂. The inclusion of the exact exchange does not significantly influence the self-interaction corrected results. The variation of the A, B, and C parameters of a hybrid DFT method influences the H(SINGLE BOND)H equilibrium bond length through a very simple linear equation, and it is possible to reproduce the experimental H(SINGLE BOND)H distance with appropriate selection of these parameters, although an infinite number of solutions exists. Similar results were obtained for the total energy and the electron density along the internuclear axis. The analysis of the exact KS potential at the bond critical point of the dissociating H₂ molecule shows that, for this property, the second order Moller–Plesset perturbation theory yields a better potential than the density functionals studied in this article. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1534–1545, 1997     

Combination of pseudopotentials and density functionals

Semilocal pseudopotentials have been determined for first–row (Li to Ne), second row (Na to Ar), and third-row atoms (K, Ca). Core–valence correlation is included by adjusting the pseudopotentials to experimental energies of ions with a single valence electron. Correlation within the valence shell is taken into account by using the spin–density functional formalism. The approximations involved in this approach are tested for atomic ionization energies as well as binding energies of monohydrides and alkali diatomics, agreement with experiment is usually satisfactory, but in certain applications density functionals should be already included in the fitting of the local part of the pseudopotential. In addition, 3s/3p and 3s/2p basis sets (for first and second row, respectively), designed for use in connection with our pseudopotentials, are given; it is shown that they yield reasonable results for both SCF and correlation energies.     

Topology of density difference and force analysis

The density difference (r) of a molecule A-B is defined as the difference of the density (r) of the molecule A-B and the density _A(r) + _B(r) put at the position of the atoms A and B. We investigate here the topological features of the density difference and define electron density flow (EDF) as representing the direction and the amount of the electron density flow in the course of the nuclear displacement processes. As such examples, we study H₂ molecule formation reaction and the interaction of two He atoms. By the topological analysis of (r), and by using the Hellmann-Feynman force and its partition into the AD, EC, and EGC forces, the characteristic behaviors of the (r) map are clarified. In particular, the electron cloud preceding and incomplete following are represented by using the concept of the EDF. The natures of the covalent bond are clarified based on the topological properties of the difference density (r) rather than that of the total density (r).On leave from the Department of Chemistry, Hebei Teachers' College, Shijiazhuang, Hebei, 050091, China     

Conduction behavior of doped polyaniline films at high current density regime

The conduction behavior at high current density at room temperature and above of polyaniline (PANI) films doped with HCl and camphor sulfonic acid (HCSA), respectively, is reported. It is found that the current density deviates strongly from the linear relation with the electric field in high current density region, and a saturation of the current density is observed. The maximum current density J_m seems to be proportional to the conductivity of the sample and hence, for PANI doped with HCl, J_m is about 200 A/cm^2, whereas for HCSA doped samples, J_m can reach more than 1,200 A/cm². The saturation of current density is interpreted as being caused by space charge accumulation at the insulating barrier regions and a dedoping effect in the conduction domains due to the detraping of the ions under high fields. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2845–2850, 1999     

Determination of density changes with expansion polymerization

Materials that polymerize without substantial volume contraction have numerous valuable applications in dentistry and other fields. Two new symmetrical aliphatic spiro-orthocar-bonates, formulated from cis- and trans-2-hydroxymethyl-cyclohexanol to give 2,3,8,9-di(tetramethylene)-1,5,7,11-tetraoxaspiro [5,5] undecanes, showed, by density determinations in dilute solution, that expansion occurs with polymerization. The cis / cis-spiroorthocarbonate racemic mixture showed a 3.9% expansion upon polymerization, and the trans/trans racemic mixture showed a 3.5% expansion. The densities were determined by pycnometry, with a precision reliable to at least four figures. This method is thus recommended for cases of soluble monomer/polymer pairs for which the density difference will be small and where the partial specific volumes are independent of concentration, thereby avoiding the complexities of variable physical states of monomer and/or polymer. © 1993 John Wiley & Sons, Inc.     

Analytical Schwartz density applied to heavy two-electron ions

An analytical expression of the electron density function p(r) due to Schwartz for two-electron atomic systems is applied to a detailed study of density-dependent properties of relatively heavy two-electron ions. Comparison of the Schwartz results with those from accurate Hartree-Fock and Hylleraas wave functions shows that despite its simple yet analytical form, the Schwartz density has a quantitative applicability in the density study of two-electron atoms within the nonrelativistic framework. © 1997 John Wiley & Sons, Inc.     

Diffusion in polymers dependence on crosslink density

Kinetics of N-methyl pyrrolidone evaporation from swollen photo-crosslinked polyacrylate was monitored thermogravimetrically at temperatures ranging from 323 to 398 K. Crosslink density dependence of evaporation kinetics was investigated in photo-crosslinked polyacrylates with crosslinked density ranging from ≈1.2 × 10² to ≈1.7 × 10⁴ mol m⁻³ and number of main chain atoms between crosslinks ranging from ≈70 atoms to ≈6 atoms, respectively. As was shown, evaporation kinetics was controlled by the solvent diffusion in polymer. Activation energies of evaporation (diffusion) were deduced from the rate measurements at different temperatures. Apparent activation energy of evaporation decreased from 48.7 to 31.1 kJ mol⁻¹ with crosslink density increase. Activation energy of pure N-methyl pyrrolidone evaporation was 50.6 kJ mol⁻¹. Decrease of the rate of solvent diffusion and unexpected decrease of diffusion activation energy with increase of crosslink density of swollen polymer matrix was explained by decrease in polymer chain segments mobility, as indicated by Eyring’s approach to diffusion in polymers.     

Geometric interpretation of density displacements and charge sensitivities

The “geometric” interpretation of the electronic density displacements in the Hilbert space is given and the associated projection-operator partitioning of the hardness and softness operators (kernels) is developed. The eigenvectors |?〉= |α〉 of the hardness operator define the complete (identity) projector P =Σ_α| α〉〈α | = 1 for general density displacements, including thecharge-transfer (CT) component, while the eigenvectors |i〉= |i〉 of the linear response operator determine thepolarizational P-projector, P_p = Σ_i |i〉〈i| . Their difference thus defines the complementary CT-projector: P_CT = 1-P_P. The complete vector space for density displacements can be also spanned by supplementing the P-modes with the homogeneous CT-mode. These subspaces separate the integral (normalization) and local aspects of density shifts in molecular systems.     

Spin density distributions in some fluorinated radical cations

A theoretical study of π-electron spin density distributions has been made for a series of fluoro-substituted hydrocarbon radical cations using unrestricted Hartree-Fock theory. Although some of the predicted proton splittings are not in very good agreement with experiment, the overall agreement with experiment can be passed as fairly satisfactory considering the approximate nature of the theory used. The experimental fluorine splittings can be well predicted by using a one-parameter relationship between the isotropic fluorine splitting (a_F) and the π-electron spin density (ρ_CC) on the attached carbon. It has been further shown that both ρ_CC and the proportionality constant (Q_eff) in the linear relation, are fairly insensitive to the parameter choice.     

Additive density functional correlation corrections to single particle theories

The correlation energy functional E_c of the Hartree-Fock density is investigated. It was previously established that E_c produces the exact ground-state energy when added to the Hartree-Fock energy. Except when certain degeneracies occur, it is here shown that E_c is bounded from below when the coordinates of the Hartree-Fock density are scaled uniformly by λ, as λ → ∞. Consequently, approximations to E_c should display this bounded property, which is a second-order perturbation energy. It is also shown that a corresponding result applies to that correlation energy functional, Ë_c, which is to be added to a completed exact exchange-only (in the OPM sense) density functional calculation. Scaling requirements are presented for each order in the perturbation expansion for Ë_c. For instance, the second-order term is dimensionless. © 1997 John Wiley & Sons, Inc.