One-electron Functions Research Articles

Doubly-charged negative ion of C60

SynopsisAn electronic structure of the negative ion C602- has been studied. It is found that in the first approximation of the variational method (when a trial wave function of the extra electrons is a product of one-electron functions) the total energy of the system is negative and that the second electron affinity (EA) of the C60 molecule within the utilized model potentials is about 1 eV. The photodetachment cross sections of the C602- ion have been calculated as well; they are found to exhibit two different threshold behaviors with their values being of the same order as the photodetachment cross sections of atomic negative ions with the same EA.

One-electron images in real space: natural adaptive orbitals.

We introduce a general procedure to construct a set of one-electron functions in chemical bonding theory, which remain physically sound both for correlated and noncorrelated electronic structure descriptions. These functions, which we call natural adaptive orbitals, decompose the n-center bonding indices used in real space theories of the chemical bond into one-electron contributions. For the n = 1 case, they coincide with the domain natural orbitals used in domain-averaged Fermi hole analyses. We examine their interpretation in the two-center case, and show how they behave and evolve in simple cases. Orbital pictures obtained through this technique converge onto the chemist's molecular orbital toolbox if electron correlation may be ignored, and provide new insight if it may not.

An Effective Description of Electron Correlation in the Green Function Approach

Beyond the usual many body perturbation theory, with the eigenfunctional theory, we directly calculate the correlation function produced by the Coulomb interaction of electrons in the equation of one-electron Green function, and give the general expression of the non-local effective interaction potential in a Hartree-type potential, which is absent in all previous many body perturbation theories. This effective interaction potential originates from the quantum many body effect of the system, and it cannot be obtained directly by the usual perturbation expression approach. Moreover, using theoretical models, we demonstrate that this effective interaction potential can be used to characterize the electron correlation strength of the system.

Communication: An efficient algorithm for evaluating the Breit and spin–spin coupling integrals

We present an efficient algorithm for evaluating a class of two-electron integrals of the form r12⊗r12/r12(n) over one-electron Gaussian basis functions. The full Breit interaction in four-component relativistic theories beyond the Gaunt term is such an operator with n = 3. Another example is the direct spin-spin coupling term in the quasi-relativistic Breit-Pauli Hamiltonian (n = 5). These integrals have been conventionally evaluated by expensive derivative techniques. Our algorithm is based on tailored Gaussian quadrature, similar to the Rys quadrature for electron repulsion integrals (ERIs), and can utilize the so-called horizontal recurrence relation to reduce the computational cost. The CPU time for computing all six Cartesian components of the Breit or spin-spin coupling integrals is found to be only 3 to 4 times that of the ERI evaluation.

Information-theoretic multiplicities of chemical bond in Shull’s model of H2

Alternative information-theoretic (IT) measures of the chemical bond multiplicity and its covalent/ionic composition in the orbital communication theory (OCT) are examined using Shull’s natural orbital (NO) model of the homopolar bond in H2. In OCT a molecule is treated as an information (probability-scattering) system, generated by the network of conditional probabilities (from the quantum mechanical superposition principle) linking elementary events of the adopted perspective. For the first time this atomic orbital (AO) invariant, two-NO description of Shull allows one to examine in several alternative representations the behavior of the previously adopted IT indices, of the channel average communication noise (OCT-covalency) and information flow (OCT-ionicity), with changing internuclear distance R, from the united atom (R = 0 ) to the separated atoms limit (SAL) (R → ∞). The adopted references include the two-electron atomic and ionic functions of the model, as well as the alternative one-electron functions, of the AO and NO sets, respectively. The numerical results for the Wang function description of H2 are reported and a general agreement with the accepted chemical intuition is tested. Joint probabilities of Shull’s reference states are linked to the energy partitioning. The incorrect SAL behavior of the OCT-ionicity index, giving rise to the constant (interaction independent) overall multiplicity measure, emphasizes a need for a revision of these IT bond descriptors. The modified set of indices is proposed, reflecting the complementary localization (determinicity) and delocalization (indeterminicity) aspects of the communication system in question. The novel IT-ionicity now reflects the diagonal (intra-orbital, additive) information propagation in the molecular channel, while the modified IT-covalency accordingly measures the effect of its off-diagonal (inter-orbital, nonadditive) probability scatterings. These components are shown to give rise to the interaction strength dependent overall IT bond-order, which adequately reflects the chemical intuition.

A hierarchy of chemical bonding indices in real space from reduced density matrices and cumulants

Real space techniques in the theory of the chemical bond may in fact be understood as means to obtain chemical information from reduced density matrices (RDMs). Much work has been devoted to examine the one particle density, but the renewed interest in electron correlation effects on bonding has progressively shifted research towards further order densities. Up to date, however, not much work has been carried out to determine higher order RDMs from standard electronic structure packages beyond the single-determinant level. We show in this work a possible efficient route to this end. We discuss how to compute RDMs from multi-determinant wavefunctions, how to construct from them nth order cumulant density matrices (n-CDMs), and how to make profit of the recursivity and extensivity of the latter to define a full hierarchy of bonding indices through exhaustive partitions of the physical space. This hierarchy includes a full set of one- to n-center quantities (multi-center bonding indices) which may be conveniently decomposed into one-electron components, each of them partnered with a one-electron function (natural adaptive orbital, NAdO). Illustrative results are obtained and discussed for a set of simple test molecules.

What are the most efficient basis set strategies for correlated wave function calculations of reaction energies and barrier heights?

As electronic structure methods are being used to obtain quantitatively accurate reaction energies and barrier heights for increasingly larger systems, the choice of an efficient basis set is becoming more critical. The optimum strategy for achieving basis set convergence can depend on the way that electron correlation is treated and can take advantage of flexibility in the order in which basis functions are added. Here we study several approaches for estimating accurate reaction energies and barrier heights from post-Hartree-Fock electronic structure calculations. First and second, we evaluate methods of estimating the basis set limit of second order Møller-Plesset perturbation theory and of coupled cluster theory with single and double excitations and a quasiperturbative treatment of connected triple excitations by using explicitly correlated basis functions (in the F12a implementation) along with valence, polarization, and diffuse one-electron basis functions. Third, we test the scheme of adding a higher-order correction to MP2 results (sometimes called MP2∕CBS + ΔCCSD(T)). Finally, we evaluate the basis set requirements of these methods in light of comparisons to Weizmann-3.2, Weizmann-4, and CCSDT(2)(Q)∕CBS+CV+R results.

Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory

Many approximations have been developed to help deal with the O(N(4)) growth of the electron repulsion integral (ERI) tensor, where N is the number of one-electron basis functions used to represent the electronic wavefunction. Of these, the density fitting (DF) approximation is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to molecular size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decomposition to obtain a low-rank approximation to density fitting (tensor hypercontraction density fitting or THC-DF). This new approximation reduces the 4th-order ERI tensor to a product of five matrices, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling reduction for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N(4)) operations. This should be compared to the usual scaling behavior of O(N(5)) and O(N(6)) for MP2 and MP3, respectively. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and configuration interaction, promising significant gains in computational efficiency and storage reduction.

EL: the new aromaticity measure based on one-electron density function

Ellipticity of the bond, being the quantity which numerically reflects how far the given chemical bond has elliptic cross-section, may be used to estimate π-electron contribution in bonding. We make use of that fact and develop a new measure of aromaticity—EL index. Since ellipticity is available from calculations on one-electron density function, EL can be used for both theoretical and experimental data. The EL measure is normalized to make interpretation of this parameter as easy and comfortable as possible. We compare EL values with the values of other commonly used aromaticity measures, such as HOMA, PDI, FLU, and NICS. It appears that the indications of EL are in agreement with indications of other indices and general expectations.

Localization scheme for relativistic spinors

A new method to determine localized complex-valued one-electron functions in the occupied space is presented. The approach allows the calculation of localized orbitals regardless of their structure and of the entries in the spinor coefficient matrix, i.e., one-, two-, and four-component Kramers-restricted or unrestricted one-electron functions with real or complex expansion coefficients. The method is applicable to localization schemes that maximize (or minimize) a functional of the occupied spinors and that use a localization operator for which a matrix representation is available. The approach relies on the approximate joint diagonalization (AJD) of several Hermitian (symmetric) matrices which is utilized in electronic signal processing. The use of AJD in this approach has the advantage that it allows a reformulation of the localization criterion on an iterative 2 × 2 pair rotating basis in an analytical closed form which has not yet been described in the literature for multi-component (complex-valued) spinors. For the one-component case, the approach delivers the same Foster-Boys or Pipek-Mezey localized orbitals that one obtains from standard quantum chemical software, whereas in the multi-component case complex-valued spinors satisfying the selected localization criterion are obtained. These localized spinors allow the formulation of local correlation methods in a multi-component relativistic framework, which was not yet available. As an example, several heavy and super-heavy element systems are calculated using a Kramers-restricted self-consistent field and relativistic two-component pseudopotentials in order to investigate the effect of spin-orbit coupling on localization.

New insights in quantum chemical topology studies using numerical grid‐based analyses

New insights in Quantum Chemical Topology of one-electron density functions have been proposed here by using a recent grid-based algorithm (Tang et al., J Phys Condens Matter 2009, 21, 084204), initially designed for the decomposition of the electron density. Beyond the charge analysis, we show that this algorithm is suitable for different scalar functions showing a more complex topology, that is, the Laplacian of the electron density, the electron localization function (ELF), and the molecular electrostatic potential (MEP). This algorithm makes use of a robust methodology enabling to numerically assign the data points of three-dimensional grids to basin volumes, and it has the advantage of requiring only the values of the scalar function without details on the wave function used to build the grid. Our implementation is briefly outlined (program named TopChem), its capabilities are examined, and technical aspects in terms of CPU requirement and accuracy of the results are discussed. Illustrative examples for individual molecules and crystalline solids obtained with gaussian and plane-wave-based density functional theory calculations are presented. Special attention was given to the MEP because its topological analysis is complex and scarce.

A stationary property of the APSG wave function

The method of antisymmetrized product of strongly orthogonal geminals (APSG) emerges by optimizing the one-electron functions used to construct two-electron functions (geminals), the latter being expanded (with coefficients selected variationally) in mutually exclusive subspaces of the former ones. Accordingly, E APSG is stationary with respect to the expansion coefficients and to unitary transformations of the one-electron orbitals. We show that the APSG energy is also stationary to the unitary transformation of identical geminals. For non-identical geminals, this statement holds only approximately.

System of matrix-vector equations in the multiconfiguration Hartree-Fock method

A method for determining radial parts of one-electron functions entering the manyelectron basis is proposed within the multiconfiguration Hartree-Fock procedure. The solution of the Hartree-Fock equations is reduced to the solution of a system of matrix-vector equations. Rules of construction of these equations are formulated, and a stable numerical scheme of their solution is found.

Polarization of ultimately short optical pulses in a semiconductor superlattice in the presence of a magnetic field

Using the Boltzmann equation in the relaxation-time approximation for the one-electron distribution function in combination with the coupled Maxwell equations for the electromagnetic field, we obtained a system of equations describing the propagation of ultimately short optical pulses through a semiconductor superlattice in an external magnetic field. It is shown that the initially linearly polarized light pulse propagating in the sample gives rise to the field of orthogonal polarization. The dynamics of propagation of the pulse in both polarizations is studied. The effects produced by the lattice geometry are revealed.

A connection between domain-averaged Fermi hole orbitals and electron number distribution functions in real space

We show in this article how for single-determinant wave functions the one-electron functions derived from the diagonalization of the Fermi hole, averaged over an arbitrary domain Omega of real space, and expressed in terms of the occupied canonical orbitals, describe coarse-grained statistically independent electrons. With these domain-averaged Fermi hole (DAFH) orbitals, the full electron number distribution function (EDF) is given by a simple product of one-electron events. This useful property follows from the simultaneous orthogonality of the DAFH orbitals in Omega, Omega(')=R(3)-Omega, and R(3). We also show how the interfragment (shared electron) delocalization index, delta(Omega,Omega(')), transforms into a sum of one-electron DAFH contributions. Description of chemical bonding in terms of DAFH orbitals provides a vivid picture relating bonding and delocalization in real space. DAFH and EDF analyses are performed on several test systems to illustrate the close relationship between both concepts. Finally, these analyses clearly prove how DAFH orbitals well localized in Omega or Omega(') can be simply ignored in computing the EDFs and/or delta(Omega,Omega(')), and thus do not contribute to the chemical bonding between the two fragments.

Dynamics of ultrashort light pulses in a semiconductor superlattice in the presence of a magnetic field

A system of equations that describes the propagation of ultrashort light pulses (optical solitons) in a semiconductor superlattice in the presence of a magnetic field is obtained using coupled Maxwell equations for the electromagnetic field and the Boltzmann equation written in the relaxation time approximation for the one-electron distribution function. It is shown that an initial linearly polarized light pulse induces a field with the orthogonal polarization in the sample. The dynamics of joint propagation of the initial and induced pulses in the sample is studied.

Some applications of the thermal single-determinant approximation

A new extension of Hartree–Fock theory to non-zero temperature, T, namely the Thermal Single-Determinant Approximation (TSDA)—based on the variational principle of statistical mechanics—has been applied to a model of a crystal of widely separated hydrogen atoms. It is found that, in this TSDA, solutions to the equations of stationarity of the free energy consist of one-electron functions which are either spatially extended (like Bloch functions), or localized. Whereas it appears that in the standard thermal Hartree–Fock approximation (THFA) only extended solutions are possible (at finite atomic separation). Furthermore, in the TSDA at T ≠ 0 the localized solutions give a lower free energy than that corresponding to the extended solutions, and the latter is less than or equal to the free energy in the THFA. As far as we know, this is the first calculation in which a strictly variational requirement has rejected extended one-electron functions in favor of localized functions for a crystal.

Single-particle spectral function of the Hubbard chain: frustration induced

The one-electron spectral function of a frustrated Hubbard chain is computed by making use of the cluster perturbation theory. The spectral weight we found turns out to be strongly dependent on the frustrating next-nearest-neighbor hopping t″. A frustration induced pseudogap arises when the system evolves from a gapful Mott insulator to a gapless conductor for an intermediate value of the frustration parameter |t″|. Furthermore, the opening of a pseudogap in the density of states already in the metallic side leads to a continuous opening of the true gap in the insulator. For the hole-doped case, the pseudogap is pinned at the Fermi energy, while the Mott gap is shifted in energy with increasing Hubbard interaction U. The separation of the pseudogap and Mott gap in the hole-doped system demonstrates the validity of the existence of a pseudogap.

Inner and outer radial density functions in singly-excited 1 snl states of the He atom

The one-electron radial density function D ( r ) has recently been found to be separable into inner D < ( r ) and outer D > ( r ) radial density functions. The inner D < ( r ) and outer D > ( r ) densities are studied for 28 singly-excited 1 s n l singlet and triplet states ( 0 ≤ l < n ≤ 5 ) of the He atom at a correlated level. Theoretical structures of D < ( r ) and D > ( r ) are discussed within the Hartree–Fock framework. Comparison of correlated D < ( r ) and D > ( r ) with hydrogenic radial densities based on the modal characteristics and Carbó’s similarity index clarifies that D < ( r ) represents the 1 s density of the helium cation, while D > ( r ) extracts the n l density of the hydrogen atom from D ( r ) . The radial separation 〈 | r 1 − r 2 | 〉 , which constitutes a lower bound to the standard deviation of D ( r ) , is shown to be estimated from the location of the outermost maximum of D > ( r ) .

All-electron CI calculations of 3d transition-metalL2,3 XANES using zeroth-order regular approximation for relativistic effects

X-ray-absorption near-edge structures (XANES) at 3d transition-metal (TM)L2,3 edges are computed using the all-electron configuration interaction (CI) method. Slaterdeterminants for the CI calculations are composed of molecular orbitals obtained bydensity functional theory (DFT) calculations of model clusters. Relativistic effects aretaken into account by the zeroth-order regular approximation (ZORA) usingtwo-component wavefunctions. The theoretical spectra are found to be strongly dependenton the quality of the one-electron basis functions. On the other hand, a different choice ofthe exchange–correlation functionals for the DFT calculations does not exhibit visiblechanges in the spectral shape. Fine details of multiplet structures in the experimental TML2,3 XANES of MnO, FeO and CoO are well reproduced by the present calculations when theone-electron basis functions are properly selected. This is consistent with ourprevious report showing good agreement between theoretical and experimental TML2,3 XANES when four-component relativistic wavefunctions were used.