Extrapolation of real-space quantum chemical calculations from finite-size super cells to the ideal infinite system. I. Theory

A method is proposed how to calculate the correct density matrix of an infinite polymeric chain from that of a standard finite supercell calculation. The density matrix of the finite supercell is transformed into k-space for all k-values allowed by the periodic boundary conditions. The k-dependent matrices are then unitarily transformed, with each unitary matrix being represented by a set of complex rotation matrices. It is shown that the corresponding angles can be interpolated and extrapolated toward the zone boundaries in a straghtforward manner and that this extrapolation can be done from any finite supercell with reasonable accuracy. This gives rise to an infinite system density matrix for which all fundamental properties are guaranteed by construction. This infinite system density matrix may be used to construct a corrected density matrix for the finite supercell calculation. © 1994 John Wiley & Sons, Inc.     

Efficient method for computation of representation matrices of the unitary group generators

The representation matrices of the unitary group generators E_k1 are equivalent to the representation matrices of cyclic permutations (k, k ± l, …, l ? 1, l). A method is presented for simultaneous computation of matrices corresponding to all different generators at a cost of less than one multiplication per nonzero element. The number of operations necessary for calculation of individual matrices for single generators or for products of two generators is at most proportional to the number of matrix elements of the final matrix. This approach eliminates the need to store the representation matrices in CI calculations.     

A density matrix variational calculation for atomic Be

The ground-state energy of the beryllium atom is calculated using a variational procedure in which the elements of the two-body reduced density matrix (particle–particle matrix) are the variational parameters. It is shown that, for this problem and with the limited number of spin-orbitals used, the trace condition and the simultaneous nonnegativity conditions on the particle–particle, the particle–hole, and the hole–hole matrices form a complete solution to the N-representability problem. The energy obtained is – 14.61425 a.u., practically identical to the value given by a configuration interaction calculation which uses the same states. The effects of weakening the nonnegativity conditions on each of the matrices in turn were also explored.     

Structure and spin-purity conditions for reduced density matrices of arbitrary order

For arbitrary k, the separation of spin variables is performed in the reduced density matrix of the kth order (RDM -k) on the basis of the Fock coordinate function method. The independent spatial components of RDM -k are analyzed. For RDM -k of the total spin eigenstate, their number is proved never to exceed its spin multiplicity 2s + 1. Integral and other nontrivial interrelations between spatial components are established which turn out to be the necessary and sufficient conditions of spin purity of a wavefunction corresponding to a given RDM -k. It is shown that the r-rank k-particle spin distribution matrix F, defined as a spatial coefficient at the spin-tensorial operator of rank r in the RDM -k expansion, can be obtained by reduction of the (k + r)-particle charge density matrix F. In particular, all spatial components of RDM -2 are explicitly expressed in terms of the four-electron charge density matrix only. This allows us to purpose some approximative formulas for the McWeeny-Mizuno spin–orbit and spin–spin coupling functions in the case of the weak spin contamination.     

Boson and fermion many-body assemblies: fingerprints of excitations in the ground-state wave functions,with examples of superfluid 4He and the homogeneous correlated electron liquid

After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid ⁴He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.     

Density matrix expansions and their application in the theory of the many-electron systems

A method of calculation of the correlation energy is proposed, which includes the superposition of configurations and the two particle approach. This method is based on the density matrix formalism. The approximate, but N-representable expressions for the reduced density matrices are used. The correlation energy of the beryllium atom is calculated as an example.     

A comparison of geometric methods with other methods for the characterization of density matrices

Some recently developed geometric methods for characterizing the subset of density matrices within the space of Hermitian matrices are compared with methods commonly used for the approximate characterization of reduced density matrices. The decomposition of a density matrix into components in terms of the reducing basis set is compared with decomposition in terms of representations of U(r).     

k0-NAA quality assessment by analysis of different certified reference materials using the KAYZERO/SOLCOI software

A suite of natural matrix reference materials (RMs) were used to assess the quality of analytical results obtained by k ₀-instrumental neutron activation analysis (k ₀-INAA) at the Joef Stefan Institute (IJS). Five certified reference materials (CRMs) from the Institute for Reference Materials and Measurements (IRMM), two standard reference materials (SRMs) from the National Institute of Standards and Technology (NIST), three RMs from the International Atomic Energy Agency (IAEA) and one RM from IJS were analyzed. Altogether, results for twenty-four elements in inorganic matrices and twenty-nine elements in organic matrices, obtained by k ₀-INAA, were compared to certified values. Results obtained show good agreement with certified or assigned values except for Fe, La, Nd, Sm and U in inorganic matrices, and Ag, Al and Cr in organic matrices.     

Sweeping with an Enhanced Electric Field of Neutral Analyte Zones in Electrokinetic Chromatography

The concept of sweeping has been used in electrokinetic chromatography to explain the concentrating mechanism in micellar electrokinetic chromatography when the sample matrix is a high resistivity non-micellar aqueous solution. Theoretical and experimental studies were undertaken. It was found that the total focusing effect is brought about by the cumulative effect of sweeping and sample stacking. For better analytical results, compounds showing low to moderate retention factors (k) and compounds showing high values of k must be dissolved in low and high conductivity matrices, respectively.     

New algorithm for high-order time-dependent hartree–fock theory for nonlinear optical properties

A new formalism and algorithm is developed for solving the general-order time-dependent Hartree–Fock (TDHF) problem. It is shown that for any order a generalization of the TDHF equations can be derived where all lower-order solutions constitute a constant term. This makes it very easy to obtain high-order solutions. As the space required for the mapping of density matrices to Fock matrices in a problem of a giver order is largely reduced, we can perform the most time-consuming steps within the core memory of the machine and easily manipulate vector products via optimum routines. The second hyperpolarizability γ is obtained from the secondorder TDHF solution via a 2n rule. The formalism also allows for expressing all terms in the equation diagrammatically, which provides additional physical insight and a more systematic evaluation of terms. To illustrate the method, TDHF results are presented for trans-butadiene and carbon monoxide for several optical processes, including correlation corrections to their static hyperpolarizabilities obtained via coupled cluster (CCSD) and many-body perturbation theory. The hybrid TDHF/CCSD method provides excellent agreement with the DC–SHG experiments (χ||⁽²⁾ = 11.4 x 10^?32 esu/mol compared to 12.9 ± 11.4 × 10^?32 esu/mol and χ||⁽³⁾ = 149 compared to 144 ± 4 × 10^?39 esu/mol).     

稀土对ZA27合金晶间腐蚀影响的电子理论研究

通过自行开发的计算机软件构造了ZA27合金中α相大角度晶界析出η相及稀土元素的原子集团模型. 采用递归法计算了Al, Zn, La, Y的局域态密度, 计算并分析了α相、η相的总态密度和费米能级, 及稀土对态密度和费米能级的影响. 计算表明: 稀土元素的局域态密度形状与Zn相近, 其与Zn结合的能力大; 稀土元素降低η相的费米能级, 减小Zn, Al电极电位差, 具有抑制晶间腐蚀的作用; 稀土元素不改变α相、η相的总态密度形状, 但使η相的态密度增大, 改变η相的电子结构; 晶格结构对原子的态密度有一定的影响, 掺杂原子的态密度趋于与基体原子态密度相同.     

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.     

Quantum properties of complete solutions for a new noncentral ring‐shaped potential

We propose a new exactly solvable ring‐shaped potential V(r,θ) = ?(α/r) + (σ/r²) + β cos²θ/(r²sin²θ). The exact bound‐state solutions are presented explicitly. The creation and annihilation operators are established directly from the normalized radial wave functions. We present two important recurrence relations among the diagonal matrix elements and obtain some explicit expressions of mean values of r^k (8 ≥ k ≥ ?11). The exact form of continuum states is also obtained analytically. Comments are made on the calculation formula of phase shifts and the analytical properties of the scattering amplitude. It is interesting to find that the exact form of continuum states will reduce to that of the bound states at the poles of the scattering amplitude. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Density functional theory study of ω phase in steel with varied alloying elements

The presence of a long-abandoned hexagonal omega (ω) phase in steel samples is recently gaining momentum owing to the advances in transmission electron microscopy (TEM) measurements, even though it is already reported in other transition-metal alloys. The stabilization of this metastable phase is mainly investigated in presence of C, even though the formation of the ω phase is attributed to the combined effect of many factors, one among which is the enrichment of solute elements such as Al, Mn, Si, C, and Cr in the nanometer-sized regimes. The present study investigates the effect of the above alloying elements in ω-Fe using density functional theory (DFT) calculations. It is seen that the magnetic states of the atoms play a major role in the stability of ω-Fe. Cohesive energy calculations show that the alloying elements affect the energetics and stabilization of ω-Fe. Further, density of states calculations reveal the variation in d-band occupancy in the presence of alloying elements, which in turn affects the cohesive energy. Phonon band structure calculations show that only ω-Fe with substitutional C shows positive frequencies and hence possess thermodynamic stability. Finally, we confirm the existence of ω-Fe using TEM measurements of a steel sample containing the same alloying elements. Our results can shed light on the stabilization of the ω in other transition-metal alloys as well, in the presence of minor alloying elements.     

Computational aspects of Laplace transform inversion of thermal unimolecular rate constant

The thermal limiting high-pressure unimolecular rate constant k∞ represents, operationally, the Laplace transform of the product of microcanonical rate constant for decomposition of molecules having specified energy E [k(E)] and the density of states [N(E)]. By inversion, it is possible to recover k(E)N(E), from which one can obtain the energy dependence of k(E) and the pressure dependence of k_uni, the thermal general-pressure unimolecular rate constant. This article examines numerical aspects of three methods of inversion, their reliability and dependence on sampling, i.e., on the number of available experimental data points, by comparing exact k(E) and k_uni with those obtained by inversion. It turns out that the method of steepest descents is the best all-round performer.     

The preparation and homogeneity study of synthetic multi-element standards (SMELS) for QC/QA of k0-NAA

Three new synthetic multi-element standards (SMELS) were developed for testing the performance of the k ₀-standardization neutron activation analysis (k ₀-NAA) method when implemented in a laboratory. SMELS consist of a phenol-formaldehyde polymer matrix spiked with different groups of elements according to the half-lives of the formed radionuclides. They are classified as Type I for short-, Type II for medium- and Type III for long-lived radionuclides. This article presents the preparation of SMELS including different aspects such as choice of the matrix and spiking elements, chemical compounds, target concentrations, purity of the matrix and spiking procedure and the homogeneity study for all the spiked elements in order to demonstrate the quality of the produced materials.     

Investigation of plasma-related matrix effects in inductively coupled plasma-atomic emission spectrometry caused by matrices with low second ionization potentials—identification of the secondary factor

Plasma-related matrix effects induced by a comprehensive list of matrix elements (a total of fifty-one matrices) in inductively coupled plasma-atomic emission spectrometry were investigated and used to confirm that matrix effects caused by elements with a low second ionization potential are more severe than those from matrix elements having a low first ionization potential. Although the matrix effect is correlated unambiguously with the second ionization potential of a matrix, the correlation is not monotonic, which suggests that at least one other factor is operative. Through study of a large pool of matrix elements, it becomes possible to identify another critical parameter that defines the magnitude of the matrix effect; namely the presence of low-lying energy levels in the doubly charged matrix ion. Penning ionization by Ar excited states is proposed as the dominant mechanism for both analyte ionization/excitation and matrix effects; matrices with a low second ionization potential can effectively quench the population of Ar excited states through successive Penning ionization followed by ion-electron recombination and lead to more severe matrix effects.