Basis Set Of Wave Functions Research Articles

Optimization of atomic density-fitting basis functions for molecular two-electron integral approximations.

A general procedure for the optimization of atomic density-fitting basis functions is designed with the balance between accuracy and numerical stability in mind. Given one-electron wavefunctions and energies, weights are assigned to the product densities, modeling their contribution to the exchange and second-order correlation energy, and a simple weighted error measure is minimized. Generally contracted Gaussian auxiliary basis sets are optimized to match the wavefunction basis sets [D. N. Laikov, Theor. Chem. Acc. 138, 40 (2019)] for all 102 elements in a scalar-relativistic approximation [D. N. Laikov, J. Chem. Phys. 150, 061103 (2019)].

Parabolic, prolate spheroidal bases and relation between bases of the nine-dimensional MICZ-Kepler problem

The nine-dimensional MICZ-Kepler problem (9D MICZ KP) considers a charged particle moving in the Coulomb field with the presence of a SO(8) monopole in a nine-dimensional space. This problem received much effort recently, for example, exact solutions of the Schrödinger equation of the 9D MICZ KP have been given in spherical coordinates. In this paper, we construct parabolic and prolate spheroidal basis sets of wave functions for the system and give the explicit interbasis transformations and relations between spherical, parabolic, and prolate spheroidal bases. To build the parabolic and prolate spheroidal bases, we show that the Schrödinger equation of the considered system is also variable separable in both parabolic and prolate spheroidal coordinates, and then, solve this equation exactly. The variable separability in different coordinate systems is actually a consequence of the superintegrability which has been proved recently for the 9D MICZ KP.

DISCUSSION OF THE CENTER OF MASS MOTION IN A SYSTEM WITH THREE DEGREES OF FREEDOM

This paper considers the problem of the center of mass motion of three fermionic particles, e. g., equal nucleons (no spin), which can move only in the x-direction in the laboratory system. This system (with three degrees of freedom only) is transformed to a system where one coordinate is proportional to the center of mass coordinate. The center of mass moves in a harmonic oscillator potential. The basis set of wave functions is constructed with Hermite polynomials in part. As example we study the case of a quadratic intrinsic potential between the particles. The problem may be important for the motion of three nucleons in a crystal channel.

Crystal structure, ab initio calculations and fingerprint plots of a new polymorph of N′, N″, N″′-triphenylbiuret

A new polymorph of N′, N″, N″′-triphenylbiuret, C 20H 17N 3O 2 (form II), has been synthesized and the structure has been solved by X-ray diffraction. The crystals are monoclinic, space group P2 1/ c, with a = 7.6966 (3) Å, b = 12.5490 (4) Å, c = 18.5996 (6) Å, β = 107.632(2)°, Mr = 331.37, V = 1712.04 (10) Å 3, Z = 4 and R = 0.0454. The hydrogen bonding of this polymorph is considerably different from that of the previously known structure. The molecules are linked in infinite chains, via C–H⋯O hydrogen bonds and there is also an intramolecular N–H⋯O hydrogen bond. The intermolecular interactions present in this polymorph, and on the previously reported polymorph, were analysed by means of the fingerprint plots derived from the Hirshfeld surfaces. The fingerprint plots evidenced the different packing modes of the two structures. Quantum–mechanical ab initio calculations for the free molecule were performed using the Hartree–Fock and DFT/B3LYP methods with the 6-31G(d,p) basis set of wave functions. The solid-state conformations compared with those obtained theoretically from DFT calculations for the isolated molecules show significant differences. Some difficulties of using quantum–mechanical calculations for the determination of relative conformational energies are also discussed.

A vibrational circular dichroism approach to the determination of the absolute configuration of flexible and transparent molecules: fluorenone ketals of 1,n-diols

The infrared absorption (IR) and vibrational circular dichroism (VCD) spectra for five ketal molecules, three of which obtained from 1,2-diols and two from 1,3-diols, were recorded in the mid-IR region. The spectra have been satisfactorily reproduced by DFT calculations, even with not too large wavefunction basis sets, especially due to the low number of conformers to be considered. The mobility of some moieties provides a recognizable signature. A characteristic couplet of VCD bands attributed to normal modes involving the methine and a phenyl ring bonded to the stereogenic carbon atom is evidenced for two ketals of the series as a signature of the absolute configuration; due comparison with existing literature is made. A relation is discussed of the present VCD data with the literature VCD data of simple alcohols and diols.

Indirect carbon–carbon couplings across one, two and three bonds in substituted benzenes: Experiment and theory

An excellent linear correlation is found between a large body of experimental spin–spin carbon–carbon couplings, J(CC), and B3PW91/6-311++G(d,p)//B3PW91/6-311++G(d,p) calculated estimates. The correlation does not differ significantly from the simplest relationship possible, J(CC) exp. = J(CC) calcd., within a small and random spread of about 1 Hz. There are 257 experimental values considered, and 126 out of these are new and come from the present work. The carbon–carbon couplings which include those across one, two and three bonds, span a considerable range of about −3 through +83 Hz. This is the first report of such a fine correspondence between experiment and theory in the field of spin–spin couplings where a large set of experimental data and a broad range of couplings is confronted with the relevant density functional theory (DFT) computations where the latter are preceded by molecular geometry optimizations using the same combination of the density functional and the basis set of wavefunctions. The quantum mechanical computations on the level of sophistication employed can reliably predict the signs of aromatic couplings J(CC) and this turned out to be important for 2 J(CC)′s which, according to the present work, span a range of about −3 through about +7 Hz. It is demonstrated for the first time that aromatic 2 J(CC)′s can bear either sign, and that substituent effects on the latter are significant. As can be inferred from the present report, substituent effects on aromatic spin–spin couplings between carbon–carbon nuclei result in the following ranges of the coupling constants: 1 J(CC)′s, +48.0 through +82.8 Hz; 2 J(CC)′s, −2.9 through +6.6 Hz; 3 J(CC)′s, +4.7 through +11.3 Hz. These can be compared with the literature experimental data for benzene: 1 J(CC) = (+)56.0 Hz; 2 J(CC) = (−)2.5 Hz; 3 J(CC) = (+)10.1 Hz. It is fairly obvious from the present work that rovibronic effects on aromatic J(CC)′s and those of nuclear motions at zero K are negligible.

An analytical expression for quantum-well tunneling lifetimes

An analytical expression is presented for the tunneling-escape lifetime of an electron occupying a quantum state localized in a quantum well in a heterostructure, using the perturbation approach. In the present derivation, an orthogonal basis set of wave functions is used, and all the terms in the perturbation expansion are included. The present formulation is exact and free from the inconsistencies in the transfer Hamiltonian method found recently by Fransson et al. [Phys. Rev. B 64, 153403 (2001)]. The tunneling lifetimes obtained by the present formulation agree with those obtained by the complex energy method. The analytical expression is useful in modeling tunneling devices in the sequential tunneling picture. We find that coupling of quantum states outside the quantum well to excited states localized in the quantum well is very important, and including these terms changes the tunneling lifetime by about six orders of magnitude.

Tight-binding model for carbon from the third-generation LMTO method: A study of transferability

The third-generation LMTO method provides a new wave function basis set in which the energy dependence of the interstitial region and inside muffin-tin (MT) spheres is treated on an equal footing. Within the improved method, basis functions in the interstitial are the screened spherical waves (SSWs) with boundary condition defined in terms of a set of ‘hard’ sphere radii a RL. Energy eigenvalues obtained from the single-particle Schrödinger equation for MT potential is energetically accurate and very useful for predicting a reliable first-principles tight-binding (TB) model of widely different systems. In this study, we investigate a possibility of the new basis sets transferability to different environment which could be crucial for TB applications to very large and complicated systems in realistic materials modelling. For the case of C where the issue of sp2 vssp 3 bonding description is primarily important, we have found that by downfolding the unwanted channels in the basis, the TB electronic structure calculations in both hexagonal graphite and diamond structures are well compared with those obtained from the full LDA schemes if we use the same choice of hard sphere radii,a RL and a fixed, arbitrary energy, εw. Moreover, the choice is robust and transferable to various situations, from different forms of graphite to a wide range of coordination. Using the obtained minimal basis set, we have been investigating the TB Hamiltonian and overlap matrices for different structure types for carbon, in particular we have predicted the on-site and hopping parameters (y1, y2, …, y6) within an orthogonal representation for Slonczewski-Weiss-McClure (S WMcC) model of the Bernal structure. Our theoretical values are in excellent agreement with experimental ones from magnetoreflection measurements of Fermi surfaces for hexagonal graphite.

Competition between shallow-impurity and T-shaped quantum-wire states

The binding energy of an electron bound to a shallow hydrogenic impurity located in a T-shaped quantum wire is calculated within the effective-mass approximation. The three-dimensional Schr\odinger equation is solved by employing a method which does not rely on expansions of basis sets of wave functions. The electron binding energy is calculated as a function of the impurity position for different quantum wires. There is a competition between the T-quantum-wire states and the shallow-impurity states. Depending on the sizes of the T structure and on the position of the impurity, the electron becomes localized at the intersection of the two wells or around the impurity.

A novel, general method of analyzing magnetic circular dichroism spectra and magnetization curves of high-spin metal ions: Application to the protein oxidized rubredoxin, Desulfovibrio gigas

We have developed a general theoretical approach for analyzing the intensities of magnetic circular dichroism (MCD) spectra of paramagnetic species with S>1/2 in the nonlinear regions of temperature and magnetic field. The method takes full advantage of the irreducible tensor method in order to obtain maximum simplification from symmetry. The approach, which is based on a detailed treatment of spin-orbit coupling and Zeeman interaction in terms of the symmetry properties of basis sets of wave functions, factorizes contributions into bands with Gaussian and derivative shapes in order to extend earlier treatments based on the so-called linear field limit. The method is applied to analyze and fit the form of the MCD spectra and the MCD magnetization curves of pseudo-tetrahedral high-spin Fe(III), S=5/2, in the protein rubredoxin from Desulfovibrio gigas, a representative of a family of iron–sulphur proteins. This treatment provides for the first time a satisfactory fit of these curves over a temperature range between 1.6 and 10 K and up to magnetic fields of 5 T. We show that the forms of the magnetization curves are strongly dependent on the polarizations of the optical transitions and on both the sign and magnitude of D, the ground state axial zero-field parameter. The sign and magnitude of D is determined to be −0.6 cm−1 with a fixed value of E/D=0.25 obtained from an analysis of electron paramagnetic resonance data. This shows that earlier, simpler fitting procedures were inadequate.

Symmetry vibronic wavefunction basis set and Hamiltonian representation for XY 3Z molecules near a doubly degenerate Jahn-Teller electronic state in the C 3v point group

Methodology based on the crude adiabatic approximation is presented for the complete linear and quadratic vibronic coupling of a degenerate E electronic state to the 3e 1 and 3e vibrational modes in the case of XY 3Z molecules with C 3v symmetry. First we present a method for building a symmetry adapted vibronic wavefunction basis set which block diagonalizes the Hamiltonian representation. Secondly we report the representation of the electronic Hamiltonian in the crude adiabatic approximation for XY 3Z molecules with C 3v symmetry. Computer codes developed for solving the vibronic coupling are also briefly reviewed. Finally some results are given with reasonable force constants and different choices of coupling constants for the limited dynamic Jahn-Teller E⊗e coupling.

Ground-state energy of singular potentials

It is shown that for singular potentials of the form \ensuremath{\lambda}/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$, the asymptotic form of the wave function both at r\ensuremath{\rightarrow}\ensuremath{\infty} and r\ensuremath{\rightarrow}0 plays an important role. Using a wave function having the correct asymptotic behavior for the potential \ensuremath{\lambda}/${\mathit{r}}^{4}$, it is shown that it gives the exact ground-state energy for this potential when \ensuremath{\lambda}\ensuremath{\rightarrow}0, as given earlier by Harrell [Ann. Phys. (NY) 105, 379 (1977)]. For other values of the coupling parameter \ensuremath{\lambda}, a trial basis set of wave functions which also satisfy the correct boundary conditions at r\ensuremath{\rightarrow}\ensuremath{\infty} ande r\ensuremath{\rightarrow}0 are used to find the ground-state energy of the singular potential \ensuremath{\lambda}/${\mathit{r}}^{4}$. It is shown that the obtained eigenvalues are in excellent agreement with their exact ones for a very large range of \ensuremath{\lambda} values.

Image potential for an electron near an impenetrable surface.

A self-energy formalism is used to determine the quantal image potential for an electron near an impenetrable solid surface. The electron is considered to be in a Coulombic potential outside a repulsive wall. The complete, orthonormal basis set of wave functions for an electron in such a model potential is employed in the calculation of the image potential.

Parametrization of 4f – 4f Transition Probabilities of Eu3+ Ions in Na5Eu(MoO4)4 and Na5Eu(WO4)4

AbstractThe simulation of the 4f—4f electric and magnetic dipole transition probabilities of Eu3+ ions in Na5Eu(MO4)4 (M = Mo and W) was carried out according to the Judd‐Ofelt theory. The experimental data consist of 29 5D0,1 → 7FJM transitions between crystal field sublevels. — The optimization of the six non‐zero Bλkg parameters allowed for the D2d symmetry yielded the following sets for Na5Eu(MoO4)4 and Na5Eu(WO4)4 (in 10−10 cm units). B232 = ‐2.01 (‐1.94), B432 = 0.66 (0.57), B452 = 3.31 (3.07), B652 = 8.47 (6.72), B672 = 2.27 (2.63) and B676 = 31.11 (30.93), respectively. These sets give reasonable simulation to the experimental transition probabilities. The conventional Judd‐Ofelt theory was, however, unable to simulate the intensity of the “hypersensitive” 5D0 → 7F2 transition. Most of the discrepancies are due to the uncertainty in experimental data although the influence of basis set truncation (only 671 |LJM> levels out of the total of 3003 were included in the basis set of wave functions) and the approximate symmetry (D2d symmetry was used instead of the real S4 one) cannot be neglected, either.

An algorithm for constructing basis sets of wave functions with required properties

An algorithm for constructing basis sets of wave functions with required properties

Electrically induced shifts of the GaAs nuclear spin levels

The application of a dc homogeneous electric field in a GaAs crystal will induce a quadrupolar splitting of the nuclear-spin levels. This splitting has been observed for $^{71}\mathrm{Ga}$, $^{69}\mathrm{Ga}$, and $^{75}\mathrm{As}$ using a pulsed NMR spectrometer. A constant (${R}_{14}$) that relates the coupling between the induced field gradient and the applied electric field has been measured. The values found were ${R}_{14}(\mathrm{Ga})=2.85\ifmmode\times\else\texttimes\fi{}{10}^{10}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ and ${R}_{14}(\mathrm{As})=3.16\ifmmode\times\else\texttimes\fi{}{10}^{10}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. These results are in disagreement with those from a similar experiment reported by Gill and Bloembergen. The measured values of ${R}_{14}$ were found to be independent of the applied electric field strength. The two-electron bond-orbital model developed by Huang, Moriarty, Sher, and Breckenridge has been used to interpret the experimental measurements. In evaluating the theory, a Hartree-Fock atomic $s{p}^{3}$ hybrid wave-function basis set is used to find the induced field gradient caused by the electrostatic distortion of the bond. With no adjustable parameters, the value of ${R}_{14}(\mathrm{As})$ calculated from the theory is smaller than the experimental number by a factor of 1.4, while the calculated ${R}_{14}(\mathrm{Ga})$ is a factor of 4 too small.