Dimensional scaling and the quantum mechanical many-body problem

Summary A growing repertoire of electronic structure methods employ the spatial dimensionD as an interpolation or scaling parameter. It is advantageous to transform the Schrödinger equation to remove all dependence onD from the Jacobian volume element and the Laplacian operator; this introduces a centrifugal term, quadratic inD, that augments the effective potential. Here we explicitly formulate this procedure forS states of an arbitrary many-particle system, in two variants. One version reduces the Laplacian to a quasicartesian form, and is particularly suited to evaluating the exactly solvableD limit and perturbation expansions about this limit. The other version casts the Jacobian and Laplacian into the familiar forms forD=3, and is particularly suited to calculations employing conventional Rayleigh-Ritz variational methods.     

Variational justification of the dimensional-scaling method in chemical physics: the H-atom

The dimensional scaling (D-scaling) method first originated from quantum chromodynamics by using the spatial dimension D as an order parameter. It later has found many useful applications in chemical physics and other fields. It enables, e.g., the calculation of the energies of the Schr?dinger equation with Coulomb potentials without having to solve the partial differential equation (PDE). This is done by imbedding the PDE in a D-dimensional space and by letting D tend to infinity. One can avoid the partial derivatives and then solve instead a reduced-order finite dimensional minimization problem. Nevertheless, mathematical proofs for the D-scaling method remain to be rigorously established. In this paper, we will establish this by examining the D-scaling procedures from the variational point of view. We show how the ground state energy of the hydrogen atom model can be calculated by justifying the singular perturbation procedures. In the process, we see in a more clear and mathematical way confirming (Herschbach J Chem Phys 85:838, 1986 Sect. II.A) how the D-dimensional electron wave function “condenses into a particle,” the Dirac delta function, located at the unit Bohr radius.     

Approximate separation of the hyperradius in the many-particle Schrödinger equation

For large values of d = 3N, the radial distribution function of an N-particle system is sharply peaked near the hyperradius r_m = (d ? 2)/2k₀, where k₀≡(2/E/)^1/2. This fact allows an approximate separation of the hyperradius, leading to many-dimensional hydrogenlike radial solutions. Kindred applications to dimensional scaling are also discussed, where d = DN, with D the spatial dimension. For the large D regime, illustrative analytic formulas are obtained giving the energy and effective nuclear charge for the lowest few S states of the helium atom.     

Large-Z and -N dependence of atomic energies from renormalization of the large-dimension limit

By combining Hartree–Fock results for nonrelativistic ground-state energies of N-electron atoms with analytic expressions for the large-dimension limit, we have obtained a simple renormalization procedure. For neutral atoms, this yields energies typically threefold more accurate than the Hartree–Fock approximation. Here, we examine the dependence on Z and N of the renormalized energies E(N, Z) for atoms and cations over the range Z, N = 2 → 290. We find that this gives for large Z = N an expansion of the same form as the Thomas–Fermi statistical model, E → Z^7/2(C₀ + C₁Z^?1/3 + C₂Z^?2/3 + C₃Z^?3/3 + ?), with similar values of the coefficients for the three leading terms. Use of the renormalized large-D limit enables us to derive three further terms. This provides an analogous expansion for the correlation energy of the form δE δZ^4/3(δC₃ + δC₅Z^?2/3 + δC₆Z^?3/3 + ?); comparison with accurate values of δE available for the range Z ? 36 indicates the mean error is only about 10%. Oscillatory terms in E and δE are also evaluated. © 1994 John Wiley & Sons, Inc.     

A quantitative study of tethered chains in various solution conditions using Langmuir diblock copolymer monolayers

This article summarizes our investigations of tethered chain systems using Langmuir monolayers of poly(dimethylsiloxane)‐polystyrene (PDMS‐PS) diblock copolymers on organic liquids. In this system, the PDMS block adsorbs strongly to the air surface while the PS block dangles into the subphase liquid. The air surface can be made either repulsive or attractive for the tethered PS chain segments by choosing a subphase liquid which has a surface tension less than or greater than that of PS, respectively. The segment profile of the PS block is determined by neutron reflection as a function of the surface density, the molecular weights of the PS and PDMS blocks, and the solution conditions. We cover the range of reduced surface density (Σ ) characteristic of the large body of data in the literature for systems of chains tethered onto solid surfaces from dilute solution in good or theta solvent conditions (Σ < 12). We emphasize quantitative comparisons with analytical profile forms and scaling predictions. We find that the strong‐stretching limit assumed in analytical self‐consistent field calculations (SCF) and scaling theories is not valid over this Σ range. On the other hand, over a large portion of this range (Σ ⪇ 5) tethered chain profiles are well described by a renormalization group theory for weakly interacting or noninteracting chains. Simultaneous with the study of the profile form, the free energy of the tethered chains is examined through the surface tension. A strong increase in the surface pressure is observed with increasing surface density which determines the maximum surface density which can be achieved. This effect is attributed to a combination of higher order osmotic interactions and configurational constraints. This effect may explain several outstanding discrepancies regarding the adsorption of end‐functionalized chains and diblock copolymers onto solid surfaces.     

Solving the low dimensional Smoluchowski equation with a singular value basis set

Reaction kinetics on free energy surfaces with small activation barriers can be computed directly with the Smoluchowski equation. The procedure is computationally expensive even in a few dimensions. We present a propagation method that considerably reduces computational time for a particular class of problems: when the free energy surface suddenly switches by a small amount, and the probability distribution relaxes to a new equilibrium value. This case describes relaxation experiments. To achieve efficient solution, we expand the density matrix in a basis set obtained by singular value decomposition of equilibrium density matrices. Grid size during propagation is reduced from (100–1000)^N to (2–4)^N in N dimensions. Although the scaling with N is not improved, the smaller basis set nonetheless yields a significant speed up for low‐dimensional calculations. To demonstrate the practicality of our method, we couple Smoluchowsi dynamics with a genetic algorithm to search for free energy surfaces compatible with the multiprobe thermodynamics and temperature jump experiment reported for the protein α₃D. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Molecular structures of acetylene derivatives of tin

The geometry and force fields of the bis(trimethylstannyl)acetylene molecule (a conformer withD _3d symmetry corresponding to a minimum of the total energy of the molecule) were calculated by the RHF and MP2(fc) methods. The effective core potential in SBK form with the optimized 31G^* valence basis set was employed in the case of Sn atoms. The 6–31G^** and 6–311G^** basis sets were used for carbon and hydrogen atoms. Vibrational spectra of the light and perdeuterated isotopomers of bis(trimethylstannyl)acetylene were interpreted using the procedure of scaling the quantum-chemical force fields. For Part 5, see Ref. 1. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 616–626, April, 2000.     

Zur Konformation des Bicyclo[2.2.2]octan-Systems

Crystals of bicyclo [2.2.2] octane-1, 4-dicarboxylic acid are monoclinic, a = 6.01 Å, b = 16.73 Å, c = 10.29 Å, β = 115.08°, space group P2₁/c, with 4 molecules in the unit cell. The structure was solved with the help of direct methods and refined by full-matrix least-squares analysis of the three-dimensional intensity data. Within experimental error the bicyclo [2.2.2]-octane (BCO) skeleton has apparent D_3h-symmetry, corresponding to the totally eclipsed conformation. Analysis of the thermal ellipsoids in terms of the translational and rotational motion of the BCO skeleton leads to an r.m.s. amplitude of 5.9 ± 0.2° for rotation about the threefold axis. On the assumption that the bond lengths remain effectively constant during a torsional vibration of BCO, the potential energy surface has been calculated for a range of semi-empirical potential functions. These calculations show that the energy minimum may be slightly displaced from D_3h symmetry, but if so the barrier between the two such equivalent minima is only about 0.1 kcal mole^?1. The energy eigenvalues and eigenfunctions for a typical variation of potential energy vs torsion angle have been calculated. From the form of the eigenfunction of the ground vibrational state we conclude that BCO has effective D_3h symmetry at all temperatures as far as diffraction methods are concerned.     

Laser light scattering of the molecular weight distribution of unfractionated phenolphthalein poly(aryl ether sulfone)

Two unfractionated samples of phenolphthalein poly(aryl ether sulfone) (PES-C) were characterized in CHCl₃ at 25°C by applying a recently developed laser light-scattering (LLS) procedure. The Laplace inversion of precisely measured intensity–intensity time correlation function lead us first to an estimate of the characteristic line-width distribution G(Λ) and then to the translational diffusion coefficient distribution G(D). A combination of static and dynamic LLS results enabled us to determine D = (2.69 × 10⁻⁴)M^−0.553, which agrees with the calibration of D = (2.45 × 10⁻⁴)M^−0.55 previously established by a set of narrowly distributed PES-C samples. Using this newly obtained scaling between D and M, we were able to convert G(D) into a differential weight distribution f_w(M) for the two PES-C samples. The weight-average molecular weights calculated from f_w(M) are comparable to that obtained directly from static LLS. Our results showed that using two broadly distributed samples instead of a set of narrowly distributed samples have provided not only similar final results, but also a more practical method for the PES-C characterization. © 1997 John Wiley & Sons, Inc.     

Chain dynamics in entangled polymers: Diffusion versus rheology and their comparison

This review article scrutinizes and reanalyzes the extensively available literature data on the tracer and self chain diffusion coefficients D_tr and D_s along with the corresponding zero‐shear viscosity η₀ to show that D_s ∝ M^?ν starts with ν > 2.0 and converges to the asymptotic scaling exhibited by D_tr ∝ M^?2.0 as the molecular weight M increases beyond M/M_e = 10–20, in contrast to the onset of the asymptotic scaling M³ for η₀ taking place typically for M/M_e ? 10–20. A coherent analysis of these observations leads to the suggestion that the observed crossover in D_s is due to the constraint release effect, which diminishes around M/M_e = 10–20 and is negligible in measurements of D_tr when the matrix molecular weight P is much greater than M. The contour length fluctuation (CLF) effect, which is believed to cause the molecular weight scaling of η₀ to deviate significantly from its limiting behavior of M³, has little direct influence on the chain diffusion. The absence of the CLF effect on D_s leads to a much stronger than linear dependence of the product η₀D_s on M, which has been observed previously. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1589–1604, 2003     

Crystal structure,thermal behaviour and detonation characterization of bis(4,5‐diamino‐1,2,4‐triazol‐3‐yl)methane monohydrate

Bis(4,5‐diamino‐1,2,4‐triazol‐3‐yl)methane monohydrate (BDATZM·H₂O or C₅H₁₀N₁₀·H₂O) was synthesized and its crystal structure characterized by single‐crystal X‐ray diffraction; it belongs to the space group P (triclinic) with Z = 2. The structure of BDATZM·H₂O can be described as a two‐dimensional ladder plane with extensive hydrogen bonding and no disorder. The thermal behaviour was studied under non‐isothermal conditions by differential scanning calorimetry (DSC) and thermogravimetric/differential thermogravimetric (TG/DTG) methods. The detonation velocity (D) and detonation pressure (P) of BDATZM were estimated using the nitrogen equivalent equation according to the experimental density. A comparison between BDATZM·H₂O and bis(5‐amino‐1,2,4‐triazol‐3‐yl)methane (BATZM) was made to determine the effect of the amino group; the results suggest that the amino group increases the hydrophilicity, space utilization and energy, and decreases the thermal stability and symmetry of the resulting compound.     

Critical properties of thin films of polymer solutions

The critical properties of polymer solutions confined in thin‐film environments is studied with simple scaling arguments and a molecular theory. For purely repulsive surfaces, the critical volume fraction is a universal function of x = N^1/2/L, where N is the chain length and L is the film thickness. The critical volume fraction is nonmonotonic in x and shows a deep minimum at a film thickness several times larger than the chain's radius of gyration. This nonmonotonic behavior results from the interplay between the surface–polymer entropic repulsion and the tendency of the film to avoid large density gradients. The critical temperature is a monotonically increasing function of L, as L goes from the two‐dimensional limit to the three‐dimensional limit. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1849–1853, 2005     

E5M7+ (E=C–Pb,M=Li–Cs): A Source of Viable Star‐Shaped Clusters

Herein we report the systematic exploration of the potential energy surfaces of a series of clusters with formula E₅M₇⁺ (E=C‐Pb and M=Li‐Cs). Fifteen of these combinations adopt a D_5h three‐dimensional seven‐pointed star‐like structure in a singlet state, where M atoms interact electrostatically with the E₅ ring. The determining factors in the relative preference of having the D_5h structure over the most competitive isomer or vice‐versa are analyzed. These star‐shaped systems satisfy the 4n+2 Hückel's rule and exhibit a strong diatropic (σ and π) response to an external magnetic field.     

Degeneracy of confined D‐dimensional harmonic oscillator

Using the mathematical properties of the confluent hypergeometric functions, the conditions for the incidental, simultaneous, and interdimensional degeneracy of the confined D‐dimensional (D > 1) harmonic oscillator energy levels are derived, assuming that the isotropic confinement is defined by an infinite potential well and a finite radius R_c. Very accurate energy eigenvalues are obtained numerically by finding the roots of the confluent hypergeometric functions that confirm the degeneracy conditions. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Dirac equation with a Coulomb plus scalar potential in D + 1 dimensions

We generalize the Dirac equation to D + 1‐dimensional spacetime. The exact solutions of the D‐dimensional radial equations with a Coulomb plus scalar potential taking the form 1/r are analytically presented by studying the Tricomi equations. The energies E(n, l, D) are exactly presented. The dependences of the energies E(n, l, D) on the dimension D are analyzed in some detail. The energies E(n, 0, D) first decrease and then increase when increasing dimension D, but the energies E(n, l, D) (l ≠ 0) increase when increasing dimension D. The energies E(n, 0, D) are symmetric with respect to D = 1 for D ∈ (0, 2). It is shown that the energies E(n, l, D) (l ≠ 0) are almost independent of the quantum number l for large D and are completely independent of it if the Coulomb potential is equal to the scalar one. The energies E(n, l, D) almost overlap for large D. The dependences of the energies E(n, l, v) and E(n, l, s) on the vector potential parameter v and scalar potential one s are also studied for D = 3. All are found to decrease when these parameters are increased. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Crystal and molecular structures of cis-1-phenyl-3-piperidinocyclohexan-1-ol hydrochloride

Mr = 295.84, triclinic, Pl, a = 6.786(1), b = 7.658(1), c = 8.561(1) Å, α = 108.17(1), β = 97.94(1), γ = 103.32(2)°, V = 400.6 ų, Z = 1,D_m = 1.23, D_x = 1.226 Mgm^?3, δ(Cu Kα) = 1.5418 Å, μ = 20.81 cm^?1, F(000) = 160. The structure has been solved by direct and Fourier methods and refined by a least-squares procedure to the final R = 0.043 for 1182 observed reflections (|F_o] >3σ(F_o)). cis-1-Phenyl-3-piperidinocyclohexan-1-ol possessing 1,3-diaxial positions between the piperidine and hydroxyl groups is converted to the isomer with 1,3-diequatorial positions in its hydrochloride. The hydrogen bond is formed between the chloride anion and the protonated nitrogen atom of piperidine instead of the intramolecular hydrogen bond in the free cis-base between the oxygen and nitrogen atoms.     

The effect of inhomogeneities on evaluation of a fractal dimension for objects on a lattice

A comparison of the covering properties of windowsm, m + 1, where the window sidel _m isl _m = 2 ^m , yields an expression for the fractal dimensionD _m which displays directly effects due to periodicity and inhomogeneities. The structure of teD _m versusm curve gives insight into the nature of the representation of the fractal. In some cases bounds forD may be obtained and, if appropriate, the effect of the inhomogeneities due to boundaries, initial conditions, the pixel limit or periodicity can be removed.     

The crystal and molecular structure of phenothiazine-10-propionitrile. The effect of crystal packing on the dihedral angle of phenothiazine heterocycles

Phenothiazine-10-propionitrile, C₁₂H₈SNC₂H₄CN, crystallizes in the centrosymmetric monoclinic space group P2₁/n, with a = 5.785(1)Å, b = 15.427(3)Å, c = 14.497(4)Å, β = 92.50(1)°, Z = 4, D_meas = 1.29(1) g cm³ and D_calc = 1.28 g cm³ at 23°. Three dimensional X-ray data were collected with a manual diffractometer using MoKα (λ 0.71069Å) radiation and by multiple film Weissenberg techniques using CuKα (λ 1.5418Å) radiation. The structure was determined by Patterson and Fourier methods and refined with 519 observed reflections by full matrix least-squares methods to an R of 0.077. The dihedral angle between the two planes of the o-phenylene rings is 135.4(3)°. In the folded heterocyclic ring the C-S-C angle is 97.8(7)° and the average C? S bond is 1.76(1)Å. A comparison of this structure to that of phenothiazine-10-propionic acid shows the two chemically similar molecules have the same dihedral angles in spite of completely different solid state packing patterns.     

Localization of Hydrogen and Content of Oxonium Cations in Alunite via Neutron Diffraction

Polycrystalline alunite‐d₆ KAl₃(OD)₆(SO₄)₂, prepared by hydrothermal reaction of Al₂(SO₄)₃, K₂SO₄ and D₂SO₄, was studied by neutron powder diffraction performed on the diffractometer E2 (HMI‐BENSC, Berlin). Rietveld refinement of the data set for T = 2 K yielded the crystallographic data: space group R3m, Z = 3, trigonal setting, a = 694.3(1) pm, c = 1722.7(2) pm, N(I/σ(I) > 1) = 172, N(Var.) = 19, R_p = 0.036, wR_p = 0.046, R_B(I/σ(I) > 1) = 0.020. The deuterium nuclei could be located precisely. Three equivalent O–D bonds with nuclear distances r(O(4)–D) = 96.6(3) pm directed to each of the terminal oxygen atoms of the SO₄ groups are found. Partial substitution of K⁺ by D₃O⁺ was also considered in the refinement procedure. In good agreement with results of other methods a site occupation fraction n(D₃O⁺) = 0.0104 was obtained.