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  • 1
    Electronic Resource
    Electronic Resource
    Dimensional expansions for two-electron atoms (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 86 (1987), S. 2114-2122 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Approximate expansions in inverse powers of the dimensionality of space D are obtained for the ground-state energies of two-electron atoms. The method involves fitting polynomials in δ=1/D to accurate eigenvalues of the generalized D-dimensional Schrödinger equation. To the maximum order obtainable from the data, about δ7, the power series for nuclear charges Z=2, 3, and 6 all diverge at D=3. Asymptotic summation yields an energy for the Z=2 atom 1% in excess of the true value at D=3. However, expansions with a shifted origin, i.e., expansions in (δ−δ0), show improved convergence. Of particular interest is the case δ0=1, because the expansion coefficients can in principle be calculated by perturbation theory applied to the one-dimensional atom. Series in powers of (δ−1) appear to converge rapidly. Also the series in (δ−1) can be evaluated even for the hydride ion, with Z=1. For helium, this series is quite comparable to the more familiar expansion in powers of λ=1/Z, with errors in the partial sums decreasing by roughly an order of magnitude per term. Thus, for Z=2 the first four terms of the expansion in (δ−1) yield an energy within 0.02% of the true value at D=3. Similar results are found in an analogous treatment of accurate eigenvalues for the Hartree–Fock approximation. This provides a rapidly convergent dimensional expansion for the correlation energy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.452109
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  • 2
    Electronic Resource
    Electronic Resource
    Vibrational frequency shifts induced by molecular compression of pyridine in solution (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 85 (1986), S. 2376-2383 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Pressure-induced vibrational frequency shifts are calculated for a diatomic oscillator immersed in a benign solvent, employing a simplified version of the Schweizer–Chandler model for solute–solvent interaction. The repulsive contribution is determined from the pair distribution function for hard-sphere cavities. Interpolative evaluation of the pair distribution function is facilitated by noting that to an excellent approximation the pertinent expansion coefficients are merely linear functions of the reduced density. The treatment is applied to the quasidiatomic ring breathing vibrations of neat liquid pyridine, benzene, and toluene and to solutions of pyridine in several solvents including H2O, D2O, CH3OH, CHCl3, dimethylformamide, and toluene. The predicted pressure dependence of the ring breathing frequency is in the range ∂ν/∂P(approximate)0.3–0.8 cm−1/kbar for all these systems. The corresponding compression of the mean ring radius is in the range 0.9 to 2.0×10−4 A(ring)/kbar. Especially for the associated solvents, the dominant contribution (〉90%) to ∂ν/∂P comes from the effective hard-sphere repulsion. Accurate values of the effective diameters thus can be evaluated from the observed pressure derivatives.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.451092
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  • 3
    Electronic Resource
    Electronic Resource
    Hartree–Fock approximation for D-dimensional two-electron atoms (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 84 (1986), S. 3893-3900 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The Hartree–Fock method for two-electron atoms is generalized to spaces of arbitrary dimensionality. The problem is exactly soluble in two limiting cases, D→1 and D→∞, for any value of the nuclear charge Z. Numerical calculations of the ground-state energy are reported for a wide range of D and for Z=1 to 6, with an accuracy typically better than one part in 1010. Together with previous variational calculations employing the Pekeris method, these results permit the correlation energy to be evaluated with an accuracy typically better than one part in 106. The correlation energy is found to be largely independent of Z for any D. For a given Z, the correlation energy decreases smoothly as D increases; to a good approximation it is simply linear in 1/D. However, the correlation energy remains appreciable even in the limit D→∞; as a fraction of the total energy, for Z=2 the correlation energy varies from 2.28% at D=1 to 1.45% at D=3 to 0.99% as D→∞.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.450100
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  • 4
    Electronic Resource
    Electronic Resource
    Electron correlation calibrated at the large dimension limit (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 86 (1987), S. 4997-5008 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Correlation energies (CEs) for two-electron atom ground states have recently been obtained to good approximation from a simple perturbation treatment using 1/D as the expansion parameter, with D the dimensionality of space. In hydrogenic units, the CE varies almost linearly with 1/D between limits at D→1 and D→∞ which are exactly calculable. However, for D→∞ the CE is only about 35% smaller than the "true-world'' value at D=3. This is in striking contrast to the analogous error in the mean field approximation of statistical mechanics, which vanishes for sufficiently large D. Here we show that the CE for D→∞ can be made to vanish by modifying the Hartree–Fock (HF) variational wave function. A separable form is retained but a factor aitch-theta(θ) is included, with θ the angle between the electron–nucleus radii r1 and r2. Likewise, the error in the HF value for the first derivative of the energy with respect to 1/D can be made to vanish by employing a suitable choice of coordinates in separate factors of the wave function. The choice is determined by the vibrational normal modes of the electrons about the rigid configuration attained in the D→∞ limit. We estimate that these improvements in the HF wave function at large D will reduce the CE for D=3 by about a factor of 10 or more for any two-electron atom. We also relate our results to those obtained with hyperspherical coordinates and show that the large-D limit accounts for the success enjoyed by the hyperspherical approximation at D=3. These findings offer prospects for reducing CEs for multielectron systems by exploiting dimensional calibration of the HF wave function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.452671
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  • 5
    Electronic Resource
    Electronic Resource
    Dimension dependence of correlation energies in two-electron atoms (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 86 (1987), S. 3512-3521 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Correlation energies (CEs) for two-electron atom ground states have been computed as a function of the dimensionality of space D. The classical limit D→∞ and hyperquantum limit D→1 are qualitatively different and especially easy to solve. In hydrogenic units, the CE for any two-electron atom is found to be roughly 35% smaller than the real-world value in the D→∞ limit, and about 70% larger in the D→1 limit. Between the limits the CE varies almost linearly in 1/D. Accurate approximations to real CEs may therefore be obtained by linear interpolation or extrapolation from the much more easily evaluated dimensional limits. We give two explicit procedures, each of which yields CEs accurate to about 1%; this is comparable to the best available configuration interaction calculations. Steps toward the generalization of these procedures to larger atoms are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.451954
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  • 6
    Electronic Resource
    Electronic Resource
    Interdimensional degeneracies, near degeneracies, and their applications (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 85 (1986), S. 4557-4562 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Recently developed approximation methods for quantum mechanical problems which treat the spatial dimension D as an expansion parameter offer approximations to energy levels at arbitrary D. Rather than simply being a detour to the D=3 case, there is physical interest in nonphysical values of D due to degeneracies between states in different dimensions. For example, such degeneracies make it possible to calculate some excited states of two-electron atoms in three dimensions from the ground state energy at nonphysical values of D. Such relationships can be exploited in a simple derivation of the hydrogen atom spectrum in arbitrary D, using only the solution at D=1 and a combination of inter- and intradimensional symmetry arguments. Applications to the Yukawa potential and an anharmonic oscillator are also presented. A large class of interdimensional degeneracies is found for two-electron atoms. Approximate degeneracies are also identified for these atoms which allow highly excited D=3 states to be treated as perturbed low-lying states in another dimension. The approximate degeneracies also serve to generalize the treatment of the hydrogen atom spectrum in a way appropriate to two-electron atoms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.451776
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  • 7
    Electronic Resource
    Electronic Resource
    Two-electron atoms near the one-dimensional limit (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 87 (1987), S. 433-442 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: If the Hamiltonian of a two-electron atom is generalized in a natural way to arbitrary spatial dimension D, an especially simple case is found in the D=1 limit. While the ground state energy is singular at this point, a scaling argument reduces the problem to a limiting Hamiltonian with only two degrees of freedom in which the Coulombic potentials all reduce to δ functions. Since the singularity at D=1 dominates the energy at nearby dimensions, this limit forms the basis for an expansion in (D−1)/D which is reasonably accurate at D=3. By combining results from this expansion with the 1/D expansion about the D→∞ limit, estimates of the energy at D=3 are obtained with accuracy orders of magnitude better than that of either series alone. The simplicity of the D=1 and large-D limits and the accuracy of this method allow some qualitative insight into the physical features contributing to correlation effects in small atoms. Analysis of other singularities suggests that the 1/D series has zero radius of convergence for two-electron atoms. We conclude with a discussion of excited states and larger atoms and make some appealing connections with the orbital picture.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.453588
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  • 8
    Electronic Resource
    Electronic Resource
    Hylleraas–Pekeris treatment of D-dimensional two-electron atoms (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 84 (1986), S. 3882-3892 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The algorithm of Pekeris for S states of two-electron atoms is generalized to spaces of arbitrary dimensionality. Numerical calculations are reported for the ground state (1 1S) and first two excited states (2 3S and 2 1S) for a wide range of dimensions, 1〈D〈∞, and nuclear charge, 1≤Z≤6. The accuracy is typically better than one part in 108. The energy eigenvalues may be continued to arbitrary real values of the parameter δ=1/D. Real atoms, with D=3, connect smoothly with simple, exactly known limits at D→1 and D→∞. Analysis of the data permits several further terms in the 1/D expansion for the ground state energy to be determined, up to order D−12. This indicates that the expansion does not converge for D=3 but terms of third to sixth order do conform approximately to a geometric series form, as previously postulated in order to carry out dimensional interpolation. The excited state data exemplify near continuum motion at D→1 and quasivibrational asymmetric and symmetric stretching modes of electron motion as D→∞.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.450099
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  • 9
    Electronic Resource
    Electronic Resource
    Relation of vibrational frequency shifts to molecular compression in liquid benzene (1985)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 83 (1985), S. 6540-6541 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.449555
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  • 10
    Electronic Resource
    Electronic Resource
    High pressure study of associated media: Raman scattering of pyridine complexes in aqueous solution (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 84 (1986), S. 1080-1088 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The ν1 and ν12 ring breathing vibrational modes of pyridine in aqueous solution were studied as functions of concentration (up to 12 M) and pressure (up to 35 kbar) in a diamond anvil cell. The pyridine isotopes -h5 and -d5 exhibited marked differences. The pressure dependence of ν1 is quite linear for pyr-h5 but has pronounced curvature for pyr-d5 and this curvature increases as the concentration decreases. This unusual isotope effect is attributed to pyridine–water complexes involving both O–H ⋅ ⋅ ⋅N and C–H ⋅ ⋅ ⋅O hydrogen bonds. For ν12, the pressure dependence does not change with concentration, but it is at least five times more steep for pyr-h5 than pyr-d5. This may occur because the mass at all six ring vertices becomes equal for the -d5 isotope, making the volume change during vibration very small. At high concentrations, ν1 appears as two distinct peaks with different pressure dependence, indicative of uncomplexed and complexed pyridine. For certain combinations of solvent and concentration, a strong pressure dependent Fermi resonance between the ν1 and ν12 modes appears. Similar experiments were done with a weakly associating solvent, dimethylformamide, and nonassociating solvents, toluene and benzene. For these, the pressure dependence did not change with concentration; this and other features indicate which aspects of the aqueous solution results can be attributed to pyridine–water and pyridine–pyridine complexes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.450550
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