Sabre Kais

Density functionals and dimensional renormalization for an exactly solvable model

We treat an analytically solvable version of the ‘‘Hooke’s Law’’ model for a two‐electron atom, in which the electron–electron repulsion is Coulombic but the electron‐nucleus attraction is replaced by a harmonic oscillator potential. Exact expressions are obtained for the ground‐state wave function and electron density, the Hartree–Fock solution, the correlation energy, the Kohn–Sham orbital, and, by inversion, the exchange and correlation functionals. These functionals pertain to the ‘‘intermediate’’ density regime (rs≥1.4) for an electron gas. As a test of customary approximations employed in density functional theory, we compare our exact density, exchange, and correlation potentials and energies with results from two approximations. These use Becke’s exchange functional and either the Lee–Yang–Parr or the Perdew correlation functional. Both approximations yield rather good results for the density and the exchange and correlation energies, but both deviate markedly from the exact exchange and correlati...

Dimensional scaling as a symmetry operation

The scaling of the Schrodinger equation with spatial dimension D is studied by an algebraic approach. For any spherically symmetric potential, the Hamiltonian is invariant under such scaling to order 1/D2. For the special family of potentials that are homogeneous functions of the radial coordinate, the scaling invariance is exact to all orders in 1/D. Explicit algebraic expressions are derived for the operators which shift D up or down. These ladder operators form an SU(1,1) algebra. The spectrum generating algebra to order 1/D2 corresponds to harmonic motion. In the D→∞ limit the ladder operators commute and yield a classical‐like continuous energy spectrum. The relation of supersymmetry and D scaling is also illustrated by deriving an analytic solution for the Hooke’s law model of a two‐electron atom, subject to a constraint linking the harmonic frequency to the nuclear charge and the dimension.

Dimensional interpolation of hard sphere virial coefficients

We examine the dependence on spatial dimension D of the Mayer cluster integrals that determine the virial coefficients Bn for a fluid of rigid hyperspheres. The integrals vary smoothly with D, and can be characterized analytically in both the low‐D and high‐D limits. Dimensional interpolation (DI) allows one to evaluate individual Mayer cluster integrals at D=2 and D=3 to within about 1%. The resulting low‐order virial coefficients have an accuracy intermediate between those of the Percus–Yevick and hypernetted chain approximations. Much higher accuracy can be achieved by combining the DI and HNC approximations, using DI to evaluate those integrals omitted by HNC. The resulting low‐order virial coefficients are more accurate than those given by any existing integral equation approximation. At higher order, the accuracy of the individual cluster integrals is insufficient to compute reliable virial coefficients from the Mayer expansion. Reasonably accurate values can still be computed, however, by taking pa...

Dimensional scaling for quasistationary states

Complex energy eigenvalues which specify the location and width of quasibound or resonant states are computed to good approximation by a simple dimensional scaling method. As applied to bound states, the method involves minimizing an effective potential function in appropriately scaled coordinates to obtain exact energies in the D→∞ limit, then computing approximate results for D=3 by a perturbation expansion in 1/D about this limit. For resonant states, the same procedure is used, with the radial coordinate now allowed to be complex. Five examples are treated: the repulsive exponential potential (e−r); a squelched harmonic oscillator (r2e−r); the inverted Kratzer potential (r−1 repulsion plus r−2 attraction); the Lennard‐Jones potential (r−12 repulsion, r−6 attraction); and quasibound states for the rotational spectrum of the hydrogen molecule (X 1∑g+, v=0, J=0 to 50). Comparisons with numerical integrations and other methods show that the much simpler dimensional scaling method, carried to second‐order ...

Large order dimensional perturbation theory for complex energy eigenvalues

Dimensional pertubation theory is applied to the calculation of complex energies for quasibound, or resonant, eigenstates of central potentials. Energy coefficients for an asymptotic expansion in powers of 1/κ, where κ=D+2l and D is the Cartesian dimensionality of space, are computed using an iterative matrix‐based procedure. For effective potentials which contain a minimum along the real axis in the κ→∞ limit, Hermite–Pade summation is employed to obtain complex eigenenergies from real expansion coefficients. For repulsive potentials, we simply allow the radial coordinate to become complex and obtain complex expansion coefficients. Results for ground and excited states are presented for squelched harmonic oscillator (V0r2e−r) and Lennard‐Jones (12‐6) potentials. Bound and quasibound rovibrational states for the hydrogen molecule are calculated from an analytic potential. We also describe the calculation of resonances for the hydrogen atom Stark effect by using the separated equations in parabolic coordin...

Electronic tunneling and exchange energy in the D‐dimensional hydrogen‐molecule ion

Dimensional scaling generates an effective potential for the electronic structure of atoms and molecules, but this potential may acquire multiple minima for certain ranges of nuclear charges or geometries that produce symmetry breaking. Tunneling among such minima is akin to resonance among valence bond structures. Here we treat the D‐dimensional H+2 molecule ion as a prototype test case. In spheroidal coordinates it offers a separable double‐minimum potential and tunneling occurs in only one coordinate; in cylindrical coordinates the potential is nonseparable and tunneling occurs in two coordinates. We determine for both cases the ground state energy splitting ΔED as a function of the internuclear distance R. By virtue of exact interdimensional degeneracies, this yields the exchange energy for all pairs of g, u states of the D=3 molecule that stem from separated atom states with m=l=n−1, for n=1→∞. We evaluate ΔED by two semiclassical techniques, the asymptotic and instanton methods, and obtain good agre...

The 1/Z expansion and renormalization of the large‐dimension limit for many‐electron atoms

Analytic expressions for the large‐dimension limit, when renormalized by introducing a suitable effective nuclear charge ζ yield accurate D=3 nonrelativistic energies for ground states of many‐electron atoms. Using Hartree–Fock data to estimate ζ, which typically differs from the actual charge Z by ∼1% or less, we find this dimensional renormalization method (denoted DR‐0) gives results substantially better than the HF input. Comparison of the 1/Z expansion for the large‐D limit with that for D=3 atoms provides expressions for the leading error terms in the renormalized total energy and correlation energy. When configuration mixing occurs in the Z→∞ limit (as for Be and many other atoms), we find the renormalization procedure is markedly improved by including the zeroth‐order mixing (denoted DR‐1); this contributes a term linear in Z. Including the Z‐independent term (DR‐2) also improves the accuracy when zeroth‐order mixing is absent (e.g., ground‐state atoms with N=2, 3, and 7–11) but not otherwise. Cor...

Atomic energies from renormalization of the large-dimension limit

By augmenting Hartree–Fock (HF) results for nonrelativistic ground‐state energies of N‐electron atoms by analytic expressions for the D→∞ limit derived by Loeser, we obtain a simple renormalization procedure which substantially enhances accuracy. A renormalized nuclear charge, Z∞, is found which renders the dimensionally scaled energy at D→∞ a good approximation to that for D=3 with the actual Z. The renormalized charge is readily evaluated by comparing the HF energy (or any other input approximation) with its D→∞ limit. For atoms with any N or Z, the computations are elementary, requiring little more than solution of a quartic equation. With only HF input in addition to the D→∞ limit, the renormalization procedure yields about 2/3 or more of the correlation energy, for neutral atoms with N=Z=2→86. Further improvements in the method seem feasible, but will require better means to incorporate shell‐structure in the large‐D limit.

Electronic tunneling in H+2 evaluated from the large-dimension limit

Abstract We derive a simple, analytic expression for the energy splitting Δ E between the lowest pair of H + 2 states (1sσ g and 2pσ u ) that results from electron exchange between two protons. The calculation employs the semiclassical instanton method, with two unorthodox features which markedly simplify the treatment: (1) The double-minimum potential and corresponding wavefunctions that govern the electronic tunneling are evaluated in the large-dimension limit. (2) The time variable is rescaled to cure divergent behavior of fluctuations about the instanton path that otherwise appears because of the potential develops sharp cusps as the internuclear distance increases. By virtue of exact interdimensional degeneracies, the large- D limit yields valid results for a 3D molecule. Indeed, a simple dimensional scaling law gives Δ E for all pairs of g, u states that stem from separated atom states with m = l = n −1, for n =1→∞. For a wide range of internuclear distances, our analytic expression for Δ E , which pertains to the leading order in l>, gives for such states good agreement with comparable semiclassical methods as well as with exact numerical calculations. It is remarkable that use of the effective potential for the large-dimension limit, which is exactly calculable from classical electrostatics, yields quantitatively results for electronic tunneling, an intrinsically quantal phenomenon.

Statistical model for delocalized π bonding in the C60 molecule

Abstract We present a model for the C 60 molecule in which the σ bond energy is given by pairwise Morse potentials and the π bond energy is given by a statistical model. Good results for the binding energy and equilibrium sphere radius are obtained when exchange and correlation energies are included in the model, whereas poor results are obtained when these effects are neglected.