Nonorthogonality corrections in the method of correlated basis functions

A set of normalized linearly independent basis functions φ₁, φ₂, …, φ_j, … generates matrix representatives $\text{H}$ and $\text{N}$ of the Hamiltonian operator and the identity. An orthonormal basis $\text{φ}{}_1$, $\text{φ}{}_2$, …, $\text{φ}{}_{\mathrm{j}}$, … generated by a Löwdin transformation is characterized by the distance in Hilbert space between $\text{φ}{}_{\mathrm{j}}$ and φ_j. The choice of positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ minimizes these distances and maximizes the diagonal elements of $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Again for positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and a finite basis, 1 ? j ? p, the analysis yields a general theorem on Trace $\text{N}$${}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ (? p for all positive and negative integral values of n except n = ?1 and ? p for n = ?1).Sufficient conditions are determined which permit the application of the binomial theorem to the evaluation of the transform of $\text{H}$. Approximate formulas for the energy eigenvalues through third order in nondiagonal matrix elements are presented in a compact form containing characteristic nonorthogonality corrections depending on the exterior or interior location of the matrix element in the perturbation formulas.     

Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ and modified Ursell correlation functions $\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ of a classical system of particles with the two-body potential $\text{q(}\text{r}\text{) + γ}{}^{\mathrm{\nu }}\text{K(γ}\text{r}\text{)}$ are considered. The limiting values of the functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ),}\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ and $\text{γ(}{}^{\mathrm{1?k}}\text{)ν}\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γ^νK(γr). The direct correlation function c? (r, γ) is also considered and it is shown that for $\text{S}\text{≠ 0}$, $\text{lim}{}_{\mathrm{\gamma \to 0}}\text{γ}{}^{\mathrm{?\nu }}\text{c}\text{?}\text{(}\text{S}\text{γ}\text{, γ) = ?βK (}\text{S}\text{)}$, for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.     

Hartree-fock wave functions and oscillator strengths for the helium isoelectronic sequence

Oscillator strengths for 1¹S?n¹P transitions for some members of the helium isoelectronic sequence have been calculated by utilizing the dipole length, velocity and acceleration forms of the transition matrix element with coupled H.F. wave functions. Good agreement has been obtained between the values of the oscillator strengths. The quantities $\text{(?}{}_{\mathrm{A}}\text{??}{}_{\mathrm{L}}\text{)}$ and $\text{(?}{}_{\mathrm{A}}\text{??}{}_{\mathrm{V}}\text{)}$ decrease slowly with nuclear charge Z, whereas $\text{(?}{}_{\mathrm{L}}\text{-?}{}_{\mathrm{V}}\text{)}$ first increase to a maximum value and then decreases slowly.     

Shell-model studies of the 90Y, 91Zr, 92Nb and 93Mo nuclei

A shell-model calculation of the N = 51, 39 ≦ Z ≦ 42 nuclei is presented. The ⁸⁸Sr nucleus is assumed to be an inert closed core. The extra-core protons are restricted to the ($\text{2}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{1}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$) configurations, and the active neutron is allowed to occupy the $\text{2}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$, $\text{3}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{2}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ and $\text{1}\text{g}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ orbits. The proton-proton effective interaction is directly taken from the previous analysis on the energy levels for N = 50 isotones by Ball et al. The proton-neutron effective interaction is assumed to be of the form of the surface δ-interaction. The energy spectra are calculated from a least-squares fit to the experimental data, varying the T = 0 and T = 1 strengths of the surface δ-interaction. Spectroscopic factors, E2 transition rates and two-body matrix elements are also calculated and compared with the observed values and the previous theoretical results.     

Electromigration in tin single crystals

Atom transport in high-purity tin single crystals due to the influence of large direct currents has been measured by the “vacancy flux” technique. Cylindrical specimens were selected with c-axis oriented with 9° perpendicular or parallel to the direction of current flow. Rates of both longitudinal and transverse dimensional changes were used to calculate the anode-directed atom drift velocity. The results gave $\text{Z}{}^1{}_6\text{?}{}_6\text{= ?18±2}$ and $\text{z}{}^1{}_{\mathrm{\perp }}\text{?}{}_{\mathrm{\perp }}\text{= ?18 ±2}$, where Z¹ is the effective charge number and ?₆ = 0.89 and ?_⊥ = 0.54 are the estimated correlation factors in the parallel and perpendicular directions. These values for Z¹ are appreciably smaller than the results reported earlier for polycrystalline tin by Kuz'memko. The activation energies for $\text{Z}{}^1\text{?}$ agree within experimental error with those of self-diffusion.     

Three-nucleon results for separable potentials modelled on the Reid soft-core potential

Using a solution to the inverse scattering problem we have generated phase-equivalent separable potentials in the ${}^1\text{S}{}_0$ and ${}^3\text{S}{}_1\text{?}{}^3\text{D}{}_1$ states, which have nearly the same singlet UPA form factors and deuteron parameters (E_D, P_D, Q_D, A_S and $\text{A}{}_{\mathrm{D}}\text{A}{}_{\mathrm{S}}$) as the Reid soft-core potential. We compare our results for the binding energy of the triton and the neutron-deuteron doublet scattering length with the corresponding values for the Reid soft-core potential.     

Relativistic corrections to the Zeeman effect in diatomic molecules

Basic molecular g-factors for the ²Π ground states of OH, SH, NO, NS, and ClO have been determined by a weighted nonlinear least squares fit including zero-field as well as Zeeman data. The combined g-factor $\text{g}{}_{\mathrm{l}}\text{+}\text{1}\text{2}\text{g}{}_{\mathrm{s}}$ accounts for the dominant contribution to the Zeeman effect of the paramagnetic ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ substate. Theoretical values of $\text{g}{}_{\mathrm{l}}\text{+}\text{1}\text{2}\text{g}{}_{\mathrm{s}}$ have been computed by considering the relativistic corrections arising from the reduction of the Breit equation to the second Pauli limit, and a contribution due to the motion of the nuclear center of mass is also included. The relativistic corrections have been computed by use of known LCAO molecular wavefunctions. Furthermore the free electron quantum electrodynamic correction ($\text{α}\text{2}\text{Π}$) to g_s is used as an ad hoc value for an electron bound to a diatomic molecule. A systematic difference is found between theoretical and fitted values of $\text{g}{}_{\mathrm{l}}\text{+}\text{1}\text{2}\text{g}{}_{\mathrm{s}}$. The difference increases linearly with the parameter $\text{Y = |}\text{A}\text{B}\text{|}$ for the molecules OH, SH, and NO. The maximum difference is encountered for NO and is about 20% of the free electron quantum electrodynamic correction. With further increase in Y (NS and ClO) the agreement seems to improve again.     

Radii of single-proton states in isotopes of tin

Sub-Coulomb (t, α) reactions have been employed in the determination of the rms radii of the $\text{1}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$, $\text{2}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{2}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ proton orbits in ^{112,116,118,120,124}Sn. The experimental values are compared to the results of Hartree-Fock calculations.     

“Forbidden” rotational spectra of symmetric-top molecules: PH3 and PD3

A millimeter-wave spectrometer having a sensitivity of 4 × 10^?10 cm^?1 in the 2-mm region has been constructed for observation of extremely weak millimeter-wave spectra of gases. It has been used to measure J → J, K = 0 ← 3 transitions in PH₃ and J → J, K = 0 ← 3 as well as K = ±1 ← ±4 transitions in PD₃. The B₀ and C₀ spectral constants (in MHz) are: for PH₃, B₀ = 133 480.15 ± 0.12 and C₀ = 117 488.85 ± 0.16; for PD₃, B₀ = 69 471.10 ± 0.03 and C₀ = 58 974.37 ± 0.05. The effective ground-state values obtained for the bond angle and bond length are: for PH₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.420}{}_0$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.34}{}_5$; for PD₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.417}{}_6$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.35}{}_9$. The corresponding zero-point-average values were calculated to be: for PH₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4269}{}_9\text{± 0.0002}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.228}{}_7$; for PD₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4226}{}_5\text{± 0.0001}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.256}{}_7\text{± 0.004}$. For both species, the equilibrium values are $\text{r}{}_{\mathrm{e}}\text{(}\text{A}\text{?}\text{) = 1.4115}{}_9\text{± 0.0006}$ and $\text{α}{}_{\mathrm{e}}\text{(}{}^{\mathrm{o}}\text{) = 93.32}{}_8\text{± 0.02}$.     

Fission and decay of excited nuclei

The statistical theory of the decay of excited nuclei including the competition between particle emission and fission is considered. Fission barriers are given by a phenomenological formula which is an analog of the known Cameron formula and approximates well experimental data on nuclei up to transuranic elements. The contribution of shell corrections and odd-even effects is discussed. The calculated ratios of evaporation and fission widths Γ_n/Γ_f agree with experiment even in the region of nuclei with Z > 90. The $\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}$ values depend weakly on excitation energy for nuclei with Z ≈ 90, While in other regions this dependence proves rather strong. The last conclusion does not contradict known experimental data since in experiments one usually measures some effective value $\text{〈}\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}\text{〉}$, which is only indirectly related to the real $\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}$, values. It is shown that the calculated values of $\text{〈}\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}\text{〉}$ depend on excitation energy either. The effects of angular momentum and uncertainty in the choice of level density parameter are discussed.     

CLL = (L,L; 0, 0) measurements and a structure due to a P partial wave in the pp system

Measurements of C_LL of pp elastic scattering near θ_c.m. = 90° at thirteen energies between 300 and 800 MeV are reported. These, together with previous values of C_NN, are used to extract values of two quantities, $\text{?}{}_{\mathrm{<mtext>s</mtext>}}$ and $\text{?}{}_{\mathrm{<mtext>t</mtext>}}$, which contain only spin-singlet and only coupled spin-triplet partial waves, respectively. The $\text{?}{}_{\mathrm{<mtext>s</mtext>}}$ curve, which is not dependent on C_LL, exhibits the behavior expected for the previously conjectured ¹D₂ resonance. The $\text{?}{}_{\mathrm{<mtext>t</mtext>}}$ curve also exhibits a resonance-like behavior, which could be due either to the ³P₀ or the ³P₂ partial wave.     

The dipole moment of thioformaldehyde in its singlet and triplet π1 − n excited states

Laser-induced fluorescence excitation has been used to measure Stark splittings of selected lines in the $\text{A}\text{?}{}^1\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ and $\text{a}\text{?}{}^3\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_2$ band systems of H₂CS in electric fields up to 13 kV/cm. The derived excited state a-axis dipole moments are 0.820 ± 0.007 D for the 4¹ level of the ¹A₂ state; 0.838 ± 0.008 D for the zeroth vibrational level of ¹A₂; and 0.534 ± 0.015 D for the zeroth vibrational level of the ³A₂ state. These results are compared with the corresponding values of H₂CO, and interpreted in terms of the changing localization of the π and $\text{π}{}^1$ orbitals accompanying electronic excitation.     

Phenomenological SU(1, 1) supergravity

We investigate the structure of phenomenological supergravity models which permit the hierarchy problem to be “solved” in the sense that $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ and m_W are determined dynamically to be exp [-O(1)/α] × m_P. Such models must have a flat hidden sector potential, which is only possible if the theory has an underlying SU(1, 1) invariance. Flat SU(1, 1) theories necessarily have a zero cosmological constant and the hidden sector is an Einstein space with . The SU(1, 1) invariance is necessarily broken down to U(1) by the gravitino mass. If $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ is the only source of SU(1, 1) breaking then the tree-level gaugino masses are small and $\text{A =}\text{3}\text{2}$, while values of A up to 3 and non-zero gaugino masses are possible if other sources of SU(1, 1) breaking are tolerated. Yukawa couplings may scale as some power of $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{m}{}_{\mathrm{<mtext>P</mtext>}}$ in these models where $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ is generated dynamically, which may explain the hierarchy of Higgs-fermion Yukawa couplings: $\text{m}{}_{\mathrm{<mtext>f</mtext>}}\text{m}{}_{\mathrm{<mtext>W</mtext>}}\text{=}\text{O}\text{(}\text{m}{}_{\mathrm{<mtext>W</mtext>}}\text{m}{}_{\mathrm{<mtext>P</mtext>}}\text{)}{}^{\mathrm{\lambda >0}}$? These models also permit the spontaneous violation of CP in the Yukawa coupling matrix. Numerical studies yield 20 GeV < m_t < 100 GeV in these phenomenological SU(1, 1) supergravity models. Speculations are presented about their relation to a fundamental theory based on extended supergravity.     

Renormalization group improved predictions for quarkonium decay

A method recently suggested to resolve the ambiguity related to scheme dependence in perturbative QCD is applied to calculations of radiative corrections to quarkonium decay. Renormalization group improved predictions are given for the ratio $\text{R=Г(}{}^1\text{S}{}_0\text{→had)/Г(}{}^3\text{S}{}_1\text{→had)}$, as a function of $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$. The values obtained are substantially larger than in conventional schemes. One finds in particular that R>182 for charmonium, and R>174 for bottomium, independently of the value of the quark mass and $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$. It follows that $\text{Г(}\text{n}{}_{\mathrm{c}}\text{→}\text{had}\text{)>6.9}\text{MeV}$ and $\text{Г(}\text{n}{}_{\mathrm{b}}\text{→}\text{had}\text{)>3.6}\text{MeV}$. Values of the wave function at the origin as a function of $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$ are also extracted from the measured gluonic width of the ³S₁ state. An upper bound for $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$ is obtained under some assumptions.     

The tensor J8 contribution to the rotational energy of a tetrahedral molecule

Two methods are outlined for calculating the tensor distortion energy of a tetrahedral molecule to the eighth degree in the rotational angular momentum. In the first, the entire distortion Hamiltonian including the eighth-rank distortion tensor $\text{Ω}{}_8$ is diagonalized. In the second, the basis which diagonalizes only the fourth rank tensor $\text{Ω}{}_4$ is used to generate tables which allow the direct determination of the octic energy terms. For J ≤ 20, numerical values are given of the diagonal matrix elements of $\text{Ω}{}_8$ and of the second-order perturbation correction due to offdiagonal matrix elements of the sixth-rank tensor $\text{Ω}{}_6$. Explicit expressions in terms of angular momentum operators are given for $\text{Ω}{}_8$ and a selection of the operator equivalents T_8,m.     

Optical absorption by neutral bound excitons

In this note we determine the oscillator strengths for the dipole absorption of neutral bound excitons in direct gap semiconductors, using our previously obtained 35-term Page and Fraser type wave function, and taking into account the detailed electronic structure as well as the electron-hole exchange interaction.The envelope part of the oscillator strengths varies considerably with the electron-hole mass ratio $\text{σ =}\text{m}{}^1{}_{\mathrm{e}}\text{m}{}^1{}_{\mathrm{h}}$, and is maximum for the (D⁰, X)- complex when σ = 0.4. For typical σ-values (σ? 0.1–0.2), . But when σ approaches zero, the overlapping of the electron and the hole envelope wave functions of the (A⁰,X)-complex decreases progressively so that the oscillator strength also decreases and tends to zero.In the case of zinc-blende materials (T_d) and positive spin-orbit coupling at $\text{k}\text{= 0}$, we confirm that the line strength for transitions to or from $\text{J =}\text{1}\text{2}\text{(Γ}{}_6\text{)}$ or $\text{J =}\text{5}\text{2}\text{(Γ}{}_7\text{+ Γ}{}_8\text{)}$ level of the (A⁰, X)-complex is equal to one quarter of the line strength to or from the $\text{J =}\text{3}\text{2}\text{(Γ}{}_8\text{)}$ level.In the case of CdS, where our computed values are only in qualitative agreement with the experimental values, we discuss the use of the phenomenological result of Rashba.     

Core polarization for M2 transitions in odd-A tin isotopes

The reduced M2 transition probabilities $\text{11}\text{2}{}^{\mathrm{?}}{}_1\text{→}\text{7}\text{2}{}^{\mathrm{+}}{}_1$ in the odd-A isotopes ^109–121Sn are found to reveal a specific behaviour. B(M2) values are calculated in the framework of the quasiparticlephonon model. The coupling of a quasineutron with the 2⁺, 3^? and 2^? one-phonon core excitation is taken into account. Inclusion of all one-phonon 2^? states up to 24 MeV in the wave functions of the excited states $\text{11}\text{2}{}^{\mathrm{?}}{}_1\text{and}\text{7}\text{2}{}^{\mathrm{+}}{}_1$ reduces the theoretical B(M2) values by 3–4 times as compared to the single-particle values. The specific B(M2) dependence on the mass number appears to be due to the pairing effect.     

Energies and orbital sizes for some Rydberg and valence states of the nitrogen molecule

Accurate SCF computations are reported on the Rydberg states of N₂ of electron configurations $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$, $\text{---1π}{}_{\mathrm{u}}{}^3\text{3σ}{}_{\mathrm{u}}$, and ---3σ_g2π_g, also on the valence states of the configuration $\text{---1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$. The Rydberg state calculations supplement those of Lefebvre-Brion and Moser. A comparison is made between the $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ states and the parallel set of states of the $\text{1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$ configuration. This comparison shows a sharp difference in the ¹Σ⁺ states of the two configurations, the ¹Σ⁺ state being very high in the latter but relatively low in the former configuration. Recknagel coefficients are given for the several states of the two configurations; as expected, these are much smaller for the $\text{1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ configuration. Also, the ¹Δ state is relatively lower for the latter configuration.     

Etude des proprietes thermodynamiques d'un groupement tetranucleaire: Application au compose Na3RuO4

The exact eigenvalues spectrum of the spin hamiltonian $\text{H}\text{= ?2}\text{∑}\text{(i, j)}\text{J}{}_{\mathrm{ij}}\text{S}\text{?}{}_{\mathrm{i}}\text{S}\text{?}{}_{\mathrm{j}}$ have been calculated for a tetranuclear cluster formed by four spins 3/2 at the vertices of a lozenge. Two isotrope exchange interactions J₁ and J₂ are able to explain the thermodynamic properties (magnetic susceptibility, entropy, specific heat). A ground state transition from singlet to triplet state occurs when the $\text{J}{}_2\text{J}{}_1$ ratio reaches the value $\text{4}\text{3}$. The magnetic susceptibility data of Na₃RuO₄ fit well with the theoretical values proposed for $\text{J}{}_1\text{K}\text{= (?19,5 K)}$ and J₂/k (?22,5 K).