Ordinary Energies Research Articles

Relationship between Ordinary, Laplacian, Randić, Incidence, and Sombor Energies of Trees

The sum of the absolute values of the eigenvalues of the graph’s adjacency matrix is known as its ordinary energy. Based on the eigenvalues of a range of other graph matrices, several other equivalent energies are being considered. In this work, we considered ordinary energy, Laplacian, Randi´c, incidence, and Sombor energy to analyze their relationship using polynomial regression. The performance of each model is exceptional with cross-validation RMSE mostly below 1.

A cosmology of a trans-Planckian theory and dark energy

We investigate a model based on a generalized version of the Fourier transform for curved spacetime manifolds. This model is possible if the metric has an asymptotic flat region which allows a duality to be implement between coordinates and momenta, hence, the model's name, trans-Planckian. The theory and the action are based on the postulate of the absolute egalitarian relation between coordinates x and momenta p. We show how to implement this construction in a cosmological setting, on a Friedman–Robertson–Walker (FRW) metric background, where the asymptotic time infinity plays the role of the required asymptotic flat region. We discuss the effect of gravity, and, in particular, of the Hubble expansion of the universe scale factor on the Fourier map. The dual-inflationary stage is responsible for making the dual-sector of the action inaccessible at ordinary low energies. We propose a scenario in which an effective positive cosmological constant is caused by how the dual-sector of the theory affects the equation of state for matter particles.

Effective Field Theory Approach to Gravitationally Induced Decoherence

Adopting the viewpoint that the standard perturbative quantization of general relativity provides an effective description of quantum gravity that is valid at ordinary energies, we show that gravity as an environment induces the rapid decoherence of stationary matter superposition states when the energy differences in the superposition exceed the Planck energy scale.

The self-consistent field for crystals

Exchange in isolated atoms and ions is investigated, to throw light on the approximate exchange to be used in crystals. An intercomparison, carried out by the author, J. B. Mann, J. H. Wood, and T. M. Wilson, for the case of Cu+ is described in detail. Here five different values of exchange are used: VXHF, the Hartree-Fock value; VXS, a free-electron exchange suggested by Kohn, Sham, and Gaspar, in which the exchange correct for a free-electron gas at the Fermi energy is assumed; VXLSW, a modification of an exchange recently suggested by Liberman, proposed by the writer and Wood, in which we use an energy-dependent exchange suggested by the free-electron gas theory; and VXa, in which the exchange VXS is multiplied by such a factor α that the total energy of the atom is minimized. Of these, VXLSW and VXα both give orbitals which are good approximations to the Hartree-Fock and VXLSW also gives eigenvalues agreeing well with the Hartree-Fock. The VXα method, however, gives eigenvalues numerically smaller than the Hartree-Fock, by amounts which increase as we go to the inner electrons. In spite of this, the eigenvalues of the VXα method empirically prove to form very good one-electron energies to use in the energy-band problem. The reason for this is then examined. A method for studying atoms with partially filled shells, such as the 3d transition atoms, is described. We call it the Hyper-Hartree-Fock method, since it deals with an average energy over many electronic states, rather than a single state as in the HF method. The criterion for minimum energy is shown to be the equality of one-electron energies E'i of two shells such as 3d and 4s, which can interchange electrons, where E'i differs from the ordinary eigenvalue of the Hartree-Fock method. The latter represents the energy difference between an atom and an ion lacking an electron; the modified energy E'i is the derivative of the energy with respect to the occupation number. It is shown that these E'i's are the energies which should be used in an energy-band calculation, not the ordinary HF energies. The two sets of energies differ by a considerable amount in the transition elements, but the quantities E'i agree well with the eigenvalues of the method using the exchange VXα. This appears to explain the success of the latter method for energy-band calculations, and suggests that it is better than a Hartree-Fock calculation, even if the latter could be carried out. It is pointed out that the quantities E'i are closely related to the electronegativities used by the chemists, and also to one-particle energies met in the theory of the Fermi liquid.

Thin Film Synthesis by Energetic Condensation

▪ Abstract The use of energetic particles (ions and atoms) has become increasingly important in physical vapor deposition techniques. These deposition processes can be divided in two main classes: ion beam-assisted deposition and energetic condensation (or deposition). This review focuses on the latter, i.e. processes in which the actual depositing species have energies that far exceed ordinary thermal energies, namely energies greater than 20 eV. The phenomenology of the effect of these high-energy particles on the growth of thin films is first broadly presented, and then specific examples of film deposition are given. The examples drawn here are of films that have been prepared by metal plasma immersion implantation and deposition. The observed microstructures and functional properties of these films are discussed in terms of processing conditions.

Low-energy supersymmetry withD-term contributions to scalar masses

We investigate how the predictions of the Minimal Supersymmetric Standard Model are modified by D-term contributions to soft scalar masses, which arise whenever the rank of the gauge group at very high energies is greater than four. We give a parameterization of the most general such contributions that can occur when the unbroken gauge symmetry is an arbitrary subgroup of E_6, and show how the D-term contributions leave their imprint on physics at ordinary energies. We impose experimental constraints on the resulting parameter space and discuss some features of the resulting supersymmetric spectrum which differ from the predictions obtained with universal boundary conditions on scalar masses near the Planck scale. These include relations between squark and slepton masses; the behavior of $\sin^2 (\beta-\alpha)$ (which determines the production cross-section for the lightest Higgs scalar boson at an e+e- collider) and the mass of the pseudoscalar Higgs bosons; R_b [the ratio $\Gamma(Z -> b\overline b)/\Gamma(Z -> hadrons)$; and mass differences between charginos and neutralinos.

General relativity as an effective field theory: The leading quantum corrections.

I describe the treatment of gravity as a quantum effective field theory. This allows a natural separation of the (known) low energy quantum effects from the (unknown) high energy contributions. Within this framework, gravity is a well behaved quantum field theory at ordinary energies. In studying the class of quantum corrections at low energy, the dominant effects at large distance can be isolated, as these are due to the propagation of the massless particles (including gravitons) of the theory and are manifested in the nonlocal/nonanalytic contributions to vertex functions and propagators. These leading quantum corrections are parameter-free and represent necessary consequences of quantum gravity. The methodology is illustrated by a calculation of the leading quantum corrections to the gravitational interaction of two heavy masses.

Black holes, wormholes, and the disappearance of global charge

One of the paradoxes associated with the theory of the formation and subsequent Hawking evaporation of a black hole is the disappearance of conserved global charges. It has long been known that metric fluctuations at short distances (wormholes) violate global-charge conservation; if global charges are apparently conserved at ordinary energies, it is only because wormhole-induced global-charge-violating terms in the low-energy effective Lagrangian are suppressed by large mass denominators. However, such suppressed interactions can become important at the high energy densities inside a collapsing star. We analyze this effect for a simple model of the black-hole singularity. (Our analysis is totally independent of any detailed theory of wormhole dynamics; in particular it does not depend on the wormhole theory of the vanishing of the cosmological constant.) We find that in general all charge is extinguished before the infalling matter crosses the singularity. No global charge appears in the outgoing Hawking radiation because it has all gone down the wormholes.

Large‐scale RPA calculations of chiroptical properties of organic molecules: Program RPAC

AbstractThe computational considerations involved in calculating ordinary and rotatory intensities and electronic excitation energies in the random phase approximation (RPA) are examined. We employ a localized orbital formulation in order to analyze the results in terms of local and charge‐transfer excitations. Occupied orbitals are localized by the Foster–Boys procedure. The virtual space is transformed into a localized “valence” set that maximizes dipole strengths with the occupied counterparts, and a delocalized remainder. The two‐electron integral transformation is performed with an efficient algorithm, based on Diercksen's, that generates only the particle–hole‐type integrals required in the RPA. The lowest solutions of the RPA equations are obtained iteratively using a modification of the Davidson‐Liu simultaneous vector expansion method. This allows the inclusion of the entire set of particle–hole states supported by a basis set of up to 102 orbitals. Calculations at this level give better excitation energies and intensities than SDCI methods, at substantial savings in computational effort. Comparative timings, computed results and analysis in terms of localized orbitals are given for planar and distorted ethylene using extended atomic orbital bases including diffuse functions. The results for planar ethylene are in excellent agreement with experiment.

Supersymmetry at ordinary energies. Masses and conservation laws

An assessment is made of the general problems encountered in formulating a realistic supersymmetric theory in which the spontaneous breakdown of supersymmetry occurs at ordinary energies accessible to accelerators. As a starting point, three problems are identified in SU(3)\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1) supersymmetric models with only quark and lepton chiral superfields: the up quarks get no masses, baryon and lepton ($B$ and $L$) conservation are violated by renormalizable and hence unsuppressed interactions, and the scalar counterparts of the quarks and leptons are too light. An interesting SU(3)\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1) model of Dimopoulos and Georgi that avoids these problems is considered; it is found that this model contains $B$- and $L$-nonconserving effective interactions of dimensionality 5 that lead to proton decay at too rapid a rate. To guarantee natural $B$ and $L$ conservation in effective interactions of dimensionality 4 and 5, it is suggested that the gauge group that describes physics at ordinary energies contains a factor, such as another U(1), in addition to SU(3)\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1). Such theories do not contain dimension-5 $L$-nonconserving interactions which could produce an observable neutrino mass, but they do allow dimension-6 $B$- and $L$-nonconserving interactions that would lead to proton decay at an observable rate. Supersymmetry is found to constrain the matrix elements for proton decay in a phenomenologically interesting way. A general explanation is given of how such theories naturally avoid the problem of light scalars, as found by Fayet. The formalism is used to derive general approximate mass relations for the scalar superpartners of the quarks and leptons. The problem of anomalies in the new U(1) current is considered, and one attractive scheme for avoiding them is offered, in which the anomalies cancel for precisely three generations of quarks and leptons.

Grand unification of effective gauge theories

An effective non-renormalizable SU(3)×SU(2)×U(1) invariant gauge theory results at ordinary energies when superheavy fields are integrated out from a grand unified theory based on a simple gauge group G. The solutions of the second-order renormalization-group equations for the gauge coupling constants of the effective theory are examined. General formulae for the superheavy vector boson mass and for sin 2 θ near M W are given in this approach to grand unification. The superheavy vector boson mass is plotted against the QCD scale parameter Λ for a certain set of grand unified models. Corrections to the prediction when the set of models is enlarged are discussed, and illustrated with examples from G≡SU(5) and O(10).

Implications of dynamical symmetry breaking

An analysis is presented of the physical implications of theories in which the masses of the intermediate vector bosons arise from a dynamical symmetry breaking. In the absence of elementary spin-zero fields or bare fermion masses, such theories are necessarily invariant to zeroth order in the weak and electromagnetic gauge interactions under a global $\mathrm{U}(N)\ensuremath{\bigotimes}\mathrm{U}(N)$ symmetry, where $N$ is the number of fermion types, not counting color. This symmetry is broken both intrinsically by the weak and electromagnetic interactions and spontaneously by dynamical effects of the strong interactions. An effective Lagrangian is constructed which allows the calculation of leading terms in matrix elements at low energy; this effective Lagrangian is used to analyze the relative direction of the intrinsic and spontaneous symmetry breakdown and to construct a unitarity gauge. Spontaneously broken symmetries which belong to the gauge group of the weak and electromagnetic interactions correspond to fictitious Goldstone bosons which are removed by the Higgs mechanism. Spontaneously broken symmetries of the weak and electromagnetic interactions which are not members of the gauge group correspond to true Goldstone bosons of zero mass; their presence makes it difficult to construct realistic models of this sort. Spontaneously broken elements of $\mathrm{U}(N)\ensuremath{\bigotimes}\mathrm{U}(N)$ which are not symmetries of the weak and electromagnetic interactions correspond to pseudo-Goldstone bosons, with mass comparable to that of the intermediate vector bosons and weak couplings at ordinary energies. Quark masses in these theories are typically less than 300 GeV by factors of order $\ensuremath{\alpha}$. These theories require the existence of "extra-strong" gauge interactions which are not felt at energies below 300 GeV.

Nonintegral Occupation Numbers in Transition Atoms in Crystals

We consider the implications of a nonintegral occupation number of $3d$ and $4s$ electrons in a $3d$ transition element or compound, in a configuration such as $3{d}^{n+x}4{s}^{2\ensuremath{-}x}$, where $x$ is a variable. In the energy-band problem, such fractional occupation numbers are common, but we pay particular attention to the atomic problem. We apply Hartree-Fock procedures to such a problem, using the formula for the average energy of all multiplets associated with the configuration, and vary not only the orbitals but the occupation numbers to minimize the energy. We consider both the non-spin-polarized and spin-polarized cases. This procedure, which is more general than the ordinary Hartree-Fock procedure, we shall call the hyper-Hartree-Fock method (HHF). We have carried through HHF calculations for fractional occupation numbers in the Co and Ni atoms, and have also treated these atoms by several schemes involving approximate statistical exchanges. We compare the results with the atomic spectra of these atoms. We find that the condition for minimum energy, in the HHF scheme, can be put in a form stating that one-electron energies $E_{i}^{}{}_{}{}^{\ensuremath{'}}$ of the $3d$ and $4s$ orbitals must be equal; these quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, which we call modified one-electron energies, are different from the ordinary one-electron energies ${E}_{i}$ of Hartree-Fock theory, involving only one-half the self-energy correction met with in HF theory. These quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, rather than the ordinary one-electron energies ${E}_{i}$, are the quantities which have the properties desired for one-electron energies in energy-band theory and Fermi statistics: The change in the total energy of the system, when an infinitesimal fraction of the electrons shifts from one orbital to another, rigorously equals the net change in the quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$ for the electrons which have made the shift. We show that the ordinary one-electron eigenvalues of the Kohn-Sham statistical exchange method form fairly good approximations to these HHF quantities $E_{i}^{}{}_{}{}^{\ensuremath{'}}$, which explains why energy-band calculations using that exchange have had considerable success in studies of transition-element crystals and compounds. Preliminary mention is made of calculations under way by one of the authors (TMW) on the antiferromagnetic crystals MnO and NiO, in which an exchange potential set up according to the ideas presented here leads to energy bands describing correctly the electrical, magnetic, and optical behavior of these crystals, including the insulating properties and the crystal-field splitting of the $3d$ orbitals into the ${e}_{g}$ and ${t}_{2g}$ components.

On the Structure of Light Nuclei

Extensive calculations based on the approximation of single particle wave functions (the Hartree method) have been made for the nuclei between ${\mathrm{He}}^{6}$ and ${\mathrm{O}}^{16}$ using the general symmetrical interaction operator given by Eq. (1). The Coulomb interaction is treated as a small perturbation. Secular equations are avoided by the construction of space wave functions in the normal state configuration belonging to irreducible representations of the symmetric group. These functions yield an energy matrix which is diagonal in the ordinary and Majorana interaction energies. The contributions of the spin exchange and Coulomb operators to the energy terms are found by a first-order perturbation calculation. Although the general symmetrical operator contains several parameters as yet undetermined, only those parameters which have been fixed by consideration of the two, three and four particle problems are involved in the energy differences within the group of low lying terms belonging to the normal state configuration. These term differences are identical with those recently computed for unsymmetrical interaction operators of the saturation type. New results for mass defects, excitation energies and energy relations between isobars are compared with experimental values.