Non-exponential time-correlation function for random physical processes

The exponential correlation function is theoretically incorrect in the entire frequency range of interest for processes described in terms of linear response theory. The Lorentzian lineshape results from an inconsistent assumption of exponential correlation at timescales smaller than the relaxation time. A new correlation function is proposed that avoids the deficiencies of the exponential function. Comparison on dielectric relaxation in gases shows that the new correlation function can be used to fit data satisfactorily instead of the exponential function. The new correlation function is theoretically consistent for all processes described in terms of linear response theory. Its additional mathematical superiority implies that it can be used instead of the exponential function for all such processes.Comment: 14 pages, 1 Table, 4 figures, submitted for publication. Manuscript rewritten to emphasise that exponential correlation is theoretically incorrect for processes described in terms of linear response theor

Probabilistic method to determine electron correlation energy

A new method to determine electron correlation energy is described. This method is based on a better representation of the potential due to interacting electrons that is obtained by specifying both the average and standard deviation. The standard deviation is determined from a probabilistic interpretation of the Coulomb interaction between electrons. This leads to a better representation of orbital energies as Ei (plus-minus) delta(Ei), where Ei is the Hartree-Fock orbital energy and delta(Ei), the spread, is an indicator of the magnitude of correlation energy. This new representation of the potential when combined with an empirical constant leads naturally to a new method to determine electron correlation energy. Correlation energy is determined within the independent electron approximation without any contribution from higher energy unoccupied states. A consistent physical interpretation that an electron occupies a given position when other electrons are farther than on average can be made. It is a general technique that can be used to determine correlation energy in any system of particles with inter-particle interaction V(r1, r2) and can be considered to be universal first step beyond mean-field theory.Comment: 9 pages. Submitted for publicatio

Analysis of long range order

A first principles analysis of order-disorder transition in alloys shows that ordering energy is a function of temperature due to thermal vibrations. The inter-nuclear potential energy term converges if zero point vibrations are incorporated and this method can replace the Ewald sum method. Core energy contributions to the ordering energy are stored exclusively in superlattice lines. The effect of electron-phonon interactions on ordering energy is of the same order of magnitude as ordering energy near transition temperatures and cannot be ignored. Ising model and variants are incorrect in explaining alloy phase transitions as they ignore the role of electron-phonon interactions without justification. A theoretical formalism that incorporates the Debye-Waller Factor component of electron-phonon interactions in electronic structure calculations already exists and must be adopted when modeling temperature dependent phenomena. It is suggested that DWF correction will account substantially for the discrepancy between experimental and theoretical ordering energy in Ni3V. Thermal vibrations alter magnetic ordering energy at finite temperatures. The role of electron-phonon interactions in alloy and magnetic phase transitions cannot be ignored and must be incorporated in all models. This will also ensure consistency with x-ray and electron diffraction (alloy transitions) and neutron diffraction (magnetic transitions) results. An isotope effect is predicted for (magnetic) phase transitions if the transition temperature is below Debye temperature. Recent observations of an isotope effect in magnetic phase transitions confirm our above conclusions and imply that the role of electron-phonon interactions must be incorporated in all theories and models of magnetism to avoid contradictions.Comment: revised manuscript, 31 pages, submitted for publicatio

Form of the exact partition function for the generalized Ising Model

The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.Comment: 17 pages, 1 figure Submitted for publication. Improved figure. Expanded discussion section adde

Role of thermal vibrations in magnetic phase transitions

We address and resolve the fundamental contradiction that has existed from the earliest studies on magnetic phase transitions between theoretical models that ignore the role of thermal vibrations and represent the exchange interaction as a constant, Jij(0), and analysis of neutron diffraction data that always incorporates thermal vibrations even though it is also possible to analyze the same data by ignoring them. Of the two possibilities, ignoring thermal vibrations in both theoretical models and analysis of diffraction data leads to the latter giving different magnetic order parameters for different reciprocal lattice lines. This appears to be the first report of a unique consequence, viz. the assumption to neglect a physical phenomenon turns a single-valued experimental observable into a multiple-valued one where all values are equally valid. This assumption is clearly unacceptable and must be rejected. The second possibility of incorporating thermal vibrations in both leads to single-valued theoretical and experimental order parameters. Thus, analysis of neutron diffraction data constrain the exchange interaction in all theoretical models to be temperature dependent and represented as Jij(T). Additional experimental and theoretical evidences in support of this conclusion are presented.Comment: Submitted for publicatio

Correct form of the electron wavefunction in periodic solids

The Bloch wavefunction leads either to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape when the geometry of large crystals is considered in detail. It is incompatible with the assumption underlying the Born-von Karman periodic boundary condition. The source of the difficulty is the incorrect dependence of the Bloch wavefunction on the wavenumber index k. The mathematically impossible consequences can be overcome if the periodic part of the electron wavefunction is represented as un(r), which is dependent only on the band index, n, and is independent of the wavenumber index k. This correct form of the wavefunction is consistent with the Bloch theorem and with all other properties of Bloch wavefunctions. The correct form is also consistent with the Born-von Karman periodic boundary condition. The correct form of the electronic wavefunction in a periodic solid has profound consequences. It simplifies the calculation of electronic structure as only one wavefunction per band, un(r), needs to be evaluated. It brings about a conceptual unification between the band picture favored by physicists and the bond picture favored by chemists. The correct form of the electron wavefunction will simplify the understanding of many phenomena involving valence electrons.Comment: 15 pages, submitted for publicatio

Role of thermal vibrations in phase transitions

All theoretical models (Heisenberg, Ising etc.) assume a negligible role for thermal vibrations in alloy and magnetic phase transitions. Analysis of diffraction data conclusively proves that this assumption is incorrect. A simple criterion emerges that theoretical models can ignore the role of thermal vibrations only if the Debye-Waller Factor is ignored in the analysis of diffraction data. Diffraction data constrain all theoretical models to incorporate the role of thermal vibrations. This conclusion is also supported by other experimental results, the effect of thermal vibrations on ordering energy that is of the same order of magnitude as ordering energy and an isotope effect on magnetic phase transitions. An electron-phonon interaction (EPI) formalism that incorporates the Debye-Waller Factor in electronic structure calculations already exists and must be adopted for a correct understanding of phase transitions as it can account for all the different experimental results mentioned above. The discrepancy between experimental and theoretical ordering energy in Ni3V is direct evidence for the role of thermal vibrations in altering ordering energy. The inter-nuclear potential energy term converges if zero point vibrations are incorporated and this method can replace the Ewald sum method. The three dimensional Ising model cannot represent order-disorder transition in beta brass, CuZn. An isotope effect is predicted for magnetic phase transitions if the transition temperature is below Debye temperature. The long range order parameter obtained from diffraction data can only be compared with predictions of models that incorporate the role of thermal vibrations and not otherwise.Comment: 37 pages, submittd for publicatio

Relation between ab initio molecular dynamics and electron-phonon interaction formalisms

The relation between ab initio molecular dynamics formalism and the electron-phonon interaction formalism [P.B. Allen and V. Heine, J. Phys. C 9, 2305 (1976)] is explored. The fundamental quantity obtained in the AIMD formalism - total energy for any configuration - is also obtained from the formalism (ES-DWF) that incorporates the role of Debye-Waller Factor in electronic structure calculations. The two formalisms are exactly equivalent and represent the direct and perturbation theory approaches to determine total energy. This equivalence allows either formalism to be used depending on the requirement - ES-DWF for a priori theoretical analysis and AIMD for ab initio modeling of the effect of thermal vibrations. Combining the two formalisms makes the ES-DWF formalism into an ab initio method and increases the range of problems that can be modeled ab initio. It is also theoretically possible to obtain self-consistent band structures from AIMD calculations. This study clarifies the incorrect assumptions regarding the two formalisms that exist in published literature. By combining the two formalisms and including self-energy effects, more accurate results can be obtained, ab initio, within the adiabatic approximation, than by using AIMD alone.Comment: 16 pages, manusript extensively rewritten, submitted for publicatio

Finite temperature electronic structure of Diamond and Silicon

The electron-phonon interaction contribution to the electronic energies is included in density functional total energy calculations with ab initio pseudopotentials via the Allen formalism [Phys. Rev. B 18, 5217 (1978)] to obtain temperature dependent electronic structure of diamond and silicon. This method allows us to obtain the thermally-averaged ab initio electronic structure in a straightforward and computationally inexpensive way. Our investigations on the finite temperature electronic structure of diamond and silicon lead to a new criterion, temperature transferability, which is required in the ab initio pseudopotentials for temperature dependent studies. The temperature transferability of the Troullier-Martins pseudopotentials used in this work is strongly dependent on the cut-off radius and the inclusion of the unbound 3d$^0$ state. The finite temperature indirect band gaps are highly sensitive to the choice of cut-off radius used in the pseudopotentials. The finite temperature band structures and density of states show that thermal vibrations affect the electron energies throughout the valence and conduction band. We compare our results on the band gap shifts with that due to the Debye-Waller term in the Allen-Heine theory and discuss the observed differences in the zero point and high temperature band gap shifts. Although, the electron energy shifts in the highest occupied valence band and lowest unoccupied conduction band enable to obtain the changes in the indirect and direct band gaps at finite temperatures, the shifts in other electronic levels with temperature enable investigations into the finite temperature valence charge distribution in the bonding region. Thus, we demonstrate that the Allen theory provides a simple and theoretically justified formalism to obtain finite temperature valence electron charge densities that go beyond the rigid pseudo-atom approximation.Comment: 15 pages, 9 figures, 8 table

Finite temperature external potential in crystalline solids

Thermal vibrations alter the external potential. Allen (Phys. Rev. B 18 (1978) 5217) proved that at finite temperatures the pseudopotential form factors are corrected by a Debye-Waller Factor (DWF). We generalize this result to the crystal potential. (The generalization to the all-electron case of the nuclear potential fails due to the breakdown of the rigid-atom approximation.) This finite temperature formalism only gives thermal-averaged properties and no dynamical information can be obtained. Hence, it is labeled the Quasi Ab Initio formalism. Analogous to the use of experimental lattice parameters in ab initio studies, experimental DWF can also be used. The justification is identical; the experimental parameters can be validated by separate ab initio studies. Our work transforms, forty years later, Kasowski's empirical study (Phys. Rev. B. 8 (1973) 1378) into the first ab initio finite temperature band structure calculation. This formalism opens the way to obtain ab initio finite temperature thermal-averaged properties from a single calculation.Comment: 16 pages, submitted for publicatio