Calculation of the energy density function by the method of steepest descent

We present a unified theory of diatomic molecules which reconciles bound state spectroscopy and atomic scattering theory. The total wave-function is expanded in a complete set of atomic channel states which is entirely equivalent to an expansion in Hund's case (e) electronic-rotational states. An analysis of the coupled radial, that is vibrational, functions places strong constraints on the asymptotic properties of the molecular wave-functions. These are presented in terms of the reactance K and scattering S matrices of atomic scattering theory which offers a uniform treatment for open channels (inelastic scattering and continuum spectroscopy), closed channels (bound state spectroscopy) and mixtures of both (predissociation). The normalization of the total wavefunction is derived and related to the asymptotic boundary conditions both for continuum and bound states.     

An analysis of energy differences in atomic multiplets in connection with the inequality formulation of Hund's rules

A detailed consideration is given of a recent reformulation of Hund's rules in terms of inequalities. The interpretation of Hund's rules is reconsidered assuming that each energy level is known exactly in the restricted Hartree-Fock approximation. Some general theorems and series expansions are derived for energy differences between atomic states. As is now well known, the state of lower energy in a multiplet has the higher expectation value for the electron-electron repulsion, especially in neutral systems. The behaviour of this repulsion is analysed in terms of the Fermi hole and the contraction of the wavefunction which one finds for the state of lowest energy. Numerical examples are given for the ¹P and ³P states of the He isoelectronic series, the ¹S, ¹D and ³P states of the C and O series and the ²P, ²D and ⁴S states of the N series.     

On the determination of small electric dipole moments of excited electronic states in diatomic molecules

A new method is proposed for the determination of small electric-dipole moments in diatomic molecules by measuring the induced transitions due to crossed electric and magnetic fields. A theoretical treatment is given for states belonging to Hund's cases (a) and (b), and for states in the intermediate coupling case between Hund's cases (a) and (b). The method is restricted to non-Σ states.     

Derivation,properties and application of Coulomb Sturmians defined in spheroidal coordinates

Coulomb Sturmian amplitude functions are derived in prolate spheroidal coordinates and are presented in a closed algebraic form. Spheroidal Sturnian functions are revealed to be related to the polynomial solutions of Heun's confluent equation. A reduction of symmetry from spherical to axial leads to the coupling of spherical polar orbitals and the formation of hybrid orbitals. The contribution of each spherical orbital into a hybrid orbital depends strongly on distance R from a nucleus to the dummy centre, and substantially alters when R varies. At two limiting cases R = 0 and R → ∞ spheroidal Sturmians are purely atomic orbitals, whereas at intermediate R they contain many features intrinsic to diatomic molecular orbitals. Applications of spheroidal Sturmian basis are discussed; Coulomb spheroidal Sturmians are asserted to be the most appropriate basis functions for diatomic molecular calculations.     

Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions

The formalism and results of truncated coupled channels evaluations of three-body models of deutron-induced nuclear reactions are reviewed. Emphasis is placed on breakup, elastic scattering and stripping. The relations of the coupled channels method to the Faddeev method, the adiabatic approximation and the distorted wave Born approximation are discussed extensively. Although the adiabatic approximation is seen to be excellent for the wavefunction in the elastic channel, it significantly underestimates the contributions of breakup states in stripping. Significant effects are associated with coupling to relative l = 2 breakup states.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

MRCISD calculations of the six lowest valence states of I2 −

Potential curves for the four Hund's case (a) basis electronic states (²Σ, ²Π_g, ²Π_u, and ²Σ⁺) correlating with the I(²P_u) + I^?(¹S_g) dissociation limit have been calculated at the MRCISD level of theory using an all-electron double-zeta basis set augmented with an extensive set of polarization and diffuse functions complete through i angular momentum functions. Transition moments for the dipole allowed transitions between these states are determined as a function of internuclear separation. The four Hund's case (a) curves are used to determine the six Hund's case (c) potential curves which arise when spin-orbit coupling is considered. The rovibrational eigenvalues for each of these six states are determined numerically, and standard spectroscopic constants are determined by fitting the energy of these levels to a Duham series. The results are compared with available experimental and theoretical information.     

Ab initio theory of complex electronic ground state of superconductors: II.: Antiadiabatic state—Ground state of superconductors

Based on Q, P-dependent modification of the Born-Oppenheimer approximation (BOA), the ab initio theory of complex electronic ground state of superconductors is presented. As an illustrative example, application of the theory to superconductors of a different character and to the corresponding nonsuperconducting analogues is presented. It is shown that due to electron-phonon (EP) interactions, which drive system from adiabatic into antiadiabatic state, adiabatic translation symmetry is broken and system is stabilized in antiadiabatic state at distorted geometry with respect to adiabatic equilibrium high-symmetry structure. Stabilization effect in the antiadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect that is neglected on the adiabatic level within the BOA. At distorted geometry, antiadiabatic ground state is geometrically degenerated with fluxional nuclear configurations in the phonon modes that drive system into this state. It has been shown that until the system remains in antiadiabatic state, nonadiabatic polaron-renormalized phonon interactions are zero in the well-defined k-region of reciprocal lattice. This, along with geometric degeneracy of the antiadiabatic state, enables formation of mobile bipolarons that can move over lattice as supercarriers without dissipation. Moreover, it has been shown that due to EP interactions at transition into antiadiabatic state, k-dependent gap in one-electron spectrum has been opened. Gap opening is associated with shift of the original adiabatic Hartree-Fock orbital energies and with the k-dependent change in density of states of particular band(s) at Fermi level. Corrected one-particle spectrum enables to derive thermodynamic properties in full agreement with corresponding thermodynamic properties of superconductors.Based on the complex ab initio theory, it has been shown that Fröhlich's effective attractive electron-electron interaction term represents correction to electron correlation energy at transition from adiabatic into antiadiabatic state due to EP interactions. It has been shown that increased electron correlation is a consequence of stabilization of the system in superconducting electronic ground state, but not the reason for its formation.     

Three-body model of deuteron breakup and stripping

For a three-body model Hamiltonian, the scattering eigenfunction that corresponds to an incident deuteron is expanded in terms of eigenfunctions of the neutron-proton relative Hamiltonian, as suggested by Johnson and Soper. In this expansion, breakup is represented by an integral over the continuum of neutron-proton scattering states. Only states of zero relative angular momentum are included; the validity and advantages of this approximation are discussed. The continuum is divided into five discrete channels, whose coupling to each other and to the deuteron channel is treated by solving coupled differential equations with appropriate boundary conditions. It is found necessary to use a simple WKB method to take account of the long-range coupling among breakup channels; this method introduces potential matrices W and S that describe local and derivative coupling of the channels. The reaction of breakup on the elastic channel is neglected.The properties of W and S and the breakup wavefunction are examined for the case of 22.9 MeV deuterons incident on a target of mass number A ≈ 40. The Coulomb interaction is ignored, and a local Gaussian shape is used for both the real and imaginary parts of the nucleon-nucleus optical potential.It is found that a rather broad spectrum of n-p continuum states is excited, especially for low center-of-mass angular momentum. This result weakens the justification for the Johnson-Soper adiabatic theory, which emphasizes breakup into states of low relative energy.The breakup part of the wavefunction at zero n-p separation is comparable with the elastic part, but is important only over a surprisingly short range in the center-of-mass coordinate, with the result that breakup cross sections are quite small. Nevertheless, breakup produces major modifications of (d, p) cross sections. These modifications can to some extent be simulated by the Johnson-Soper method. The breakup wavefunctions show several interesting effects in their dependence on angular momentum and radius.     

Critical binding of diatomic molecules

The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α₀)², where α₀ is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k ² is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.     

P-Matrix and J-Matrix Approaches: Coulomb Asymptotics in the Harmonic Oscillator Representation of Scattering Theory

The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use a well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. The P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.     

Ab initio theory of complex electronic ground state of superconductors: I. Nonadiabatic modification of the Born-Oppenheimer approximation

The ARPES of high-T_c cuprates and theoretical results of low-Fermi energy band structure fluctuation for different groups of superconductors indicate that electron coupling to pertinent phonon modes drive system from adiabatic into anti-adiabatic state (ω>E_F). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P.In the present paper—part I, it has been shown that Q, P-dependent modification of the BOA for ground electronic state can be derived by sequence of canonical transformations of the basis functions. The effect of nuclear coordinates and momenta on electronic structure is presented in the form of corrections to zero-, one- and two-particle terms of clamped nuclear Hamiltonian. In the anti-adiabatic state, correction to electronic ground state energy (zero-particle term correction) is negative and system can be stabilized in the anti-adiabatic state at distorted geometry with respect to adiabatic equilibrium structure and gap in one-particle spectrum of quasi-continuum states at Fermi level can be opened. Stabilization effect is solely the consequence of nuclear dynamics (P) that is crucial in anti-adiabatic state. It has been shown that nuclear dynamics also increases electron correlation until system at nuclear motion remains in a bound state. Corresponding corrections to electronic wave function are also specified.On the other hand, when system remains at vibration motion of nuclei in adiabatic state, the influence of nuclear dynamics (P-dependence) is negligible. In this case, all basic effects are covered through nuclear coordinates (Q-dependence) within the adiabatic BOA and standard results of solid-state (or molecular) physics are recovered.     

Elastic scattering, muon transfer, bound states and resonances in the three-body mesic molecular systems in the reduced adiabatic hyperspherical approach

The uniform method of numerical investigation of bound states and scattering processes 2→ 2 (including resonance states) in the Coulomb three-body (CTB) systems is developed. It is based on the adiabatic hyperspherical approach (AHSA) and includes the numerical realization and applications to the three-body mesic atomic systems. The results of calculations of bound states of these systems (including the local characteristics of the wave functions) and the scattering processes 2→ 2 (including the characteristics of the resonance states) are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Discussion of the He-He interaction energy in the average energy approximation

Using the Unsöld average energy method a double perturbation theory expression for the interaction energy through second order is obtained which includes intra-atomic correction terms arising from the use of trial wavefunctions to represent the non-interacting molecules. These formal expressions are applied to the ground state He-He interaction and results for the interaction energy are obtained that compare favourably with recent semiempirical and ab initio calculations. The Unsöld calculations are used to investigate approximations that have been introduced into the average energy calculation by other workers, and as a model for discussing the relative importance of second-order charge overlap and exchange effects and the convergence of the multipole treatment of the interaction energy for this typical non-bonded interaction. The results illustrate the importance of knowing as many terms as possible in the R ^-1 expansion of the energy. Finally a portion of this work required the recalculation of the He-He repulsive energy obtained by using the Eckart wavefunction to represent the helium atom. The resultant energy is approximately 25 per cent lower than that previously obtained using this wavefunction for most values of R.     

Comparative analysis of perturbations of the energy, radiative, and magnetic characteristics of electronic-vibrational-rotational states of the hydrogen molecule

A comparative analysis of perturbations of various characteristics of electronic-vibrational-rotational (EVR) levels of the singlet 3d and triplet 3s, 3d complexes of terms of the hydrogen molecule obtained by spectroscopic and semiempirical methods is performed. A quantitative hierarchy of the influence of perturbations is established: for most states under study, perturbations of the probabilities of EVR radiative transitions, the g factors of EVR levels, and the radiative lifetimes of EVR levels are larger, respectively, by 3–5, 3–4, and 1–3 orders of magnitude than the perturbations of EVR terms with respect to the corresponding adiabatic values. It is established that the dependences of the ratios of the perturbed and adiabatic values of the characteristics of EVR levels on the rotational quantum number are significantly different for (i) different characteristics of the same electronic-vibrational states and (ii) the same characteristics of the same vibrational levels but different electronic states (Λ^? and Λ⁺). These differences are explained by the significantly different role of the interference effects of electronic-rotational and electronic-vibrational interactions of EVR states in the perturbation of these characteristics. It is shown that the probabilities of EVR radiative transitions and the g factors of EVR levels may be much more informative for studying these effects than the data on EVR terms.     

Hund's rules and the interpretation and common misinterpretation of energy differences

With a Z expansion formalism which is exact in non-relativistic quantum mechanics it has been shown that for multiplets of neutral atoms and of many positive ions the state of the highest energy has the lowest expectation value for the interelectronic repulsion energy. This reversed order of the repulsion energy occurs for cases which obey Hund's first rule as well as for cases which obey the second of Hund's rules. It can be shown that the energy differences are in all cases dominated by the difference in electron nuclear attraction energy and not by the difference in electron repulsion energy.

The lowest ¹Π _u and ³Π _u states of the H₂ molecule have similar features. At many internuclear distances, including the equilibrium ones, the ¹Π _u state has the highest energy but the lower kinetic energy and electron repulsion energy but again the higher electron nuclear attraction energy. These results contradict clearly the usual theories for energy differences between spectroscopic states in atoms and molecules.