Analytical Wave Functions Research Articles

Self consistent calculation of dynamic multipole polarizabilities and excited state wave functions of open shell ions: Li sequence

AbstractA variational formulation of the Hartree–Fock–Roothaan (HFR) theory for open shell ions in presence of time dependent perturbations is presented. The theory has been used to calculate the dynamic polarizabilities of the first three ions of the Li sequence. The polarizability values, extrapolated to zero frequency, show good shell by shell agreement with the corresponding static results. The polarizability graphs display resonance behaviour at the transition frequencies of the ions, and a study of these points leads to analytic HF wave functions for their low lying excited states. The calculated transition frequencies are in excellent accord with the experimental values. The calculated oscillator strengths for the 2s → np transitions are in reasonable agreement with the extensive multiconfiguration calculations of Weiss and the available experimental results.

Improved lower bounds to energy levels (of 1s ns1S states of He)

Improved lower bounds to the energy levels of the 1s ns 1S states of helium through n=6 have been calculated with analytical orbital wavefunctions using an extension of the methods of Cohen and Feldmann (see abstr. A51946 of 1970). For the excited states, these bounds are all within 1.25% of the observed levels, and the error reduces to 0.15% for the highest levels. These results confirm the reliability of the frozen core wavefunctions for excited states.

Length and velocity matrix elements in quadrupole transition calculations (for transitions in He and Li+)

Electric quadrupole multiplet strengths have been calculated for a large number of transitions in He and Li+ using analytical orbital wavefunctions of frozen core type. Very good agreement between length and velocity forms of the transition matrix element is obtained when the orbitals have the correct asymptotic form. The velocity form is far less sensitive to small changes in the orbitals than is the length form. The bound-state contributions to the quadrupole polarizabilities of the ions He through B+3 have been estimated from 1S-1D multiplet strengths, and it is concluded that the continuum contributions are very large.

Accurate Analytical Self-Consistent-Field Wavefunctions for Dy2+

Accurate analytical self-consistent-field (SCF) wavefunctions were calculated for the dysprosium laseractive ion, namely for Dy2+, (Z = 66), 4f10, 5I. These ab initio calculations were done by the analytical SCF expansion method, with full exchange effects for all the 64 electrons included. All of the basis function exponents in the analytical expansions were exhaustively optimized. These results represent the only analytical wavefunctions for Dy2+ available at this time. Our usual accuracy criteria were satisfied in these calculations.

Lower bounds to energy levels of two-electron atoms

Lower bounds to the energy levels of the ground states of the helium isoelectronic sequence and of excited singlet and triplet S states of He and Li+ have been calculated, using analytical orbital wavefunction of 'frozen core' type and a variety of lower bound formulae. Only the bounds which result from the formulae of Cohen and Feldmann are generally useful. The numerical results indicate that the 'frozen core' wavefunctions for two-electron atoms are of satisfactory quality.

Analytic Solutions of the Reference Spectrum Equation

Analytic solutions of the fundamental equation of the reference spectrum method for nuclear matter are obtained in the cases of the square well and exponential potentials with hard cores. In the former case analytic wavefunctions are obtained for every partial wave L. These wavefunctions are expressed in terms of spherical Bessel and Hankel functions or in terms of spherical Bessel, Hankel, and Newmann functions of order L. In the latter case an analytic expression is obtained for the s wavefunction only. This is given in terms of the transcendental functions G±v(r) and M±v(r) which are defined as series of well-known functions. Finally an application is made of our results to the calculation of the well depth of a Λ particle in its ground state in nuclear matter.

Theoretical Considerations for Auto-Ionizing States with Microsecond Lifetimes

Employing analytic Hartree-Fock wave functions, model calculations have been performed on the structure and lifetimes of the slowly auto-ionizing states arising from $(1s, 2s, 2p)^{4}P$ in ${\mathrm{He}}^{\ensuremath{-}}$ and Li. The effect of mixing with the associated $^{2}P'\mathrm{s}$ is considered and the phenomenon of differential metastability discussed. A framework is provided to interpret experimental observations.

Analytic SCF wave functions for transition-metal atoms

Analytic self-consistent field was functions have been obtained for the transition-metal atoms in their ground-state configurations. The atomic orbitals have been expanded in terms of linear combinations of orbitals from basis sets of 10 and 11 Slater-type functions.

High-Accuracy Wavefunctions and Matrix Elements for Vibration–Rotation States of Diatomic Molecules Using the Dunham Potential

An iterative procedure for deriving accurate analytic wavefunctions for the Dunham molecular oscillator is described, and the wavefunctions so obtained are tabulated for the 10 lowest vibrational states. These wavefunctions are more accurate than those which are available for simpler models (e.g., the simple harmonic oscillator with an added cubic term or the Morse oscillator) and, in fact, can be credited with accuracies at least as great as those associated with the Rydberg–Klein–Rees numerical methods. The inclusion of rotational motion is also described. Finally, a method for the rapid derivation of analytic expressions for off-diagonal radial matrix elements between vibration–rotation states is detailed. This method makes use of only certain parts of the analytic wavefunctions, relaying on a recursion relation to complete the development.

Potential curves for one-electron heteronuclear diatomic systems in one-center treatment

Calculations were carried out for the heteronuclear diatomic systems HeH++, LiH3+, HBe4+, HeLi4+, HeBe5+, LiBe6+, LiB7+ in the 1sσ and 2pπ unstable states with one-center one-electron analytic trial wave functions. The total energy values, as a function of the internuclear distanceR, are improved by the variation of the center of the wave function.

Oscillator strengths for helium

Oscillator strengths for 95 singlet and triplet s–p and p–d transitions in neutral helium have been calculated using analytical wave functions of the "frozen-core" type. Both length and velocity formulations of the oscillator strengths have been employed. Our results are generally in good agreement with the most accurate values available.

Analytical orbital wave functions for the two-electron atom

Orbital wave functions for a number of singlet and triplet P and D states of neutral helium have been calculated within the framework of the `frozen core' approximation. The radial equation has been solved using a matrix method, which leads to analytical wave functions more convenient in many applications than numerical Hartree-Fock orbitals of similar accuracy.

Inner electron binding energy in atoms and ions

The 1s-electron binding energies of first and second row atoms were obtained from Clementi's analytical Hartree-Fock wavefunctions with the use of Koopman's theorem. These were compared with the differences of the expectation values of √ 1 between various low lying states of the charged and neutral' atoms, and the ground state of the neutral atom. Good agreement was found. Extension of this analysis to molecules and crystals should make possible a better interpretation of the chemical shift observed in photoelectron spectroscopy.

The L I/K orbital electron capture ratio of low- Z elements

The L I/K orbital electron capture ratio has been calculated for allowed transitions using analytical Hartree-Fock wave functions and taking account of relativistic corrections.

Approximate Analytical Wavefunctions for Certain States of Nd3 +

Approximate analytical self-consistent field wavefunctions were calculated for the ground state and for certain excited states of the laser-active ion Nd3 +. The following states of the 4f3 configuration of Nd3 + were covered by our calculations: S4,D4,F4,G4,, and I4. For the purpose of this work we combined the minimum and the saturated basis set approximations. All the orbital exponents were accurately optimized. The d orbitals, and particularly the 4f orbital, were calculated unusually accurately. Except the optically unaffected orbitals 1s, 2s, and 2p, all the orbitals of the s, p, d, and f symmetries exhibited zero deviation (to five decimals) from the tail nodal extinction. The virial theorem was always satisfied to at least five significant figures.

Approximate analytical wave functions for Pr3+

Approximate analytical self-consistent field wave functions were calculated for Pr3+. For the purpose of this work we combined the minimum and the saturated basis set approximations. All the orbital exponents were optimized.

Perturbational-Variational Approach to the Calculation of Variational Wave Functions. I. Theory

Certain aspects of Rayleigh-Schr\odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.

Magnetoelectric Susceptibilities ofS-State Atomic Systems

A study is made of the effect on $^{1}S_{0}$ atomic systems of the simultaneous interaction of uniform electric and magnetic fields. It is found that in addition to the conventional magnetic-susceptibilities and electric-polarizability terms there exists an interaction term in which the energy depends quadratically on both fields. The coefficient of this term in the energy is defined here as the magnetoelectric susceptibility. The magnetoelectric susceptibilities of a large number of two-, four-, ten-, twelve-, eighteen-, and twenty- electron atomic systems are calculated. Analytical Hartree-Fock wave functions are used. The calculations are carried out using double-perturbation theory within the Hartree-Fock framework. The results show that the energy terms described by the magnetoelectric susceptibilities are of the same order of magnitude as those corresponding to the hyperpolarizabilities for magnetic fields of the order of ${10}^{6}$ G. The magnetoelectric susceptibility effect is dependent on the relative orientation of the two fields, and its behavior with respect to field orientation is described by a property defined as the anisotropy. It is found that the magnetoelectric susceptibilities of the inert-gas isoelectronic-series systems are all negative and decrease in magnitude as the fields are rotated from perpendicular to parallel. The alkaline-earth systems show considerable variation: some are positive, some are negative, and in some the sign depends upon the relative orientation of the fields.

Analytical wave functions for the 1snp1P and3P states of the helium isoelectronic sequence

Approximate analytical Hartree-Fock wave functions have been obtained for 1snp 1P and 3P states of the He-like isoelectronic sequence up to N5+ for n = 2, 3, 4 and 5. Calculated energy eigenvalues compare well with previous results obtained by Davidson for n = 2, 3 and 4 for helium and Perrin and Stewart for n = 2. New results for higher members of the isoelectronic sequence are presented.

An elementary model for calculations of effective nuclear charge values in many electron atoms

One can calculate the potential energies of electrons in atoms through two Jiffereut routes with the help of analytical single electron wave functions. Equating the two analytical expressions for the potential energies we obtain quite easily the effective nuclear charge values (Z eff). The correlation between experimentally determined ionization energy values and those energy values calculated by the described method is surprisingly good.