Cusp Condition Research Articles

Asymptotic behaviour of incoherent scattering functions

The exact asymptotic behaviour of the incoherent scattering function SK is found to be related to the behaviour of the weighted pair distribution near the origin rather than the pair distribution function as previously reported in the literature. The expression may be simplified by employing the electron-electron cusp condition to show that the asymptotic behaviour of SK is determined by the wavefunction at the electron-electron singularity.

Perturbation theory of short-range atomic interactions

A new united atom perturbation theory of the interaction of two atoms at small separations is described. The key feature is a similarity transformation of the Schrödinger equation which enables the cusp conditions to be satisfied at both nuclei and preserves the correct molecular symmetry. The first-order theory is examined in detail and compared with other united atom theories. Numerical calculations are presented for the ground states of the systems H + 2 , HeH 2+ HeH, He 2 and Li + He, based mainly on Hartree-Fock wavefunctions for the united atoms, and are compared with accurate molecular calculations. The agreement is remarkably good for separations up to 1 bohr.

Asymptotic Properties of Atomic Form Factors and Incoherent Scattering Functions

By use of the cusp condition for electronic wavefunctions the exact asymptotic behavior of the atomic form factor is determined. The relation between the pair distribution function and the incoherent scattering function for large wave vector is pointed out.

A discussion of integral transform radial basis functions

An improved transform function is given for the helium atom ground state. This function satisfies the nuclear cusp condition and yields an energy very close to the Hartree-Fock value. Applications of this transform are discussed.

Self-Consistent-Field Atomic Wavefunctions from Efficient Nested Basis Sets

Constrained self-consistent-field wavefunctions are reported for the ground-state neutral atoms, He–Ne, in which each atomic orbital is a linear combination of Slater-type orbitals. A more efficient use of the basis set chosen reduces the number of parameters to be optimized while maintaining the energy equal to within five significant figures to the energy of the fully optimized basis set. The reduction in the number of independent linear and nonlinear parameters is achieved by forcing a certain long-range behavior and a nuclear cusp condition. In one set of calculations the orbitals are forced to behave asymptotically as φi ∼ exp[−r(− 2εi)1/2], where εi is the orbital energy. In the second set of calculations, the orbitals are also forced to exactly obey the nuclear cusp condition.

Hulthén Approximations to 1s and 2p Orbitals of Atoms

It is shown that a substantial energy improvement over the Slater-orbital description is obtained by the variational use of 0s and 1p functions in the 1s and 1p shells of atoms, where 0s and 1p represent differences of two Slater 0s and Slater 1p functions, respectively. Reduction of the number of parameters by enforcing nuclear cusp conditions does not affect the variational energy significantly. The reduction yields one-parameter 0s and 1p orbitals giving a better energy than both one-parameter Slater functions and two-parameter variable-principal-quantum-number functions. It is also shown that in correlated functions of the form ψ = φ (r1) φ (r2) χ (r12), the orbital 0s when used as a one-term approximation to the best possible φ gives an excellent result for the energy of heliumlike systems and the correlation function 1 + c exp(− ηr12) gives a very good approximation to the best χ(r12). Reduction of the number of parameters by requiring both nuclear and correlation cusp conditions yields a two-parameter trial function which for helium has the energy − 2.90002 a.u., vs the Roothaan–Weiss best energy, − 2.90039 a.u., for a function of this form.

Towards an Accurate Wave Function for Positronium Hydride

A variational wave function for the ground state of positronium hydride is presented. The wave function is considerably more accurate than any previously reported. The only simplifications used are (a) only kinetic and Coulombic potential energy terms are included in the Hamiltonian, (b) the motion of the proton is ignored, and (c) the nucleus-positron distance is excluded from the wave function. The last assumption is the major source of error, but it enables us to evaluate all integrals exactly. Several trial wave functions are examined, the one giving the lowest energy being an open-shell exponential function times a 12-term power series through second-degree terms in the five remaining interparticle coordinates. Positronium hydride is found to be stable with respect to separation into hydrogen and positronium by at least 0.657 eV. The annihilation rate for two-photon events is found to be 2.095 ${\mathrm{nsec}}^{\ensuremath{-}1}$ for the trial wave function giving the lowest energy, and 2.265 ${\mathrm{nsec}}^{\ensuremath{-}1}$ for the trial wave function which most nearly satisfies the positron-electron cusp condition. A search for bound excited states of PsH and a bound state for the system ${e}^{+}\mathrm{He}$ was unsuccessful.

Perturbation Treatment of the Helium Ground State with Correlation in Zeroth Order

Pluvinage's method for calculating correlation energies of two-electron atoms is modified so that a consistent perturbation calculation becomes possible. Correlation is thereby introduced into the wavefunction in zeroth order, and the cusp conditions between all pairs of particles are satisfied in zeroth order. The first-order wavefunction is estimated variationally, and the resulting energy to third order is compared with that of Scherr and Knight and Midtdal for the systems H−, He, and Li+. The sensitivity of certain calculated properties to the accuracy of the first-order wavefunction is examined in the case of He.

New Type of Wave Function for Li,Be+, andB++

A recently developed method for electronic wave functions has been applied to Li, Be+, and B++. This method, the G1 method, leads to much better energies than does the Hartree-Fock (HF) method (for Li the G1 energy is -7.447560 as compared to the HF energy of -7.432727); however, the independent-particle interpretation is retained. A significant difference is that although the HF valence orbital has a node (being orthogonal to the core orbital), none of the three G1 orbitals has a node. The valence G1 orbital satisfies the cusp condition at the nucleus, has a minimum in the core region, and outside the core region is very similar to the HF valence orbital. The G1 wave function has the form of a paired-electron wave function but strong orthogonality is not imposed. Since the resulting orbitals have significant overlaps and since the energy is made much worse by imposing strong orthogonality, it appears that strong orthogonality is a significant constraint for orbital-type paired-electron wave functions (such as in the extended valence bond method). Since the G1 wave function is nearly at the radial limit and does not include instantaneous correlation among the electrons, the dynamic correlation is essentially all due to angular correlation just as for two electron atoms and ions. Since the G1 valence orbitals need not be orthogonal to the core orbitals, the use of this method for solids could eliminate these problems in band-structure calculations which are due to the orthogonality conditions inherent in the HF method.

Studies on Rydberg Orbitals. II. Calculation of the Rydberg Orbitals of Li(I)–F(I) and Their Isoelectronic Series

The 3s, 3p, and 3d orbitals of the atoms Li(I)–F(I) and their isoelectronic ions were calculated by optimizing the one-electron energy with the core electrons being fixed. These orbitals are for the valence states of the lowest Rydberg excited configurations, e.g., 1s22s22p3l, V2 for the C(I) series. First, the wave-functions were expanded with Slater orbitals of the least basis (3—l STO). Except for the case of 3p orbitals of the Li(I)–B(I) series, they are different from hydrogenlike expansions in that the coefficients and orbital exponents are independently optimized. Secondly, a 3l-STO was added to improve the shape of the inner loop. These Rydberg wavefunctions satisfy the following restrictions: (i) normalization, (ii) orthogonalization to the core orbitals, and (iii) cusp condition. The orbital exponents of the terms for the inner loops were set equal to the values calculated by Slater's rule for the precursors (the core orbitals of the same l with the Rydberg orbital). “Energy-optimization” and “virial-optimization” were applied separately, case by case. The results are generally good with respect to the orbital energy or quantum defect when compared to the values observed and calculated by others, the error being in the order of d ≪ p < s. Perturbations of the core by the Rydberg electron were discussed. It was revealed that for d orbitals the Slater–Condon parameters are almost constant for the different spectroscopic terms and the calculated and observed values coincide with each other, while for 3p and 3s orbitals the Slater–Condon parameters change drastically from term to term and due caution is necessary for assigning their values.

On the use of integral electron cusp conditions as constraints

Eleven approximate configuration-interaction wavefunctions for the ground state of helium are used to test the application of the integral electron cusp condition as a constraint. Our results indicate that, if the approximate wavefunction is flexible enough, the calculated electron density at the nucleus is improved when the cusp constraint is imposed. However, the expectation values of γ −2 and γ −1 do not change significantly.

Accurate Wavefunctions for Fe3+ and Its Excited Configurations

The accurate analytical self-consistent-field wavefunctions were calculated for the Fe3+ ground state, and for the Fe3+ excited configurations with two and three open shells of different symmetries. All the expansion basis sets were selected so as to satisfy precisely the cusp condition. The deviations from the tail nodal extinction for the radial functions Piλ(r) never exceeded 0.00010. Our wavefunctions for Fe3+ are the most accurate ones available. For the two- and three-open-shell states of Fe3+ the calculated wavefunctions and total energies are presented for the first time. The experimental energy levels were available for comparison with one of the excited states calculated.

A physical interpretation of the cusp conditions for molecular wave functions

The cusp conditions for the Coulomb- and Correlation-cusp of molecular wave functions are derived directly in integrated form from the Schrodinger equation. For the Coulomb-cusp the slope of the wave function at nucleus α is given by the directional derivative $$ - \left( {\frac{1}{\psi }\frac{{\partial \psi }}{{\partial r1\alpha }}} \right)_{r1\alpha = 0} = Z_\alpha + d \cdot \cos \vartheta _1 .$$ The magnitude of d remains undetermined, the direction of steepest slope (ϑ 1= 0) coincides with that of the vector of the intramolecular electric field E produced by the other electrons and nuclei at the position of nucleus α. For the correlation cusp the corresponding directional derivative is shown to be $$\left( {\frac{1}{\psi }\frac{{\partial \psi }}{{\partial r12}}} \right)_{r12 = 0} = \tfrac{1}{2} + c \cdot \cos \alpha .$$ Here both the magnitude of c and the direction of the polar axis for the angle α remain undetermined. The special cases of an 1-electron molecule and a 2-electron atom are considered.

Integral coalescence conditions

The general coalescence conditions of Pack and Byers Brown for spatial eigenfunctions are used to derive integral coalescence conditions for exact wavefunctions Ψ for atoms and molecules with definite spin quantum numbers. These conditions for the special case of nodeless Ψ are equivalent to the cusp conditions for the electron and spin densities of Steiner and Bingel. Finally, the integral coalescence conditions are applied to approximate wavefunctions.

Cusp Conditions for Molecular Wavefunctions

The conditions on the behavior of a wavefunction for a system of charged particles near the coalescence of any two of them are derived. The new features of the derivation are: (a) the fixed-nucleus approximation is not required; (b) the wavefunction is not spherically averaged; (c) the wavefunction can have a node at the singularity; and (d) the consequences of the two particles being identical are examined. In addition to the general treatment, the electron—nucleus cusp conditions for diatomic molecules are discussed in detail.

Analytical Wave Functions for Excited States of Al*

The self-consistent-field (SCF) expansion method was applied to the calculation of the analytical wave functions for several higher excited states of atomic aluminum. The Hartree–Fock total energies are represented to about five significant figures. The cusp condition was precisely satisfied for all the orbitals. The nodal condition was well satisfied for the outermost orbitals of each symmetry. The results supplement previous data, inter alia, for the calculation of transition probabilities.

Accurate Analytical Self-Consistent-Field Functions for Atoms. VIII. The Ground States ofK+, K, and Ca

Self-consistent-field calculations by the expansion method were carried out for the ground states of ${\mathrm{K}}^{+}$, K, and Ca. The merit of this calculation is in its accuracy. Compared with solutions of the Hartree-Fock equations by the numerical-integration technique, the wave functions are considered accurate to three decimal places. The deviations from the nodal condition for the radial functions do not exceed 0.00006. The cusp condition is exactly satisfied.

Asymptotic Large-ZAtomic Wave Functions

A new for calculating wave functions for atoms and ions is developed. Formal asymptotic expansions with largeness parameter $Z$ (nuclear charge) of the form $\ensuremath{\psi}={e}^{\ensuremath{-}Zh}\ensuremath{\Sigma}{a}_{n}{Z}^{\ensuremath{-}n}$, where $h$ and the set of ${a}_{n}'\mathrm{s}$ are functions of the electron coordinates, are determined through first order in $\frac{1}{Z}$. In so doing, the Schr\odinger equation for a general atomic system with $N$ electrons is reduced to a set of first-order partial differential equations for successive ${a}_{n}$. These equations are then solved recursively. Screening and correlation are exhibited explicitly in the resulting asymptotic atomic wave functions. Applications made to the ground state of 2-electron systems show that the asymptotic wave function obeys the virial theorem through first order. Magnetic susceptibilities within 5% of the accepted values are obtained for helium and singly ionized lithium. Other expectation values, $\frac{1}{2}({s}_{1}+{s}_{2})$, ${{s}_{12}}^{2}$, and ${s}_{12}$, are found to be in excellent agreement with Pekeris's variational calculations utilizing many parameters. In the neighborhood of certain singular points the large-$Z$ asymptotic solutions obtained by a matching technique are shown to satisfy the correct cusp conditions. In certain of these regions, $\frac{(\mathrm{ln}Z)}{Z}$ terms enter. The omission of such terms in ordinary variational-perturbation wave functions may result in a loss of accuracy when computing expectation values other than the energy.

The behaviour of the second-order density matrix at the coulomb singularities of the Schr�dinger equation

The cusp conditions of kato for a spinless n-electron wave function are used to derive corresponding conditions for the second-order density matrix Γ (2)(r1r2,r'1r'2).

Analytical Self-Consistent Field Calculations forMn3+and Its Excited Configurations

Analytical self-consistent field functions were calculated for the ${\mathrm{Mn}}^{3+}$ ground state, and for several ${\mathrm{Mn}}^{3+}$ excited configurations with two and three open shells of different symmetries. The cusp condition is precisely satisfied. The deviations from the nodal condition for the radial functions ${P}_{i\ensuremath{\lambda}}(r)$ never exceeded 0.00010. The energy levels of the three-open-shell states, calculated in this work, have been neither measured nor calculated before.