Slater's Rules Research Articles

Importance of the kinetic energy expression for the bound electron in the binary collision theory of ionization

Electron impact ionization cross sections of alkali metal atoms have been calculated using Gryzinski (1965), Vriens (1964) and Burgess formulae. Kinetic energy of the bound electron has been estimated by Slater's rule. Comparison with experimental results has been made.

Auger electron emission spectra: a simple treatment of energies and intensities

The energies of group Auger transitions are calculated using Slater's rules to allow for the effects of ionisation on orbital relaxation. The calculated intensities of Auger spectra, based on the Bethe expression for the ionisation cross-sections of the various electronic levels, are in fair agreement with the observed data. Assignment of the transitions show the predominance of the contribution of valence electrons to the Auger process.

Importance of the kinetic-energy expression for a bound electron in classical theory

Electron-impact ionization cross-sections of alkali-metal atoms and noble-gas atoms have been calculated using Gryzinski's, Vriens' and Burgess' formulae. The kinetic energy of the bound electron has been estimated by using Slater's rule. Comparisons have been made with experimental results.

Theory of the Hyperfine Splittings of Pi-Electron Free Radicals. III. Methyl Radical in a Pyramidal Configuration: Temperature Dependence of the Hyperfine Splittings

Nonempirical calculations for methyl radical (CH3·) in a pyramidal configuration (C3υ) were performed using two minimum basis sets of Slater-type orbitals, one in which orbital exponents were chosen according to Slater's rules (unoptimized) and the other in which they were optimized by minimization of the SCF energy. The spin-restricted SCF plus configuration-interaction method, including all spin-adapted configurations with single and double excitations of space orbitals, was employed. The temperature dependence of the contact hyperfine splittings was computed assuming that the variation with temperature arises from the out-of-plane bending motion. Agreement with experiment at various temperatures is good. The optimized-basis values of the temperature coefficient of the proton splitting, daH / dT, are about 20%–50% too large and the unoptimized-basis results are about 10%–25% too small. Temperature coefficients of the carbon-13 splitting calculated using the optimized and unoptimized basis sets also bracket the experimental values, from which they deviate by less than 10%, which is within the experimental uncertainty. The variation of the spin densities at the magnetic nuclei with the out-of-plane angle is analyzed in detail. Other results of the calculation are discussed, namely, the equilibrium molecular geometry, an approximate frequency for the bending motion, and the incomplete orbital following.

Theory of the Hyperfine Splittings of Pi-Electron Free Radicals. II. Nonempirical Calculations of Methyl Radical (Planar)

Nonempirical calculations of the ground-state electronic wavefunction, energy, and proton and carbon-13 contact hyperfine splittings of methyl radical in its planar configuration are performed using the spin-restricted SCF method and configuration interaction including all spin configurations involving up to double replacement of space orbitals. Two minimum basis sets of Slater-type orbitals are employed, one in which orbital exponents are chosen according to Slater's rules (unoptimized) and the other in which they are optimized through minimization of the SCF energy. Some calculations are made which involve double-zeta basis sets. Considerable progress toward the goal of accurate ab initio calculations of hyperfine splittings is reported [S. Y. Chang, E. R. Davidson, and G. Vincow, J. Chem. Phys. 49, 529 (1968), paper I in this series]. The proton splittings computed using the unoptimized (unopt.) (1) and optimized (opt.) (2) minimum basis sets and a double-zeta set (3) (free atom exponents on carbon and ζh = 1.00, 1.26) are − 33.1, − 43.8, and − 38.2 G, respectively (− 23 G exptl.). A “hybrid” minimum-basis calculation using the optimized exponents on carbon and ζh = 1.0 yields − 27.5 G. Carbon-13 hyperfine splittings computed with the minimum basis sets are + 142.4 (unopt.) and + 140.5 G (opt.) (+ 38 G exptl). Spin polarization of the valence shell dominates the splitting. Much better agreement with experiment is obtained using the double-zeta set, i.e., + 25.7 G. In this case the contributions arising from inner- and valence-shell excitations are comparable in magnitude and opposite in sign. Other features of this work include (1) calculation of SCF energies and contributions to the correlation energy for the three basis sets identified above, namely − 39.415 (1), − 39.487 (2), and − 39.547 a.u. (3) (− 39.60 ± 0.02 a.u. estimated HF limit), and − 0.054 (1), − 0.061 (2), and − 0.106 a.u. (3) (− 0.2 a.u. estimated correlation energy), (2) a population analysis of the minimum basis wavefunctions, and (3) transformation of the minimum basis canonical SCF–MO yielding equivalent sigma bonding (Bi) and antibonding (Ai) orbitals. Bi hybridizations deviate significantly from sp2, namely sp1.36 (unopt.), and sp1.45 (opt.). A detailed analysis of the contributions of the various configurations to the hyperfine splittings is presented. This analysis leads to the conclusion that a good approximation in computing the proton splitting can be obtained from the SCF results for Bi and Ai and a first-order perturbation theory calculation of the mixing coefficient for the configuration representing the Bi→Ai excitation. Calculations of the electron-nuclear coalescence cusps for the SCF–MO's and spin and charge densities are reported. The spin-density cusp at the proton is correlated with the proton hyperfine splitting. Comparison is made with previous calculations of CH3· and with our approximate calculation of the proton splitting of CH· fragment (paper I). The results of I are found to be in very good agreement with those of the ab initio calculations of CH3·.

Valence Bond Study of the Hydrogen Bond. I

A valence bond calculation for the four electrons involved in the straight hydrogen bond O–H…O without excess charges is performed nonempirically. Five valence bond structures proposed by Tsubomura are adopted, i. e. , (I) O–H O, (II) O - H + O, (III) O + H - O, (IV) O - H–O + and (V) OH - O + . The calculations are done for both trigonal and tetrahedral hybrid atomic orbitals of oxygen atoms. The adiabatic potential of the proton is calculated as a function of the O…O and the O–H distances. The shifts of the equilibrium O–H distance and of the O–H frequency with the formation of the hydrogen bond are studied. When different orbital exponents are used in the atomic orbitals of O, O - and O + , and H and H - , according to Slater's rule, the obtained shifts agree qualitatively with experiment, although the absolute values deviate from experimental ones in the short hydrogen bond.

D Orbitals in the sp3d, sp2d2, and p3d2 Configurations of Phosphorus

Wavefunctions and energy matrices for spectroscopic terms of phosphorus in the sp3d configuration are reported. Using a two-term Slater function for 3d orbitals and orthogonalized Slater functions for other orbitals, term energies are optimized with respect to 3s, 3p, and 3d orbital parameters. The core wavefunction is kept unchanged. Calculations on the lowest terms and configuration averages of sp2d2 and p3d2 of phosphorus are also reported. d Orbitals for the multiplets of phosphorus in the sp3d configuration are diffuse and have mean radii upwards of 3.66 Å (6D0), while in the lowest terms of sp2d2 and p3d2 they are compact, the mean radius being 2.03 Å. d Orbitals in the excited configurations of phosphorus are compared with those of sulfur. The applicability of Slater's rules to excited configurations is examined. A computer method for performing all the calculations, including the algebra, is given in an Appendix.

Screening constants for a relativistic description of atomic systems

Relativistic one-electron wave functions for atomic systems are approximated by Dirac’s screened hydrogenic wave functions. Our screening constants are calculated starting from the well-known Slater rules by means of a relativistic generalization. It is shown that one has no longer any general rule to apply, but it is necessary to carry out some simple calculation for each atomic system in order to obtain the screening constants. Calculations were performed for the elements from Be to Ne, from Mg to A and from Zn to Kr, but they could easily be extended to all the atomic systems. The results are briefly discussed.

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.

A simple semi-empirical procedure for determining the parameters in the MO LCAO method

A simple method is described for evaluating the exchange and Coulomb integrals with account for the change in the properties of the atoms on entry into a molecule. Generalizations of Slater's rule permit the calculation of the ionization potentials of atoms in a molecule which make the main contribution to the Coulomb integrals. Analytical expressions are given for the Coulomb integrals in the δ-function approximation.

The theory of separated electronic groups

Usually, many-electron wave-functions for molecules are constructed from antisymmetrized products of one-electron orbitals. Configuration interaction and energy level calculations are based on variational principles, and matrix elements are obtained using Slater's rules. In this paper this method is generalized, the orbitals being replaced by group functions, each one describing any number of electrons. It is assumed that wave-functions satisfy separability conditions between groups similar to those already given by Lykos and Parr, and Parks and Parr. This work should be considered as an extension of theirs. Corrections to orbital energies and operators for the interaction between electrons belonging to different groups are given, the expressions are simplified using Mulliken's approximations. The method here developed is compared with McWeeny's density matrix formalism, and similarities and differences pointed out. Possibilities of application in actual molecules are compared with those of other methods.

Studies on Rydberg Orbitals. I. Slater—Condon Parameters and Valence‐State Energies for Rydberg Excitations Determined from Atomic Spectra of Be–F

The observed atomic spectral terms of 2sm2pn3l (m = 0, 1, 2; n = 0, 1, ···, 5; l = s, p, d) were analyzed for the isoelectronic series from Be I to F I. By use of graphical inspection for deleting the effects of configuration interactions, nine series of the Slater—Condon parameters pertinent to the Rydberg orbitals were obtained together with F2pp and G1sp. The selected values are tabulated, most of which are accurate enough to be compared with and criticized by the authentic values of F2pp and G1sp. It was shown from these observed parameters that the spatial distribution of the 2p (or 2s) electron in the 2sm2pn3l state is very close to the spatial distribution in the 2sm2pn state as implied in the Slater rule. The energy expressions for the terms of spml′ (m = 1, 2, ···, 5; l = s, p) configurations are given. With these F and G values, valence‐state energies for Rydberg excitations were obtained.

Calculation of the Barrier to Internal Rotation in Ethane with Improved Exponential Wavefunctions

A minimum set of Slater exponential orbitals is used to construct molecular orbital wavefunctions for staggered and eclipsed ethane. The orbital exponents chosen are those which are optimum for methane, rather than those obtained from Slater's rules, as were used previously. The value of the internal rotation barrier is calculated to be 3.5 kcal/mole. Comparison with other calculations indicates that the barrier is fairly insensitive to choice of basis functions and to the value of the total energy.

Modifications of the Impulse Approximation for Ionization and Detachment Cross Sections

A bound-state correction is suggested for the Thomson-Gryzinski impulse approximation. Also, it is noted that Slater's rules yield estimates of kinetic-energy expectation values which are more accurate than those used in previous applications of this approximation. These corrections are important for neutral atoms with outer electrons in a $p$ state and for all negative ions. Comparison is made with experiment and previous theoretical results.

A calculation of the spin-orbit splitting of the X 2? state of NO

Using Brion, Moser and Yamazaki's SCF LCAO MO the spin orbit splitting of the X2Π state of NO has been calculated using various potentials. Experimentally W (2Π3/2) −W (2Π1/2) is known to be 122 cm−1. Using an “unscreened” nuclear field we find a value of 328 cm−1. Inclusion of screening due to electronic repulsions reduces this value to 283 cm−1. Agreement with the experimental value may be obtained by reducing the orbital exponents in the 2π molecular orbital by 30% over the value given by Slater's rules.

Homopolar Dipole

Numerical tables for the homopolar dipole or overlap moment μh.d. associated with various molecular orbitals (MO's) are presented. The present tables include values of μh.d./R for the common MO's constructed from linear combinations of atomic orbitals (the LCAO approximation), one of which is an 1s, 2s, 2pσ, 2pπ, 3s, or 3pσ orbital. The calculations employ approximate atomic orbitals of the Slater type. Formulas for x̄AB, the x coordinate of the centroid of negative charge for an electron in an MO, are given as functions of the parameters p=12R(ζA+ζB) and t=(ζA−ζB)/(ζA+ζB), where R is the internuclear distance, ζ (n—δ) is the effective nuclear charge found by the Slater rules, and (n—δ) is an effective quantum number. Values of μh.d./R are also given for some MO's in which one AO is a hybrid. The variation of μh.d. with the parameters p and t is shown graphically for certain MO's. The overlap moment μh.d. has been computed for various chemical bonds by use of the tables. Overlap integrals for the AO combinations 1s—3dσ, 2s—5s, 2s—5pσ, 2pπ—3dπ, and 3s—3s were computed since these were not available in the literature over the necessary range of the variables.

Crystal Field Splitting and Covalent Bonding in Fe++ Silicate Garnets

A point-ion model and Slater free-ion wave functions have been used to calculate the crystal field splitting of the Fe++ ion 5D ground term in Fe3Al2(SiO4)3 garnet for various assumed charge distributions within the (SiO4)4— ion. The best fit with Clark's experimental data on absorption spectra is obtained with about 50% covalent bonding between Si and O, and with an effective nuclear charge for the Fe++ ion about 10% less than that given by Slater's rules. This estimate of the degree of covalency is consistent with that given by Pauling. The calculated pressure dependence of the optical absorption is in satisfactory agreement with the experiments of Balchan and Drickamer.

Relativistic K Electron Wave Functions by the Variational Principle

The variational principle is applied to obtain the Dirac wave functions of the K electrons of heavy atoms, using the method of variable parameters. In formulation, three parameters are used. The Coulomb and spin-spin interactions of the two K electrons are taken into consideration as well as the effect of the rest of the atom. A numerical example for mercury, Z=80, is given. Its K absorption edge is calculated and relativistic wave functions obtained, using one variable parameter. The results justify the use of screened hydrogenic wave functions although the screening constant (~0.5) is somewhat higher than that given by Slater's rules. Calculations using two variable parameters are also given and the total energy is found to have a saddle point and not a true minimum. This is briefly discussed in the light of the hydrogen atom.

Use of 4p Orbitals in Metal-Aromatic Complexes

The use of 4p orbitals of the central metal atom in metal-aromatic complexes is discussed. It is shown that, within the usual assumptions of the semiempirical molecular-orbital method, it is necessary to use Slater orbitals of a nonintegral power and with an exponent close to that calculated by the Slater rules. The electroneutrality principle of Pauling is also discussed and a large charge transfer within these complexes is found to contradict the original assumptions of the molecular-orbital theory.

Electronic Structures of the Methyl Radical: Effect of 3s Atomic Orbital of Carbon

By taking the 3s orbital of carbon into account, the self-consistent-field (SCF) LCAO—MO method has been applied to the ground state of the methyl radical. The 3s AO used has been determined for the valence state V4(1s22s2pπ+2pπ—3s) by a variational procedure, so that the orbital exponent of the 3s simple Slater AO obtained is 0.6374 instead of Slater's value 0.4833. For the other states with 3s AO, it is similarly concluded that good approximate 3s orbitals cannot be obtained by using Slater's rule, and more satisfactory shielding constants for the extra-valence-shell electron are given. Interactions between all electrons have been taken into consideration but the orthogonality relationship between 1s orbitals of carbon and hydrogen has been assumed. For the MOs occupied in the ground state, the forms are hardly changed by the inclusion of the 3s AO, and the contribution of the 3s AO is very small. For the lowest vacant orbital, however, this contribution is exceedingly great and qualitatively the conventional approximation by the 3s AO is rather more satisfactory than the use of the antibonding LCAO—MO constructed by the valence-shell orbitals. In the lower electronic states calculated, such properties are prominent and the agreements with the observations are relatively good, especially by the inclusion of the 3s AO. The influence of the other extra-valence-shell orbitals is discussed.