On the extraction of optical sums from experimental generalized oscillator strengths and the experimental determination of the x-ray incoherent scattering factory by electron scattering

The connection between an experimentally measured Bethe surface and certain optical sum rules is discussed. The simplest procedure for constructing optical sums involves summing (integrating) the generalized oscillator strength over energy loss at a fixed scattering angle. This involves an approximation and an error formula is developed to estimate the magnitude of the correction involved. The approximation is shown to be extremely accurate in the limit of high incident energy and large momentum transfer. The sum S(?1,K) which is simply related to the X-ray incoherent scattering factor is found to have a small first order correction in the case of electron scattering from He with 25 keV electrons even at angles as small as 1°.     

Calculations on the singlet-triplet energy separations of silaethylene

The total energy and the conformational hypersurface of the lowest singlet and triplet states of silaethylene, CH₂SiH₂ have been studied using ab initio SCF MO calculations with unrestricted and restricted Hartree-Fock methods. A minimal and an extended basis set was employed. The ground state is predicted to be a singlet and the lowest triplet state to lie 9.6 kcal/mole above. The estimated correlation energy correction would raise ΔE(T₁S₀) to ≈ 16 kcal/mole.     

Spectroscopy of complex molecules in the gas phase

A brief review is presented of theoretical and experimental research into the role of statistical factors in the formation of the characteristics of absorption and emission processes of light, observed for low-density and rarefied vapours of complex molecules. It is shown, in particular, that the average energy of molecules excited per unit time differs from the mean energy of molecules in the ground state, not by the energy of the exciting photon hv, but by the sum of hv and the selective energy which is the result of different absorption probabilities for molecules of different energy. This correction is the most important in rotational—vibrational absorption bands. This was established when the selective energy was calculated using the experimental data with the help of the formulae obtained. Average energies of the initial and final combining states reduced by the average energy of molecules in the ground state are calculated. Correlation curves similar to those of Condon are plotted according to the calculated data. The electron transition frequencies and the identities of absorption and emission transitions are determined through these curves; whereas for rotational—vibrational bands the value of the rotational constant and the variation of the latter upon excitation are estimated.     

Second order of perturbation theory correction to the radial distribution function of a liquid with the interaction potential of hard spheres plus a square-well

A liquid with the interaction potential of hard spheres plus a square-well is analyzed using the Monte-Carlo technique. Numerical results for the perturbation theory series over a square-well potential are obtained in the form of the Barker and Henderson discrete representation. Approximating expressions for the correction to a liquid radial distribution function in the second order of perturbation theory are presented. The obtained results allow us to define this correction with a root-mean-square deviation of about 0.007. It is shown that the given approach provides a complete calculation in the second order of perturbation theory, and also the determination of the third order correction to the free energy for a liquid interacting with the potential of the Lennard-Jones type.     

Ab initio base-pairing energies of an oxidized thymine product, 5-formyluracil, with standard DNA bases at the BSSE-free DFT and MP2 theory levels

Oxidation of the thymine methyl group produces two stable products, non-mutagenic 5-hydroxymethyluracil and highly mutagenic 5-formyluracil. We have calculated the interaction energy of base-pair formation involving 5-formyluracil bound to the natural DNA bases adenine (A), cytosine (C), guanine (G), and thymine (T), and discuss the effects of the 5-formyl group with respect to similar base-pairs containing uracil, 5-hydroxyuracil, thymine (5-methyluracil), and 5-hydroxycytosine. The interaction geometries and energies were calculated four ways: (a) using density functional theory (DFT) without basis set super-position error (BSSE) corrections, (b) using DFT with BSSE correction of geometries and energies, (c) using M?ller-Plesset second order perturbation theory (MP2) without BSSE correction, and (d) using MP2 with BSSE geometry and energy correction. All calculations used the 6-311G(d,p) basis set. Notably, we find that the A:5-formyluracil base-pair is more stable than the precursor A:T base-pair. The relative order of base-pair stabilities is A:5-Fo-U > G:5-Fo-U > C:5-Fo-U > T:5-Fo-U.     

Evaluation of the individual hydrogen bonding energies in N-methylacetamide chains

The individual hydrogen bonding energies in N-methylacetamide chains were evaluated at the MP2/6-31+G** level including BSSE correction and at the B3LYP/6-311++G(3df,2pd) level including BSSE and van der Waals correction. The calculation results indicate that compared with MP2 results, B3LYP calculations without van der Waals correction underestimate the individual hydrogen bonding energies about 5.4 kJ mol^?1 for both the terminal and central hydrogen bonds, whereas B3LYP calculations with van der Waals correction produce almost the same individual hydrogen bonding energies as MP2 does for those terminal hydrogen bonds, but still underestimate the individual hydrogen bonding energies about 2.5 kJ mol^?1 for the hydrogen bonds near the center. Our calculation results show that the individual hydrogen bonding energy becomes more negative (more attractive) as the chain becomes longer and that the hydrogen bonds close to the interior of the chain are stronger than those near the ends. The weakest individual hydrogen bonding energy is about ?29.0 kJ mol^?1 found in the dimer, whereas with the growth of the N-methylacetamide chain the individual hydrogen bonding energy was estimated to be as large as ?62.5 kJ mol^?1 found in the N-methylacetamide decamer, showing that there is a significant hydrogen bond cooperative effect in N-methylacetamide chains. The natural bond orbital analysis indicates that a stronger hydrogen bond corresponds to a larger positive charge for the H atom and a larger negative charge for the O atom in the N-H?O=C bond, corresponds to a stronger second-order stabilization energy between the oxygen lone pair and the N-H antibonding orbital, and corresponds to more charge transfer between the hydrogen bonded donor and acceptor molecules.     

Further development and application of the ECS scaling theory: Non-linear molecules

The energy corrected sudden (ECS) scaling theory is applied to rotational energy transfer (RR,T) in the HeNH₃ system: one of the largest and most complex systems studied thus far. A test of the scaling and an accurate prediction of available cross sections required the development of two new results: (1) a scaling relation between properly symmetrized symmetric-top cross sections; and (2) a higher-order ECS adiabaticity correction factor than was previously available. The final scaling relation, through the incorporation of ortho—para symmetry effects, exhibits two interesting features: (1) the fundamental input to the scaling, used to generate both ortho and para results, consists of only ortho transitions; and (2) these ortho fundamentals divide into two distinct groups — one used to predict only Δk = 0 transitions and the other only Δk ?= 0 transitions. The scaling is tested using “exact” cross sections calculated on a short-range Gordon—Kim potential surface and on a more realistic, longer-range Hartee—Fock surface. Scaling predictions are in very good agreement in the short-range case with the kinetic-energy shift being the only adiabaic correction necessary. In the second, longer-range case, considerably more adiabatic behavior is encountered and the scaling results are in only moderately good agreement even with the incorporation of a higher-order adiabatic correction factor. These two systems serve to illustrate some of the virtues and limitations of the scaling approach.     

Complex Franck—Condon overlap integral in nonradiative decays

Considering the nuclear coordinate (Q) dependence of the electronic energy denominator appearing in the virbonic coupling matrix element, a complex Franck—Condon overlap integral which is needed in order to evaluate the nonradiative decay rate constant not only in the weak coupling but also in the strong coupling case is derived. The real part of the overlap integral plays an important role in the weak coupling case. The imaginary part is originated from the potential energy surface crossing regions and, consequently, contributes to the nonradiative decay rate constant in the strong coupling case. When the Q-dependence of the electronic energy denominator is neglected, the complex overlap integral leads to the results ontained by using the usual Herzberg—Teller expansion method. It is shown that the complex integral is expressed by the optical Franck—Condon overlap integral multiplied by a correction factor when the nonradiative decay from the vibrationless state is considered.     

Thermodynamic representation of ternary liquid–liquid equilibria near-to and far-from the plait point

Common classical expressions for the molar excess Gibbs energy of mixing g^E do not contain a contribution from composition fluctuations that are significant in the vicinity of the plait point for a ternary system. We propose a correction to g^E based on reasonable phenomenological assumptions. This correction requires three ternary constants, but two are obtained from stability relations provided that we know the composition of the plait point. While the method proposed here is not predictive, it provides a consistent thermodynamic procedure for calculating the liquid–liquid phase diagram of a ternary system with a plait point. The proposed method is illustrated for three ternaries. When calculations are based on the classical expressions for g^E alone, calculated results are in serious disagreement with experiment near critical conditions. Inclusion of the proposed correction for g^E provides dramatic improvement.     

Gamma-ray spectrometry for the self-attenuation correction factor of the sand samples from Antalya in Turkey

In this paper, we attempt to determine the self-attenuation correction factor for 37 different sand samples collected from Antalya region of Turkey with densities changing from 2.205 to 2.679 g  \(\hbox {cm}^{-3}.\) Transmission method has been used in order to obtain self-attenuation correction factor in comparison with the air and ultrapure water samples for each case. Self-attenuation correction factor versus energy fit curve is obtained. While the self-attenuation correction factor has large values at low energies, it becomes smaller at high energies and tends to become constant thereafter.     

A simple correction to final state energies of doublet radicals described by equation-of-motion coupled cluster theory in the singles and doubles approximation

A computationally inexpensive energy correction is suggested for radicals described by the equation-of-motion coupled cluster method for ionized states in the singles and doubles approximation (EOMIP-CCSD). The approach is primarily intended for doublet states that are qualitatively described by Koopmans' approximation. Following a strategy similar to those used in multireference coupled cluster theory, the proposed correction accounts for all correlation effects through third order in perturbation theory and also includes selected contributions to higher-order energies. As an initial test of the numerical performance of the method, total energies and energy splittings are calculated for some small prototype radicals.     

A new approach to theab initio energy of the homodesmic reaction for the resonance energy of benzene

Summary A scheme of the basis set superposition error (BSSE) correction is first proposed and introduced to determine theab initio energy of the homodesmic reaction for the resonance energy of benzene. Calculations with 6-31G*(5D) and 6-31G*(6D) basis sets at the complete fourth-order Møller-Plesset perturbation level furnish the energy value of 21.35 kcal/mol after the correction, which is in complete agreement with the experimental value of 21.3±0.2 kcal/mol. The energy values at the lower theoretical levels are generally underestimated but they are superior to the uncorrected values. The inclusion of triple excitations displays the dominant contribution of the correlation energy. Detailed analysis of the results reveals some of the similarities between the homodesmic reaction of benzene and the interaction of van der Waals molecule, which provides further justification of the BSSE correction scheme presented in this study.     

Shell effects on symmetric fragmentations of alkali-metal clusters

Symmetric fragmentation of multiply charged alkali-metal clusters consisting of several tens of atoms is studied. The energy variation during the fragmentation process is calculated using the theory ofshell corrections, in which total energy is written as a sum of the liquid-droplet and shell correction terms. It is found that the variation of the shell correction term is much larger than that of the liquid-droplet one if the parent cluster is metastable. Fragmentation into nearly-magic cluster is most favored regardless of parent size since the barrier height for fragmentation is mainly determined by the shell configuration of fragments rather than that of the parent.     

Perturbation theory for the density matrix in the MO LCAO method

The problem of calculating the density matrix of perturbed electron shells in complex molecules has been solved in the MO LCAO [molecular orbital linear combination atomic orbitals] approximation. Considering the electron interaction in correcting the density matrix in each order of the perturbation theory yields one matrix equation uniquely defining the correction. This equation is formulated as an ordinary, linear nonhomogeneous system with respect to the individual matrix elements; this provided the possibility of establishing the convergence region of the self-consistent method. The solution of this system is accomplished by the method of steepest descent which, as has been shown, is always convergent in this case. It was established that just as in the conventional perturbation theory, the (2i + 1)th correction for the energy in terms of the Hartree-Fock method may be expressed by a correction of the density matrix not exceeding the i-th order. Detailed expressions are presented up to the eighth energy correction.     

Triaxial shapes of sodium clusters

A modified Nilsson-Clemenger model is combined with Strutinsky's shell correction method. For spherical clusters, the model potential is fitted to the single-particle spectra obtained from selfconsistent Kohn-Sham calculations. The deformation energy surfaces of sodium clusters with sizes of up toN=270 atoms are calculated for a combination of triaxial, quadrupole and hexadecapole deformations. The ground state shapes and energies are determined by simultaneous minimization with respect to the three shape parameters. A significant fraction of the clusters is predicted to be triaxial. The deviations from the axial shape do not generate any systematic odd-even staggering of the binding energies.     

Accurate ab initio determination of the adiabatic potential energy function and the Born-Oppenheimer breakdown corrections for the electronic ground state of LiH isotopologues

High level ab initio potential energy functions have been constructed for LiH in order to predict vibrational levels up to dissociation. After careful tests of the parameters of the calculation, the final adiabatic potential energy function has been composed from: (a) an ab initio nonrelativistic potential obtained at the multireference configuration interaction with singles and doubles level including a size-extensivity correction and quintuple-sextuple ζ extrapolations of the basis, (b) a mass-velocity-Darwin relativistic correction, and (c) a diagonal Born-Oppenheimer (BO) correction. Finally, nonadiabatic effects have also been considered by including a nonadiabatic correction to the kinetic energy operator of the nuclei. This correction is calculated from nonadiabatic matrix elements between the ground and excited electronic states. The calculated vibrational levels have been compared with those obtained from the experimental data [J. A. Coxon and C. S. Dickinson, J. Chem. Phys. 134, 9378 (2004)]. It was found that the calculated BO potential results in vibrational levels which have root mean square (rms) deviations of about 6-7 cm(-1) for LiH and ～3 cm(-1) for LiD. With all the above mentioned corrections accounted for, the rms deviation falls down to ～1 cm(-1). These results represent a drastic improvement over previous theoretical predictions of vibrational levels for all isotopologues of LiH.     

Fission of metal clusters: A comparison of jellium model calculations and shell correction method calculations

We investigate the fission process Ag²⁺₂₃ → Ag⁺₁₂ + Ag⁺₁₁ in order to compare total energies that calculated by the shell correction method and jellium models. A cavity potential and a Woods-Saxon-type potential are used as the shell potential for the shell correction method, with which the single-particle energy levels are calculated. Shell corrections are obtained by using the Balian-Bloch formula and by smoothing the discrete energy levels in the shell potentials. The jellium model calculations are performed in the framework of the local spin density functional approximation. The conventional jellium model and stabilized jellium model are used. Although the qualitative agreement between the shell correction method calculations and the stabilized jellium calculations is very good, an improvement of the liquid drop energy will be necessary for the satisfactory quantitative agreement. © 1996 John Wiley & Sons, Inc.     

On the validity of using the TKR sum rule to place zero angle electron impact spectra on an absolute scale

The use of the Thomas—Kuhn—Reiche (TKR) sum rule to place electron impact spectra on an absolute scale is only rigorously correct if the observed intensities can be extrapolated to their first Born zero momentum transfer limit. This note investigates the error involved in using the TKR sum rule at zero scattering angle instead of zero momentum transfer. By considering an expansion of the generalized oscillator strength at zero angle it is shown that the first order correction to the TKR sum rule can be written as ($\text{1}\text{3}\text{k}{}^2$)[|E_o|?($\text{4}\text{5}$)$\text{V}$_ee] for target systems randomly oriented in space where k² is the incident electron energy and |E_o| and $\text{V}$_ee are the magnitude of the total electronic energy and the electron—electron repulsive energy of the ground state of the target system respectively. Estimates based on Hartree—Fock calculations indicate a 0.27% error in the use of the TKR sum rule for F and 0.79% for Ar at an incident electron energy of 25 keV.     

Large configuration interaction calculations of nuclear spin—spin coupling constants. I. HD molecule

A method for perturbation calculations of NMR spin—spin coupling constants is presented. The zeroth order wave-function includes correlation by means of large configuration interaction. The rust order correction to the wavefunction is computed by expansion in all singly and doubly excited triplets. Expansion coefficients are found iteratively. The method is applied to the hydrogen molecule yielding, after vibrational averaging, J_HD = 43.48 Hz. The experimental value is 42.94±0.1 Hz. Comparisons are made with other calculations.