A simple model to calculate total and ionization energies of any atom

In order to systemically evaluate the reliability of the method of predicting ionization energy with density functional method in quantum chemistry [B3LYP/6-311++G(d,p)] and composite ab initio(G3B3),70 chemical portions of known ionization energy in cigarette smoke were calculated and analyzed as the validation set,and some smoke portions of unknown ionization energy were calculated.Meanwhile,by synchrotron radiation photoionization mass spectrometry assisted with an in-house cigarette smoke sampler,the gas phase components in cigarette smoke were on-line analyzed.The photoionization mass spectra and photoionization efficiency spectra were measured,also obtained were the experimental values of the ionization energy of some smoke portions.The results showed that G3B3 method was competent for calculating ionization energy of various smoke portions,the accuracy was up to 0.06 eV,and the calculated results were suitable for the qualitative analysis of cigarette smoke components.

The use of fractional occupation of energy levels in the self-interaction-corrected local spin density (SIC-LSD) theory of electronic energy structure is studied with reference to calculation of ionization and excitation energies. With the original form of the SIC-LSD energy functional for fractional occupation, the one-electron eigenvalues exhibit nonlinear dependence on the occupation number. A new SIC-LSD density functional for fractional occupation based on the general behavior of the universal functional is proposed. The one-electron eigenvalues derived from this new functional vary quite linearly with the occupation number. This makes it possible to obtain ionization and excitation energies by a simple numerical integration over the occupation number. A one-point integration gives ionization energies that agree on the average with the results on taking the difference in self-consistent-field total energy between the atom and the ion to within 0.4% for the atoms He, Li, Be,...,Ar. Improvement can be made by using a three-point integration over the occupation number. This method is also applied to calculate excitation energies of selected transitions for He through Ar and the results are in good agreement with the corresponding total-energy differences.

PART A: GVB and GVB-Cl wavefunctions (using a double zeta basis) have been obtained as a function of internuclear distance for the lowest three states of NiCO. The wavefunctions lead to a qualitative description in which the Ni atom is neutral with a (4s)^1(3d)^9 atomic configuration. The CO lone pair delocalizes slightly onto the Ni, leading to the 4s-like orbital hybridizing away from the CO. The dir pairs on the Ni are slightly back-bonding to the CO. The three bound states are ^3Σ^+, ^3∏, and ^3Δ consisting of the singly occupied 4s-like orbital plus a single d hole in a σ, π, or δ orbital, respectively. The ground state is found to be ^3Δ with calculated R_e = 1.90 A, D_e = 1.15 eV = 26.5 kcal/mol, and ω_c(Ni-C) = 428 cm^(-1), all reasonable values, although direct information on NiCO is not yet available. The adiabatic excitation energies are calculated as 0.240 eV to ^3Σ^+, and 0.293 eV to ^3∏. The states with (4s)^2(3d)^8 configurations on the Ni lead to repulsive potential curves with vertical excitation energies in the range of 3.0 to 5.0 eV. PART B: Configuration interaction calculations have been carried out for a number of positive ion states of NiCO. These calculations indicate that there are two distinct groups of ionization potentials. The first group involves ionizations out of Ni-like orbitals. The lowest states of this group involve ionization out of a Ni 4s-like orbital leading to a 3d^9 configuration and states of symmetry ^2Σ^+, ^2∏, and ^2Δ depending on whether the 3d-hole is taken in a σ , π) or δ orbital. At the optimum geometry of NiCO, the dissociation energy of NiCO^+ to Ni^+(^2D) and CO is calculated to be 2.26, 2.03 and 2.50 eV for the ^2Σ^+, ^2∏, and ^2Δ states, respectively, in reasonable agreement with the value of 2.10 eV calculated from the experimental heat of formation of NiCO^+. Other states in the first group involve ionization out of Ni 3d orbitals leading to a group of ion states with a width of 3.1 eV. This is in good agreement with the Ni d bandwidth as observed in photoemission experiments. The second group of ion states correlates at large Ni-C separation with the ground state of the Ni atom and various states of CO^+. The principal change as compared with free CO is that the 5σ ionization (lone pair on the CO) increases in energy by about 2.5 eV, whereas the 4σ and 1π ionizations change only slightly. This leads to the 5σ and 1π ionizations being nearly degenerate, with the 4σ ionization about 3.0 eV higher, in agreement with the currently accepted interpretation of the photoelectron spectrum of CO chemisorbed on Ni. PART C: Geometries for 0 and S overlayers on the (100) and (110) surfaces of Ni have been calculated using ab initio wavefunctions for 0 and S bonded to small clusters of Ni atoms (1 to 5 Ni atoms). The calculated geometries are within 0.07A of the results of dynamic LEED intensity calculations, indicating that accurate geometries of chemisorbed atoms may be obtained from calculations using clusters including only those metal atoms within bonding distance. PART D: Electronic wavefunctions have been obtained as a function of geometry fora S atom bonded to Ni clusters consisting of l to 4 atoms de-signed to model bonding to the Ni(100) and Ni(110) surfaces. Electron correlation effects were included using the generalized valence bond and configuration interaction methods. Modeling the (100) surface with four Ni atoms, we find the optimum S position to be 1.33A above the surface, in good agreement with the value (1.30 ± 0.10A) from dynamic LEED intensity calculations. The bonding is qualitatively like that in H_2S with two covalent bonds to one diagonal pair of Ni atoms. There is a S pπ pair overlapping the other diagonal pair of Ni atoms. [Deleting this pair the S moves in to a position 1.04A from the surface.] There are two equivalent such structures, the resonance leading to equivalent S atoms and a c(2x2) structure for the S overlayer. The Ni in the layer beneath the surface seems to have little effect (~0.03A) on the calculated geometry. The above model of the bonding suggests that for the (110) surface the S lies along the long edge of the rectangular unit cell (2 coordinate) rather than at the four coordinate site usually assumed. Our calculated position for the S of 1.04A is in reasonable agreement with the value from dynamic LEED intensity calculations, 0.93 ±0.10A. Bonding the S directly above a single Ni atom leads to a much weaker bond (D_e = 3.32 eV) than does bonding in a bridge position (D_e = 5.37 eV). PART E: Electronic wavefunctions have been obtained as a function of geometry for an 0 atom bonded to Ni clusters (consisting of 1 to 5 atoms) designed to model bonding to the Ni(100) and Ni(110) surfaces. Electron correlation effects were included using the generalized valence bond and configuration interaction methods. For the (100) surface, we find that the charge distribution for the full 0 overlayer is consistent with taking a positively charged cluster. The four surface atoms in the surface unit cell and the atom beneath the surface are important in determining the geometry, leading to a Ni^+_50 cluster as the model for the (100) surface. The optimum oxygen position with this model is 0.96A above the surface (four-fold coordinate site) in good agreement with the value (0.90± 0.10A) from dynamic LEED intensity analysis. The atom beneath the surface allows important polarization effects for the positively charged cluster. The bonding to the surface involves bridging two diagonal surface Ni atoms. There is an 0(2pπ)pair which overlaps the other diagonal pair of Ni atoms leading to nonbonded repulsions which increase the distance above the surface. There are two equivalent such structures, the resonance leading to a c(2 x 2) structure for the 0 over-layer. The above model suggests that for the (110) surface the 0 lies along the long edge of the rectangular unit cell. For this registry with the surface, calculations based on Ni_20 and Ni_30 models indicate that the oxygen is only 0.1A above the plane of the surface. PART F: Generalized valence bond and configuration interaction wave-functions have been obtained as a function of R for numerous electronic states of NiO. All the lower states are found to involve the (4s)^1(3d)^9 Ni atom configuration and 0 in the (2s)^2(2p)^4 configuration. There are two groups of states. The lower group of states involves pairing singly occupied Mi(4s).and 0(2p σ) orbitals into a (somewhat ionic) sigma bond pair with various pairings of the Ni(3d)^9 and 0(2pπ)^3 configurations. This leads to a number of states including the ground state which we find to be x^3Σ^-. (The electronic structure is analogous to that of O_2.) The calculated D_o and R_e for the x^3Σ^- state of Ni0 are 89.9 Kcal/mole and 1.60 A respectively. The bond energy is in good agreement with the experimental value 86.5 ±5 Kcal/mole, while the R_e value is not known experimentally. The higher group of states involve a doubly occupied 0(2p σ) orbital., The Ni(4s) orbital in this case is non-bonding and builds in 4p character to move away from the oxygen orbitals. The bonding mainly involves stabilization of the oxygen orbitals by the Ni(3d)^9 core (somewhat analogously to the bonding in NiC0). Numerous allowed transitions between these states and the states of the lower group are calculated to be in the range 1.0 to 3.0 eV where numerous bands are seen in emission.

We have developed accurate Gaussian basis functions obtained with the polynomial generator coordinate Hartree-Fock (p-GCHF) method for H, Zn, and Ga-Kr atoms. These basis sets have been applied in the calculation of nonrelativistic energies for neutral atoms, monovalent cations, monovalent anions, ionization potential (IP), and electron affinity (EA), with the objective of proving the quality of the basis set generated by the p-GCHF method. The total energies calculated for neutral atoms and monovalent cations and respective IP were minimally affected by the addition of polarization functions and their precision was comparable to the values reported in the literature. The relative errors were lower than 6.0 \texttimes\ 10$^{-5}$% and 7.0 \texttimes\ 10$^{-5}$% for neutral atoms and monovalent cations, respectively. The IP results were strictly equal to numerical Hartree-Fock (NHF) calculations and comparable to some experimental values. For monovalent anions, the nonrelativistic total energies were better than the Slater-type functions results and the relative errors were lower than 0.05% when compared to NHF. The EA results were the same as those obtained with NHF calculations reported in the literature for heavier elements. For IP and EA, our results followed the same periodic tendency when compared with experimental data.

Numerical errors in total energy values in large-scale Hartree–Fock calculations are discussed. To obtain total energy values within chemical accuracy, 0.01 kcal/mol, stricter numerical accuracy is required as basis size increases. In molecules with 10,000 basis sizes, such as proteins, numerical accuracy for total energy values must be retained to at least 11 digits (i.e., to the order of 1.0D-10) to keep accumulation of numerical errors less than the chemical accuracy (0.01 kcal/mol). With this criterion, we examined the sensitivity analysis of numerical accuracy in Hartree–Fock calculation by uniformly replacing the last bit of the mantissa part of a double-precision real number by zero in the Fock matrix construction step, the total energy calculation step, and the Fock matrix diagonalization step. Using a partial summation technique in the Fock matrix generation step, the numerical error for total energy value of molecules with basis size greater than 10,000 was within chemical accuracy (0.01 kcal/mol), whereas with the conventional method the numerical error with several thousand basis sets was larger than chemical accuracy. Computation of one Fock matrix element with parallel machines can include the partial summation technique automatically, so that parallel calculation yields not only high-performance computing but also more precise numerical solutions than the conventional sequential algorithm. We also found that the numerical error of the Householder-QR diagonalization routine is equal to or less than chemical accuracy, even with a matrix size of 10,000. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 443–454, 1999

Near Hartree-Fock wavefunctions have been calculated for the ground state of the water molecule using both Slater and contracted Gaussian basis sets. Total energies of −76.063 hartree were obtained with a (5s4p1d/3s1p) Slater basis and a [6s5p2d/3s1p] contracted Gaussian basis derived from an (11s7p2d/5s1p) primitive set; these energies are estimated to be within 0.003±0.002 hartree of the Hartree-Fock limit. The Hartree-Fock wavefunctions account for ∼70% of the dissociation energy of water. The Hartree-Fock vertical ionization potentials (in electron volts), 11.1(2B1), 13.3(2A1), and 17.6(2B2), are too low by 1–1.5 eV as expected. With the Gaussian basis set a potential surface was computed and the equilibrium geometry and harmonic force constants were calculated. The calculated bond length, 0.941 Å, and bond angle 106.6°, are in good agreement with the experimental values, 0.957 Å and 104.52°. In spite of the rather good agreement for the geometry, the force constants are in error by 15%–20%. This is attributed to an inadequancy of the Hartree-Fock model. A number of one-electron properties were also computed; they differ only slightly from those reported in earlier work and are in satisfactory agreement with experiment. Plots of the valence (canonical) molecular orbitals are given.

The ionization energies (IEs) of TiO and TiO2 and the 0 K bond dissociation energies (D0) and the heats of formation at 0 K (ΔH°f0) and 298 K (ΔH°f298) for TiO/TiO+ and TiO2/TiO2+ are predicted by the wave-function-based CCSDTQ/CBS approach. The CCSDTQ/CBS calculations involve the approximation to the complete basis set (CBS) limit at the coupled cluster level up to full quadruple excitations along with the zero-point vibrational energy (ZPVE), high-order correlation (HOC), core-valence (CV) electronic, spin-orbit (SO) coupling, and scalar relativistic (SR) effect corrections. The present calculations yield IE(TiO) = 6.815 eV and are in good agreement with the experimental IE value of 6.819 80 ± 0.000 10 eV determined in a two-color laser-pulsed field ionization-photoelectron (PFI-PE) study. The CCSDT and MRCI+Q methods give the best predictions to the harmonic frequencies: ωe (ωe+) = 1013 (1069) and 1027 (1059) cm-1 and the bond lengths re (re+) = 1.625 (1.587) and 1.621 (1.588) Å, for TiO (TiO+) compared with the experimental values. Two nearly degenerate, stable structures are found for TiO2 cation: TiO2+(C2v) structure has two equivalent TiO bonds, while the TiO2+(Cs) structure features a long and a short TiO bond. The IEs for the TiO2+(C2v)←TiO2 and TiO2+(Cs)←TiO2 ionization transitions are calculated to be 9.515 and 9.525 eV, respectively, giving the theoretical adiabatic IE value in good agreement with the experiment IE(TiO2) = 9.573 55 ± 0.000 15 eV obtained in the previous vacuum ultraviolet (VUV)-PFI-PE study of TiO2. The potential energy surface of TiO2+ along the normal vibrational coordinates of asymmetric stretching mode (ω3+) is nearly flat and exhibits a double-well potential with the well of TiO2+ (Cs) situated around the central well of TiO2+(C2v). This makes the theoretical calculation of ω3+ infeasible. For the symmetric stretching (ω1+), the current theoretical predictions overestimate the experimental value of 829.1 ± 2.0 cm-1 by more than 100 cm-1. This work together with the previous experimental and theoretical investigations supports the conclusion that the CCSDTQ/CBS approach is capable of providing reliable IE and D0 predictions for TiO/TiO+ and TiO2/TiO2+ with error limits less than or equal to 60 meV. The CCSDTQ/CBS calculations give the predictions of D0(Ti+-O) - D0(Ti-O) = 0.004 eV and D0(O-TiO) - D0(O-TiO+) = 2.699 eV, which are also consistent with the respective experimental determination of 0.008 32 ± 0.000 10 and 2.753 75 ± 0.000 18 eV.

The vibrational spectroscopic studies and molecular property analysis of l-Phenylalanine using quantum chemical method

The $Z$ dependence of the total electronic energy for an atomic isoelectronic series, was investigated in several different ways, first by the consideration of a Taylor-series expansion of ${\overline{V}}_{\mathrm{en}}$, the average electron-nuclear attractive potential energy, used in conjunction with the virial and Hellmann-Feynman theorems and, secondly, a Taylor-series expansion of ${\overline{V}}_{\mathrm{ee}}$, the average electron-electron repulsive potential energy, used in conjunction with the same two theorems. A direct Taylor-series expansion of the total energy $E$ was also studied. Convergence of these expansions was inferred from the rate of convergence of least-squares fits of tabulated values of the total electronic energies as a function of the increasing number of terms in the expansion. Theoretical nonrelativistic Hartree-Fock energies, accurate theoretical nonrelativistic energies, and experimental total energies, the latter approximately corrected for relativistic and quantum electrodynamic effects, were used in the fits. These expansions were compared with $\frac{1}{Z}$ expansion results. All Taylor-series variants studied showed comparable or superior convergence properties to the $\frac{1}{Z}$ expansion. The choice of a ${\overline{V}}_{\mathrm{en}}$ or a ${\overline{V}}_{\mathrm{ee}}$ expansion forces certain behavior on the total electronic energy $E$ as a function of $Z$. From a least-squares rate-of-convergence criterion and other considerations it appears that the Taylor-series expansion of ${\overline{V}}_{\mathrm{en}}$ offers slightly better results. It is pointed out that by the use of the virial and Hellmann-Feynman theorems it is possible to obtain accurate estimates of ${\overline{V}}_{\mathrm{ee}}$ and ${\overline{V}}_{\mathrm{en}}$ from a knowledge of total electronic energies of isoelectronic series alone. The technique is applied to accurate values for the He isoelectronic series and experimental data corrected for relativistic and radiative contributions for the He through Ne isoelectronic series in order to determine the effect of electron correlation on the quantities ${\overline{V}}_{\mathrm{en}}$ and ${\overline{V}}_{\mathrm{ee}}$. Comparisons are made with available experimental and theoretical results in the case of Ne. The $Z$ expansions for the energy were also used to fit individual ionization potentials across an isoelectronic series. These fits were used to predict electron affinities and improved values for certain ionization potentials.

Chapter 17 - Equations of motion methods for computing electron affinities and ionization potentials

The ab initio calculation of molecular electron affinities (EA) and ionization potentials (IP) is a difficult task because the energy of interest is a very small fraction of the total electronic energy of the parent species. For example, EAs typically lie in the 0.01-10 eV range, but the total electronic energy of even a small molecule, radical, or ion is usually several orders of magnitude larger. Moreover, the EA or IP is an intensive quantity but the total energy is an extensive quantity, so the difficulty in evaluating EAs and IPs to within a fixed specified (e.g., ±0.1 eV) accuracy becomes more and more difficult as the system's size and number of electrons grows. The situation becomes especially problematic when studying extended systems such as solids, polymers, or surfaces for which the EA or IP is an infinitesimal fraction of the total energy. EOM methods such as the author developed in the 1970s offer a route to calculating the intensive EAs and IPs directly as eigenvalues of a set of working equations. A history of the development of EOM theories as applied to EAs and IPs, their numerous practical implementations, and their relations to Greens function or propagator theories are given in this contribution. EOM methods based upon Møller-Plesset, multiconfiguration self-consistent field, and coupled-cluster reference wave functions are included in the discussion as is the application of EOM methods to metastable states of anions.

Response of a Molecule to Adding or Removing an Electron

The Roothaan–Hartree–Fock (HF) method has been implemented in deMon–DynaRho within the resolution-of-the-identity (RI) auxiliary-function approximation. While previous studies have focused primarily upon the effect of the RI approximation on total energies, very little information has been available regarding the effect of the RI approximation on orbital energies, even though orbital energies play a central role in many theories of ionization and excitation. We fill this gap by testing the accuracy of the RI approximation against non-RI-HF calculations using the same basis sets, for the occupied orbital energies and an equal number of unoccupied orbital energies of five small molecules, namely CO, N2, CH2O, C2H4, and pyridine (in total 102 orbitals). These molecules have well-characterized excited states and so are commonly used to test and validate molecular excitation spectra computations. Of the deMon auxiliary basis sets tested, the best results are obtained with the (44) auxiliary basis sets, yielding orbital energies to within 0.05 eV, which is adequate for analyzing typical low resolution polyatomic molecule ionization and excitation spectra. Interestingly, we find that the error in orbital energies due to the RI approximation does not seem to increase with the number of electrons. The absolute RI error in the orbital energies is also roughly related to their absolute magnitude, being larger for the core orbitals where the magnitude of orbital energy is large and smallest where the molecular orbital energy is smallest. Two further approximations were also considered, namely uniterated (“zero-order”) and single-iteration (“first-order”) calculations of orbital energies beginning with a local density approximation initial guess. We find that zero- and first-order orbital energies are very similar for occupied but not for unoccupied orbitals, and that the first-order orbital energies are fairly close to the corresponding fully converged values. Typical root mean square errors for first-order calculations of orbital energies are about 0.5 eV for occupied and 0.05 eV for unoccupied orbitals. Also reported are a few tests of the effect of the RI approximation on total energies using deMon basis sets, although this was not the primary objective of the present work.

Impact of dietary fiber energy on the calculation of food total energy value in the Brazilian Food Composition Database

The total-energy and pressure-volume relations are calculated nonrelativistically for Si, Ge, and \ensuremath{\alpha}-Sn within the local-density-functional formalism, with use of first-principles nonlocal pseudopotentials. Ground-state static structural properties (total energy, lattice constant, and bulk modulus and its pressure derivative) are obtained and are in good agreement with experimental values. A prediction of these for \ensuremath{\alpha}-Sn from pressure determinations has not yet been reported. Its bulk modulus from both total-energy and pressure calculations is much smaller than that determined by experimental measurement. This confirms a recent theoretical prediction from other authors casting doubts on the experimental value. An assessment has been made on the advantages and disadvantages of pressure calculations over total-energy calculations and on the influence on the results of the size of all cutoff parameters and perturbative schemes used.