The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Calculations of the 13C nuclear screening tensors for coumarin and some methoxycoumarins

INDO calculations are reported of the ¹³C nuclear screening tensors in coumarin and some of its methoxy derivatives. Calculations based upon a linear combination of gauge dependent atomic orbitals and those using an uncoupled Hartree–Fock approach are described. Comparison with the results of less sophisticated calculations shows that changes in the excitation energies, bond orders and the 〈r^?3〉_2p term are together responsible for determining the ¹³C chemical shifts.     

A variation principle for energy differences between states of two different Hamiltonians

A modification of a variation principle due to Delves, is derived which permits the direct calculation of energy differences between states of two different Hamiltonians: [Δ ??] = 〈X₀| ??_x – W_x|X₁〉 – 〈Y₀|??_y – W_y|y₁〉 + 〈X₀| Δ ??|Y₀〉 · 〈X₀| Y₀〉^?1. Δ ?? = ??_y – ??_x, |X₀〉 and |Y₀〉 are the wave functions for the X and Y states and |X₁〉 and |Y₁〉 are functions defined in the text. The principle is applied to a few simple examples.     

Large-Z and -N dependence of atomic energies from renormalization of the large-dimension limit

By combining Hartree–Fock results for nonrelativistic ground-state energies of N-electron atoms with analytic expressions for the large-dimension limit, we have obtained a simple renormalization procedure. For neutral atoms, this yields energies typically threefold more accurate than the Hartree–Fock approximation. Here, we examine the dependence on Z and N of the renormalized energies E(N, Z) for atoms and cations over the range Z, N = 2 → 290. We find that this gives for large Z = N an expansion of the same form as the Thomas–Fermi statistical model, E → Z^7/2(C₀ + C₁Z^?1/3 + C₂Z^?2/3 + C₃Z^?3/3 + ?), with similar values of the coefficients for the three leading terms. Use of the renormalized large-D limit enables us to derive three further terms. This provides an analogous expansion for the correlation energy of the form δE δZ^4/3(δC₃ + δC₅Z^?2/3 + δC₆Z^?3/3 + ?); comparison with accurate values of δE available for the range Z ? 36 indicates the mean error is only about 10%. Oscillatory terms in E and δE are also evaluated. © 1994 John Wiley & Sons, Inc.     

Resonance-enhanced multiphoton ionization of molecular hydrogen via the E,F1Σg+ state: Photoelectron energy and angular distributions

Photoelectron energy and angular distributions are measured for the 2+1 multiphoton ionization process H₂ X¹Σ_g⁺ (ν = 0,J) + 2hv → E,F¹Σ_g⁺(ν_E,J_E = J) + hν → H₂⁺X²Σ_g⁺ (ν⁺) + e^?, for ν_E = 0, 1, or 2 and for J_E = 0 or 1 of the inner well of the double-minimum E,F state. Although a strong preference is found for ν⁺ = ν_E, the detailed H₂⁺ vibrational distribution does not exhibit Franck-Condon behavior, and the photoelectron angular distributions vary markedly with both the J_E value of the intermediate state and the ν⁺ value of the ion.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Some sum-over-states perturbation calculations of 19F?13C coupling constants for some carbocations and related fluorobenzenes

Sum-over-states perturbation calculations within the INDO framework are reported for 24 ¹J(FC) and 34 ³J(FC) couplings. In general, satisfactory agreement with the experimental data is obtained when the integral products S_F²(O) S_C²(O) and 〈r^?3〉_F 〈r^?3〉_C take the values of 136.543 au^?6 and 58.352 au^?6, respectively, for the ¹J(FC) couplings. The corresponding values for the ³J(FC) couplings are 29.520 au^?6 and 44.340 au^?6, respectively. All of the ¹J(FC) values are predicted to be negative, whereas all of the ³J(FC) values are calculated with positive signs. The results indicate the importance of including the contact, orbital and dipolar contributions in the calculations.     

Correlation effects to the expectation values of atomic systems

The Hartree-Fock and first natural spin determinants were compared as reference determinants for calculating various one-electron properties such as ρ(0), 〈½?〉, 〈r^?2〉,…, 〈r³〉, and r^?1₁₂〉. Calculations were made on various small atoms and their positive and negative ions. For nearly all the expectation values studied, the first natural spin orbital determinant gave consistently superior results. In particular, the Hartree-Fock functions gave markedly inferior results for some long range properties such as the magnentic susceptibilities of negative ions. The major correlation error in the expectation values is primarily an orbital effect which may be accounted for by including correlation terms in the one-particle Hamiltonian. Such approximate Brueckner or best overlap orbitals should reproduce most one-electron expectation values accurately.     

Some INDO perturbation calculations of nJ(NC) values

Standard INDO parameters are used in ‘sum-over-states’ perturbation calculations of ⁿJ(NC) in a variety of molecular environments. Good agreement with the experimental data is, in general, obtained when the integral products S_N²(o)S_C²(o) and 〈r^?3〉_N〈r^?3〉_C assume the values of 35.167 a.u.^?3 and 4.980 a.u.^?3, respectively. For ‘pyridine-type’ nitrogen atoms the major contribution to ⁿJ(NC) usually arises from the orbital term whereas the contact term dominates the values of ⁿJ(NC) for ‘pyrrole-type’ and amino nitrogen atoms.     

Radial and angular correlation in heliumlike ions

In the framework of nonrelativistic variational formalism a new type of basis set is proposed, to estimate separately the effect of radial and angular correlations on the ground‐state energy for helium isoelectronic sequence H^? to Ar¹⁶⁺. Effect of radial correlation is incorporated by using multiexponential functions arising from product basis sets suitably formed out of Slater‐type one‐particle orbitals. The angular correlation can be switched on by incorporating an expansion in terms of basis involving interparticle coordinates. With a set of six‐term Slater‐type one‐particle basis and five‐term interparticle expansion, the ground‐state energy of helium is estimated as ?2.9037236 (a.u.) compared with the multiterm variational estimates ?2.9037244 (a.u.) due to Pekeris and Thakkar and Smith and Drake. Matrix elements of different operators in the ground state have been calculated and found to be in good agreement with available accurate results. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

On the Controversial Fitting of Susceptibility Curves of Ferromagnetic CuII Cubanes: Insights from Theoretical Calculations

This paper reports a theoretical analysis of the electronic structure and magnetic properties of a tetranuclear Cu^II complex, [Cu₄(HL)₄], which has a 4+2 cubane‐like structure (H₃L=N,N′‐(2‐hydroxypropane‐1,3‐diyl)bis(acetylacetoneimine)). These theoretical calculations indicate a quintet (S=2) ground state; the energy‐level distribution of the magnetic states confirm Heisenberg behaviour and correspond to an S₄ spin–spin interaction model. The dominant interaction is the ferromagnetic coupling between the pseudo‐dimeric units (J₁=22.2 cm^?1), whilst a weak and ferromagnetic interaction is found within the pseudo‐dimeric units (J₂=1.4 cm^?1). The amplitude and sign of these interactions are consistent with the structure and arrangement of the magnetic Cu 3d orbitals; they accurately simulate the thermal dependence of magnetic susceptibility, but do not agree with the reported J values (J₁=38.4 cm^?1, J₂=?18.0 cm^?1) that result from the experimental fitting. This result is not an isolated case; many other polynuclear systems, in particular 4+2 Cu^II cubanes, have been reported in which the fitted magnetic terms are not consistent with the geometrical features of the system. In this context, theoretical evaluation can be considered as a valuable tool in the interpretation of the macroscopic behaviour, thus providing clues for a rational and directed design of new materials with specific properties.     

Construction of orthogonal subspaces

An orthogonalization procedure is presented that allows construction of at least (n?m) vectors orthogonal to {X_j}, j equals; 1, m, by linear combinations solely among {η_i}, i equals; 1, n, n>m, and 〈X_j/η_i〉≠0. An important application of the procedure is in effective core potential methods for which valence orbitals can be constructed that are orthogonal to the core orbitals and yet involve no component of the core. Thus, a separate calculation for only the valence electrons can be performed without any explicit reference to the core electrons (orbitals).     

Some self-consistent perturbation calculations of 15N?13C coupling constants

Self-consistent perturbation calculations within the INDO framework are reported for 63 ¹⁵N? ¹³C coupling constants. Examples are presented for which each of the contact, orbital and dipolar terms provides the dominant contribution to the observed coupling constant. In general, good agreement with the experimental data is obtained when the integral products S_N²(O)S_C²(O) and 〈r^?3〉_N〈r^?3〉_C take the values 14.480 au^?6 and 2.446 au^?6 for ¹J(¹⁵N? ¹³C), and the corresponding values of 10.444 au^?6 and 17.664 au^?6 for ¹J(¹⁵N?¹³C). All 19 of the ¹J(¹⁵N?¹³C) couplings considered are predicted to have a negative sign.     

MRCI study on spectroscopic parameters and molecular constants of the ground state of AsP(X1Σ+) molecule

The potential energy curve (PEC) for the ground state of AsP(X¹Σ⁺) has been investigated by the highly accurate valence internally contracted multireference configuration interaction method in the Molpro2008 program package with the correlation consistent basis set. The PEC is fitted to the analytic Murrrell–Sorbie function (M–S function) from which the spectroscopic constants are determined. The present D_e, B_e, α_e, ω_eχ_e, R_e, and ω_e values are of 4.2823 eV, 0.188622 cm^?1, 0.000749 cm^?1, 1.984427 cm^?1, 2.0194 Å, and 598.60 cm^?1, respectively. In addition, by numerically solving the radial Schrödinger equation of nuclear motion in the adiabatic approximation, the total of 96 vibration states is predicted when the rotational quantum number J = 0. The complete vibration levels, classical turning points, inertial rotation, and centrifugal distortion constants are reproduced. Comparison has been made with recent theoretical and experimental data. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Gaussian vs. Slater representations of d orbitals: An information theoretic appraisal based on both position and momentum space properties

A detailed appraisal of Gaussian-type orbital (GTO) and Slater-type orbital (STO) expansions of 3d orbitals is carried out for the ²S state of copper—a case that should be maximally unfavorable for STOs. The appraisal is based on a wide variety of both position and momentum space properties and utilizes an information theoretic quality assessment technique. It is found that GTO expansions are not as useful as STO expansions for the prediction of 〈p⁸〉, 〈p⁷〉, and 〈r^?6〉 because these properties probe the functional deficiencies of GTOs at small r and large p. On the other hand, GTO expansions can predict accurate values of large r properties like 〈r⁸〉 despite the fact that their position space asymptotic decay is too fast. Unlike the case of s orbitals in helium, there does not seem to be any consistent ordering between accuracy in position space and accuracy in momentum space. The quality measures are found to be very useful for pinpointing the deficiencies of various expansions. This information enables us to construct easily a new GTO and a new STO expansion that are more accurate than any of the others in the literature. It is suggested that one STO is worth no more than two GTOs in the case of d orbitals.     

Electronic states and nature of the chemical bond in the molecule CrC by all-electron ab initio calculations

All-electron ab initio Hartree–Fock (HF ), valence configuration interaction (CI ), and multiconfiguration self-consistent-field (CASSCF ) calculations have been applied to investigate the electronic states of the CrC molecule. The molecule is predicted as having four low-lying electronic states, ³∑^?, ⁵∑^?, ⁷∑^?, and ⁹∑^?, separated by an energy gap of 0.55 eV from the next higher-lying state, ¹∑^?, which is followed by the states ⁵Π and ⁷Π. The four lowest-lying electronic states are due to the coupling of the angular momenta of the ⁶S_g Cr⁺ ion with those of the ⁴S_u C^? anion. The chemical bond in the ³∑^? ground state can be viewed as a quadruple bond composed of two σ and two π bonds. One σ bond is due to the formation of a molecular orbital that is doubly occupied. The remaining bonds, i.e., one σ and two π bonds, arise from valence-bond couplings. The π bonds originate from the valence-bond couplings of the electrons in the C 2pπ orbitals with those in the Cr 3dπ orbitals. The σ bond originates from the valence-bond coupling of the C 2pσ electron with an electron in the Cr 4s, 4p hybrid that is polarized away from the C atom.     

Thermodynamics of Lanthanum Fluoride in Aqueous Sodium Perchlorate

The solubility-product constant of LaF,(s) was obtained by potentiometric titration of sodium fluoride with lanthanum perchlorate in sodium perchlorate solutions of various ionic strengths. The limiting value of pK_s.p were found to be 17.65 and 17.95 at 20° and 25°C, respectively. Thermodynamic calculation from these values gives δG° =-102.46 kJ mol^?1, δH° = 100.40 kJ mol^?1 and δS° = 680.40 JK^?1 for the reaction, La⁺³ (aq)+3F-(aq)=LaF,(s), and δG° =-1613.8kJ mol-1, δH₁° =-1594.0kJ mol^?1 and S° =433.1 J K^?1 mol^?1 for the formation of LaF₂,(s) at 25°C. The mean activity coefficients and solubilities of LaF₂,(s) in NaCIO₄ solution at various ionic strengths at 25°C were evaluated.     

Multireference configuration interaction study on spectroscopic parameters and molecular constants of PO and PO+

The highly accurate valence internally contracted multireference configuration interaction (MRCI) approach has been employed to investigate the potential energy curves (PECs) for the X²Π, b⁴Σ^?, C²Σ^? states of PO and the X¹Σ⁺ state of PO⁺. For these electronic states, the spectroscopic parameters of the isotopes (P¹⁶O, P¹⁸O, P¹⁶O⁺, and P¹⁸O⁺) have been determined and compared with those of the investigations reported in the literature. The comparison shows that excellent agreement exists between the present results and the available experiments. With the PECs determined here, the first 30 vibrational states for P¹⁶O(X²Π, b⁴Σ^?), P¹⁸O(X²Π, b⁴Σ^?), P¹⁶O⁺(X¹Σ⁺), and P¹⁸O⁺(X¹Σ⁺) are computed when the rotational quantum number J equals zero (J = 0). The vibrational level G(υ), inertial rotation constant B_υ and centrifugal distortion constant D_υ are determined when J = 0. All the results of vibrational states except for P¹⁶O (X²Π) are reported for the first time. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Unusually long, multicenter, cation(δ+)···anion(δ-) bonding observed for several polymorphs of [TTF][TCNE

The α, β, and δ polymorphs of [TTF][TCNE] (TTF=tetrathiafulvalene; TCNE=tetracyanoethylene) exhibit a new type of long, multicenter bonding between the [TTF]^δ+ and [TCNE]^δ? moieties, demonstrating the existence of long, hetero‐multicenter bonding with a cationic^δ+???anionic^δ? zwitterionic‐like structure. These diamagnetic π‐[TTF]^δ+[TCNE]^δ? heterodimers exhibit a transfer of about 0.5 e^? from the TTF to the TCNE fragments, as observed from experimental studies, in accord with theoretical predictions, that is, [TTF^δ+???TCNE^δ?] (δ?0.5). They have several interfragment distances <3.4 Å, and a computed interaction energy of ?21.2 kcal mol^?1, which is typical of long, multicenter bonds. The lower stability of [TTF]^δ+[TCNE]^δ? with respect to typical ionic bonds is due, in part, to the partial electron transfer that reduces the electrostatic bonding component. This reduced electrostatic interaction, and the large interfragment dispersion stabilize the long, heterocationic/anionic multicenter interaction, which in [TTF^δ+???TCNE^δ?] always involves two electrons, but have ten, eight, and eight bond critical points (bcps) involving C? C, N? S, and sometimes C? S and C? N components for the α, β, and δ polymorphs, respectively. In contrast, γ‐[TTF][TCNE] possesses [TTF]₂²⁺ and [TCNE]₂^2? dimers, each with long, homo‐multicenter 2e^?/12c (c=center, 2 C+4 S) [TTF]₂²⁺ cationic⁺???cationic⁺ bonds, as well as long, homo‐multicenter 2e^?/4c [TCNE]₂^2? anionic^????anionic^? bonding. The MO diagrams for the α, β, and δ polymorphs have all of the features found for conventional covalent C? C bonds, and for all of the previously studied multicenter long bonds, for example, π‐[TTF]₂²⁺ and π‐[TCNE]₂^2?. The HOMOs for α‐, β‐, and δ‐[TTF][TCNE] have 2c C? S and 3c C? C? C orbital‐overlap contributions between the [TTF]^δ+? and [TCNE]^δ? moieties; these are the shortest intra [TTF???TCNE] separations. Thus, from an orbital‐overlap perspective, the bonding has 2c and 3c components residing over one S and four C atoms.